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Brian J. Olson1Applied Physics Laboratory,

Air and Missile Defense Department,The Johns Hopkins University,

Laurel, MD 20723-6099e-mail: [email protected]

Steven W. ShawUniversity Distinguished Professor

Department of Mechanical Engineering,Michigan State University,

East Lansing, MI 48824-1226e-mail: [email protected]

Chengzhi ShiDepartment of Mechanical Engineering,

University of California, Berkeley,Berkeley, CA 94720

e-mail: [email protected]

Christophe PierreProfessor and Vice President

for Academic AffairsDepartment of Mechanical Science

and Engineering,University of Illinois at Urbana-Champaign,

Urbana, IL 61801e-mail: [email protected]

Robert G. ParkerL.S. Randolph Professor and Department Head

Department of Mechanical Engineering,Virginia Polytechnic Institute

and State University,Blacksburg, VA 24061

e-mail: [email protected]

Circulant Matrices and TheirApplication to Vibration AnalysisThis paper provides a tutorial and summary of the theory of circulant matrices and their application to the modeling and analysis of the free and forced vibration of mechanicalstructures with cyclic symmetry. Our presentation of the basic theory is distilled from theclassic book of Davis (1979, Circulant Matrices , 2nd ed., Wiley, New York) with results, proofs, and examples geared specically to vibration applications. Our aim is to collect the most relevant results of the existing theory in a single paper, couch the mathematicsin a form that is accessible to the vibrations analyst, and provide examples to highlight key concepts. A nonexhaustive survey of the relevant literature is also included, which

can be used for further examples and to point the reader to important extensions, applica-tions, and generalizations of the theory. [DOI: 10.1115/1.4027722]

1 IntroductionThe theory of matrices exhibits much that is visually attractive.Thus, diagonal matrices, symmetric matrices, (0, 1) matrices, and the like are attractive independently of their applications. In thesame category are the circulants.

Philip J. Davis

The modeling and analysis of structural vibration is amature eld, especially when vibration amplitudes are smalland linear models apply. For such models, the powerful toolsof modal analysis and superposition allow one to decomposethe system and its response into a set of uncoupled single

degree-of-freedom (DOF) systems, each of which captures themotion of the overall system in a given normal mode. For geometrically simple continua, the natural frequencies, vibra-tion modes, and response to known excitations can be analyti-cally determined. When discrete models are developed, matrixmethods are readily applied for both natural frequency andresponse analyses. For general system models with no specialproperties, which occur in the majority of applications, large-scale computational models must be developed, typicallyusing nite element methods. However, certain classes of

systems possess special properties, such as symmetries, whichaid in the analysis by enabling signicant reduction of the -nite element models (Fig. 1(a)). This is particularly true for systems with cyclic symmetry.

The main goal of this tutorial is to consider the vibrations of structural systems with cyclic symmetry, also known as rotation-ally periodic systems. A useful geometric view of these systems isthat of a circular disk (a pie) split into N equal sectors (i.e., equallysized pieces of the pie), each of which contains an identicalmechanical structure with identical coupling to forward-nearest-neighbors and to ground (Fig. 1(b)).

The theory of circulants also applies to more general forms

of coupling with non-nearest-neighbors, for example througha base substructure, as long as the rotational symmetry ispreserved. These structural arrangements arise naturally incertain types of rotating machines. Turbomachinery examplesinclude bladed disks, such as fans, compressors, turbines, andimpeller stages of aircraft, helicopter engines, power plants,as well as propellers, pumps, and the like. Rotational perio-dicity also arises in some stationary structures such as satel-lite antennae. In some systems, such as planetary gears androtors with pendulum vibration absorbers, the overall systemis not necessarily cyclic, but subcomponents of it may be.When perfect cyclic symmetry of a model is assumed, specialmodal properties exist that signicantly facilitate its vibrationanalysis.

1Corresponding author.Manuscript received January 12, 2014; nal manuscript received May 7, 2014;

published online June 19, 2014. Editor: Harry Dankowicz.

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The nature of rotationally periodic systems imposes a cyclicstructure on their mass and stiffness matrices, which are block cir-culant for systems with many DOFs per sector (Fig. 1(a)) and cir-culant for the special case of a single DOF per sector (Fig. 1(b)).By denoting the stiffness of the internal elements of each sector by K 0 and the coupling stiffness between sectors as K 1 , the stiff-ness matrices of rotationally periodic structures with nearest-neighbor coupling have the general form

K

K 0 K 1 0 0 K 1K 1 K0 K 1 0 00 K 1 K0 0 0... ..

