Research ArticleTetris Tight Frames Construction via Hadamard Matrices

A Abdollahi and M Monfaredpour

Department of Mathematics College of Sciences Shiraz University Shiraz 71454 Iran

Correspondence should be addressed to A Abdollahi abdollahishirazuacir

Received 30 April 2014 Accepted 23 September 2014 Published 27 October 2014

Academic Editor Hossein Jafari

Copyright copy 2014 A Abdollahi and M Monfaredpour This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We present a new method to construct unit norm tight frames by applying altered Hadamard matrices Also we determine anelementary construction which can be used to produce a unit norm frame with prescribed spectrum of frame operator

1 Introduction and Preliminaries

Frames were first introduced in 1952 by Duffin and Schaeffer[1] in the context of nonharmonic Fourier seriesThey are sys-tem of functions in Hilbert spaces that provide numericallystable methods for finding overcomplete decompositions ofvectors and such are useful tools in various signal processingapplications data compression wireless communicationsand so on [2ndash4] Frames in finite dimensional Hilbert spaceshave become of interests for many of researches [5 6]One of the important subjects in this area is the way forconstructing such frames Some methods of constructionfinite tight frames are stated by researchers [7ndash9]

Let 119873 be a positive integer A sequence of vectors 119891119894119894isin119868

in Hilbert spaceR119873 is said to be a frame (see also [10]) forR119873if there exist constants119860 and 119861 such that 0 lt 119860 le 119861 lt infin and

11986010038171003817100381710038171198911003817100381710038171003817

2le sum

119894isin119868

1003816100381610038161003816⟨119891 119891119894⟩1003816100381610038161003816

2le 11986110038171003817100381710038171198911003817100381710038171003817

2forall119891 isin R

119873 (1)

The numbers 119860 and 119861 are called frame bounds and they arenot unique The frame 119891

119894119894isin119868

is said to be tight (or 119860-tight) if119860 = 119861 In this case 119860 is said to be the frame constant and it isa Parsval frame if 119860 = 119861 = 1 When the index set 119868 is a finiteset the frame will be called finite A normalized frame or unitnorm frame is the one in which elements have the norm one

If only the right side of the inequalities (1) holds then119891119894119894isin119868

is called a Bessel sequence and if 119891119894119894isin119868

is a normalizedBessel sequence in R119873 then 119891

119894119894isin119868

is finite sequence [5]According to this our framewill be the form 119891

119894119872

119894=1whenever

119872 is some positive integer Also we will replace 1198972(119868) by R119872To each Bessel sequence 119891

119894119872

119894=1corresponds an operator

119879 R119873997888rarr R

119872119879 (119891) = ⟨119891 119891

119894⟩119872

119894=1(2)

called analysis operator which is well-defined and boundedoperator Its adjoint is the operator

119879lowast R119872997888rarr R

119873119879lowast(119888119894119872

119894=1) =

119872

sum

119894=1

119888119894119891119894 (3)

called the synthesis operator If 119891119894119872

119894=1is a frame with frame

bounds 119860 and 119861 then the operator

119879lowast119879 R119873997888rarr R

119873 119879

lowast119879 (119891) =

119872

sum

119894=1

⟨119891 119891119894⟩ 119891119894

(4)

is called the frame operator of the frame 119891119894119872

119894=1 It is a positive

self-adjoint bounded and hence invertible operator with theinverse (119879lowast119879)minus1 Benedetto and Fickus in [5] have proved thatif 119891119894119872

119894=1is a normalized 119860-tight frame for a 119873-dimensional

Hilbert space H then 119860 = 119872119873In this paper we address the question of how to effi-

ciently construct unit norm tight frame After reviewingknown results about the Spectral Tetris Construction andintroducing the Spectral Hadamard-Tetris Construction inSection 2 our main result in Section 3 presents a way toconstruct unit norm tight frames that we call Hadamard-Tetris Construction In Section 4 we introduce another

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 917491 8 pageshttpdxdoiorg1011552014917491

2 Abstract and Applied Analysis

version of Hadamard-Tetris which can be used to constructunit norm frames with different eigenvalues for the frameoperator We give necessary condition for this version ofHadamard-Tetris to produce the desired frames

2 Special Hadamard-Tetris Construction

In [11] by using the Schur-Horn Theorem the authors haveshown how to construct every possible frame in which frameoperator has a given spectrum and in which vectors are ofgiven prescribed norms but their method is complex In[12] the authors have presented another construction of unitnorm frames which is called ldquoSpectral Tetris Constructionrdquo(STC) Spectral Tetris is a flexible and elementary methodto construct unit norm frames with a given frame operatorhaving all of its eigenvalues greater than or equal to two Butthat way is limited to the case of frames with119872 elements forR119873 in which119872 ge 2119873 However they extended the existingconstruction to unit norm tight frame of redundancy lessthan two [13] but those frames are in C119873 not in R119873

In this paper we implement an algorithmwhich is namedHadamard-Tetris for the construction of unit norm tightframes Hadamard-Tetris constructs synthesis matrix 119873 times

119872 with unit norm columns in which rows are pairwiseorthogonal and square sum to 119872119873 By this constructionthe frame operator is (119872119873)119868 The main advantage here isgiven an elementary and easily implementable algorithm forconstructing frames

In our way the assumption on the spectrum of the unitnorm tight frame (119872119873 ge 2) in STC can be dropped andconstruct unit norm tight frame for R119873 with 119872 elementswhere119872 and119873 are positive integers and satisfy the followingcondition

If we can decompose119872 as 119870 = 119872 minus 119873 + 1 summation119889119894(1 le 119894 le 119870) that is 119872 = sum

119870

119894=1119889119894 which there exist

Hadamardmatrices of sizes 119889119894rsquos and they satisfy the condition

(119894 minus 1)119872

119872 minus119873le

119894

sum

119895=1

119889119895lt 119894

119872

119872 minus1198731 le 119894 le 119870 minus 1 (5)

then there exists a unit norm tight frame forR119873 with sparsityat most sum119870

119895=11198892

119895

A frame constructed via the Hadamard-Tetris construc-tion is called Hadamard-Tetris frame

Aside from the fact that Hadamard-Tetris frames are easyto construct their major advantage for applications is thesparsity of their synthesis matricesThis sparsity is dependenton the decomposition of119872

In our construction for any positive integers 119872 and119873 (119872 gt 119873) we put 119870 = 119872 minus 119873 + 1 and if we coulddecompose 119872 as 119870 summation of power of two then wegive a unit normalized tight frame for R119873 with119872 elementsHence 119870 has minimum value If119873 = sum

119905

119894=01198861198942119894 is the binary

representation of119873 where 119886119894rsquos are 0 or 1 then sum119905

119894=0119886119894le 119870

HTC(I) Hadamard-Tetris constructionParameter

(i) Dimension119873 isin NAlgorithm

(1) For 119895 = 1 to119873 minus 1

(2) 119909119895= 1 minus

119895

119873

(3) 1198912119895minus1

= radic119909119895119890119895 + radic1 minus 119909119895119890119895+1

(4) 1198912119895= radic119909119895119890119895 minus radic1 minus 119909119895119890119895+1

(5) endOutput

(i) unit norm tight frame 1198911198952119873minus2

119895=1for R119873

Algorithm 1 The HTC(I) algorithm for constructing a unit normtight frame with (2119873 minus 2)-elements for R119873

In special case when 119872 = 2119873 minus 2 we can constructsynthesismatrix119879lowast of size119873times119872which has119891

119894rsquos as its columns

as in Algorithm 1

Theorem 1 Let 119873 be a positive integer number Then thesparsest synthesis matrix of the (2119873 minus 2)-element Hadamard-Tetris unit norm tight frame for R119873 which consists of 119873 minus 1

blocks of size 2 can be constructed by HTC(I) and its sparsity is4(119873 minus 1)

Proof Let 119872 = 2119873 minus 2 By the construction of HTC(I) inlines (3) and (4) 119891

119894rsquos are normalized We should show that

the square norm of each row of synthesis matrix is119872119873 Itis clear that the square norm of the first row of the matrix is119872119873 and in the last row since 119909

119873minus1= 1119873 the square norm

of the last row is

2 (1 minus1

119873) =

2119873 minus 2

119873=119872

119873 (6)

Also for 2 le 119895 le 119873 minus 1 the square norm of 119895th row is

2 (1 minus 119909119895minus1) + 2119909

119895= 2 (

119895 minus 1

119873) + 2 (1 minus

119895

119873)

=2119873 minus 2

119873=119872

119873

(7)

In this construction synthesis matrix of frame consists of119873 minus 1 blocks of size 2 so the sparsity of the matrix is 4(119873 minus1)

In [14] Casazza et al proved that the algorithm STCcan be performed to generate a unit norm tight frame of119872vectors in R119873 if and only if 120582 = 119872119873 or 120582 is of the form120582 = (2119871 minus 1)119871 for some positive integer 119871 Whereas weexhibit an algorithm that utilizes matrices of size 2 such as

[

[

radic119909 radic119909

radic1 minus 119909 minusradic1 minus 119909

]

]

(8)

Abstract and Applied Analysis 3

used in the construction of STC For example for119872 = 4 and119873 = 3 the matrix

[[[[[[[[[[

[

radic2

3radic2

30 0

radic1

3minusradic

1

3radic1

3radic1

3

0 0 radic2

3minusradic

2

3

]]]]]]]]]]

]

(9)

is synthesis matrix of R3

Example 2 If119873 = 4 and119872 = 6 then

[[[[[[[[[[[[[[[[

[

radic3

4radic3

40 0 0 0

radic1

4minusradic

1

4radic1

2radic1

20 0

0 0 radic1

2minusradic

1

2radic1

4radic1

4

0 0 0 0 radic3

4minusradic

3

4

]]]]]]]]]]]]]]]]

]

(10)

is synthesis of 6-elements for R4

3 Hadamard-Tetris Frame

In this section we provide a new method for constructingfinite normalized tight frames (FNTFs) In brief we want toconstruct119873 times119872 synthesis matrix 119879lowast which has

(i) columns of unit norm

(ii) orthogonal rows meaning that the frame operator119879lowast119879 is diagonal

(iii) rows of constant norm meaning that 119879lowast119879 is a con-stant multiple of the identity matrix

The Hadamard-Tetris Construction (HTC) is capable ofconstructing unit norm tight frames with the number ofelements that decompose as 119889

119894(1 le 119894 le 119872 minus 119873 +

1) It constructs the synthesis matrices of such frames bysuccessively filling of size 119889

119894of 119872 minus 119873 + 1 blocks In this

construction we use altered Hadamard matricesThe importance of Hadamard matrices to our construc-

tion stems from the fact that they have orthogonal rows andcolumns and that all entries have the same modulus In thecourse of the construction we will have to alter the rownorms of the Hadamard matrices by multiplying rows withappropriate constants While this will destroy the pairwiseorthogonality of the columns it will preserve the pairwiseorthogonality of the rows which is the crucial feature for ourconstruction to work

Definition 3 Given positive integers 119872 and 119873 we denotea matrix by 119867

119889or call it a 119867

119889block if it is derived from a

Hadamard matrix of size 119889 by multiplying the entries of the119895th row of Hadamard matrix byradic119872119873119889 for 119895 = 2 119889 minus 1and the entries of the first and the last row of Hadamardmatrix by 119903

1and 119903119889and if it has normalized columns We

call 1199031and 119903119889the first and the last correction factor of 119867

119889

respectively

Note that the row norm of a 119867119889block equals radic119872119873

except possibly for the first and the last rows We wantto present an algorithm following the lines of Example 4that is we want to compose the desired synthesis matrixwhich consists of square blocks in which first and last rowssuccessively overlap If we use 119870 square matrix as buildingblocks of the 119873 times 119872 synthesis matrix then due to theoverlapping we have119872 = 119873 + (119870 minus 1)

It is perhapsmost instructive to first look at the example ofthe construction thatwe are going to introduce in this section

Example 4 We construct a 6-elements unit norm tight framein R5 We can start filling the desired 5 times 6 synthesis matrixwith an altered 4 times 4 Hadamard matrix in the upper leftcorner The alteration we make is to multiply the entries ofthe first row by radic310 in size to make the first row have thedesired norm radic65 We multiply the second and the thirdrows of 4 times 4 Hadamard matrix by radic310 to make the normof those equal toradic65

To get normalized column we multiply the forth row ofthe 4 times 4 Hadamard matrix by radic110 At this point we haveconstructed the first three rows and the first four columns ofthe desired synthesis matrix

[[[[[[[[[[[[[[[[[[[[

[

radic3

10radic3

10minusradic

3

10radic3

100 0

minusradic3

10radic3

10minusradic

3

10minusradic

3

100 0

radic3

10minusradic

3

10minusradic

3

10minusradic

3

100 0

radic1

10radic1

10radic1

10minusradic

1

10sdot sdot

0 0 0 0 sdot sdot

]]]]]]]]]]]]]]]]]]]]

]

(11)

Note that so far we have constructed a matrix in whichfirst four rows are orthogonal no matter how we keep fillingthe first four rows The fourth row at this point has normradic25 while we need to make it have normradic65

We can insert altered 2 times 2Hadamard matrix in the samefashion as above To do this we wouldmultiply its first row bythe factor radic25 in size to have the forth row of the synthesis

4 Abstract and Applied Analysis

matrix square sum to 65 and its second row by the factorradic35 to get

[[[[[[[[[[[[[[[[[[

[

radic3

10radic3

10minusradic

3

10radic3

100 0

minusradic3

10radic3

10minusradic

3

10minusradic

3

100 0

radic3

10minusradic

3

10minusradic

3

10minusradic

3

100 0

radic1

10radic1

10radic1

10minusradic

1

10radic2

5minusradic

2

5

0 0 0 0 radic3

5radic3

5

]]]]]]]]]]]]]]]]]]

]

(12)

The latter matrix is the synthesis matrix of the desired framesince its columns are normalized and its rows are pairwiseorthogonal and square sum to 65

The next theorem is the main theorem of this section

Theorem 5 (main theorem) Let 119872 and 119873 (119872 gt 119873) bepositive integers and 119870 = 119872 minus 119873 + 1 Furthermore assumethat119872 can be decomposed as119872 = sum

119870

119895=1119889119895such that for each

1 le 119894 le 119870 minus 1

(119894 minus 1)119872

119872 minus119873le

119894

sum

119895=1

119889119895lt 119894

119872

119872 minus119873 (13)

and there exist Hadamard matrices of sizes 119889119894rsquos Then HTC(II)

gives a unit normalized tight frame for R119873 with119872 elements

In special case when119872 minus 119873 = 1 then the condition on119889119894rsquos is confirmed automatically Indeed if 119889

1is selected such

that 0 lt 1198891lt 119872 then 0 lt 119889

2= 119872 minus 119889

1lt 119872

Corollary 6 If119872 = 1198891+ 1198892such that there exist Hadamard

matrices of sizes 1198891and 119889

2 then the synthesis matrix of the119872

elements for R119872minus1 can be constructed via HTC(II)

To proveTheorem 5 we need to collect some informationabout the first and the last correction factors of119867

119889119894rsquos

In construction of synthesis matrix of size119873times119872 we use119870 = 119872 minus 119873 + 1 blocks of sizes 119889

119894rsquos (1 le 119894 le 119870) where119872 =

sum119870

119894=1119889119894

Aswasmentioned each block is119867119889119894that derived from the

Hadamard matrix of size 119889119894 If the factor that multiplies the

entries of the 119895th row of Hadamard matrix of size 119889119894is shown

by 119903119895119894 where 1 le 119895 le 119889

119894 then for each block of size 119889

119894 the

correction factor is

1199032

119895119894=119872

119873119889119894

(2 le 119895 le 119889119894minus 1) (14)

The last row of a block and the first row of the followingblock appear in the same row of the synthesis matrix andwhile the last correction factor is chosen to ensure normalizedcolumns the following first correction factor is chosen to

guarantee that the rows of the synthesis matrix have squarenorm119872119873

Hence we have the following relationships for 1199031119894

and119903119889119894 119894 where 1 le 119894 le 119870

Lemma 7 If 1199031119894

and 119903119889119894 119894

are the first and the last correctionfactors respectively then

1199032

119889119894 119894= 1 minus 119903

2

1119894minus119872

119873+ 2

119872

119873119889119894

1199032

1119894+1=

1

119889119894+1

(119872

119873minus 1198891198941199032

119889119894 119894)

(15)

Proof The correction factors of second row to (119889119894minus 1)th row

of119867119889119894are

119903119895119894= radic

119872

119873119889119894

(2 le 119895 le 119889119894minus 1) (16)

Since norm of each column of119867119889119894must be one we have

1199032

1119894+ (119889119894minus 2)

119872

119873119889119894

+ 1199032

119889119894 119894= 1 (17)

and so on

1199032

119889119894 119894= 1 minus 119903

2

1119894minus119872

119873+ 2

119872

119873119889119894

(18)

Also the last correction factor of 119867119889119894

and the firstcorrection factor of119867

119889119894+1must be chosen such that the square

norm of that row is119872119873 Hence

1198891198941199032

119889119894 119894+ 119889119894+11199032

1119894+1=119872

119873(19)

and so on

1199032

1119894+1=

1

119889119894+1

(119872

119873minus 1198891198941199032

119889119894 119894) (20)

By combining formulas (15) in Lemma 7 we get thefollowing recursive relations

1199032

1119894+1=119889119894

119889119894+1

(119872

119873+ 1199032

1119894minus 1) minus

119872

119873119889119894+1

1199032

119889119894+1 119894+1=119889119894

119889119894+1

1199032

119889119894 119894minus119872 minus119873

119873+

119872

119873119889119894+1

(21)

for 1 le 119894 le 119870 minus 1

Lemma 8 If 1199031119894

and 119903119889119894 119894

are the first and the last correctionfactors of119867

119889119894 respectively then

1199032

1119894=

1

119873119889119894

((119872 minus119873)

119894minus1

sum

119895=1

119889119895minus (119894 minus 2)119872) (119894 ge 2)

1199032

119889119894 119894=

1

119873119889119894

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895) (119894 ge 1)

(22)

Abstract and Applied Analysis 5

Proof We prove these by induction on 119894 The first steps of theinduction are trivially true Assume that the identities are truefor 119894 For the case 119894 + 1 the following identities hold by usingformulas (21) and the induction hypothesis

1199032

1119894+1=119889119894

119889119894+1

(119872

119873+ 1199032

1119894minus 1) minus

119872

119873119889119894+1

=119889119894

119889119894+1

(119872

119873+(

sum119894minus1

119895=1119889119895

119889119894

)(119872 minus119873

119873)

minus(119894 minus 2

119889119894

)(119872

119873) minus 1) minus

119872

119873119889119894+1

=1

119873119889119894+1

(119889119894119872+ (119872 minus119873)

119894minus1

sum

119895=1

119889119895

minus (119894 minus 2)119872 minus 119873119889119894minus119872)

=1

119873119889119894+1

((119872 minus119873)

119894

sum

119895=1

119889119895minus (119894 minus 1)119872)