. ... . .

. ... ..

.

0 0 0 K0 K 1K 1 0 0 K 1 K0

266666664

377777775

where K 0 and K 1 are themselves matrices for the complex modelshown in Fig. 1(a) and scalars for the simplest prototypical modelshown in Fig. 1(b). A key property of K is that the elements of each row are obtained from the previous row by cyclically per-muting its entries. That is, for j 2; 3; ; N , row j is obtainedfrom row j 1 by shifting the elements of row j 1 to the right byone position and wrapping the right-end element of row j 1 intothe rst position. This is precisely the form of a circulant matrix,which is formally dened in Sec. 2.2 . The mass matrix of a rota-tionally periodic structure with nearest-neighbor coupling is block

diagonal and also shares this cyclic property. The size of the ele-ments of K is equal to the number of DOFs per sector, and isdenoted by M . Thus, a system with N sectors and M DOFs per sec-tor has a total of NM DOFs. The most important utility of thetheory of circulants in analyzing rotationally periodic systems isthat they enable a NM -DOF system to be decomposed to a set of NM -DOF uncoupled systems using the appropriate coordinatetransformation. Admittedly, the same can be accomplished usingbrute-force methods to uncouple the entire system using modalanalysis, but such an approach overlooks fundamental properties

that are crucial to understanding the free and forced response of these systems and requires signicantly more computationalpower. This is the central motivation for understanding and utiliz-ing circulants to analyze cyclic systems.

The vibration modes of rotationally periodic systems consist of multiple pairs of repeated natural frequencies (eigenvalues) thatlead to pairs of degenerate normal modes (eigenvectors). Thenumber and nature of such pairs depend on whether N is even or odd. Each mode pair is characterized as a pair of standing waves(SWs) with different spatial phases, or a pair of traveling waves,labeled as a forward traveling wave (FTW) and backward travel-ing wave (BTW) when following the terminology used in applica-tions to rotating machinery. The choice of formulation is based onconvenience for a given application, which depends on the natureof the system excitation. For example, the excitation frequency isproportional to the engine speed for many cyclic rotating systems,which leads to the so-called engine order (e.o.) excitation , andoften the spatial nature of the excitation (in the rotating frame of reference) is in the form of a traveling wave. When such excita-tion is applied to systems with cyclic symmetry, the response alsohas special properties that can be easily uncovered by making useof the system traveling wave vibration modes.

The strength of the intersector coupling is an important parame-ter in rotationally periodic systems. When the intersector couplingis strong, the frequencies of the mode pairs are well separated. Incontrast, weak intersector coupling yields closely spaced frequen-cies, high modal density, and large sensitivity to cyclic-symme-try-breaking imperfections. A wave representation of the response[2] shows that the strength of the coupling determines frequencypassband widths, wherein unattenuated propagation of wavestakes place. Weak intersector coupling leads to narrow passbands,and the passbands widen as the coupling strength increases.Another important parameter for cyclically symmetric structuresis the total number of sectors. The modal density is larger for large N , which corresponds to more natural frequencies within each fre-quency passband. In all cases, the modes are spatially distributed,or extended, for models of cyclic systems. That is, the pattern of displacements in a modal response is uniformly spread around thecircumference of the structure.

Systems with cyclic symmetry have been studied in the contextof vibration analysis for over 40 yr. Early work considered proper-ties of the vibration modes [ 3,4] and the steady-state response toharmonic excitation [ 5 7] of tuned and mistuned turbomachineryrotors. Many of these contributions were motivated by vibrationstudies of general rotationally periodic systems [ 8 20], bladeddisks [1,3,21 30], planetary gear systems [ 31 43], rings [ 44,45],

circular plates [ 46 48], disk spindle systems [ 49 51], centrifugalpendulum vibration absorbers [ 52 56], space antennae [ 57], andmicroelectromechanical system frequency lters [ 58]. Implicit inthese investigations is the assumption of perfect symmetry which,of course, is an idealization. Perfect symmetry gives rise to well-structured vibration modes [ 9,31,33 37,39,53,56], which are char-acterized by certain phase indices that dene specic phase rela-tionships between cyclic components in each vibration mode [ 18].This vibration mode structure is critical in the investigation of dynamic response of cyclic systems using modal analysis[54]. These special properties of rotationally periodic structuressave tremendous calculation effort in the analysis of the systemdynamics [ 59 61]. The properties of cyclic symmetry are not onlyused in the study of mechanical vibrations, they are also important

Fig. 1 ( a ) Finite element model of a bladed disk assembly [ 1]and ( b ) general cyclic system with N identical sectors andnearest-neighbor coupling

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to the analysis of elastic stress [ 62,63] and coupled cellnetworks [ 64].