(23)

1199032

119889119894+1119894+1=119889119894

119889119894+1

1199032

119889119894 119894minus119872 minus119873

119873+

119872

119873119889119894+1

=119889119894

119889119894+1

((119894

119889119894

)(119872

119873) minus (

sum119894

119895=1119889119895

119889119894

)(119872 minus119873

119873))

minus119872 minus119873

119873+

119872

119873119889119894+1

=1

119873119889119894+1

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895

minus 119889119894+1 (119872 minus 119873) +119872)

=1

119873119889119894+1

((119894 + 1)119872 minus (119872 minus119873)

119894+1

sum

119895=1

119889119895)

(24)

Now we are ready to prove the mainTheorem 5

Proof We check that synthesis matrix constructed byHTC(II) has row norm equal radic119872119873 and columns arenormalized First we show that each row of synthesis matrixwhich is constructed via HTC(II) has square norm119872119873 Itis enough to show that the last row of 119867

119889119894together with the

first row of119867119889119894+1

makes square norm119872119873 In other words

1198891198941199032

119889119894 119894+ 119889119894+11199032

1119894+1=119872

119873 (25)

By Lemma 8 we have

1198891198941199032

119889119894 119894+ 119889119894+11199032

1119894+1=1

119873(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895

+ (119872 minus119873)

119894

sum

119895=1

119889119895minus (119894 minus 1)119872)

=119872

119873

(26)

Now we show each column of synthesis matrix that isconstructed with HTC(II) has norm 1 that is

1199032

1119894+ (119889119894minus 2)

119872

119873119889119894

+ 1199032

119889119894 119894= 1 (27)

By Lemma 8 we have

1199032

1119894+ 1199032

119889119894 119894=

1

119873119889119894

((119872 minus119873)(

119894minus1

sum

119895=1

119889119895minus

119894

sum

119895=1

119889119895)

+ 119872(minus119894 + 2 + 119894))

= minus(119872 minus119873

119873) + 2

119872

119873119889119894

= 1 minus (119889119894minus 2)

119872

119873119889119894

(28)

We have to ensure that the last correction factor of thefinal block inserted into the synthesismatrix equalsradic119872119873119889

119870

(in other words the next first correction factor would bezero but we have arrived at this point where the algorithmterminates)

In Algorithm 2 line (9) when 119894 = 119870 and 119905 = 119889119870 we have

119903119889119870119870

= radic1

119873119889119870

(119870119872 minus (119872 minus119873)

119870

sum

119895=1

119889119895) = radic

119872

119873119889119870

(29)

and in line (4) for 119894 = 119870 + 1 we have 1199031119889119870+1

= 0

Note that if we do a HTC(II) as in Example 4 with119870 altered Hadamard matrices of sizes 119889

1 119889

119870 then the

sparsity of the synthesis matrix is sum119870119894=11198892

119894

The next lemma states that if we construct synthesismatrix with blocks of sizes 119889

1 1198892 119889

119870 each condition on

119889119894must be established for 119889

119870minus119894+1and vice versa

Lemma 9 Let 119872 and 119873 be positive integers with 119872 gt 119873119870 = 119872 minus 119873 + 1 and119872 = sum

119870

119895=1119889119895such that 119889

119895rsquos are satisfied

in inequalities

(119894 minus 1)119872

119872 minus119873lt

119894

sum

119895=1

119889119895lt 119894

119872

119872 minus119873

(0 le 119894 le 119870 minus 1)

(30)

6 Abstract and Applied Analysis

HTC(II) Hadamard-Tetris ConstructionParameters

(i) Dimension119873 isin N(ii) Number of frame elements119872 isin N such that decompose as119872 = sum

119870

119894=1119889119894 where 119870 = 119872 minus119873 + 1

(iii) Hadamard matrices119867119889119894= [119886(119889119894)

119896119895]1le119896119895le119889119894

of size 119889119894for 119894 = 1 119870

Algorithm(1) 119904 = 0119898 = 0(2) For 119894 = 1 to 119870 do(3) For 119905 = 1 to 119889

119894do

(4) 1199031119894= radic

1

119873119889119894

((119872 minus119873)

119894minus1

sum

119895=1

119889119895minus (119894 minus 2)119872)

(5) 11990311= radic

119872

1198731198891

(6) For 119895 = 2 to 119889119894minus 1 do

(7) 119903119895119894= radic

119872

119873119889119894

(8) end

(9) 119903119889119894 119894= radic

1

119873119889119894

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895)

(10) 119891119905+119898

=

119889119894

sum

119895=1

119886(119889119894)

119905119895119903119895119894119890119904+119895

(11) end(12) 119898 = 119898 + 119889

119894

(13) 119904 = 119904 + 119889119894minus 1

(14) endOutput

(i) unit norm tight frame 119891119895119872

119895=1for R119873

Algorithm 2 The HTC(II) algorithm for constructing a unit norm tight frame with119872 elements for R119873

Then

(119894 minus 1)119872

119872 minus119873lt

119894

sum

119895=1

119889119870minus119895+1

lt 119894119872

119872 minus119873

(0 le 119894 le 119870 minus 1)

(31)

Proof Since sum119870119895=1119889119895= 119872 so

119894

sum

119895=1

119889119870minus119895+1

=

119870

sum

119895=1

119889119895minus

119870minus119894

sum

119895=1

119889119895= 119872 minus

119870minus119894

sum

119895=1

119889119895 (32)

In the other hand by replacing 119894 with 119870 minus 119894 in the inequality(30) we have

119872

119872minus119873(119870 minus 119894 minus 1) lt

119870minus119894

sum

119895=1

119889119895lt (119870 minus 119894)

119872

119872 minus119873(33)

and so on

119872minus (119870 minus 119894)119872

119872 minus119873lt 119872 minus

119870minus119894

sum

119895=1

119889119895lt 119872 minus

119872

119872minus119873

times (119870 minus 119894 minus 1)

(34)

Hence we have

119872

119872minus119873(119872 minus119873 + 119894 minus 119870) lt

119894

sum

119895=1

119889119870minus119895+1

lt119872

119872 minus119873

times (119872 minus119873 minus 119870 + 119894 + 1)

(35)

which give inequalities (31)

Following Lemma 9 if 119879lowast = [119886119894119895] is a synthesis matrix of

size119873times119872 then 119865lowast = [119887119894119895] where 119887

119894119895= 119886(119873minus119894+1)(119872minus119895+1)

is alsohaving the same situation

Since Hadamard matrices of the size power of 2 exist sowe use altered Hadamard matrices of size 2119904119894 where

2119904119894 = max

2119904| (119894 minus 1)

119872

119872 minus119873le

119894minus1

sum

119895=1

2119904119895 + 2119904lt 119894

119872

119872 minus119873

(36)

for 1 le 119894 le 119870 minus 1 However it is provided that there exists aHadamard matrix of size 119889

119870times 119889119870 with 119889

119870= 119872 minus sum

119870minus1

119895=12119904119895

In this case the algorithmHTC builds the synthesis matrix ofthe tight frame as in Example 4 by inserting119867

119889119894 where 119889

119894=

2119904119894 (1 le 119894 le 119870 minus 1) and 119889

119870= 119872 minus sum

119870minus1

119895=12119904119895

Abstract and Applied Analysis 7

SHTC Spectral Hadamard-Tetris constructionParameters

(i) Dimension119873 isin N(ii) Number of frame elements119872 isin N such that decompose as119872 = sum

119870

119894=1119889119894where 119870 = 119872 minus119873 + 1

(iii) eigenvalues 120582119899119873

119899=1which sum119873

119899=1120582119899= 119872 and sum119905119894minus119894

119895=1120582119895le 119905119894le sum119905119894minus119894+1

119895=1120582119895where 119905

119894= sum119894

119895=1119889119895and 1 le 119894 le 119870

(iv) Hadamard matrices119867119889119894= [119886(119889119894)

119896119895]1le119896119895le119889119894

of size 119889119894for 119894 = 1 119870

Algorithm(1) 119904 = 0119898 = 0 119905

0= 0

(2) For 119894 = 1 to 119870 do

(3) 119905119894=

119894

sum

119895=1

119889119895

(4) For 119905 = 1 to 119889119894do

(5) 1199031119894= radic

1

119889119894

(

119905119894minus1minus(119894minus2)

sum

119895=1

120582119895minus 119905119894minus1)

(6) For 119895 = 2 to 119889119894minus 1 do

(7) 119903119895119894= radic

1

119889119894

120582119905119894minus1minus(119894minus1)+119895

(8) end

(9) 119903119894119894= radic

1

119889119894

(119905119894minus

119905119894minus119894

sum

119895=1

120582119895)

(10) 119891119905+119898

=

119889119894

sum

119895=1

119886(119889119894)

119905119895119903119895119894119890119904+119895

(11) end(12) 119898 = 119898 + 119889

119894

(13) 119904 = 119904 + 119889119894minus 1

(14) endOutput

(i) unit norm frame 119891119895119872

119895=1for R119873 with eigenvalues 120582

119899119873

119899=1

Algorithm 3 The SHTC algorithm for constructing a unit norm tight frame with119872 elements for R119873

4 Spectral Hadamard Tetris Construction

Another case of HTC is the case of unit norm but notnecessarily tight frames Such frames are known to existprovided that the eigenvalues of the frame operator sumup to the number of frame vectors [15] In this section wegive a version of Hadamard-Tetris Construction which usesaltered Hadamard matrices to construct unit norm frameswith a given spectrum of frame operator Ourmethod cannotconstruct all such framesWe give necessary condition underwhich this constructionworks To construct unit norm framewith prescribed spectrum we are looking for a matrix withcolumns square summing up to 1 and rows square sum-ming up to 120582

119899119873

119899=1 In Algorithm 3 we present a modified

Hadamard-Tetris algorithm HTC allowing the constructionof such frames

Given a fixed dimension 119873 and frame cardinality 119872 anecessary condition on the prescribed eigenvalues 120582

119899119873

119899=1of

the frame operator for SHTC towork is given in the followingtheorem

Theorem 10 Let sum119873119899=1120582119899= 119872 = sum

119870

119894=1119889119894 where 119870 = 119872 minus

119873 + 1 and there exist Hadamard matrices of sizes 119889119894rsquos Then

SHTC can be used to produce unit norm frame for R119873 with

eigenvalues 120582119899119873

119899=1if there are some permutations of 120582

119899119873

119899=1

and 119889119894119870

119894=1such that for each 119894 = 1 119870 we have

119905119894minus119894

sum

119895=1

120582119895le 119905119894le

119905119894minus119894+1

sum

119895=1

120582119895 (37)

where 119905119894= sum119894

119895=1119889119895

Here we present an example in which STC does not workwhereas SHTC works

Example 11 There are choices of prescribed eigenvalueswhich satisfy the conditions of Theorem 10 that unit normframe with given eigenvalues exists but STC cannot be usedto construct such a frame because no rearrangement ofeigenvalues satisfies in their conditions An example of thiskind is 120582

1198994

119899=1= 04 24 11 11 as eigenvalues of frame

with 5 elements forR4

Sometimes one rearrangement of eigenvalues anddimensions of Hadamard matrices is satisfied in desiredcondition while another is not For example the SHTCcannot be performed for the sequence of 120582

119894rsquos is the given

8 Abstract and Applied Analysis

rearrangement 04 11 11 24 and 1198891= 4 119889

2= 1

However rearranging the eigenvalues to the rearrangement04 24 11 11 allows SHTC to construct the desired frameIn this case its synthesis matrix will be

[[[[[[[[[[[[[[

[

radic1

10radic1

10minusradic

1

10radic1

100

minusradic6

10radic6

10minusradic

6

10minusradic

6

100

radic11

40minusradic

11

40minusradic

11

40minusradic

11

400

radic1

40radic1

40radic1

40minusradic

1

401

]]]]]]]]]]]]]]

]

(38)

So there is no reason that SHTC works when theeigenvalues are in the monotonic rearrangement

Note that since we make no assumption about the order-ing of the sequences 120582

119899119873

119899=1and 119889

119894119870

119894=1 we may permute the

elements of the sequences in order to make them ready tosatisfy in the inequality (37)

Given sequences of eigenvalues and 119889119894119870

119894=1 where 119870 =

119872 minus 119873 + 1 and there exist Hadamard matrices of sizes119889119894rsquos it may be time-consuming to find permutation of these

sequences which satisfy in Theorem 10 Until now we couldnot find simple conditions on these sequences that SHTC canbe performed It has been a mystery as to when HadamardTetris works and when fails

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthors are grateful to ProfessorHossein Jafari the Editorof AAA and the referee for their valuable comments

References

[1] R J Duffin andA C Schaeffer ldquoA class of nonharmonic Fourierseriesrdquo Transactions of the American Mathematical Society vol72 pp 341ndash366 1952

[2] J Kovacevic and A Chebira ldquoLife beyond bases the advent offrames (Parts I and II)rdquo IEEE Signal Processing Magazine vol24 no 5 pp 115ndash125 2007

[3] E Soljanin ldquoTight frames in quantum information theoryrdquo inProceedings of the DIMACS Workshop on Source Coding andHarmonic Analysis Rutgers NJ USA May 2002

[4] T Strohmer and J Heath ldquoGrassmannian frames with appli-cations to coding and communicationrdquo Applied and Computa-tional Harmonic Analysis vol 14 no 3 pp 257ndash275 2003

[5] J J Benedetto and M Fickus ldquoFinite normalized tight framesrdquoAdvances in Computational Mathematics vol 18 no 2-4 pp357ndash385 2003

[6] N Cotfas and J P Gazeau ldquoFinite tight frames and some appli-cationsrdquo Journal of Physics AMathematical andTheoretical vol43 no 19 Article ID 193001 2010

[7] D F Li and W C Sun ldquoExpansion of frames to tight framesrdquoActa Mathematica Sinica (English Series) vol 25 no 2 pp 287ndash292 2009

[8] J C Tremain ldquoAlgorithmic constructions of unitary matricesand tight framesrdquo httparxivorgabs11044539v1

[9] D-J Feng L Wang and Y Wang ldquoGeneration of finitetight frames by Householder transformationsrdquo Advances inComputational Mathematics vol 24 no 1ndash4 pp 297ndash309 2006

[10] O Christensen Frames and Bases An Introductory CourseBirkhauser Boston Mass USA 2008

[11] J Cahill M Fickus D G Mixon M J Poteet and N StrawnldquoConstructing finite frames of a given spectrum and set oflengthsrdquoApplied and Computational Harmonic Analysis vol 35no 1 pp 52ndash73 2013

[12] P G Casazza A Heinecke F Krahmer and G KutyniokldquoOptimally sparse framesrdquo IEEE Transactions on InformationTheory vol 57 no 11 pp 7279ndash7287 2011

[13] P G Casazza M Fickus A Heinecke Y Wang and Z ZhouldquoSpectral tetris fusion frame constructionsrdquo The Journal ofFourier Analysis and Applications vol 18 no 4 pp 828ndash8512012

[14] P G Casazza AHeinecke K Kornelson YWang andZ ZhouldquoNecessary and sufficient conditions to perform spectral tetrisrdquoLinear Algebra and its Applications vol 438 no 5 pp 2239ndash2255 2013

[15] K Dykema D Freeman K Kornelson D Larson M Ordowerand E Weber ldquoEllipsoidal tight frames and projection decom-positions of operatorsrdquo Illinois Journal of Mathematics vol 48no 2 pp 477ndash489 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

version of Hadamard-Tetris which can be used to constructunit norm frames with different eigenvalues for the frameoperator We give necessary condition for this version ofHadamard-Tetris to produce the desired frames

2 Special Hadamard-Tetris Construction

In [11] by using the Schur-Horn Theorem the authors haveshown how to construct every possible frame in which frameoperator has a given spectrum and in which vectors are ofgiven prescribed norms but their method is complex In[12] the authors have presented another construction of unitnorm frames which is called ldquoSpectral Tetris Constructionrdquo(STC) Spectral Tetris is a flexible and elementary methodto construct unit norm frames with a given frame operatorhaving all of its eigenvalues greater than or equal to two Butthat way is limited to the case of frames with119872 elements forR119873 in which119872 ge 2119873 However they extended the existingconstruction to unit norm tight frame of redundancy lessthan two [13] but those frames are in C119873 not in R119873

In this paper we implement an algorithmwhich is namedHadamard-Tetris for the construction of unit norm tightframes Hadamard-Tetris constructs synthesis matrix 119873 times

119872 with unit norm columns in which rows are pairwiseorthogonal and square sum to 119872119873 By this constructionthe frame operator is (119872119873)119868 The main advantage here isgiven an elementary and easily implementable algorithm forconstructing frames

In our way the assumption on the spectrum of the unitnorm tight frame (119872119873 ge 2) in STC can be dropped andconstruct unit norm tight frame for R119873 with 119872 elementswhere119872 and119873 are positive integers and satisfy the followingcondition

If we can decompose119872 as 119870 = 119872 minus 119873 + 1 summation119889119894(1 le 119894 le 119870) that is 119872 = sum

119870

119894=1119889119894 which there exist

Hadamardmatrices of sizes 119889119894rsquos and they satisfy the condition

(119894 minus 1)119872

119872 minus119873le

119894

sum

119895=1

119889119895lt 119894

119872

119872 minus1198731 le 119894 le 119870 minus 1 (5)

then there exists a unit norm tight frame forR119873 with sparsityat most sum119870

119895=11198892

119895

A frame constructed via the Hadamard-Tetris construc-tion is called Hadamard-Tetris frame

Aside from the fact that Hadamard-Tetris frames are easyto construct their major advantage for applications is thesparsity of their synthesis matricesThis sparsity is dependenton the decomposition of119872

In our construction for any positive integers 119872 and119873 (119872 gt 119873) we put 119870 = 119872 minus 119873 + 1 and if we coulddecompose 119872 as 119870 summation of power of two then wegive a unit normalized tight frame for R119873 with119872 elementsHence 119870 has minimum value If119873 = sum

119905

119894=01198861198942119894 is the binary

representation of119873 where 119886119894rsquos are 0 or 1 then sum119905

119894=0119886119894le 119870

HTC(I) Hadamard-Tetris constructionParameter

(i) Dimension119873 isin NAlgorithm

(1) For 119895 = 1 to119873 minus 1

(2) 119909119895= 1 minus

119895

119873

(3) 1198912119895minus1

= radic119909119895119890119895 + radic1 minus 119909119895119890119895+1

(4) 1198912119895= radic119909119895119890119895 minus radic1 minus 119909119895119890119895+1

(5) endOutput

(i) unit norm tight frame 1198911198952119873minus2

119895=1for R119873

Algorithm 1 The HTC(I) algorithm for constructing a unit normtight frame with (2119873 minus 2)-elements for R119873

In special case when 119872 = 2119873 minus 2 we can constructsynthesismatrix119879lowast of size119873times119872which has119891

119894rsquos as its columns

as in Algorithm 1

Theorem 1 Let 119873 be a positive integer number Then thesparsest synthesis matrix of the (2119873 minus 2)-element Hadamard-Tetris unit norm tight frame for R119873 which consists of 119873 minus 1

blocks of size 2 can be constructed by HTC(I) and its sparsity is4(119873 minus 1)

Proof Let 119872 = 2119873 minus 2 By the construction of HTC(I) inlines (3) and (4) 119891

119894rsquos are normalized We should show that

the square norm of each row of synthesis matrix is119872119873 Itis clear that the square norm of the first row of the matrix is119872119873 and in the last row since 119909

119873minus1= 1119873 the square norm

of the last row is

2 (1 minus1

119873) =

2119873 minus 2

119873=119872

119873 (6)

Also for 2 le 119895 le 119873 minus 1 the square norm of 119895th row is

2 (1 minus 119909119895minus1) + 2119909

119895= 2 (

119895 minus 1

119873) + 2 (1 minus

119895

119873)

=2119873 minus 2

119873=119872

119873

(7)

In this construction synthesis matrix of frame consists of119873 minus 1 blocks of size 2 so the sparsity of the matrix is 4(119873 minus1)

In [14] Casazza et al proved that the algorithm STCcan be performed to generate a unit norm tight frame of119872vectors in R119873 if and only if 120582 = 119872119873 or 120582 is of the form120582 = (2119871 minus 1)119871 for some positive integer 119871 Whereas weexhibit an algorithm that utilizes matrices of size 2 such as

[

[

radic119909 radic119909

radic1 minus 119909 minusradic1 minus 119909

]

]

(8)

Abstract and Applied Analysis 3

used in the construction of STC For example for119872 = 4 and119873 = 3 the matrix

[[[[[[[[[[

[

radic2

3radic2

30 0

radic1

3minusradic

1

3radic1

3radic1

3

0 0 radic2

3minusradic

2

3

]]]]]]]]]]

]

(9)

is synthesis matrix of R3

Example 2 If119873 = 4 and119872 = 6 then

[[[[[[[[[[[[[[[[

[

radic3

4radic3

40 0 0 0

radic1

4minusradic

1

4radic1

2radic1

20 0

0 0 radic1

2minusradic

1

2radic1

4radic1

4

0 0 0 0 radic3

4minusradic

3

4

]]]]]]]]]]]]]]]]

]

(10)

is synthesis of 6-elements for R4

3 Hadamard-Tetris Frame

In this section we provide a new method for constructingfinite normalized tight frames (FNTFs) In brief we want toconstruct119873 times119872 synthesis matrix 119879lowast which has

(i) columns of unit norm

(ii) orthogonal rows meaning that the frame operator119879lowast119879 is diagonal

(iii) rows of constant norm meaning that 119879lowast119879 is a con-stant multiple of the identity matrix

The Hadamard-Tetris Construction (HTC) is capable ofconstructing unit norm tight frames with the number ofelements that decompose as 119889

119894(1 le 119894 le 119872 minus 119873 +

1) It constructs the synthesis matrices of such frames bysuccessively filling of size 119889

119894of 119872 minus 119873 + 1 blocks In this

construction we use altered Hadamard matricesThe importance of Hadamard matrices to our construc-

tion stems from the fact that they have orthogonal rows andcolumns and that all entries have the same modulus In thecourse of the construction we will have to alter the rownorms of the Hadamard matrices by multiplying rows withappropriate constants While this will destroy the pairwiseorthogonality of the columns it will preserve the pairwiseorthogonality of the rows which is the crucial feature for ourconstruction to work

Definition 3 Given positive integers 119872 and 119873 we denotea matrix by 119867

119889or call it a 119867

119889block if it is derived from a

Hadamard matrix of size 119889 by multiplying the entries of the119895th row of Hadamard matrix byradic119872119873119889 for 119895 = 2 119889 minus 1and the entries of the first and the last row of Hadamardmatrix by 119903

1and 119903119889and if it has normalized columns We

call 1199031and 119903119889the first and the last correction factor of 119867

119889

respectively

Note that the row norm of a 119867119889block equals radic119872119873

except possibly for the first and the last rows We wantto present an algorithm following the lines of Example 4that is we want to compose the desired synthesis matrixwhich consists of square blocks in which first and last rowssuccessively overlap If we use 119870 square matrix as buildingblocks of the 119873 times 119872 synthesis matrix then due to theoverlapping we have119872 = 119873 + (119870 minus 1)

It is perhapsmost instructive to first look at the example ofthe construction thatwe are going to introduce in this section

Example 4 We construct a 6-elements unit norm tight framein R5 We can start filling the desired 5 times 6 synthesis matrixwith an altered 4 times 4 Hadamard matrix in the upper leftcorner The alteration we make is to multiply the entries ofthe first row by radic310 in size to make the first row have thedesired norm radic65 We multiply the second and the thirdrows of 4 times 4 Hadamard matrix by radic310 to make the normof those equal toradic65

To get normalized column we multiply the forth row ofthe 4 times 4 Hadamard matrix by radic110 At this point we haveconstructed the first three rows and the first four columns ofthe desired synthesis matrix

[[[[[[[[[[[[[[[[[[[[

[

radic3

10radic3

10minusradic

3

10radic3

100 0

minusradic3

10radic3

10minusradic

3

10minusradic

3

100 0

radic3

10minusradic

3

10minusradic

3

10minusradic

3

100 0

radic1

10radic1

10radic1

10minusradic

1

10sdot sdot

0 0 0 0 sdot sdot

]]]]]]]]]]]]]]]]]]]]

]

(11)

Note that so far we have constructed a matrix in whichfirst four rows are orthogonal no matter how we keep fillingthe first four rows The fourth row at this point has normradic25 while we need to make it have normradic65

We can insert altered 2 times 2Hadamard matrix in the samefashion as above To do this we wouldmultiply its first row bythe factor radic25 in size to have the forth row of the synthesis

4 Abstract and Applied Analysis

matrix square sum to 65 and its second row by the factorradic35 to get

[[[[[[[[[[[[[[[[[[

[

radic3

10radic3

10minusradic

3

10radic3

100 0

minusradic3

10radic3

10minusradic

3

10minusradic

3

100 0

radic3

10minusradic

3

10minusradic

3

10minusradic

3

100 0

radic1

10radic1

10radic1

10minusradic

1

10radic2

5minusradic

2

5

0 0 0 0 radic3

5radic3

5

]]]]]]]]]]]]]]]]]]

]

(12)

The latter matrix is the synthesis matrix of the desired framesince its columns are normalized and its rows are pairwiseorthogonal and square sum to 65

The next theorem is the main theorem of this section

Theorem 5 (main theorem) Let 119872 and 119873 (119872 gt 119873) bepositive integers and 119870 = 119872 minus 119873 + 1 Furthermore assumethat119872 can be decomposed as119872 = sum

119870

119895=1119889119895such that for each

1 le 119894 le 119870 minus 1

(119894 minus 1)119872

119872 minus119873le

119894

sum

119895=1

119889119895lt 119894

119872

119872 minus119873 (13)

and there exist Hadamard matrices of sizes 119889119894rsquos Then HTC(II)

gives a unit normalized tight frame for R119873 with119872 elements

In special case when119872 minus 119873 = 1 then the condition on119889119894rsquos is confirmed automatically Indeed if 119889

1is selected such

that 0 lt 1198891lt 119872 then 0 lt 119889

2= 119872 minus 119889

1lt 119872

Corollary 6 If119872 = 1198891+ 1198892such that there exist Hadamard

matrices of sizes 1198891and 119889

2 then the synthesis matrix of the119872

elements for R119872minus1 can be constructed via HTC(II)

To proveTheorem 5 we need to collect some informationabout the first and the last correction factors of119867

119889119894rsquos

In construction of synthesis matrix of size119873times119872 we use119870 = 119872 minus 119873 + 1 blocks of sizes 119889

119894rsquos (1 le 119894 le 119870) where119872 =

sum119870

119894=1119889119894

Aswasmentioned each block is119867119889119894that derived from the

Hadamard matrix of size 119889119894 If the factor that multiplies the

entries of the 119895th row of Hadamard matrix of size 119889119894is shown

by 119903119895119894 where 1 le 119895 le 119889

119894 then for each block of size 119889

119894 the

correction factor is

1199032

119895119894=119872

119873119889119894

(2 le 119895 le 119889119894minus 1) (14)

The last row of a block and the first row of the followingblock appear in the same row of the synthesis matrix andwhile the last correction factor is chosen to ensure normalizedcolumns the following first correction factor is chosen to

guarantee that the rows of the synthesis matrix have squarenorm119872119873

Hence we have the following relationships for 1199031119894

and119903119889119894 119894 where 1 le 119894 le 119870

Lemma 7 If 1199031119894

and 119903119889119894 119894

are the first and the last correctionfactors respectively then

1199032

119889119894 119894= 1 minus 119903

2

1119894minus119872

119873+ 2

119872

119873119889119894

1199032

1119894+1=

1

119889119894+1

(119872

119873minus 1198891198941199032

119889119894 119894)

(15)

Proof The correction factors of second row to (119889119894minus 1)th row

of119867119889119894are

119903119895119894= radic

119872

119873119889119894

(2 le 119895 le 119889119894minus 1) (16)

Since norm of each column of119867119889119894must be one we have

1199032

1119894+ (119889119894minus 2)

119872

119873119889119894

+ 1199032

119889119894 119894= 1 (17)

and so on

1199032

119889119894 119894= 1 minus 119903

2

1119894minus119872

119873+ 2

119872

119873119889119894

(18)

Also the last correction factor of 119867119889119894

and the firstcorrection factor of119867

119889119894+1must be chosen such that the square

norm of that row is119872119873 Hence

1198891198941199032

119889119894 119894+ 119889119894+11199032

1119894+1=119872

119873(19)

and so on

1199032

1119894+1=

1

119889119894+1

(119872

119873minus 1198891198941199032

119889119894 119894) (20)

By combining formulas (15) in Lemma 7 we get thefollowing recursive relations

1199032

1119894+1=119889119894

119889119894+1

(119872

119873+ 1199032

1119894minus 1) minus

119872

119873119889119894+1

1199032

119889119894+1 119894+1=119889119894

119889119894+1

1199032

119889119894 119894minus119872 minus119873

119873+

119872

119873119889119894+1

(21)

for 1 le 119894 le 119870 minus 1

Lemma 8 If 1199031119894

and 119903119889119894 119894

are the first and the last correctionfactors of119867

119889119894 respectively then

1199032

1119894=

1

119873119889119894

((119872 minus119873)

119894minus1

sum

119895=1

119889119895minus (119894 minus 2)119872) (119894 ge 2)

1199032

119889119894 119894=

1

119873119889119894

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895) (119894 ge 1)

(22)

Abstract and Applied Analysis 5

Proof We prove these by induction on 119894 The first steps of theinduction are trivially true Assume that the identities are truefor 119894 For the case 119894 + 1 the following identities hold by usingformulas (21) and the induction hypothesis

1199032

1119894+1=119889119894

119889119894+1

(119872

119873+ 1199032

1119894minus 1) minus

119872

119873119889119894+1

=119889119894

119889119894+1

(119872

119873+(

sum119894minus1

119895=1119889119895

119889119894

)(119872 minus119873

119873)

minus(119894 minus 2

119889119894

)(119872

119873) minus 1) minus

119872

119873119889119894+1

=1

119873119889119894+1

(119889119894119872+ (119872 minus119873)

119894minus1

sum

119895=1

119889119895

minus (119894 minus 2)119872 minus 119873119889119894minus119872)

=1

119873119889119894+1

((119872 minus119873)

119894

sum

119895=1

119889119895minus (119894 minus 1)119872)

(23)

1199032

119889119894+1119894+1=119889119894

119889119894+1

1199032

119889119894 119894minus119872 minus119873

119873+

119872

119873119889119894+1

=119889119894

119889119894+1

((119894

119889119894

)(119872

119873) minus (

sum119894

119895=1119889119895

119889119894

)(119872 minus119873

119873))

minus119872 minus119873

119873+

119872

119873119889119894+1

=1

119873119889119894+1

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895

minus 119889119894+1 (119872 minus 119873) +119872)

=1

119873119889119894+1

((119894 + 1)119872 minus (119872 minus119873)

119894+1

sum

119895=1

119889119895)

(24)

Now we are ready to prove the mainTheorem 5

Proof We check that synthesis matrix constructed byHTC(II) has row norm equal radic119872119873 and columns arenormalized First we show that each row of synthesis matrixwhich is constructed via HTC(II) has square norm119872119873 Itis enough to show that the last row of 119867

119889119894together with the

first row of119867119889119894+1

makes square norm119872119873 In other words

1198891198941199032

119889119894 119894+ 119889119894+11199032

1119894+1=119872

119873 (25)

By Lemma 8 we have

1198891198941199032

119889119894 119894+ 119889119894+11199032

1119894+1=1

119873(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895

+ (119872 minus119873)

119894

sum

119895=1

119889119895minus (119894 minus 1)119872)

=119872

119873

(26)

Now we show each column of synthesis matrix that isconstructed with HTC(II) has norm 1 that is

1199032

1119894+ (119889119894minus 2)

119872

119873119889119894

+ 1199032

119889119894 119894= 1 (27)

By Lemma 8 we have

1199032

1119894+ 1199032

119889119894 119894=

1

119873119889119894

((119872 minus119873)(

119894minus1

sum

119895=1

119889119895minus

119894

sum

119895=1

119889119895)

+ 119872(minus119894 + 2 + 119894))

= minus(119872 minus119873

119873) + 2

119872

119873119889119894

= 1 minus (119889119894minus 2)

119872

119873119889119894

(28)

We have to ensure that the last correction factor of thefinal block inserted into the synthesismatrix equalsradic119872119873119889

119870

(in other words the next first correction factor would bezero but we have arrived at this point where the algorithmterminates)

In Algorithm 2 line (9) when 119894 = 119870 and 119905 = 119889119870 we have

119903119889119870119870

= radic1

119873119889119870

(119870119872 minus (119872 minus119873)

119870

sum

119895=1

119889119895) = radic

119872

119873119889119870

(29)

and in line (4) for 119894 = 119870 + 1 we have 1199031119889119870+1

= 0

Note that if we do a HTC(II) as in Example 4 with119870 altered Hadamard matrices of sizes 119889

1 119889

119870 then the

sparsity of the synthesis matrix is sum119870119894=11198892

119894

The next lemma states that if we construct synthesismatrix with blocks of sizes 119889

1 1198892 119889

119870 each condition on

119889119894must be established for 119889

119870minus119894+1and vice versa

Lemma 9 Let 119872 and 119873 be positive integers with 119872 gt 119873119870 = 119872 minus 119873 + 1 and119872 = sum

119870

119895=1119889119895such that 119889

119895rsquos are satisfied

in inequalities

(119894 minus 1)119872

119872 minus119873lt

119894

sum

119895=1

119889119895lt 119894

119872

119872 minus119873

(0 le 119894 le 119870 minus 1)

(30)

6 Abstract and Applied Analysis

HTC(II) Hadamard-Tetris ConstructionParameters

(i) Dimension119873 isin N(ii) Number of frame elements119872 isin N such that decompose as119872 = sum

119870

119894=1119889119894 where 119870 = 119872 minus119873 + 1

(iii) Hadamard matrices119867119889119894= [119886(119889119894)

119896119895]1le119896119895le119889119894

of size 119889119894for 119894 = 1 119870

Algorithm(1) 119904 = 0119898 = 0(2) For 119894 = 1 to 119870 do(3) For 119905 = 1 to 119889

119894do

(4) 1199031119894= radic

1

119873119889119894

((119872 minus119873)

119894minus1

sum

119895=1

119889119895minus (119894 minus 2)119872)

(5) 11990311= radic

119872

1198731198891

(6) For 119895 = 2 to 119889119894minus 1 do

(7) 119903119895119894= radic

119872

119873119889119894

(8) end

(9) 119903119889119894 119894= radic

1

119873119889119894

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895)

(10) 119891119905+119898

=

119889119894

sum

119895=1

119886(119889119894)

119905119895119903119895119894119890119904+119895

(11) end(12) 119898 = 119898 + 119889

119894

(13) 119904 = 119904 + 119889119894minus 1

(14) endOutput

(i) unit norm tight frame 119891119895119872

119895=1for R119873

Algorithm 2 The HTC(II) algorithm for constructing a unit norm tight frame with119872 elements for R119873

Then

(119894 minus 1)119872

119872 minus119873lt

119894

sum

119895=1

119889119870minus119895+1

lt 119894119872

119872 minus119873

(0 le 119894 le 119870 minus 1)

(31)

Proof Since sum119870119895=1119889119895= 119872 so

119894

sum

119895=1

119889119870minus119895+1

=

119870

sum

119895=1

119889119895minus

119870minus119894

sum

119895=1

119889119895= 119872 minus

119870minus119894

sum

119895=1

119889119895 (32)

In the other hand by replacing 119894 with 119870 minus 119894 in the inequality(30) we have

119872

119872minus119873(119870 minus 119894 minus 1) lt

119870minus119894

sum

119895=1

119889119895lt (119870 minus 119894)

119872

119872 minus119873(33)

and so on

119872minus (119870 minus 119894)119872

119872 minus119873lt 119872 minus

119870minus119894

sum

119895=1

119889119895lt 119872 minus

119872

119872minus119873

times (119870 minus 119894 minus 1)

(34)

Hence we have

119872

119872minus119873(119872 minus119873 + 119894 minus 119870) lt

119894

sum

119895=1

119889119870minus119895+1

lt119872

119872 minus119873

times (119872 minus119873 minus 119870 + 119894 + 1)

(35)

which give inequalities (31)

Following Lemma 9 if 119879lowast = [119886119894119895] is a synthesis matrix of

size119873times119872 then 119865lowast = [119887119894119895] where 119887

119894119895= 119886(119873minus119894+1)(119872minus119895+1)

is alsohaving the same situation

Since Hadamard matrices of the size power of 2 exist sowe use altered Hadamard matrices of size 2119904119894 where

2119904119894 = max

2119904| (119894 minus 1)

119872

119872 minus119873le

119894minus1

sum

119895=1

2119904119895 + 2119904lt 119894

119872

119872 minus119873

(36)

for 1 le 119894 le 119870 minus 1 However it is provided that there exists aHadamard matrix of size 119889

119870times 119889119870 with 119889

119870= 119872 minus sum

119870minus1

119895=12119904119895

In this case the algorithmHTC builds the synthesis matrix ofthe tight frame as in Example 4 by inserting119867

119889119894 where 119889

119894=

2119904119894 (1 le 119894 le 119870 minus 1) and 119889

119870= 119872 minus sum

119870minus1

119895=12119904119895

Abstract and Applied Analysis 7

SHTC Spectral Hadamard-Tetris constructionParameters

(i) Dimension119873 isin N(ii) Number of frame elements119872 isin N such that decompose as119872 = sum

119870

119894=1119889119894where 119870 = 119872 minus119873 + 1

(iii) eigenvalues 120582119899119873

119899=1which sum119873

119899=1120582119899= 119872 and sum119905119894minus119894

119895=1120582119895le 119905119894le sum119905119894minus119894+1

119895=1120582119895where 119905

119894= sum119894

119895=1119889119895and 1 le 119894 le 119870

(iv) Hadamard matrices119867119889119894= [119886(119889119894)