The special properties of systems with cyclic symmetry extendto nonlinear systems, where they are expressed quite naturally interms of symmetry groups: the cyclic group, in particular [ 65 68].The group theoretic formulation can also be applied to the linear vibration problems considered in this paper [ 69,70], but theapproach presented here is more approachable to readers with astandard engineering background in linear algebra.

The extension to systems with small imperfections that perturb

the cyclic symmetry has led to important results related to modelocalization , which arises in systems with high modal densitycaused by weak intersector coupling or a large number of sectors.In particular parameter regimes, the mode shapes are highly sensi-tive to small, symmetry-breaking imperfections among the nomi-nally identical sectors, and the spatial nature of the vibrationmodes can become highly localized. For these cases, the vibrationenergy is focused in a small number of sectors, and sometimeseven a single sector. This behavior, which stems from the seminalwork of Anderson on lattices [ 71], was originally recognized to berelevant to structural vibrations by Hodges and Woodhouse[72,73] and Pierre and Dowell [ 74], and has been extensivelystudied from both fundamental [ 75] and applied [ 76 78] points of view. The phenomenon of mode localization is also observed inthe forced response and has practical implications for the fatiguelife of bladed disks in turbomachinery [ 79,80]. It is interesting tonote that localization also arises in nonlinear systems with perfectsymmetry, where the dependence of the system natural frequen-cies on the amplitudes of vibration naturally leads to the possibil-ity of mistuning of frequencies between sectors if their amplitudesare different [ 81 88].

Another topic central to vibration analysis that relies on thetheory of circulants is the discrete Fourier transform (DFT)[89,90]. The DFT was known to Gauss [ 91], and is the most com-mon tool used to process vibration signals from experimentalmeasurements and numerical simulations. The DFT and inverseDFT (IDFT) provide a computationally convenient means of determining the frequency content of a given signal. Because themathematics of circulants is at the heart of the computation of theDFT, we include a brief introduction to the relationship betweenthe DFT and IDFT, and its connection to the theory of circulants.

The goal of this paper is to provide a detailed theory of circu-lant matrices as it applies to the analysis of free and forced struc-tural vibrations. Much of the material was developed as part of thePh.D. research of the lead author [ 21,22,92 95]. References toother relevant work are included throughout this paper, but we donot claim to provide an exhaustive survey of the relevantliterature. The remainder of the paper is organized as follows.Section 2 gives a quite exhaustive and self-contained treatment of the theory of circulants, which is distilled from the seminal workby Davis [ 96]. We adopt a presentation style similar to that of Ottarsson [ 97], one that should be familiar to an analyst in thevibrations engineering community. This section is meant to actsimultaneously as a detailed reference and tutorial, includingproofs of the main results and simple illustrative examples.Section 3 provides three examples that make use of the theory,

including ordinary circulants and the more general block circulantmatrices. Particular attention is given to cyclic systems under trav-eling wave engine order excitation because this type of systemforcing appears naturally in many relevant applications of rotatingmachinery. The apprised reader, or the reader who wishes to learnby example, can skip directly to Sec. 3, depending on their back-ground, and revisit Sec. 2 as warranted. The paper closes with abrief summary in Sec. 4.

2 The Theory of CirculantsThis section details the theory and mathematics of circulant

matrices that are relevant to vibration analysis of mechanicalstructures with cyclic symmetry. The basic theory is distilled from

the seminal work by Davis [ 96] and is presented using mathemat-ics and notation that should be familiar to the vibrations engineer.Selected topics from linear algebra are reviewed in Sec. 2.1to introduce relevant notion and support the theoreticaldevelopment of circulant matrices in Secs. 2.2 2.8 . This materialis included for completeness; the apprised reader can skip directlyto Secs. 2.2 and 2.3 , where circulant and block circulant matrices(also referred to as circulants and block circulants ) are dened.Representations of circulants are discussed in Sec. 2.4 . Diagonal-ization of circulants and block circulants is discussed at length in

Sec. 2.5 , which begins with a treatment of the N th roots of unityin Sec. 2.5.1 and the Fourier matrix in Sec. 2.5.2 . It is subse-quently shown how to diagonalize the cyclic forward shift matrixin Sec. 2.5.3 a circulant in Sec. 2.5.4 , and a block circulant inSec. 2.5.5 . Some generalizations of the theory are discussed inSec. 2.6, including the diagonalization of block circulants withcirculant blocks. Relevant mathematics of the DFT and IDFT aresummarized in Sec. 2.7 . Finally, the circulant eigenvalue problem(cEVP) is discussed in Sec. 2.8, including the eigenvalues andeigenvectors of circulants and block circulants, their symmetrycharacteristics, and connection to the DFT process.

2.1 Mathematical Preliminaries. Denitions and relevantproperties of special operators and matrices are discussed inSecs. 2.1.1 and 2.1.2 , respectively, including the direct (Kro-

necker) product, and Hermitian, unitary, cyclic forward shift, andip matrices. This is followed in Sec. 2.1.3 with a treatment of matrix diagonalizability.

2.1.1 Special Operators. Let C denote the set of complexnumbers and Z be the set of positive integers.DEFINITION 1 (Direct Sum). For each i 1; 2; ; N and pi 2Z , let A i 2C pi pi . Then the direct sum of A i is denoted by

N i1 A i A 1 A2 A N

and results in the block diagonal square matrix

A

A1 0 00 A2 0

..

....

. ..

..

.

0 0 A N

26664

37775

of order p 1 p2 p N , where each zero matrix 0 has theappropriate dimension. It is convenient to dene the operator diag that takes as itsargument the ordered set of matrices A 1; A2; ; A N and results in

the block diagonal matrix given in Denition 1, that is,

A diagA1; A2; ; A N diagi1; ; N Ai

For the case when each A i a i is a scalar (1 1), the direct sumof a i is denoted by the diagonal matrixdiaga1; a2; ; a N diagi1;; N a i

DEFINITION 2 (Direct Product). Let a; b 2C n . Then the direct prod-uct (or Kronecker product) of a and bT is the square matrix

a bT a1b1 a1b2 a1bna2b1 a2b2 a2bn

..

. ... . .

. ...

anb1 anb2 anbn26664

37775where T denotes transposition. If A 2C m n and B 2C p q arematrices, then the direct product of A and B is the matrix

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A B a11 B a12 B a1nBa21 B a22 B a2nB

..

. ... . .

. ...

am1B am2B amnB26664

37775of dimension mp nq.

Example 1 . Consider the matrices

A

1 2 3

and B

1 2

3 4 Then the direct product of A and B is given byA B 1 2 3

1 2

3 4 1

1 2

3 4 ; 2 1 23 4 ; 3 1 23 4

1 2

3 4 ; 2 46 8 ; 3 69 12

1 2 2 4 3 6

3 4 6 8 9 12

Because A is 1 3 and B is 2 2, the direct product A B hasdimension 1 2 3 2, or 2 6.Some important properties of the direct product are as follows:

(1) The direct product is a bilinear operator. If A and B aresquare matrices and a is a scalar, then

aA B aA B A aB (1)(2) The direct product distributes over addition. If A , B, and C

are square matrices with the same dimension, then

A B C A C B C (2a)A B C A B A C (2b)

(3) The direct product is associative. If A , B, and C are squarematrices, then

A B C A B C (3)(4) The product of two direct products yields another direct

product. If A, B, C, and D are square matrices such that ACand BD exist, then

A BC D AC BD (4)(5) The inverse of a direct product yields the direct product of

two matrix inverses. If A and B are invertible matrices,then

A B

1

A 1 B 1 (5)

where 1 denotes the matrix inverse.

(6) The transpose or conjugate transpose of a direct productyields the direct product of two transposes or conjugatetransposes. If A and B are square matrices, then

A BT A T BT (6a)A BH AH BH (6b)

where H T is the conjugate transpose and denotescomplex conjugation.(7) If A and B are square matrices with dimensions n and m,respectively, then

detA B det Amdet Bn (7a)tr A B tr Atr B (7b)

where det and tr denote the matrix determinant andtrace. 2.1.2 Special Matrices. The denitions and relevant proper-

ties of selected special matrices are summarized. Hermitian andunitary matrices are dened rst (see Table 1), followed by a brief treatment of two important permutation matrices: the cyclic for-ward shift matrix and the ip matrix. The details of circulant mat-rices and the Fourier matrix, which are employed extensivelythroughout this work, are deferred to Secs. 2.2, 2.3, and 2.5.2 .