119896119895]1le119896119895le119889119894

of size 119889119894for 119894 = 1 119870

Algorithm(1) 119904 = 0119898 = 0 119905

0= 0

(2) For 119894 = 1 to 119870 do

(3) 119905119894=

119894

sum

119895=1

119889119895

(4) For 119905 = 1 to 119889119894do

(5) 1199031119894= radic

1

119889119894

(

119905119894minus1minus(119894minus2)

sum

119895=1

120582119895minus 119905119894minus1)

(6) For 119895 = 2 to 119889119894minus 1 do

(7) 119903119895119894= radic

1

119889119894

120582119905119894minus1minus(119894minus1)+119895

(8) end

(9) 119903119894119894= radic

1

119889119894

(119905119894minus

119905119894minus119894

sum

119895=1

120582119895)

(10) 119891119905+119898

=

119889119894

sum

119895=1

119886(119889119894)

119905119895119903119895119894119890119904+119895

(11) end(12) 119898 = 119898 + 119889

119894

(13) 119904 = 119904 + 119889119894minus 1

(14) endOutput

(i) unit norm frame 119891119895119872

119895=1for R119873 with eigenvalues 120582

119899119873

119899=1

Algorithm 3 The SHTC algorithm for constructing a unit norm tight frame with119872 elements for R119873

4 Spectral Hadamard Tetris Construction

Another case of HTC is the case of unit norm but notnecessarily tight frames Such frames are known to existprovided that the eigenvalues of the frame operator sumup to the number of frame vectors [15] In this section wegive a version of Hadamard-Tetris Construction which usesaltered Hadamard matrices to construct unit norm frameswith a given spectrum of frame operator Ourmethod cannotconstruct all such framesWe give necessary condition underwhich this constructionworks To construct unit norm framewith prescribed spectrum we are looking for a matrix withcolumns square summing up to 1 and rows square sum-ming up to 120582

119899119873

119899=1 In Algorithm 3 we present a modified

Hadamard-Tetris algorithm HTC allowing the constructionof such frames

Given a fixed dimension 119873 and frame cardinality 119872 anecessary condition on the prescribed eigenvalues 120582

119899119873

119899=1of

the frame operator for SHTC towork is given in the followingtheorem

Theorem 10 Let sum119873119899=1120582119899= 119872 = sum

119870

119894=1119889119894 where 119870 = 119872 minus

119873 + 1 and there exist Hadamard matrices of sizes 119889119894rsquos Then

SHTC can be used to produce unit norm frame for R119873 with

eigenvalues 120582119899119873

119899=1if there are some permutations of 120582

119899119873

119899=1

and 119889119894119870

119894=1such that for each 119894 = 1 119870 we have

119905119894minus119894

sum

119895=1

120582119895le 119905119894le

119905119894minus119894+1

sum

119895=1

120582119895 (37)

where 119905119894= sum119894

119895=1119889119895

Here we present an example in which STC does not workwhereas SHTC works

Example 11 There are choices of prescribed eigenvalueswhich satisfy the conditions of Theorem 10 that unit normframe with given eigenvalues exists but STC cannot be usedto construct such a frame because no rearrangement ofeigenvalues satisfies in their conditions An example of thiskind is 120582

1198994

119899=1= 04 24 11 11 as eigenvalues of frame

with 5 elements forR4

Sometimes one rearrangement of eigenvalues anddimensions of Hadamard matrices is satisfied in desiredcondition while another is not For example the SHTCcannot be performed for the sequence of 120582

119894rsquos is the given

8 Abstract and Applied Analysis

rearrangement 04 11 11 24 and 1198891= 4 119889

2= 1

However rearranging the eigenvalues to the rearrangement04 24 11 11 allows SHTC to construct the desired frameIn this case its synthesis matrix will be

[[[[[[[[[[[[[[

[

radic1

10radic1

10minusradic

1

10radic1

100

minusradic6

10radic6

10minusradic

6

10minusradic

6

100

radic11

40minusradic

11

40minusradic

11

40minusradic

11

400

radic1

40radic1

40radic1

40minusradic

1

401

]]]]]]]]]]]]]]

]

(38)

So there is no reason that SHTC works when theeigenvalues are in the monotonic rearrangement

Note that since we make no assumption about the order-ing of the sequences 120582

119899119873

119899=1and 119889

119894119870

119894=1 we may permute the

elements of the sequences in order to make them ready tosatisfy in the inequality (37)

Given sequences of eigenvalues and 119889119894119870

119894=1 where 119870 =

119872 minus 119873 + 1 and there exist Hadamard matrices of sizes119889119894rsquos it may be time-consuming to find permutation of these

sequences which satisfy in Theorem 10 Until now we couldnot find simple conditions on these sequences that SHTC canbe performed It has been a mystery as to when HadamardTetris works and when fails

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthors are grateful to ProfessorHossein Jafari the Editorof AAA and the referee for their valuable comments

References

[1] R J Duffin andA C Schaeffer ldquoA class of nonharmonic Fourierseriesrdquo Transactions of the American Mathematical Society vol72 pp 341ndash366 1952

[2] J Kovacevic and A Chebira ldquoLife beyond bases the advent offrames (Parts I and II)rdquo IEEE Signal Processing Magazine vol24 no 5 pp 115ndash125 2007

[3] E Soljanin ldquoTight frames in quantum information theoryrdquo inProceedings of the DIMACS Workshop on Source Coding andHarmonic Analysis Rutgers NJ USA May 2002

[4] T Strohmer and J Heath ldquoGrassmannian frames with appli-cations to coding and communicationrdquo Applied and Computa-tional Harmonic Analysis vol 14 no 3 pp 257ndash275 2003

[5] J J Benedetto and M Fickus ldquoFinite normalized tight framesrdquoAdvances in Computational Mathematics vol 18 no 2-4 pp357ndash385 2003

[6] N Cotfas and J P Gazeau ldquoFinite tight frames and some appli-cationsrdquo Journal of Physics AMathematical andTheoretical vol43 no 19 Article ID 193001 2010

[7] D F Li and W C Sun ldquoExpansion of frames to tight framesrdquoActa Mathematica Sinica (English Series) vol 25 no 2 pp 287ndash292 2009

[8] J C Tremain ldquoAlgorithmic constructions of unitary matricesand tight framesrdquo httparxivorgabs11044539v1

[9] D-J Feng L Wang and Y Wang ldquoGeneration of finitetight frames by Householder transformationsrdquo Advances inComputational Mathematics vol 24 no 1ndash4 pp 297ndash309 2006

[10] O Christensen Frames and Bases An Introductory CourseBirkhauser Boston Mass USA 2008

[11] J Cahill M Fickus D G Mixon M J Poteet and N StrawnldquoConstructing finite frames of a given spectrum and set oflengthsrdquoApplied and Computational Harmonic Analysis vol 35no 1 pp 52ndash73 2013

[12] P G Casazza A Heinecke F Krahmer and G KutyniokldquoOptimally sparse framesrdquo IEEE Transactions on InformationTheory vol 57 no 11 pp 7279ndash7287 2011

[13] P G Casazza M Fickus A Heinecke Y Wang and Z ZhouldquoSpectral tetris fusion frame constructionsrdquo The Journal ofFourier Analysis and Applications vol 18 no 4 pp 828ndash8512012

[14] P G Casazza AHeinecke K Kornelson YWang andZ ZhouldquoNecessary and sufficient conditions to perform spectral tetrisrdquoLinear Algebra and its Applications vol 438 no 5 pp 2239ndash2255 2013

[15] K Dykema D Freeman K Kornelson D Larson M Ordowerand E Weber ldquoEllipsoidal tight frames and projection decom-positions of operatorsrdquo Illinois Journal of Mathematics vol 48no 2 pp 477ndash489 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

used in the construction of STC For example for119872 = 4 and119873 = 3 the matrix

[[[[[[[[[[

[

radic2

3radic2

30 0

radic1

3minusradic

1

3radic1

3radic1

3

0 0 radic2

3minusradic

2

3

]]]]]]]]]]

]

(9)

is synthesis matrix of R3

Example 2 If119873 = 4 and119872 = 6 then

[[[[[[[[[[[[[[[[

[

radic3

4radic3

40 0 0 0

radic1

4minusradic

1

4radic1

2radic1

20 0

0 0 radic1

2minusradic

1

2radic1

4radic1

4

0 0 0 0 radic3

4minusradic

3

4

]]]]]]]]]]]]]]]]

]

(10)

is synthesis of 6-elements for R4

3 Hadamard-Tetris Frame

In this section we provide a new method for constructingfinite normalized tight frames (FNTFs) In brief we want toconstruct119873 times119872 synthesis matrix 119879lowast which has

(i) columns of unit norm

(ii) orthogonal rows meaning that the frame operator119879lowast119879 is diagonal

(iii) rows of constant norm meaning that 119879lowast119879 is a con-stant multiple of the identity matrix

The Hadamard-Tetris Construction (HTC) is capable ofconstructing unit norm tight frames with the number ofelements that decompose as 119889

119894(1 le 119894 le 119872 minus 119873 +

1) It constructs the synthesis matrices of such frames bysuccessively filling of size 119889

119894of 119872 minus 119873 + 1 blocks In this

construction we use altered Hadamard matricesThe importance of Hadamard matrices to our construc-

tion stems from the fact that they have orthogonal rows andcolumns and that all entries have the same modulus In thecourse of the construction we will have to alter the rownorms of the Hadamard matrices by multiplying rows withappropriate constants While this will destroy the pairwiseorthogonality of the columns it will preserve the pairwiseorthogonality of the rows which is the crucial feature for ourconstruction to work

Definition 3 Given positive integers 119872 and 119873 we denotea matrix by 119867

119889or call it a 119867

119889block if it is derived from a

Hadamard matrix of size 119889 by multiplying the entries of the119895th row of Hadamard matrix byradic119872119873119889 for 119895 = 2 119889 minus 1and the entries of the first and the last row of Hadamardmatrix by 119903

1and 119903119889and if it has normalized columns We

call 1199031and 119903119889the first and the last correction factor of 119867

119889

respectively

Note that the row norm of a 119867119889block equals radic119872119873

except possibly for the first and the last rows We wantto present an algorithm following the lines of Example 4that is we want to compose the desired synthesis matrixwhich consists of square blocks in which first and last rowssuccessively overlap If we use 119870 square matrix as buildingblocks of the 119873 times 119872 synthesis matrix then due to theoverlapping we have119872 = 119873 + (119870 minus 1)

It is perhapsmost instructive to first look at the example ofthe construction thatwe are going to introduce in this section

Example 4 We construct a 6-elements unit norm tight framein R5 We can start filling the desired 5 times 6 synthesis matrixwith an altered 4 times 4 Hadamard matrix in the upper leftcorner The alteration we make is to multiply the entries ofthe first row by radic310 in size to make the first row have thedesired norm radic65 We multiply the second and the thirdrows of 4 times 4 Hadamard matrix by radic310 to make the normof those equal toradic65

To get normalized column we multiply the forth row ofthe 4 times 4 Hadamard matrix by radic110 At this point we haveconstructed the first three rows and the first four columns ofthe desired synthesis matrix

[[[[[[[[[[[[[[[[[[[[

[

radic3

10radic3

10minusradic

3

10radic3

100 0

minusradic3

10radic3

10minusradic

3

10minusradic

3

100 0

radic3

10minusradic

3

10minusradic

3

10minusradic

3

100 0

radic1

10radic1

10radic1

10minusradic

1

10sdot sdot

0 0 0 0 sdot sdot

]]]]]]]]]]]]]]]]]]]]

]

(11)

Note that so far we have constructed a matrix in whichfirst four rows are orthogonal no matter how we keep fillingthe first four rows The fourth row at this point has normradic25 while we need to make it have normradic65

We can insert altered 2 times 2Hadamard matrix in the samefashion as above To do this we wouldmultiply its first row bythe factor radic25 in size to have the forth row of the synthesis

4 Abstract and Applied Analysis

matrix square sum to 65 and its second row by the factorradic35 to get

[[[[[[[[[[[[[[[[[[

[

radic3

10radic3

10minusradic

3

10radic3

100 0

minusradic3

10radic3

10minusradic

3

10minusradic

3

100 0

radic3

10minusradic

3

10minusradic

3

10minusradic

3

100 0

radic1

10radic1

10radic1

10minusradic

1

10radic2

5minusradic

2

5

0 0 0 0 radic3

5radic3

5

]]]]]]]]]]]]]]]]]]

]

(12)

The latter matrix is the synthesis matrix of the desired framesince its columns are normalized and its rows are pairwiseorthogonal and square sum to 65

The next theorem is the main theorem of this section

Theorem 5 (main theorem) Let 119872 and 119873 (119872 gt 119873) bepositive integers and 119870 = 119872 minus 119873 + 1 Furthermore assumethat119872 can be decomposed as119872 = sum

119870

119895=1119889119895such that for each

1 le 119894 le 119870 minus 1

(119894 minus 1)119872

119872 minus119873le

119894

sum

119895=1

119889119895lt 119894

119872

119872 minus119873 (13)

and there exist Hadamard matrices of sizes 119889119894rsquos Then HTC(II)

gives a unit normalized tight frame for R119873 with119872 elements

In special case when119872 minus 119873 = 1 then the condition on119889119894rsquos is confirmed automatically Indeed if 119889

1is selected such

that 0 lt 1198891lt 119872 then 0 lt 119889

2= 119872 minus 119889

1lt 119872

Corollary 6 If119872 = 1198891+ 1198892such that there exist Hadamard

matrices of sizes 1198891and 119889

2 then the synthesis matrix of the119872

elements for R119872minus1 can be constructed via HTC(II)

To proveTheorem 5 we need to collect some informationabout the first and the last correction factors of119867

119889119894rsquos

In construction of synthesis matrix of size119873times119872 we use119870 = 119872 minus 119873 + 1 blocks of sizes 119889

119894rsquos (1 le 119894 le 119870) where119872 =

sum119870

119894=1119889119894

Aswasmentioned each block is119867119889119894that derived from the

Hadamard matrix of size 119889119894 If the factor that multiplies the

entries of the 119895th row of Hadamard matrix of size 119889119894is shown

by 119903119895119894 where 1 le 119895 le 119889

119894 then for each block of size 119889

119894 the

correction factor is

1199032

119895119894=119872

119873119889119894

(2 le 119895 le 119889119894minus 1) (14)

The last row of a block and the first row of the followingblock appear in the same row of the synthesis matrix andwhile the last correction factor is chosen to ensure normalizedcolumns the following first correction factor is chosen to

guarantee that the rows of the synthesis matrix have squarenorm119872119873

Hence we have the following relationships for 1199031119894

and119903119889119894 119894 where 1 le 119894 le 119870

Lemma 7 If 1199031119894

and 119903119889119894 119894

are the first and the last correctionfactors respectively then

1199032

119889119894 119894= 1 minus 119903

2

1119894minus119872

119873+ 2

119872

119873119889119894

1199032

1119894+1=

1

119889119894+1

(119872

119873minus 1198891198941199032

119889119894 119894)

(15)

Proof The correction factors of second row to (119889119894minus 1)th row

of119867119889119894are

119903119895119894= radic

119872

119873119889119894

(2 le 119895 le 119889119894minus 1) (16)

Since norm of each column of119867119889119894must be one we have

1199032

1119894+ (119889119894minus 2)

119872

119873119889119894

+ 1199032

119889119894 119894= 1 (17)

and so on

1199032

119889119894 119894= 1 minus 119903

2

1119894minus119872

119873+ 2

119872

119873119889119894

(18)

Also the last correction factor of 119867119889119894

and the firstcorrection factor of119867

119889119894+1must be chosen such that the square

norm of that row is119872119873 Hence

1198891198941199032

119889119894 119894+ 119889119894+11199032

1119894+1=119872

119873(19)

and so on

1199032

1119894+1=

1

119889119894+1

(119872

119873minus 1198891198941199032

119889119894 119894) (20)

By combining formulas (15) in Lemma 7 we get thefollowing recursive relations

1199032

1119894+1=119889119894

119889119894+1

(119872

119873+ 1199032

1119894minus 1) minus

119872

119873119889119894+1

1199032

119889119894+1 119894+1=119889119894

119889119894+1

1199032

119889119894 119894minus119872 minus119873

119873+

119872

119873119889119894+1

(21)

for 1 le 119894 le 119870 minus 1

Lemma 8 If 1199031119894

and 119903119889119894 119894

are the first and the last correctionfactors of119867

119889119894 respectively then

1199032

1119894=

1

119873119889119894

((119872 minus119873)

119894minus1

sum

119895=1

119889119895minus (119894 minus 2)119872) (119894 ge 2)

1199032

119889119894 119894=

1

119873119889119894

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895) (119894 ge 1)

(22)

Abstract and Applied Analysis 5

Proof We prove these by induction on 119894 The first steps of theinduction are trivially true Assume that the identities are truefor 119894 For the case 119894 + 1 the following identities hold by usingformulas (21) and the induction hypothesis

1199032

1119894+1=119889119894

119889119894+1

(119872

119873+ 1199032

1119894minus 1) minus

119872

119873119889119894+1

=119889119894

119889119894+1

(119872

119873+(

sum119894minus1

119895=1119889119895

119889119894

)(119872 minus119873

119873)

minus(119894 minus 2

119889119894

)(119872

119873) minus 1) minus

119872

119873119889119894+1

=1

119873119889119894+1

(119889119894119872+ (119872 minus119873)

119894minus1

sum

119895=1

119889119895

minus (119894 minus 2)119872 minus 119873119889119894minus119872)

=1

119873119889119894+1

((119872 minus119873)

119894

sum

119895=1

119889119895minus (119894 minus 1)119872)

(23)

1199032

119889119894+1119894+1=119889119894

119889119894+1

1199032

119889119894 119894minus119872 minus119873

119873+

119872

119873119889119894+1

=119889119894

119889119894+1

((119894

119889119894

)(119872

119873) minus (

sum119894

119895=1119889119895

119889119894

)(119872 minus119873

119873))

minus119872 minus119873

119873+

119872

119873119889119894+1

=1

119873119889119894+1

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895

minus 119889119894+1 (119872 minus 119873) +119872)

=1

119873119889119894+1

((119894 + 1)119872 minus (119872 minus119873)

119894+1

sum

119895=1

119889119895)

(24)

Now we are ready to prove the mainTheorem 5

Proof We check that synthesis matrix constructed byHTC(II) has row norm equal radic119872119873 and columns arenormalized First we show that each row of synthesis matrixwhich is constructed via HTC(II) has square norm119872119873 Itis enough to show that the last row of 119867

119889119894together with the

first row of119867119889119894+1

makes square norm119872119873 In other words

1198891198941199032

119889119894 119894+ 119889119894+11199032

1119894+1=119872

119873 (25)

By Lemma 8 we have

1198891198941199032

119889119894 119894+ 119889119894+11199032

1119894+1=1

119873(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895

+ (119872 minus119873)

119894

sum

119895=1

119889119895minus (119894 minus 1)119872)

=119872

119873

(26)

Now we show each column of synthesis matrix that isconstructed with HTC(II) has norm 1 that is

1199032

1119894+ (119889119894minus 2)

119872

119873119889119894

+ 1199032

119889119894 119894= 1 (27)

By Lemma 8 we have

1199032

1119894+ 1199032

119889119894 119894=

1

119873119889119894

((119872 minus119873)(

119894minus1

sum

119895=1

119889119895minus

119894

sum

119895=1

119889119895)

+ 119872(minus119894 + 2 + 119894))

= minus(119872 minus119873

119873) + 2

119872

119873119889119894

= 1 minus (119889119894minus 2)

119872

119873119889119894

(28)

We have to ensure that the last correction factor of thefinal block inserted into the synthesismatrix equalsradic119872119873119889

119870

(in other words the next first correction factor would bezero but we have arrived at this point where the algorithmterminates)

In Algorithm 2 line (9) when 119894 = 119870 and 119905 = 119889119870 we have

119903119889119870119870

= radic1

119873119889119870

(119870119872 minus (119872 minus119873)

119870

sum

119895=1

119889119895) = radic

119872

119873119889119870

(29)

and in line (4) for 119894 = 119870 + 1 we have 1199031119889119870+1

= 0

Note that if we do a HTC(II) as in Example 4 with119870 altered Hadamard matrices of sizes 119889

1 119889

119870 then the

sparsity of the synthesis matrix is sum119870119894=11198892

119894

The next lemma states that if we construct synthesismatrix with blocks of sizes 119889

1 1198892 119889

119870 each condition on

119889119894must be established for 119889

119870minus119894+1and vice versa

Lemma 9 Let 119872 and 119873 be positive integers with 119872 gt 119873119870 = 119872 minus 119873 + 1 and119872 = sum

119870

119895=1119889119895such that 119889

119895rsquos are satisfied

in inequalities

(119894 minus 1)119872

119872 minus119873lt

119894

sum

119895=1

119889119895lt 119894

119872

119872 minus119873

(0 le 119894 le 119870 minus 1)

(30)

6 Abstract and Applied Analysis

HTC(II) Hadamard-Tetris ConstructionParameters

(i) Dimension119873 isin N(ii) Number of frame elements119872 isin N such that decompose as119872 = sum

119870

119894=1119889119894 where 119870 = 119872 minus119873 + 1

(iii) Hadamard matrices119867119889119894= [119886(119889119894)

119896119895]1le119896119895le119889119894

of size 119889119894for 119894 = 1 119870

Algorithm(1) 119904 = 0119898 = 0(2) For 119894 = 1 to 119870 do(3) For 119905 = 1 to 119889

119894do

(4) 1199031119894= radic

1

119873119889119894

((119872 minus119873)

119894minus1

sum

119895=1

119889119895minus (119894 minus 2)119872)

(5) 11990311= radic

119872

1198731198891

(6) For 119895 = 2 to 119889119894minus 1 do

(7) 119903119895119894= radic

119872

119873119889119894

(8) end

(9) 119903119889119894 119894= radic

1

119873119889119894

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895)

(10) 119891119905+119898

=

119889119894

sum

119895=1

119886(119889119894)

119905119895119903119895119894119890119904+119895

(11) end(12) 119898 = 119898 + 119889

119894

(13) 119904 = 119904 + 119889119894minus 1

(14) endOutput

(i) unit norm tight frame 119891119895119872

119895=1for R119873

Algorithm 2 The HTC(II) algorithm for constructing a unit norm tight frame with119872 elements for R119873

Then

(119894 minus 1)119872

119872 minus119873lt

119894

sum

119895=1

119889119870minus119895+1

lt 119894119872

119872 minus119873

(0 le 119894 le 119870 minus 1)

(31)

Proof Since sum119870119895=1119889119895= 119872 so

119894

sum

119895=1

119889119870minus119895+1

=

119870

sum

119895=1

119889119895minus

119870minus119894

sum

119895=1

119889119895= 119872 minus

119870minus119894

sum

119895=1

119889119895 (32)

In the other hand by replacing 119894 with 119870 minus 119894 in the inequality(30) we have

119872

119872minus119873(119870 minus 119894 minus 1) lt

119870minus119894

sum

119895=1

119889119895lt (119870 minus 119894)

119872

119872 minus119873(33)

and so on

119872minus (119870 minus 119894)119872

119872 minus119873lt 119872 minus

119870minus119894

sum

119895=1

119889119895lt 119872 minus

119872

119872minus119873

times (119870 minus 119894 minus 1)

(34)

Hence we have

119872

119872minus119873(119872 minus119873 + 119894 minus 119870) lt

119894

sum

119895=1

119889119870minus119895+1

lt119872

119872 minus119873

times (119872 minus119873 minus 119870 + 119894 + 1)

(35)

which give inequalities (31)

Following Lemma 9 if 119879lowast = [119886119894119895] is a synthesis matrix of

size119873times119872 then 119865lowast = [119887119894119895] where 119887

119894119895= 119886(119873minus119894+1)(119872minus119895+1)

is alsohaving the same situation

Since Hadamard matrices of the size power of 2 exist sowe use altered Hadamard matrices of size 2119904119894 where

2119904119894 = max

2119904| (119894 minus 1)

119872

119872 minus119873le

119894minus1

sum

119895=1

2119904119895 + 2119904lt 119894

119872

119872 minus119873

(36)

for 1 le 119894 le 119870 minus 1 However it is provided that there exists aHadamard matrix of size 119889

119870times 119889119870 with 119889

119870= 119872 minus sum

119870minus1

119895=12119904119895

In this case the algorithmHTC builds the synthesis matrix ofthe tight frame as in Example 4 by inserting119867

119889119894 where 119889

119894=

2119904119894 (1 le 119894 le 119870 minus 1) and 119889

119870= 119872 minus sum

119870minus1

119895=12119904119895

Abstract and Applied Analysis 7

SHTC Spectral Hadamard-Tetris constructionParameters

(i) Dimension119873 isin N(ii) Number of frame elements119872 isin N such that decompose as119872 = sum

119870

119894=1119889119894where 119870 = 119872 minus119873 + 1

(iii) eigenvalues 120582119899119873

119899=1which sum119873

119899=1120582119899= 119872 and sum119905119894minus119894

119895=1120582119895le 119905119894le sum119905119894minus119894+1

119895=1120582119895where 119905

119894= sum119894

119895=1119889119895and 1 le 119894 le 119870

(iv) Hadamard matrices119867119889119894= [119886(119889119894)

119896119895]1le119896119895le119889119894

of size 119889119894for 119894 = 1 119870

Algorithm(1) 119904 = 0119898 = 0 119905

0= 0

(2) For 119894 = 1 to 119870 do

(3) 119905119894=

119894

sum

119895=1

119889119895

(4) For 119905 = 1 to 119889119894do

(5) 1199031119894= radic

1

119889119894

(

119905119894minus1minus(119894minus2)

sum

119895=1

120582119895minus 119905119894minus1)

(6) For 119895 = 2 to 119889119894minus 1 do

(7) 119903119895119894= radic

1

119889119894

120582119905119894minus1minus(119894minus1)+119895

(8) end

(9) 119903119894119894= radic

1

119889119894

(119905119894minus

119905119894minus119894

sum

119895=1

120582119895)

(10) 119891119905+119898

=

119889119894

sum

119895=1

119886(119889119894)

119905119895119903119895119894119890119904+119895

(11) end(12) 119898 = 119898 + 119889

119894

(13) 119904 = 119904 + 119889119894minus 1

(14) endOutput

(i) unit norm frame 119891119895119872

119895=1for R119873 with eigenvalues 120582

119899119873

119899=1

Algorithm 3 The SHTC algorithm for constructing a unit norm tight frame with119872 elements for R119873

4 Spectral Hadamard Tetris Construction

Another case of HTC is the case of unit norm but notnecessarily tight frames Such frames are known to existprovided that the eigenvalues of the frame operator sumup to the number of frame vectors [15] In this section wegive a version of Hadamard-Tetris Construction which usesaltered Hadamard matrices to construct unit norm frameswith a given spectrum of frame operator Ourmethod cannotconstruct all such framesWe give necessary condition underwhich this constructionworks To construct unit norm framewith prescribed spectrum we are looking for a matrix withcolumns square summing up to 1 and rows square sum-ming up to 120582

119899119873

119899=1 In Algorithm 3 we present a modified

Hadamard-Tetris algorithm HTC allowing the constructionof such frames

Given a fixed dimension 119873 and frame cardinality 119872 anecessary condition on the prescribed eigenvalues 120582

119899119873

119899=1of

the frame operator for SHTC towork is given in the followingtheorem

Theorem 10 Let sum119873119899=1120582119899= 119872 = sum

119870

119894=1119889119894 where 119870 = 119872 minus

119873 + 1 and there exist Hadamard matrices of sizes 119889119894rsquos Then

SHTC can be used to produce unit norm frame for R119873 with

eigenvalues 120582119899119873

119899=1if there are some permutations of 120582

119899119873

119899=1

and 119889119894119870

119894=1such that for each 119894 = 1 119870 we have

119905119894minus119894

sum

119895=1

120582119895le 119905119894le

119905119894minus119894+1

sum

119895=1

120582119895 (37)

where 119905119894= sum119894

119895=1119889119895

Here we present an example in which STC does not workwhereas SHTC works

Example 11 There are choices of prescribed eigenvalueswhich satisfy the conditions of Theorem 10 that unit normframe with given eigenvalues exists but STC cannot be usedto construct such a frame because no rearrangement ofeigenvalues satisfies in their conditions An example of thiskind is 120582

1198994

119899=1= 04 24 11 11 as eigenvalues of frame

with 5 elements forR4

Sometimes one rearrangement of eigenvalues anddimensions of Hadamard matrices is satisfied in desiredcondition while another is not For example the SHTCcannot be performed for the sequence of 120582

119894rsquos is the given

8 Abstract and Applied Analysis

rearrangement 04 11 11 24 and 1198891= 4 119889

2= 1

However rearranging the eigenvalues to the rearrangement04 24 11 11 allows SHTC to construct the desired frameIn this case its synthesis matrix will be

[[[[[[[[[[[[[[

[

radic1

10radic1

10minusradic

1

10radic1

100

minusradic6

10radic6

10minusradic

6

10minusradic

6

100

radic11

40minusradic

11

40minusradic

11

40minusradic

11

400

radic1

40radic1

40radic1

40minusradic

1

401

]]]]]]]]]]]]]]

]

(38)

So there is no reason that SHTC works when theeigenvalues are in the monotonic rearrangement

Note that since we make no assumption about the order-ing of the sequences 120582

119899119873

119899=1and 119889

119894119870

119894=1 we may permute the

elements of the sequences in order to make them ready tosatisfy in the inequality (37)

Given sequences of eigenvalues and 119889119894119870

119894=1 where 119870 =

119872 minus 119873 + 1 and there exist Hadamard matrices of sizes119889119894rsquos it may be time-consuming to find permutation of these

sequences which satisfy in Theorem 10 Until now we couldnot find simple conditions on these sequences that SHTC canbe performed It has been a mystery as to when HadamardTetris works and when fails

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthors are grateful to ProfessorHossein Jafari the Editorof AAA and the referee for their valuable comments

References

[1] R J Duffin andA C Schaeffer ldquoA class of nonharmonic Fourierseriesrdquo Transactions of the American Mathematical Society vol72 pp 341ndash366 1952

[2] J Kovacevic and A Chebira ldquoLife beyond bases the advent offrames (Parts I and II)rdquo IEEE Signal Processing Magazine vol24 no 5 pp 115ndash125 2007

[3] E Soljanin ldquoTight frames in quantum information theoryrdquo inProceedings of the DIMACS Workshop on Source Coding andHarmonic Analysis Rutgers NJ USA May 2002

[4] T Strohmer and J Heath ldquoGrassmannian frames with appli-cations to coding and communicationrdquo Applied and Computa-tional Harmonic Analysis vol 14 no 3 pp 257ndash275 2003

[5] J J Benedetto and M Fickus ldquoFinite normalized tight framesrdquoAdvances in Computational Mathematics vol 18 no 2-4 pp357ndash385 2003

[6] N Cotfas and J P Gazeau ldquoFinite tight frames and some appli-cationsrdquo Journal of Physics AMathematical andTheoretical vol43 no 19 Article ID 193001 2010

[7] D F Li and W C Sun ldquoExpansion of frames to tight framesrdquoActa Mathematica Sinica (English Series) vol 25 no 2 pp 287ndash292 2009

[8] J C Tremain ldquoAlgorithmic constructions of unitary matricesand tight framesrdquo httparxivorgabs11044539v1

[9] D-J Feng L Wang and Y Wang ldquoGeneration of finitetight frames by Householder transformationsrdquo Advances inComputational Mathematics vol 24 no 1ndash4 pp 297ndash309 2006

[10] O Christensen Frames and Bases An Introductory CourseBirkhauser Boston Mass USA 2008

[11] J Cahill M Fickus D G Mixon M J Poteet and N StrawnldquoConstructing finite frames of a given spectrum and set oflengthsrdquoApplied and Computational Harmonic Analysis vol 35no 1 pp 52ndash73 2013

[12] P G Casazza A Heinecke F Krahmer and G KutyniokldquoOptimally sparse framesrdquo IEEE Transactions on InformationTheory vol 57 no 11 pp 7279ndash7287 2011

[13] P G Casazza M Fickus A Heinecke Y Wang and Z ZhouldquoSpectral tetris fusion frame constructionsrdquo The Journal ofFourier Analysis and Applications vol 18 no 4 pp 828ndash8512012

[14] P G Casazza AHeinecke K Kornelson YWang andZ ZhouldquoNecessary and sufficient conditions to perform spectral tetrisrdquoLinear Algebra and its Applications vol 438 no 5 pp 2239ndash2255 2013

[15] K Dykema D Freeman K Kornelson D Larson M Ordowerand E Weber ldquoEllipsoidal tight frames and projection decom-positions of operatorsrdquo Illinois Journal of Mathematics vol 48no 2 pp 477ndash489 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

matrix square sum to 65 and its second row by the factorradic35 to get

[[[[[[[[[[[[[[[[[[

[

radic3

10radic3

10minusradic

3

10radic3

100 0

minusradic3

10radic3

10minusradic

3

10minusradic

3

100 0

radic3

10minusradic

3

10minusradic

3

10minusradic

3

100 0

radic1

10radic1

10radic1

10minusradic

1

10radic2

5minusradic

2

5

0 0 0 0 radic3

5radic3

5

]]]]]]]]]]]]]]]]]]

]

(12)

The latter matrix is the synthesis matrix of the desired framesince its columns are normalized and its rows are pairwiseorthogonal and square sum to 65

The next theorem is the main theorem of this section

Theorem 5 (main theorem) Let 119872 and 119873 (119872 gt 119873) bepositive integers and 119870 = 119872 minus 119873 + 1 Furthermore assumethat119872 can be decomposed as119872 = sum

119870

119895=1119889119895such that for each

1 le 119894 le 119870 minus 1

(119894 minus 1)119872

119872 minus119873le

119894

sum

119895=1

119889119895lt 119894

119872

119872 minus119873 (13)

and there exist Hadamard matrices of sizes 119889119894rsquos Then HTC(II)

gives a unit normalized tight frame for R119873 with119872 elements

In special case when119872 minus 119873 = 1 then the condition on119889119894rsquos is confirmed automatically Indeed if 119889

1is selected such

that 0 lt 1198891lt 119872 then 0 lt 119889

2= 119872 minus 119889

1lt 119872

Corollary 6 If119872 = 1198891+ 1198892such that there exist Hadamard

matrices of sizes 1198891and 119889

2 then the synthesis matrix of the119872

elements for R119872minus1 can be constructed via HTC(II)

To proveTheorem 5 we need to collect some informationabout the first and the last correction factors of119867

119889119894rsquos

In construction of synthesis matrix of size119873times119872 we use119870 = 119872 minus 119873 + 1 blocks of sizes 119889

119894rsquos (1 le 119894 le 119870) where119872 =

sum119870

119894=1119889119894

Aswasmentioned each block is119867119889119894that derived from the

Hadamard matrix of size 119889119894 If the factor that multiplies the

entries of the 119895th row of Hadamard matrix of size 119889119894is shown

by 119903119895119894 where 1 le 119895 le 119889

119894 then for each block of size 119889

119894 the

correction factor is

1199032

119895119894=119872

119873119889119894

(2 le 119895 le 119889119894minus 1) (14)

The last row of a block and the first row of the followingblock appear in the same row of the synthesis matrix andwhile the last correction factor is chosen to ensure normalizedcolumns the following first correction factor is chosen to

guarantee that the rows of the synthesis matrix have squarenorm119872119873

Hence we have the following relationships for 1199031119894

and119903119889119894 119894 where 1 le 119894 le 119870

Lemma 7 If 1199031119894

and 119903119889119894 119894

are the first and the last correctionfactors respectively then

1199032

119889119894 119894= 1 minus 119903

2

1119894minus119872

119873+ 2

119872

119873119889119894

1199032

1119894+1=

1

119889119894+1

(119872

119873minus 1198891198941199032

119889119894 119894)

(15)

Proof The correction factors of second row to (119889119894minus 1)th row

of119867119889119894are

119903119895119894= radic

119872

119873119889119894

(2 le 119895 le 119889119894minus 1) (16)

Since norm of each column of119867119889119894must be one we have

1199032

1119894+ (119889119894minus 2)

119872

119873119889119894

+ 1199032

119889119894 119894= 1 (17)

and so on

1199032

119889119894 119894= 1 minus 119903

2

1119894minus119872

119873+ 2

119872

119873119889119894

(18)

Also the last correction factor of 119867119889119894

and the firstcorrection factor of119867

119889119894+1must be chosen such that the square

norm of that row is119872119873 Hence

1198891198941199032

119889119894 119894+ 119889119894+11199032

1119894+1=119872

119873(19)

and so on

1199032

1119894+1=

1

119889119894+1

(119872

119873minus 1198891198941199032

119889119894 119894) (20)

By combining formulas (15) in Lemma 7 we get thefollowing recursive relations

1199032

1119894+1=119889119894

119889119894+1

(119872

119873+ 1199032

1119894minus 1) minus

119872

119873119889119894+1

1199032

119889119894+1 119894+1=119889119894

119889119894+1

1199032

119889119894 119894minus119872 minus119873

119873+

119872

119873119889119894+1

(21)

for 1 le 119894 le 119870 minus 1

Lemma 8 If 1199031119894

and 119903119889119894 119894

are the first and the last correctionfactors of119867

119889119894 respectively then

1199032

1119894=

1

119873119889119894

((119872 minus119873)

119894minus1

sum

119895=1

119889119895minus (119894 minus 2)119872) (119894 ge 2)

1199032

119889119894 119894=

1

119873119889119894

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895) (119894 ge 1)