DEFINITION 3 (Hermitian Matrix). A matrix H 2C N N is Hermi-tian if H HH. The elements of a Hermitian matrix H satisfy hik hki for alli; k 1; 2; ; N . Thus, the diagonal elements hii of a Hermitianmatrix must be real, while the off-diagonal elements may be com-plex. If H H T then H is said to be symmetric .DEFINITION 4 (Unitary Matrix). A matrix U 2C N N is unitary if UHU I , where I is the N N identity matrix. Real unitary matrices are orthogonal matrices. If a matrix U isunitary, then so too is UH. To see this, consider UHHUH UUH I , from which it follows that

UHU UUH I (8)

Finally, if U is unitary and nonsingular, then UH U1.

A general permutation matrix is formed from the identitymatrix by reordering its columns or rows. Here, we introduce twosuch matrices: the cyclic forward shift matrix and the ip matrix.

DEFINITION 5 (Cyclic Forward Shift Matrix). The N N cyclic forward shift matrix is given by

r N

0 1 0 0 00 0 1 0 0... ... ... . .. ... ...0 0 0 1 00 0 0 0 11 0 0 0 0

2666666437777775 N N

which is populated with ones along the superdiagonal and in the(N, 1) position, and zeros otherwise.

The cyclic forward shift matrix plays a key role in the represen-tation and diagonalization of circulant matrices, which are dis-cussed in Secs. 2.4 and 2.5.

Example 2 . Let a (a , b, c) be a three-vector. Then theoperation

ar 3 a b c 0 1 0

0 0 1

1 0 0264 375

c; a; b

Table 1 Selected special matrices

Type Condition

Symmetric AATHermitian A AHOrthogonal ATA I (or) AT A 1Unitary AHA I (or) AH A 1



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C circc1; c2; ; c N (9)An N N circulant is also characterized in terms of its ( i, k ) entryby Cik ck i1mod N for i; k 1; 2; ; N . Example 3 . The circulant array formed by the generating ele-ments a, b, c, d can be written as

circa ; b; c; d a b c d d a b cc d a bb c d a

2664

3775

2C 4

which is a circulant matrix of type 4.If a matrix is both circulant and symmetric, its generating ele-

ments are

c1; ; c N 2; c N 22 ; c N 2 ; ; c3; c2; N even

c1; ; c N 12

; c N 12 ; c N 12 ; c N 12 ; ; c3 ; c2; N odd( (10)which are necessarily repeated. Only ( N 2)/2 generating ele-ments are distinct if N is even and ( N 1)/2 are distinct if N isodd. The set of all N N symmetric circulants is denoted bySC N . A matrix contained in SC N is said to be a symmetric circu-lant of type N .

Example 4 . The 5 5 matrix

circa; b; c; c; b

a b c c bb a b c cc b a b cc c b a bb c c b a

266664377775

2SC 5

is both symmetric and circulant. Because N 5 is odd, it has( N 1)/2 3 distinct elements.The matrix dened in Example 3 is not a symmetric circulantbecause its generating elements are distinct. Next, we give a nec-essary and sufcient condition for a square matrix to be circulant.

THEOREM 4. Let r N be the cyclic forward shift matrix. Then a N N matrix C is circulant if and only if Cr N r N C. Proof. Let C be an N N matrix with arbitrary elements cik for i; k 1; 2; ; N . Then

Cr N

c1 N c11 c12 c1 N 1c2 N c21 c22 c2 N 1... ..

. ... . .

. ...

c NN c N 1 c N 2 c N N 1

2666437775

and

r N C

c21 c22 c23 c2 N c31 c32 c33

c3 N ... ..

. ... . .

. ...

c11 c12 c13 c1 N 26664 37775These matrices are equal if and only if the equalities

c1 N c21 ; c11 c22 ; c1 N 1 c2 N c2 N c31 ; c21 c32 ; c2 N 1 c3 N ... ..

. . .. ..

.

c NN c11 ; c N 1 c12 ; c N N 1 c1 N are satised. Then C can be written as

C c11 c12 c1 N c21 c22 c2 N

..

. ... . .

. ...

c N 1 c N 2 c NN 26664

37775c11 c12 c1 N c1 N c11 c1 N 1

..

. ... . .

. ...

c12 c13 c11

2666437775

which is a N N circulant matrix with generating elementsc11 ; c12 ; ; c1 N .

Any matrix that commutes with the cyclic forward shift matrixis, therefore, a circulant. Theorem 4 also says that circulant

matrices are invariant under similarity transformations involvingthe cyclic forward shift matrix. That is, C is similar to itself for asimilarity transformation using r N .