(22)

Abstract and Applied Analysis 5

Proof We prove these by induction on 119894 The first steps of theinduction are trivially true Assume that the identities are truefor 119894 For the case 119894 + 1 the following identities hold by usingformulas (21) and the induction hypothesis

1199032

1119894+1=119889119894

119889119894+1

(119872

119873+ 1199032

1119894minus 1) minus

119872

119873119889119894+1

=119889119894

119889119894+1

(119872

119873+(

sum119894minus1

119895=1119889119895

119889119894

)(119872 minus119873

119873)

minus(119894 minus 2

119889119894

)(119872

119873) minus 1) minus

119872

119873119889119894+1

=1

119873119889119894+1

(119889119894119872+ (119872 minus119873)

119894minus1

sum

119895=1

119889119895

minus (119894 minus 2)119872 minus 119873119889119894minus119872)

=1

119873119889119894+1

((119872 minus119873)

119894

sum

119895=1

119889119895minus (119894 minus 1)119872)

(23)

1199032

119889119894+1119894+1=119889119894

119889119894+1

1199032

119889119894 119894minus119872 minus119873

119873+

119872

119873119889119894+1

=119889119894

119889119894+1

((119894

119889119894

)(119872

119873) minus (

sum119894

119895=1119889119895

119889119894

)(119872 minus119873

119873))

minus119872 minus119873

119873+

119872

119873119889119894+1

=1

119873119889119894+1

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895

minus 119889119894+1 (119872 minus 119873) +119872)

=1

119873119889119894+1

((119894 + 1)119872 minus (119872 minus119873)

119894+1

sum

119895=1

119889119895)

(24)

Now we are ready to prove the mainTheorem 5

Proof We check that synthesis matrix constructed byHTC(II) has row norm equal radic119872119873 and columns arenormalized First we show that each row of synthesis matrixwhich is constructed via HTC(II) has square norm119872119873 Itis enough to show that the last row of 119867

119889119894together with the

first row of119867119889119894+1

makes square norm119872119873 In other words

1198891198941199032

119889119894 119894+ 119889119894+11199032

1119894+1=119872

119873 (25)

By Lemma 8 we have

1198891198941199032

119889119894 119894+ 119889119894+11199032

1119894+1=1

119873(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895

+ (119872 minus119873)

119894

sum

119895=1

119889119895minus (119894 minus 1)119872)

=119872

119873

(26)

Now we show each column of synthesis matrix that isconstructed with HTC(II) has norm 1 that is

1199032

1119894+ (119889119894minus 2)

119872

119873119889119894

+ 1199032

119889119894 119894= 1 (27)

By Lemma 8 we have

1199032

1119894+ 1199032

119889119894 119894=

1

119873119889119894

((119872 minus119873)(

119894minus1

sum

119895=1

119889119895minus

119894

sum

119895=1

119889119895)

+ 119872(minus119894 + 2 + 119894))

= minus(119872 minus119873

119873) + 2

119872

119873119889119894

= 1 minus (119889119894minus 2)

119872

119873119889119894

(28)

We have to ensure that the last correction factor of thefinal block inserted into the synthesismatrix equalsradic119872119873119889

119870

(in other words the next first correction factor would bezero but we have arrived at this point where the algorithmterminates)

In Algorithm 2 line (9) when 119894 = 119870 and 119905 = 119889119870 we have

119903119889119870119870

= radic1

119873119889119870

(119870119872 minus (119872 minus119873)

119870

sum

119895=1

119889119895) = radic

119872

119873119889119870

(29)

and in line (4) for 119894 = 119870 + 1 we have 1199031119889119870+1

= 0

Note that if we do a HTC(II) as in Example 4 with119870 altered Hadamard matrices of sizes 119889

1 119889

119870 then the

sparsity of the synthesis matrix is sum119870119894=11198892

119894

The next lemma states that if we construct synthesismatrix with blocks of sizes 119889

1 1198892 119889

119870 each condition on

119889119894must be established for 119889

119870minus119894+1and vice versa

Lemma 9 Let 119872 and 119873 be positive integers with 119872 gt 119873119870 = 119872 minus 119873 + 1 and119872 = sum

119870

119895=1119889119895such that 119889

119895rsquos are satisfied

in inequalities

(119894 minus 1)119872

119872 minus119873lt

119894

sum

119895=1

119889119895lt 119894

119872

119872 minus119873

(0 le 119894 le 119870 minus 1)

(30)

6 Abstract and Applied Analysis

HTC(II) Hadamard-Tetris ConstructionParameters

(i) Dimension119873 isin N(ii) Number of frame elements119872 isin N such that decompose as119872 = sum

119870

119894=1119889119894 where 119870 = 119872 minus119873 + 1

(iii) Hadamard matrices119867119889119894= [119886(119889119894)

119896119895]1le119896119895le119889119894

of size 119889119894for 119894 = 1 119870

Algorithm(1) 119904 = 0119898 = 0(2) For 119894 = 1 to 119870 do(3) For 119905 = 1 to 119889

119894do

(4) 1199031119894= radic

1

119873119889119894

((119872 minus119873)

119894minus1

sum

119895=1

119889119895minus (119894 minus 2)119872)

(5) 11990311= radic

119872

1198731198891

(6) For 119895 = 2 to 119889119894minus 1 do

(7) 119903119895119894= radic

119872

119873119889119894

(8) end

(9) 119903119889119894 119894= radic

1

119873119889119894

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895)

(10) 119891119905+119898

=

119889119894

sum

119895=1

119886(119889119894)

119905119895119903119895119894119890119904+119895

(11) end(12) 119898 = 119898 + 119889

119894

(13) 119904 = 119904 + 119889119894minus 1

(14) endOutput

(i) unit norm tight frame 119891119895119872

119895=1for R119873

Algorithm 2 The HTC(II) algorithm for constructing a unit norm tight frame with119872 elements for R119873

Then

(119894 minus 1)119872

119872 minus119873lt

119894

sum

119895=1

119889119870minus119895+1

lt 119894119872

119872 minus119873

(0 le 119894 le 119870 minus 1)

(31)

Proof Since sum119870119895=1119889119895= 119872 so

119894

sum

119895=1

119889119870minus119895+1

=

119870

sum

119895=1

119889119895minus

119870minus119894

sum

119895=1

119889119895= 119872 minus

119870minus119894

sum

119895=1

119889119895 (32)

In the other hand by replacing 119894 with 119870 minus 119894 in the inequality(30) we have

119872

119872minus119873(119870 minus 119894 minus 1) lt

119870minus119894

sum

119895=1

119889119895lt (119870 minus 119894)

119872

119872 minus119873(33)

and so on

119872minus (119870 minus 119894)119872

119872 minus119873lt 119872 minus

119870minus119894

sum

119895=1

119889119895lt 119872 minus

119872

119872minus119873

times (119870 minus 119894 minus 1)

(34)

Hence we have

119872

119872minus119873(119872 minus119873 + 119894 minus 119870) lt

119894

sum

119895=1

119889119870minus119895+1

lt119872

119872 minus119873

times (119872 minus119873 minus 119870 + 119894 + 1)

(35)

which give inequalities (31)

Following Lemma 9 if 119879lowast = [119886119894119895] is a synthesis matrix of

size119873times119872 then 119865lowast = [119887119894119895] where 119887

119894119895= 119886(119873minus119894+1)(119872minus119895+1)

is alsohaving the same situation

Since Hadamard matrices of the size power of 2 exist sowe use altered Hadamard matrices of size 2119904119894 where

2119904119894 = max

2119904| (119894 minus 1)

119872

119872 minus119873le

119894minus1

sum

119895=1

2119904119895 + 2119904lt 119894

119872

119872 minus119873

(36)

for 1 le 119894 le 119870 minus 1 However it is provided that there exists aHadamard matrix of size 119889

119870times 119889119870 with 119889

119870= 119872 minus sum

119870minus1

119895=12119904119895

In this case the algorithmHTC builds the synthesis matrix ofthe tight frame as in Example 4 by inserting119867

119889119894 where 119889

119894=

2119904119894 (1 le 119894 le 119870 minus 1) and 119889

119870= 119872 minus sum

119870minus1

119895=12119904119895

Abstract and Applied Analysis 7

SHTC Spectral Hadamard-Tetris constructionParameters

(i) Dimension119873 isin N(ii) Number of frame elements119872 isin N such that decompose as119872 = sum

119870

119894=1119889119894where 119870 = 119872 minus119873 + 1

(iii) eigenvalues 120582119899119873

119899=1which sum119873

119899=1120582119899= 119872 and sum119905119894minus119894

119895=1120582119895le 119905119894le sum119905119894minus119894+1

119895=1120582119895where 119905

119894= sum119894

119895=1119889119895and 1 le 119894 le 119870

(iv) Hadamard matrices119867119889119894= [119886(119889119894)

119896119895]1le119896119895le119889119894

of size 119889119894for 119894 = 1 119870

Algorithm(1) 119904 = 0119898 = 0 119905

0= 0

(2) For 119894 = 1 to 119870 do

(3) 119905119894=

119894

sum

119895=1

119889119895

(4) For 119905 = 1 to 119889119894do

(5) 1199031119894= radic

1

119889119894

(

119905119894minus1minus(119894minus2)

sum

119895=1

120582119895minus 119905119894minus1)

(6) For 119895 = 2 to 119889119894minus 1 do

(7) 119903119895119894= radic

1

119889119894

120582119905119894minus1minus(119894minus1)+119895

(8) end

(9) 119903119894119894= radic

1

119889119894

(119905119894minus

119905119894minus119894

sum

119895=1

120582119895)

(10) 119891119905+119898

=

119889119894

sum

119895=1

119886(119889119894)

119905119895119903119895119894119890119904+119895

(11) end(12) 119898 = 119898 + 119889

119894

(13) 119904 = 119904 + 119889119894minus 1

(14) endOutput

(i) unit norm frame 119891119895119872

119895=1for R119873 with eigenvalues 120582

119899119873

119899=1

Algorithm 3 The SHTC algorithm for constructing a unit norm tight frame with119872 elements for R119873

4 Spectral Hadamard Tetris Construction

Another case of HTC is the case of unit norm but notnecessarily tight frames Such frames are known to existprovided that the eigenvalues of the frame operator sumup to the number of frame vectors [15] In this section wegive a version of Hadamard-Tetris Construction which usesaltered Hadamard matrices to construct unit norm frameswith a given spectrum of frame operator Ourmethod cannotconstruct all such framesWe give necessary condition underwhich this constructionworks To construct unit norm framewith prescribed spectrum we are looking for a matrix withcolumns square summing up to 1 and rows square sum-ming up to 120582

119899119873

119899=1 In Algorithm 3 we present a modified

Hadamard-Tetris algorithm HTC allowing the constructionof such frames

Given a fixed dimension 119873 and frame cardinality 119872 anecessary condition on the prescribed eigenvalues 120582

119899119873

119899=1of

the frame operator for SHTC towork is given in the followingtheorem

Theorem 10 Let sum119873119899=1120582119899= 119872 = sum

119870

119894=1119889119894 where 119870 = 119872 minus

119873 + 1 and there exist Hadamard matrices of sizes 119889119894rsquos Then

SHTC can be used to produce unit norm frame for R119873 with

eigenvalues 120582119899119873

119899=1if there are some permutations of 120582

119899119873

119899=1

and 119889119894119870

119894=1such that for each 119894 = 1 119870 we have

119905119894minus119894

sum

119895=1

120582119895le 119905119894le

119905119894minus119894+1

sum

119895=1

120582119895 (37)

where 119905119894= sum119894

119895=1119889119895

Here we present an example in which STC does not workwhereas SHTC works

Example 11 There are choices of prescribed eigenvalueswhich satisfy the conditions of Theorem 10 that unit normframe with given eigenvalues exists but STC cannot be usedto construct such a frame because no rearrangement ofeigenvalues satisfies in their conditions An example of thiskind is 120582

1198994

119899=1= 04 24 11 11 as eigenvalues of frame

with 5 elements forR4

Sometimes one rearrangement of eigenvalues anddimensions of Hadamard matrices is satisfied in desiredcondition while another is not For example the SHTCcannot be performed for the sequence of 120582

119894rsquos is the given

8 Abstract and Applied Analysis

rearrangement 04 11 11 24 and 1198891= 4 119889

2= 1

However rearranging the eigenvalues to the rearrangement04 24 11 11 allows SHTC to construct the desired frameIn this case its synthesis matrix will be

[[[[[[[[[[[[[[

[

radic1

10radic1

10minusradic

1

10radic1

100

minusradic6

10radic6

10minusradic

6

10minusradic

6

100

radic11

40minusradic

11

40minusradic

11

40minusradic

11

400

radic1

40radic1

40radic1

40minusradic

1

401

]]]]]]]]]]]]]]

]

(38)

So there is no reason that SHTC works when theeigenvalues are in the monotonic rearrangement

Note that since we make no assumption about the order-ing of the sequences 120582

119899119873

119899=1and 119889

119894119870

119894=1 we may permute the

elements of the sequences in order to make them ready tosatisfy in the inequality (37)

Given sequences of eigenvalues and 119889119894119870

119894=1 where 119870 =

119872 minus 119873 + 1 and there exist Hadamard matrices of sizes119889119894rsquos it may be time-consuming to find permutation of these

sequences which satisfy in Theorem 10 Until now we couldnot find simple conditions on these sequences that SHTC canbe performed It has been a mystery as to when HadamardTetris works and when fails

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthors are grateful to ProfessorHossein Jafari the Editorof AAA and the referee for their valuable comments

References

[1] R J Duffin andA C Schaeffer ldquoA class of nonharmonic Fourierseriesrdquo Transactions of the American Mathematical Society vol72 pp 341ndash366 1952

[2] J Kovacevic and A Chebira ldquoLife beyond bases the advent offrames (Parts I and II)rdquo IEEE Signal Processing Magazine vol24 no 5 pp 115ndash125 2007

[3] E Soljanin ldquoTight frames in quantum information theoryrdquo inProceedings of the DIMACS Workshop on Source Coding andHarmonic Analysis Rutgers NJ USA May 2002

[4] T Strohmer and J Heath ldquoGrassmannian frames with appli-cations to coding and communicationrdquo Applied and Computa-tional Harmonic Analysis vol 14 no 3 pp 257ndash275 2003

[5] J J Benedetto and M Fickus ldquoFinite normalized tight framesrdquoAdvances in Computational Mathematics vol 18 no 2-4 pp357ndash385 2003

[6] N Cotfas and J P Gazeau ldquoFinite tight frames and some appli-cationsrdquo Journal of Physics AMathematical andTheoretical vol43 no 19 Article ID 193001 2010

[7] D F Li and W C Sun ldquoExpansion of frames to tight framesrdquoActa Mathematica Sinica (English Series) vol 25 no 2 pp 287ndash292 2009

[8] J C Tremain ldquoAlgorithmic constructions of unitary matricesand tight framesrdquo httparxivorgabs11044539v1

[9] D-J Feng L Wang and Y Wang ldquoGeneration of finitetight frames by Householder transformationsrdquo Advances inComputational Mathematics vol 24 no 1ndash4 pp 297ndash309 2006

[10] O Christensen Frames and Bases An Introductory CourseBirkhauser Boston Mass USA 2008

[11] J Cahill M Fickus D G Mixon M J Poteet and N StrawnldquoConstructing finite frames of a given spectrum and set oflengthsrdquoApplied and Computational Harmonic Analysis vol 35no 1 pp 52ndash73 2013

[12] P G Casazza A Heinecke F Krahmer and G KutyniokldquoOptimally sparse framesrdquo IEEE Transactions on InformationTheory vol 57 no 11 pp 7279ndash7287 2011

[13] P G Casazza M Fickus A Heinecke Y Wang and Z ZhouldquoSpectral tetris fusion frame constructionsrdquo The Journal ofFourier Analysis and Applications vol 18 no 4 pp 828ndash8512012

[14] P G Casazza AHeinecke K Kornelson YWang andZ ZhouldquoNecessary and sufficient conditions to perform spectral tetrisrdquoLinear Algebra and its Applications vol 438 no 5 pp 2239ndash2255 2013

[15] K Dykema D Freeman K Kornelson D Larson M Ordowerand E Weber ldquoEllipsoidal tight frames and projection decom-positions of operatorsrdquo Illinois Journal of Mathematics vol 48no 2 pp 477ndash489 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Proof We prove these by induction on 119894 The first steps of theinduction are trivially true Assume that the identities are truefor 119894 For the case 119894 + 1 the following identities hold by usingformulas (21) and the induction hypothesis

1199032

1119894+1=119889119894

119889119894+1

(119872

119873+ 1199032

1119894minus 1) minus

119872

119873119889119894+1

=119889119894

119889119894+1

(119872

119873+(

sum119894minus1

119895=1119889119895

119889119894

)(119872 minus119873

119873)

minus(119894 minus 2

119889119894

)(119872

119873) minus 1) minus

119872

119873119889119894+1

=1

119873119889119894+1

(119889119894119872+ (119872 minus119873)

119894minus1

sum

119895=1

119889119895

minus (119894 minus 2)119872 minus 119873119889119894minus119872)

=1

119873119889119894+1

((119872 minus119873)

119894

sum

119895=1

119889119895minus (119894 minus 1)119872)

(23)

1199032

119889119894+1119894+1=119889119894

119889119894+1

1199032

119889119894 119894minus119872 minus119873

119873+

119872

119873119889119894+1

=119889119894

119889119894+1

((119894

119889119894

)(119872

119873) minus (

sum119894

119895=1119889119895

119889119894

)(119872 minus119873

119873))

minus119872 minus119873

119873+

119872

119873119889119894+1

=1

119873119889119894+1

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895

minus 119889119894+1 (119872 minus 119873) +119872)

=1

119873119889119894+1

((119894 + 1)119872 minus (119872 minus119873)

119894+1

sum

119895=1

119889119895)

(24)

Now we are ready to prove the mainTheorem 5

Proof We check that synthesis matrix constructed byHTC(II) has row norm equal radic119872119873 and columns arenormalized First we show that each row of synthesis matrixwhich is constructed via HTC(II) has square norm119872119873 Itis enough to show that the last row of 119867

119889119894together with the

first row of119867119889119894+1

makes square norm119872119873 In other words

1198891198941199032

119889119894 119894+ 119889119894+11199032

1119894+1=119872

119873 (25)

By Lemma 8 we have

1198891198941199032

119889119894 119894+ 119889119894+11199032

1119894+1=1

119873(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895

+ (119872 minus119873)

119894

sum

119895=1

119889119895minus (119894 minus 1)119872)

=119872

119873

(26)

Now we show each column of synthesis matrix that isconstructed with HTC(II) has norm 1 that is

1199032

1119894+ (119889119894minus 2)

119872

119873119889119894

+ 1199032

119889119894 119894= 1 (27)

By Lemma 8 we have

1199032

1119894+ 1199032

119889119894 119894=

1

119873119889119894

((119872 minus119873)(

119894minus1

sum

119895=1

119889119895minus

119894

sum

119895=1

119889119895)