Example 5 . Consider the 3 3 matrix

A a b cc a bb c a24 35

Then

a b c

c a bb c a264

375

0 1 0

0 0 1

1 0 0264

375

c a b

b c aa b c264

375 0 1 00 0 1

1 0 0264 375

a b cc a b

b c a264 375

which implies that Ar 3 r 3A. Thus, A circa; b; c 2C 3 is acirculant matrix of type N 3.Next we introduce block circulant matrices, which are naturalgeneralizations of ordinary circulants.

2.3 Block Circulant Matrices. A block circulant matrixis obtained from a circulant matrix by replacing each entry ck inDenition 9 by the M M matrix C i for i 1; 2; ; N .DEFINITION 11 (Block Circulant Matrix). Let C i be a M M matrix for each i 1; 2; ; N. Then a NM NM block circulant matrix (or block circulant) is generated from the ordered set fC 1; C 2; ; C N g, and is of the form

C C 1 C2 C N C N C1 C N 1... ..

. . .. ..

.

C 2 C3 C126664

37775 D

DEFINITION 12 (Generating Matrices). Let the NM NM block circulant C be given by Denition 11. Then the elements of theordered set

fC1; C 2; ; C N

gare said to be the generating matrices of C. A block circulant is therefore dened completely by its generat-

ing matrices. The matrix array given by Denition 11 is said to bea block circulant of type ( M , N ). The set of all such matrices isdenoted by BC M ; N . A matrix C 2BC M ; N is not necessarily a cir-culant, as the following example demonstrates.

Example 6 . Let

A 2 11 2 and B 1 00 1

Then



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C

A B 0 B

B A B 0

0 B A B

B 0 B A

2666437775

2 1 1 0 0 0 1 01 2 0 1 0 0 0 11 0 2 1 1 0 0 0

0 1 1 2 0 1 0 00 0 1 0 2 1 1 00 0 0 1 1 2 0 11 0 0 0 1 0 2 1

0 1 0 0 0 1 1 2

377777777777775

266666666666664

is a block circulant of type (2, 4), but it is not a circulant.Next we give a necessary and sufcient condition for a matrix

to be block circulant.THEOREM 5. Let r N be the cyclic forward shift matrix of dimen-

sion N and I M be the identity matrix of dimension M. Then a NM NM matrix C is a block circulant of type (M, N) if and onlyif Cr N I M r N I M C. The proof of Theorem 5 follows similarly to that of Theorem 4by replacing the scalar elements cik with M M matrices C ik for i; k 1; 2; ; N . The reader can verify that the matrix C inExample 6 satises the condition in Theorem 5, but not that inTheorem 4.

A block circulant, block symmetric matrix of type ( M , N ) hasgenerating matrices of the same form as Eq. (10), and is obtainedby replacing each entry ck by the M M matrix Ck for k 1; 2; ; N . The set of all such matrices is denoted byBCBS M ; N . The matrix C in Example 6 is recognized to be ablock symmetric, block circulant matrix of type (2, 4), that is, it iscontained in BCBS 2;4 , which is a subset of BC 2;4 .

2.4 Representations of Circulants. It is clear from Deni-tion 5 that the N N cyclic forward shift matrix is a circulant with

N generating elements 0 ; 1; 0; ; 0; 0. The integer powers of r N can be written as

r 0 N circ1; 0; 0; 0; 0; ; 0; 0 I N r 1 N r N circ0; 1; 0; 0; 0; ; 0; 0

r 2 N circ0; 0; 1; 0; 0; ; 0; 0...

r N 1 N circ0; 0; 0; 0; 0; ; 0; 1r N N circ1; 0; 0; 0; 0; ; 0; 0 r 0 N I N

9>>>>>>>>>>>=>>>>>>>>>>>;

(11)

where each successive power cyclically permutes the generatingelements. This enables the representation of a general circulant interms of a nite matrix polynomial involving the cyclic forwardshift matrix and its powers.

COROLLARY 3. Let C 2C N be a circulant matrix of type N withgenerating elements c 1; c2; ; c N . Then C can be represented bythe matrix sum

C c1r 0 N c2r 1 N c3r 2 N c N r N 1 N

where r N is the N N cyclic forward shift matrix . Example 7 . The matrix A circa; b; c from Example 5 canbe represented by the matrix sum

A ar 03 br 13 cr 23 aI3 br 3 cr 23

a1 0 0

0 1 0

0 0 1

264 375b0 1 0

0 0 1

1 0 0

264 375c0 0 1

1 0 0

0 1 0

264 375

a b c

c a b

b c a

264

375where I3 and r 3 are the 3 3 identity and cyclic forward shift

matrices.Corollary 3 is exploited in Sec. 2.5.4 to diagonalize a general

circulant matrix, and can be generalized to represent a generalblock circulant matrix in terms of the cyclic forward shift matrixand its powers.

COROLLARY 4. Let C 2BC M ; N be a block circulant matrix of type (M, N) with generating matrices C 1; C2; ; C N . Then C canbe represented by the matrix sum

C r 0 N C 1 r 1 N C 2 r N 1 N C N where r

N is the cyclic forward shift matrix.

Corollaries 3 and 4 motivate the following result, which cap-tures the representation of circulant and block circulant matricesin terms of the cyclic forward shift matrix, and facilitates their diagonalization in Secs. 2.5.4 and 2.5.5 .

DEFINITION 13. Let t and s be arbitrary square matrices. Thenthe function

qt; s X N

k 1tk 1 s

is a nite sum of direct products . COROLLARY 5 . Let r N be the N N cyclic forward shift matrix.

Then a circulant matrix with generating elements c 1; c2; ; c N and a block circulant matrix with generating elements

C1; C 2; ; C N can be represented by the matrix sums

qr N ; ck X N

k 1r k 1 N ck circc1; c2; ; c N

qr N ; Ck X N

k 1r k 1 N C k circC1; C 2; ; C N

where the function q is given by Denition 13 . What is meant by the notation qr N ; ck , for example, is to sub-stitute t with r N and s with ck in Denition 13 and then performthe summation observing any indices k introduced by thesubstitution .

2.5 Diagonalization of Circulants. Any circulant or blockcirculant matrix can be represented in terms of the cyclic forwardshift matrix according to Corollary 5. The diagonalization of ageneral circulant begins, therefore, by nding a matrix that diago-nalizes r N . Together with some basic results from linear algebra(these are summarized in Sec. 2.1), this leads naturally to thediagonalization of an arbitrary circulant. Regarding a suitablediagonalizing matrix, there are a number of candidates[10 12,16,17,100 ], but all feature powers of the N th roots of unityor their real/imaginary parts. In this work, we employ an arraycomposed of the distinct N th roots of unity (Sec. 2.5.1 ) and their integer powers in the form of the complex Fourier matrix(Sec. 2.5.2 ). A unitary transformation involving the Fourier matrixis used to diagonalize the cyclic forward shift matrix (Sec. 2.5.3 ),



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general circulant matrices (Sec. 2.5.4 ), and general block circulantmatrices (Sec. 2.5.5 ).

2.5.1 Nth Roots of Unity. A root of unity is any complex num-ber that results in 1 when raised to some integer N 2 Z [101 ].More generally, the N th roots of a complex number zo r oe j ho aregiven by a nonzero number z re j h such that

z N zo or r N e jN h r oe j ho (12)

where j ffiffiffiffiffiffiffi1p . Equation (12) holds if and only if r N

r o and N h ho 2pk with k 2Z . Therefore,r ffiffiffiffir o N p h ho 2pk N 9=;

; k 2Z (13)

and the N th roots are

z ffiffiffiffir o N p exp j ho 2pk N ; k 2Z (14)Equation (14) shows that the roots all lie on a circle of radius ffiffiffiffir o N p centered at the origin in the complex plane, and that they areequally distributed every 2 p= N radians. Thus, all of the distinct roots correspond to k 0; 1; 2; ; N 1.

DEFINITION 14 (Distinct N th Roots of Unity). The distinct N throots of unity follow from Eq. (14) by setting r o 1 and ho 0and are denoted bywk N e j

2pk N

for integers k 0; 1; 2; ; N 1. DEFINITION 15 (Primitive N th Root of Unity). The primitive N throot of unity is denoted by

w N e j 2p N

which corresponds to k 1 in Denition 14 . COROLLARY 6. The integer powers w k N of the primitive Nth root of unity are equivalent to the distinct N th roots of unity w k N for k 1; 2; ; N .

Proof . Consider wk N e

j 2p N

k

e j 2pk N

wk

N , which followsfrom Denition 14. Thus, the powers

1; w1 N ; w2 N ; ; w

N 1 N

are equivalent to the distinct N th roots of unity

1; w1 N ;w2 N ;; w

N 1 N

where w0 N w0 N 1. Example plots of the distinct N th roots of unity are shown in

Fig. 2, where wk N are arranged on the unit circle in the complexplane (centered at the origin) for N 1; 2; ; 9. Note thatw0 N

1 is real, as is w N =2 N

1 if N is even. The remaining

roots appear in complex conjugate pairs. Thus, the distinct N throots of unity are symmetric about the real axis in the complexplane.