+ 119872(minus119894 + 2 + 119894))

= minus(119872 minus119873

119873) + 2

119872

119873119889119894

= 1 minus (119889119894minus 2)

119872

119873119889119894

(28)

We have to ensure that the last correction factor of thefinal block inserted into the synthesismatrix equalsradic119872119873119889

119870

(in other words the next first correction factor would bezero but we have arrived at this point where the algorithmterminates)

In Algorithm 2 line (9) when 119894 = 119870 and 119905 = 119889119870 we have

119903119889119870119870

= radic1

119873119889119870

(119870119872 minus (119872 minus119873)

119870

sum

119895=1

119889119895) = radic

119872

119873119889119870

(29)

and in line (4) for 119894 = 119870 + 1 we have 1199031119889119870+1

= 0

Note that if we do a HTC(II) as in Example 4 with119870 altered Hadamard matrices of sizes 119889

1 119889

119870 then the

sparsity of the synthesis matrix is sum119870119894=11198892

119894

The next lemma states that if we construct synthesismatrix with blocks of sizes 119889

1 1198892 119889

119870 each condition on

119889119894must be established for 119889

119870minus119894+1and vice versa

Lemma 9 Let 119872 and 119873 be positive integers with 119872 gt 119873119870 = 119872 minus 119873 + 1 and119872 = sum

119870

119895=1119889119895such that 119889

119895rsquos are satisfied

in inequalities

(119894 minus 1)119872

119872 minus119873lt

119894

sum

119895=1

119889119895lt 119894

119872

119872 minus119873

(0 le 119894 le 119870 minus 1)

(30)

6 Abstract and Applied Analysis

HTC(II) Hadamard-Tetris ConstructionParameters

(i) Dimension119873 isin N(ii) Number of frame elements119872 isin N such that decompose as119872 = sum

119870

119894=1119889119894 where 119870 = 119872 minus119873 + 1

(iii) Hadamard matrices119867119889119894= [119886(119889119894)

119896119895]1le119896119895le119889119894

of size 119889119894for 119894 = 1 119870

Algorithm(1) 119904 = 0119898 = 0(2) For 119894 = 1 to 119870 do(3) For 119905 = 1 to 119889

119894do

(4) 1199031119894= radic

1

119873119889119894

((119872 minus119873)

119894minus1

sum

119895=1

119889119895minus (119894 minus 2)119872)

(5) 11990311= radic

119872

1198731198891

(6) For 119895 = 2 to 119889119894minus 1 do

(7) 119903119895119894= radic

119872

119873119889119894

(8) end

(9) 119903119889119894 119894= radic

1

119873119889119894

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895)

(10) 119891119905+119898

=

119889119894

sum

119895=1

119886(119889119894)

119905119895119903119895119894119890119904+119895

(11) end(12) 119898 = 119898 + 119889

119894

(13) 119904 = 119904 + 119889119894minus 1

(14) endOutput

(i) unit norm tight frame 119891119895119872

119895=1for R119873

Algorithm 2 The HTC(II) algorithm for constructing a unit norm tight frame with119872 elements for R119873

Then

(119894 minus 1)119872

119872 minus119873lt

119894

sum

119895=1

119889119870minus119895+1

lt 119894119872

119872 minus119873

(0 le 119894 le 119870 minus 1)

(31)

Proof Since sum119870119895=1119889119895= 119872 so

119894

sum

119895=1

119889119870minus119895+1

=

119870

sum

119895=1

119889119895minus

119870minus119894

sum

119895=1

119889119895= 119872 minus

119870minus119894

sum

119895=1

119889119895 (32)

In the other hand by replacing 119894 with 119870 minus 119894 in the inequality(30) we have

119872

119872minus119873(119870 minus 119894 minus 1) lt

119870minus119894

sum

119895=1

119889119895lt (119870 minus 119894)

119872

119872 minus119873(33)

and so on

119872minus (119870 minus 119894)119872

119872 minus119873lt 119872 minus

119870minus119894

sum

119895=1

119889119895lt 119872 minus

119872

119872minus119873

times (119870 minus 119894 minus 1)

(34)

Hence we have

119872

119872minus119873(119872 minus119873 + 119894 minus 119870) lt

119894

sum

119895=1

119889119870minus119895+1

lt119872

119872 minus119873

times (119872 minus119873 minus 119870 + 119894 + 1)

(35)

which give inequalities (31)

Following Lemma 9 if 119879lowast = [119886119894119895] is a synthesis matrix of

size119873times119872 then 119865lowast = [119887119894119895] where 119887

119894119895= 119886(119873minus119894+1)(119872minus119895+1)

is alsohaving the same situation

Since Hadamard matrices of the size power of 2 exist sowe use altered Hadamard matrices of size 2119904119894 where

2119904119894 = max

2119904| (119894 minus 1)

119872

119872 minus119873le

119894minus1

sum

119895=1

2119904119895 + 2119904lt 119894

119872

119872 minus119873

(36)

for 1 le 119894 le 119870 minus 1 However it is provided that there exists aHadamard matrix of size 119889

119870times 119889119870 with 119889

119870= 119872 minus sum

119870minus1

119895=12119904119895

In this case the algorithmHTC builds the synthesis matrix ofthe tight frame as in Example 4 by inserting119867

119889119894 where 119889

119894=

2119904119894 (1 le 119894 le 119870 minus 1) and 119889

119870= 119872 minus sum

119870minus1

119895=12119904119895

Abstract and Applied Analysis 7

SHTC Spectral Hadamard-Tetris constructionParameters

(i) Dimension119873 isin N(ii) Number of frame elements119872 isin N such that decompose as119872 = sum

119870

119894=1119889119894where 119870 = 119872 minus119873 + 1

(iii) eigenvalues 120582119899119873

119899=1which sum119873

119899=1120582119899= 119872 and sum119905119894minus119894

119895=1120582119895le 119905119894le sum119905119894minus119894+1

119895=1120582119895where 119905

119894= sum119894

119895=1119889119895and 1 le 119894 le 119870

(iv) Hadamard matrices119867119889119894= [119886(119889119894)

119896119895]1le119896119895le119889119894

of size 119889119894for 119894 = 1 119870

Algorithm(1) 119904 = 0119898 = 0 119905

0= 0

(2) For 119894 = 1 to 119870 do

(3) 119905119894=

119894

sum

119895=1

119889119895

(4) For 119905 = 1 to 119889119894do

(5) 1199031119894= radic

1

119889119894

(

119905119894minus1minus(119894minus2)

sum

119895=1

120582119895minus 119905119894minus1)

(6) For 119895 = 2 to 119889119894minus 1 do

(7) 119903119895119894= radic

1

119889119894

120582119905119894minus1minus(119894minus1)+119895

(8) end

(9) 119903119894119894= radic

1

119889119894

(119905119894minus

119905119894minus119894

sum

119895=1

120582119895)

(10) 119891119905+119898

=

119889119894

sum

119895=1

119886(119889119894)

119905119895119903119895119894119890119904+119895

(11) end(12) 119898 = 119898 + 119889

119894

(13) 119904 = 119904 + 119889119894minus 1

(14) endOutput

(i) unit norm frame 119891119895119872

119895=1for R119873 with eigenvalues 120582

119899119873

119899=1

Algorithm 3 The SHTC algorithm for constructing a unit norm tight frame with119872 elements for R119873

4 Spectral Hadamard Tetris Construction

Another case of HTC is the case of unit norm but notnecessarily tight frames Such frames are known to existprovided that the eigenvalues of the frame operator sumup to the number of frame vectors [15] In this section wegive a version of Hadamard-Tetris Construction which usesaltered Hadamard matrices to construct unit norm frameswith a given spectrum of frame operator Ourmethod cannotconstruct all such framesWe give necessary condition underwhich this constructionworks To construct unit norm framewith prescribed spectrum we are looking for a matrix withcolumns square summing up to 1 and rows square sum-ming up to 120582

119899119873

119899=1 In Algorithm 3 we present a modified

Hadamard-Tetris algorithm HTC allowing the constructionof such frames

Given a fixed dimension 119873 and frame cardinality 119872 anecessary condition on the prescribed eigenvalues 120582

119899119873

119899=1of

the frame operator for SHTC towork is given in the followingtheorem

Theorem 10 Let sum119873119899=1120582119899= 119872 = sum

119870

119894=1119889119894 where 119870 = 119872 minus

119873 + 1 and there exist Hadamard matrices of sizes 119889119894rsquos Then

SHTC can be used to produce unit norm frame for R119873 with

eigenvalues 120582119899119873

119899=1if there are some permutations of 120582

119899119873

119899=1

and 119889119894119870

119894=1such that for each 119894 = 1 119870 we have

119905119894minus119894

sum

119895=1

120582119895le 119905119894le

119905119894minus119894+1

sum

119895=1

120582119895 (37)

where 119905119894= sum119894

119895=1119889119895

Here we present an example in which STC does not workwhereas SHTC works

Example 11 There are choices of prescribed eigenvalueswhich satisfy the conditions of Theorem 10 that unit normframe with given eigenvalues exists but STC cannot be usedto construct such a frame because no rearrangement ofeigenvalues satisfies in their conditions An example of thiskind is 120582

1198994

119899=1= 04 24 11 11 as eigenvalues of frame

with 5 elements forR4

Sometimes one rearrangement of eigenvalues anddimensions of Hadamard matrices is satisfied in desiredcondition while another is not For example the SHTCcannot be performed for the sequence of 120582

119894rsquos is the given

8 Abstract and Applied Analysis

rearrangement 04 11 11 24 and 1198891= 4 119889

2= 1

However rearranging the eigenvalues to the rearrangement04 24 11 11 allows SHTC to construct the desired frameIn this case its synthesis matrix will be

[[[[[[[[[[[[[[

[

radic1

10radic1

10minusradic

1

10radic1

100

minusradic6

10radic6

10minusradic

6

10minusradic

6

100

radic11

40minusradic

11

40minusradic

11

40minusradic

11

400

radic1

40radic1

40radic1

40minusradic

1

401

]]]]]]]]]]]]]]

]

(38)

So there is no reason that SHTC works when theeigenvalues are in the monotonic rearrangement

Note that since we make no assumption about the order-ing of the sequences 120582

119899119873

119899=1and 119889

119894119870

119894=1 we may permute the

elements of the sequences in order to make them ready tosatisfy in the inequality (37)

Given sequences of eigenvalues and 119889119894119870

119894=1 where 119870 =

119872 minus 119873 + 1 and there exist Hadamard matrices of sizes119889119894rsquos it may be time-consuming to find permutation of these

sequences which satisfy in Theorem 10 Until now we couldnot find simple conditions on these sequences that SHTC canbe performed It has been a mystery as to when HadamardTetris works and when fails

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthors are grateful to ProfessorHossein Jafari the Editorof AAA and the referee for their valuable comments

References

[1] R J Duffin andA C Schaeffer ldquoA class of nonharmonic Fourierseriesrdquo Transactions of the American Mathematical Society vol72 pp 341ndash366 1952

[2] J Kovacevic and A Chebira ldquoLife beyond bases the advent offrames (Parts I and II)rdquo IEEE Signal Processing Magazine vol24 no 5 pp 115ndash125 2007

[3] E Soljanin ldquoTight frames in quantum information theoryrdquo inProceedings of the DIMACS Workshop on Source Coding andHarmonic Analysis Rutgers NJ USA May 2002

[4] T Strohmer and J Heath ldquoGrassmannian frames with appli-cations to coding and communicationrdquo Applied and Computa-tional Harmonic Analysis vol 14 no 3 pp 257ndash275 2003

[5] J J Benedetto and M Fickus ldquoFinite normalized tight framesrdquoAdvances in Computational Mathematics vol 18 no 2-4 pp357ndash385 2003

[6] N Cotfas and J P Gazeau ldquoFinite tight frames and some appli-cationsrdquo Journal of Physics AMathematical andTheoretical vol43 no 19 Article ID 193001 2010

[7] D F Li and W C Sun ldquoExpansion of frames to tight framesrdquoActa Mathematica Sinica (English Series) vol 25 no 2 pp 287ndash292 2009

[8] J C Tremain ldquoAlgorithmic constructions of unitary matricesand tight framesrdquo httparxivorgabs11044539v1

[9] D-J Feng L Wang and Y Wang ldquoGeneration of finitetight frames by Householder transformationsrdquo Advances inComputational Mathematics vol 24 no 1ndash4 pp 297ndash309 2006

[10] O Christensen Frames and Bases An Introductory CourseBirkhauser Boston Mass USA 2008

[11] J Cahill M Fickus D G Mixon M J Poteet and N StrawnldquoConstructing finite frames of a given spectrum and set oflengthsrdquoApplied and Computational Harmonic Analysis vol 35no 1 pp 52ndash73 2013

[12] P G Casazza A Heinecke F Krahmer and G KutyniokldquoOptimally sparse framesrdquo IEEE Transactions on InformationTheory vol 57 no 11 pp 7279ndash7287 2011

[13] P G Casazza M Fickus A Heinecke Y Wang and Z ZhouldquoSpectral tetris fusion frame constructionsrdquo The Journal ofFourier Analysis and Applications vol 18 no 4 pp 828ndash8512012

[14] P G Casazza AHeinecke K Kornelson YWang andZ ZhouldquoNecessary and sufficient conditions to perform spectral tetrisrdquoLinear Algebra and its Applications vol 438 no 5 pp 2239ndash2255 2013

[15] K Dykema D Freeman K Kornelson D Larson M Ordowerand E Weber ldquoEllipsoidal tight frames and projection decom-positions of operatorsrdquo Illinois Journal of Mathematics vol 48no 2 pp 477ndash489 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

HTC(II) Hadamard-Tetris ConstructionParameters

(i) Dimension119873 isin N(ii) Number of frame elements119872 isin N such that decompose as119872 = sum

119870

119894=1119889119894 where 119870 = 119872 minus119873 + 1

(iii) Hadamard matrices119867119889119894= [119886(119889119894)

119896119895]1le119896119895le119889119894

of size 119889119894for 119894 = 1 119870

Algorithm(1) 119904 = 0119898 = 0(2) For 119894 = 1 to 119870 do(3) For 119905 = 1 to 119889

119894do

(4) 1199031119894= radic

1

119873119889119894

((119872 minus119873)

119894minus1

sum

119895=1

119889119895minus (119894 minus 2)119872)

(5) 11990311= radic

119872

1198731198891

(6) For 119895 = 2 to 119889119894minus 1 do

(7) 119903119895119894= radic

119872

119873119889119894

(8) end

(9) 119903119889119894 119894= radic

1

119873119889119894

(119894119872 minus (119872 minus119873)

119894

sum

119895=1

119889119895)

(10) 119891119905+119898

=

119889119894

sum

119895=1

119886(119889119894)

119905119895119903119895119894119890119904+119895

(11) end(12) 119898 = 119898 + 119889

119894

(13) 119904 = 119904 + 119889119894minus 1

(14) endOutput

(i) unit norm tight frame 119891119895119872

119895=1for R119873

Algorithm 2 The HTC(II) algorithm for constructing a unit norm tight frame with119872 elements for R119873

Then

(119894 minus 1)119872

119872 minus119873lt

119894

sum

119895=1

119889119870minus119895+1

lt 119894119872

119872 minus119873

(0 le 119894 le 119870 minus 1)

(31)

Proof Since sum119870119895=1119889119895= 119872 so

119894

sum

119895=1

119889119870minus119895+1

=

119870

sum

119895=1

119889119895minus

119870minus119894

sum

119895=1

119889119895= 119872 minus

119870minus119894

sum

119895=1

119889119895 (32)

In the other hand by replacing 119894 with 119870 minus 119894 in the inequality(30) we have

119872

119872minus119873(119870 minus 119894 minus 1) lt

119870minus119894

sum

119895=1

119889119895lt (119870 minus 119894)

119872

119872 minus119873(33)

and so on

119872minus (119870 minus 119894)119872

119872 minus119873lt 119872 minus

119870minus119894

sum

119895=1

119889119895lt 119872 minus

119872

119872minus119873

times (119870 minus 119894 minus 1)

(34)

Hence we have

119872

119872minus119873(119872 minus119873 + 119894 minus 119870) lt

119894

sum

119895=1

119889119870minus119895+1

lt119872

119872 minus119873

times (119872 minus119873 minus 119870 + 119894 + 1)

(35)

which give inequalities (31)

Following Lemma 9 if 119879lowast = [119886119894119895] is a synthesis matrix of

size119873times119872 then 119865lowast = [119887119894119895] where 119887

119894119895= 119886(119873minus119894+1)(119872minus119895+1)

is alsohaving the same situation

Since Hadamard matrices of the size power of 2 exist sowe use altered Hadamard matrices of size 2119904119894 where

2119904119894 = max

2119904| (119894 minus 1)

119872

119872 minus119873le

119894minus1

sum

119895=1

2119904119895 + 2119904lt 119894

119872

119872 minus119873

(36)

for 1 le 119894 le 119870 minus 1 However it is provided that there exists aHadamard matrix of size 119889

119870times 119889119870 with 119889

119870= 119872 minus sum

119870minus1

119895=12119904119895

In this case the algorithmHTC builds the synthesis matrix ofthe tight frame as in Example 4 by inserting119867

119889119894 where 119889

119894=

2119904119894 (1 le 119894 le 119870 minus 1) and 119889

119870= 119872 minus sum

119870minus1

119895=12119904119895

Abstract and Applied Analysis 7

SHTC Spectral Hadamard-Tetris constructionParameters

(i) Dimension119873 isin N(ii) Number of frame elements119872 isin N such that decompose as119872 = sum

119870

119894=1119889119894where 119870 = 119872 minus119873 + 1

(iii) eigenvalues 120582119899119873

119899=1which sum119873

119899=1120582119899= 119872 and sum119905119894minus119894

119895=1120582119895le 119905119894le sum119905119894minus119894+1

119895=1120582119895where 119905

119894= sum119894

119895=1119889119895and 1 le 119894 le 119870

(iv) Hadamard matrices119867119889119894= [119886(119889119894)

119896119895]1le119896119895le119889119894

of size 119889119894for 119894 = 1 119870

Algorithm(1) 119904 = 0119898 = 0 119905

0= 0

(2) For 119894 = 1 to 119870 do

(3) 119905119894=

119894

sum

119895=1

119889119895

(4) For 119905 = 1 to 119889119894do

(5) 1199031119894= radic

1

119889119894

(

119905119894minus1minus(119894minus2)

sum

119895=1

120582119895minus 119905119894minus1)

(6) For 119895 = 2 to 119889119894minus 1 do

(7) 119903119895119894= radic

1

119889119894

120582119905119894minus1minus(119894minus1)+119895

(8) end

(9) 119903119894119894= radic

1

119889119894

(119905119894minus

119905119894minus119894

sum

119895=1

120582119895)

(10) 119891119905+119898

=

119889119894

sum

119895=1

119886(119889119894)

119905119895119903119895119894119890119904+119895

(11) end(12) 119898 = 119898 + 119889

119894

(13) 119904 = 119904 + 119889119894minus 1

(14) endOutput

(i) unit norm frame 119891119895119872

119895=1for R119873 with eigenvalues 120582

119899119873

119899=1

Algorithm 3 The SHTC algorithm for constructing a unit norm tight frame with119872 elements for R119873

4 Spectral Hadamard Tetris Construction

Another case of HTC is the case of unit norm but notnecessarily tight frames Such frames are known to existprovided that the eigenvalues of the frame operator sumup to the number of frame vectors [15] In this section wegive a version of Hadamard-Tetris Construction which usesaltered Hadamard matrices to construct unit norm frameswith a given spectrum of frame operator Ourmethod cannotconstruct all such framesWe give necessary condition underwhich this constructionworks To construct unit norm framewith prescribed spectrum we are looking for a matrix withcolumns square summing up to 1 and rows square sum-ming up to 120582

119899119873

119899=1 In Algorithm 3 we present a modified

Hadamard-Tetris algorithm HTC allowing the constructionof such frames

Given a fixed dimension 119873 and frame cardinality 119872 anecessary condition on the prescribed eigenvalues 120582

119899119873

119899=1of

the frame operator for SHTC towork is given in the followingtheorem

Theorem 10 Let sum119873119899=1120582119899= 119872 = sum

119870

119894=1119889119894 where 119870 = 119872 minus

119873 + 1 and there exist Hadamard matrices of sizes 119889119894rsquos Then

SHTC can be used to produce unit norm frame for R119873 with

eigenvalues 120582119899119873

119899=1if there are some permutations of 120582

119899119873

119899=1

and 119889119894119870

119894=1such that for each 119894 = 1 119870 we have

119905119894minus119894

sum

119895=1

120582119895le 119905119894le

119905119894minus119894+1

sum

119895=1

120582119895 (37)

where 119905119894= sum119894

119895=1119889119895

Here we present an example in which STC does not workwhereas SHTC works

Example 11 There are choices of prescribed eigenvalueswhich satisfy the conditions of Theorem 10 that unit normframe with given eigenvalues exists but STC cannot be usedto construct such a frame because no rearrangement ofeigenvalues satisfies in their conditions An example of thiskind is 120582

1198994

119899=1= 04 24 11 11 as eigenvalues of frame

with 5 elements forR4

Sometimes one rearrangement of eigenvalues anddimensions of Hadamard matrices is satisfied in desiredcondition while another is not For example the SHTCcannot be performed for the sequence of 120582

119894rsquos is the given

8 Abstract and Applied Analysis

rearrangement 04 11 11 24 and 1198891= 4 119889

2= 1

However rearranging the eigenvalues to the rearrangement04 24 11 11 allows SHTC to construct the desired frameIn this case its synthesis matrix will be

[[[[[[[[[[[[[[

[

radic1

10radic1

10minusradic

1

10radic1

100

minusradic6

10radic6

10minusradic

6

10minusradic

6

100

radic11

40minusradic

11

40minusradic

11

40minusradic

11

400

radic1

40radic1

40radic1

40minusradic

1

401

]]]]]]]]]]]]]]

]

(38)

So there is no reason that SHTC works when theeigenvalues are in the monotonic rearrangement

Note that since we make no assumption about the order-ing of the sequences 120582

119899119873

119899=1and 119889

119894119870

119894=1 we may permute the

elements of the sequences in order to make them ready tosatisfy in the inequality (37)

Given sequences of eigenvalues and 119889119894119870

119894=1 where 119870 =

119872 minus 119873 + 1 and there exist Hadamard matrices of sizes119889119894rsquos it may be time-consuming to find permutation of these

sequences which satisfy in Theorem 10 Until now we couldnot find simple conditions on these sequences that SHTC canbe performed It has been a mystery as to when HadamardTetris works and when fails

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthors are grateful to ProfessorHossein Jafari the Editorof AAA and the referee for their valuable comments

References

[1] R J Duffin andA C Schaeffer ldquoA class of nonharmonic Fourierseriesrdquo Transactions of the American Mathematical Society vol72 pp 341ndash366 1952

[2] J Kovacevic and A Chebira ldquoLife beyond bases the advent offrames (Parts I and II)rdquo IEEE Signal Processing Magazine vol24 no 5 pp 115ndash125 2007

[3] E Soljanin ldquoTight frames in quantum information theoryrdquo inProceedings of the DIMACS Workshop on Source Coding andHarmonic Analysis Rutgers NJ USA May 2002

[4] T Strohmer and J Heath ldquoGrassmannian frames with appli-cations to coding and communicationrdquo Applied and Computa-tional Harmonic Analysis vol 14 no 3 pp 257ndash275 2003

[5] J J Benedetto and M Fickus ldquoFinite normalized tight framesrdquoAdvances in Computational Mathematics vol 18 no 2-4 pp357ndash385 2003

[6] N Cotfas and J P Gazeau ldquoFinite tight frames and some appli-cationsrdquo Journal of Physics AMathematical andTheoretical vol43 no 19 Article ID 193001 2010

[7] D F Li and W C Sun ldquoExpansion of frames to tight framesrdquoActa Mathematica Sinica (English Series) vol 25 no 2 pp 287ndash292 2009

[8] J C Tremain ldquoAlgorithmic constructions of unitary matricesand tight framesrdquo httparxivorgabs11044539v1

[9] D-J Feng L Wang and Y Wang ldquoGeneration of finitetight frames by Householder transformationsrdquo Advances inComputational Mathematics vol 24 no 1ndash4 pp 297ndash309 2006

[10] O Christensen Frames and Bases An Introductory CourseBirkhauser Boston Mass USA 2008

[11] J Cahill M Fickus D G Mixon M J Poteet and N StrawnldquoConstructing finite frames of a given spectrum and set oflengthsrdquoApplied and Computational Harmonic Analysis vol 35no 1 pp 52ndash73 2013

[12] P G Casazza A Heinecke F Krahmer and G KutyniokldquoOptimally sparse framesrdquo IEEE Transactions on InformationTheory vol 57 no 11 pp 7279ndash7287 2011

[13] P G Casazza M Fickus A Heinecke Y Wang and Z ZhouldquoSpectral tetris fusion frame constructionsrdquo The Journal ofFourier Analysis and Applications vol 18 no 4 pp 828ndash8512012

[14] P G Casazza AHeinecke K Kornelson YWang andZ ZhouldquoNecessary and sufficient conditions to perform spectral tetrisrdquoLinear Algebra and its Applications vol 438 no 5 pp 2239ndash2255 2013

[15] K Dykema D Freeman K Kornelson D Larson M Ordowerand E Weber ldquoEllipsoidal tight frames and projection decom-positions of operatorsrdquo Illinois Journal of Mathematics vol 48no 2 pp 477ndash489 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

SHTC Spectral Hadamard-Tetris constructionParameters

(i) Dimension119873 isin N(ii) Number of frame elements119872 isin N such that decompose as119872 = sum

119870

119894=1119889119894where 119870 = 119872 minus119873 + 1

(iii) eigenvalues 120582119899119873

119899=1which sum119873

119899=1120582119899= 119872 and sum119905119894minus119894

119895=1120582119895le 119905119894le sum119905119894minus119894+1

119895=1120582119895where 119905

119894= sum119894

119895=1119889119895and 1 le 119894 le 119870

(iv) Hadamard matrices119867119889119894= [119886(119889119894)

119896119895]1le119896119895le119889119894

of size 119889119894for 119894 = 1 119870

Algorithm(1) 119904 = 0119898 = 0 119905

0= 0

(2) For 119894 = 1 to 119870 do

(3) 119905119894=

119894

sum

119895=1

119889119895

(4) For 119905 = 1 to 119889119894do

(5) 1199031119894= radic

1

119889119894

(

119905119894minus1minus(119894minus2)

sum

119895=1

120582119895minus 119905119894minus1)

(6) For 119895 = 2 to 119889119894minus 1 do

(7) 119903119895119894= radic

1

119889119894

120582119905119894minus1minus(119894minus1)+119895

(8) end

(9) 119903119894119894= radic

1

119889119894

(119905119894minus

119905119894minus119894

sum

119895=1

120582119895)

(10) 119891119905+119898

=

119889119894

sum

119895=1

119886(119889119894)

119905119895119903119895119894119890119904+119895

(11) end(12) 119898 = 119898 + 119889

119894

(13) 119904 = 119904 + 119889119894minus 1

(14) endOutput

(i) unit norm frame 119891119895119872

119895=1for R119873 with eigenvalues 120582

119899119873

119899=1

Algorithm 3 The SHTC algorithm for constructing a unit norm tight frame with119872 elements for R119873

4 Spectral Hadamard Tetris Construction

Another case of HTC is the case of unit norm but notnecessarily tight frames Such frames are known to existprovided that the eigenvalues of the frame operator sumup to the number of frame vectors [15] In this section wegive a version of Hadamard-Tetris Construction which usesaltered Hadamard matrices to construct unit norm frameswith a given spectrum of frame operator Ourmethod cannotconstruct all such framesWe give necessary condition underwhich this constructionworks To construct unit norm framewith prescribed spectrum we are looking for a matrix withcolumns square summing up to 1 and rows square sum-ming up to 120582

119899119873

119899=1 In Algorithm 3 we present a modified

Hadamard-Tetris algorithm HTC allowing the constructionof such frames

Given a fixed dimension 119873 and frame cardinality 119872 anecessary condition on the prescribed eigenvalues 120582

119899119873

119899=1of

the frame operator for SHTC towork is given in the followingtheorem

Theorem 10 Let sum119873119899=1120582119899= 119872 = sum

119870

119894=1119889119894 where 119870 = 119872 minus

119873 + 1 and there exist Hadamard matrices of sizes 119889119894rsquos Then

SHTC can be used to produce unit norm frame for R119873 with

eigenvalues 120582119899119873

119899=1if there are some permutations of 120582

119899119873

119899=1

and 119889119894119870

119894=1such that for each 119894 = 1 119870 we have

119905119894minus119894

sum

119895=1

120582119895le 119905119894le

119905119894minus119894+1

sum

119895=1

120582119895 (37)

where 119905119894= sum119894

119895=1119889119895

Here we present an example in which STC does not workwhereas SHTC works

Example 11 There are choices of prescribed eigenvalueswhich satisfy the conditions of Theorem 10 that unit normframe with given eigenvalues exists but STC cannot be usedto construct such a frame because no rearrangement ofeigenvalues satisfies in their conditions An example of thiskind is 120582

1198994

119899=1= 04 24 11 11 as eigenvalues of frame

with 5 elements forR4

Sometimes one rearrangement of eigenvalues anddimensions of Hadamard matrices is satisfied in desiredcondition while another is not For example the SHTCcannot be performed for the sequence of 120582

119894rsquos is the given

8 Abstract and Applied Analysis

rearrangement 04 11 11 24 and 1198891= 4 119889

2= 1

However rearranging the eigenvalues to the rearrangement04 24 11 11 allows SHTC to construct the desired frameIn this case its synthesis matrix will be

[[[[[[[[[[[[[[

[

radic1

10radic1

10minusradic

1

10radic1

100

minusradic6

10radic6

10minusradic

6

10minusradic

6

100

radic11

40minusradic

11

40minusradic

11

40minusradic

11

400

radic1

40radic1

40radic1

40minusradic

1

401

]]]]]]]]]]]]]]

]

(38)

So there is no reason that SHTC works when theeigenvalues are in the monotonic rearrangement

Note that since we make no assumption about the order-ing of the sequences 120582

119899119873

119899=1and 119889

119894119870

119894=1 we may permute the

elements of the sequences in order to make them ready tosatisfy in the inequality (37)

Given sequences of eigenvalues and 119889119894119870

119894=1 where 119870 =

119872 minus 119873 + 1 and there exist Hadamard matrices of sizes119889119894rsquos it may be time-consuming to find permutation of these

sequences which satisfy in Theorem 10 Until now we couldnot find simple conditions on these sequences that SHTC canbe performed It has been a mystery as to when HadamardTetris works and when fails

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthors are grateful to ProfessorHossein Jafari the Editorof AAA and the referee for their valuable comments

References

[1] R J Duffin andA C Schaeffer ldquoA class of nonharmonic Fourierseriesrdquo Transactions of the American Mathematical Society vol72 pp 341ndash366 1952

[2] J Kovacevic and A Chebira ldquoLife beyond bases the advent offrames (Parts I and II)rdquo IEEE Signal Processing Magazine vol24 no 5 pp 115ndash125 2007

[3] E Soljanin ldquoTight frames in quantum information theoryrdquo inProceedings of the DIMACS Workshop on Source Coding andHarmonic Analysis Rutgers NJ USA May 2002

[4] T Strohmer and J Heath ldquoGrassmannian frames with appli-cations to coding and communicationrdquo Applied and Computa-tional Harmonic Analysis vol 14 no 3 pp 257ndash275 2003

[5] J J Benedetto and M Fickus ldquoFinite normalized tight framesrdquoAdvances in Computational Mathematics vol 18 no 2-4 pp357ndash385 2003

[6] N Cotfas and J P Gazeau ldquoFinite tight frames and some appli-cationsrdquo Journal of Physics AMathematical andTheoretical vol43 no 19 Article ID 193001 2010

[7] D F Li and W C Sun ldquoExpansion of frames to tight framesrdquoActa Mathematica Sinica (English Series) vol 25 no 2 pp 287ndash292 2009

[8] J C Tremain ldquoAlgorithmic constructions of unitary matricesand tight framesrdquo httparxivorgabs11044539v1

[9] D-J Feng L Wang and Y Wang ldquoGeneration of finitetight frames by Householder transformationsrdquo Advances inComputational Mathematics vol 24 no 1ndash4 pp 297ndash309 2006

[10] O Christensen Frames and Bases An Introductory CourseBirkhauser Boston Mass USA 2008

[11] J Cahill M Fickus D G Mixon M J Poteet and N StrawnldquoConstructing finite frames of a given spectrum and set oflengthsrdquoApplied and Computational Harmonic Analysis vol 35no 1 pp 52ndash73 2013

[12] P G Casazza A Heinecke F Krahmer and G KutyniokldquoOptimally sparse framesrdquo IEEE Transactions on InformationTheory vol 57 no 11 pp 7279ndash7287 2011

[13] P G Casazza M Fickus A Heinecke Y Wang and Z ZhouldquoSpectral tetris fusion frame constructionsrdquo The Journal ofFourier Analysis and Applications vol 18 no 4 pp 828ndash8512012

[14] P G Casazza AHeinecke K Kornelson YWang andZ ZhouldquoNecessary and sufficient conditions to perform spectral tetrisrdquoLinear Algebra and its Applications vol 438 no 5 pp 2239ndash2255 2013

[15] K Dykema D Freeman K Kornelson D Larson M Ordowerand E Weber ldquoEllipsoidal tight frames and projection decom-positions of operatorsrdquo Illinois Journal of Mathematics vol 48no 2 pp 477ndash489 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

rearrangement 04 11 11 24 and 1198891= 4 119889

2= 1

However rearranging the eigenvalues to the rearrangement04 24 11 11 allows SHTC to construct the desired frameIn this case its synthesis matrix will be

[[[[[[[[[[[[[[

[

radic1

10radic1

10minusradic

1

10radic1

100

minusradic6

10radic6

10minusradic

6

10minusradic

6

100

radic11

40minusradic

11

40minusradic

11

40minusradic

11

400

radic1

40radic1

40radic1

40minusradic

1

401

]]]]]]]]]]]]]]

]

(38)

So there is no reason that SHTC works when theeigenvalues are in the monotonic rearrangement

Note that since we make no assumption about the order-ing of the sequences 120582

119899119873

119899=1and 119889

119894119870

119894=1 we may permute the

elements of the sequences in order to make them ready tosatisfy in the inequality (37)

Given sequences of eigenvalues and 119889119894119870

119894=1 where 119870 =

119872 minus 119873 + 1 and there exist Hadamard matrices of sizes119889119894rsquos it may be time-consuming to find permutation of these

sequences which satisfy in Theorem 10 Until now we couldnot find simple conditions on these sequences that SHTC canbe performed It has been a mystery as to when HadamardTetris works and when fails

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthors are grateful to ProfessorHossein Jafari the Editorof AAA and the referee for their valuable comments

References

[1] R J Duffin andA C Schaeffer ldquoA class of nonharmonic Fourierseriesrdquo Transactions of the American Mathematical Society vol72 pp 341ndash366 1952

[2] J Kovacevic and A Chebira ldquoLife beyond bases the advent offrames (Parts I and II)rdquo IEEE Signal Processing Magazine vol24 no 5 pp 115ndash125 2007

[3] E Soljanin ldquoTight frames in quantum information theoryrdquo inProceedings of the DIMACS Workshop on Source Coding andHarmonic Analysis Rutgers NJ USA May 2002

[4] T Strohmer and J Heath ldquoGrassmannian frames with appli-cations to coding and communicationrdquo Applied and Computa-tional Harmonic Analysis vol 14 no 3 pp 257ndash275 2003

[5] J J Benedetto and M Fickus ldquoFinite normalized tight framesrdquoAdvances in Computational Mathematics vol 18 no 2-4 pp357ndash385 2003

[6] N Cotfas and J P Gazeau ldquoFinite tight frames and some appli-cationsrdquo Journal of Physics AMathematical andTheoretical vol43 no 19 Article ID 193001 2010

[7] D F Li and W C Sun ldquoExpansion of frames to tight framesrdquoActa Mathematica Sinica (English Series) vol 25 no 2 pp 287ndash292 2009

[8] J C Tremain ldquoAlgorithmic constructions of unitary matricesand tight framesrdquo httparxivorgabs11044539v1

[9] D-J Feng L Wang and Y Wang ldquoGeneration of finitetight frames by Householder transformationsrdquo Advances inComputational Mathematics vol 24 no 1ndash4 pp 297ndash309 2006

[10] O Christensen Frames and Bases An Introductory CourseBirkhauser Boston Mass USA 2008

[11] J Cahill M Fickus D G Mixon M J Poteet and N StrawnldquoConstructing finite frames of a given spectrum and set oflengthsrdquoApplied and Computational Harmonic Analysis vol 35no 1 pp 52ndash73 2013

[12] P G Casazza A Heinecke F Krahmer and G KutyniokldquoOptimally sparse framesrdquo IEEE Transactions on InformationTheory vol 57 no 11 pp 7279ndash7287 2011

[13] P G Casazza M Fickus A Heinecke Y Wang and Z ZhouldquoSpectral tetris fusion frame constructionsrdquo The Journal ofFourier Analysis and Applications vol 18 no 4 pp 828ndash8512012

[14] P G Casazza AHeinecke K Kornelson YWang andZ ZhouldquoNecessary and sufficient conditions to perform spectral tetrisrdquoLinear Algebra and its Applications vol 438 no 5 pp 2239ndash2255 2013

[15] K Dykema D Freeman K Kornelson D Larson M Ordowerand E Weber ldquoEllipsoidal tight frames and projection decom-positions of operatorsrdquo Illinois Journal of Mathematics vol 48no 2 pp 477ndash489 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of