Example 8 . Let N 4. Then the distinct N th roots of unity aregiven by the set

fw04; w14; w24; w34g fe j 2p4 0; e j

2p4 1; e j

2p4 2; e j

2p4 3g

fe0; e j p2; e j p ; e j

3p2 g

f1; j ; 1; j gThese four distinct roots of unity can be visualized in Fig. 2 for the case of N 4.

DEFINITION 16. Let w N be the primitive N th root of unity. Then

X N

1 0

w N w2 N

. ..

0 w N 1 N

266666664377777775 N N

diag1; w N ; w2 N ; ; w N 1 N is the N N diagonal matrix formed by placing the distinct N throots of unity 1; w N ; w2 N ; ; w N 1 N along its diagonal .

The matrix X N appears naturally in the diagonalization of cir-culants, which is discussed in Secs. 2.5.3 2.5.5 .

2.5.2 The Fourier Matrix. This section introduces the com-plex Fourier matrix and its relevant properties, including the sym-metric structure of the N -vectors that compose its columns. A keyresult is that the Fourier matrix is unitary, which is systematicallydeveloped and proved.

DEFINITION 17 (Fourier Matrix). The N N Fourier matrix isdened as

E N 1

ffiffiffiffi N p 1 1 1 11 w N w2 N w N 1 N 1 w

2 N w

4 N w

2

N 1

N ... ..

. ... . .

. ...

1 w N 1 N w2 N 1 N w

N 1 N 1 N

266666664

377777775 N N

where w N is the primitive N th root of unity and N 2Z .

Fig. 2 Example plots of the distinct N th roots of unity



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Example 9 . For the special case of N 4, the Fourier matrix isgiven by

E 4 1

ffiffiffi4p 1 1 1 1

1 j 1 j 1 1 1 11 j 1 j

2666437775

Clearly the Fourier matrix is symmetric, but generally it is not

Hermitian. It can be written element wise as

E N ik 1

ffiffiffiffi N p wi 1k 1 N

1

ffiffiffiffi N p e j i 1u k

1

ffiffiffiffi N p e j k 1u i ; i; k 1; 2; ; N (15)

where

u i 2p N i 1 (16)

is the angle subtended from the positive real axis in the complex

plane to the ith power of w N , which is also the i 1th of the N roots of unity according to Denition 14 and the numberingscheme in Fig. 2.

COROLLARY 7. The matrix EH N is obtained from E N by changingthe signs of the powers of each element .

Proof . The Fourier matrix is unaffected by transpositionbecause it is symmetric. Thus, the ( i, k ) element of EH N is

EH N ik E N ik 1 ffiffiffiffi N p wi 1k 1 N

1

ffiffiffiffi N p w i 1k 1 N

where the identity

wk N e j 2p N k e j

2p N k e j

2p N k w k N ; k 2Z

is employed. It follows that EH N

can be obtained from E N bychanging the sign of the powers of the N th roots of unity.

It is shown in Sec. 2.8 that all circulant matrices contained inC N share the same linearly independent eigenvectors, the ele-ments of which compose the N columns (or rows) of E N .

DEFINITION 18. Let w N be the primitive Nth root of unity. Then for i 1; 2; ; N the columns of the Fourier matrix E N aredened by

ei 1

ffiffiffiffi N p 1; wi 1 N ;w

2i 1 N ; ; w N 1i 1 N T

1

ffiffiffiffi N p 1; e j u i ; e j 2u i ; ; e j N 1u i T

where u i is given by Eq. (16) . Example 10 . The third column of the Fourier matrix E4 is given by

e3 1

ffiffiffi4p 1; w3 14 ;w

23 14 ;w33 14 T

12

1; w24; w44; w

64 T

12

1; e j 2p4 2; e j

2p4 4; e j

2p4 6 T

12

1; e j p ; e j 2p ; e j 3p T

12 1; 1; 1; 1

T

for the special case of N 4.

The columns e i of the Fourier matrix E N e1; ; e N exhibita symmetric structure with respect to the index i( N 2)/2 for even N . In Example 9, for instance, the vectors e1 ande N 2=2 e3 are real and distinct, and the vectors e2 and e4appear in complex conjugate pairs. This same structure generallyholds for any E N with even N . To see this, consider

e N 22 6 q 1

ffiffiffiffi N p 1; w

N 2

6 q N 1; ; w N 2 6 q N N 1 T

1

ffiffiffiffi N p 1; w6

q N 1; ; w

6q N

N 1

T

(17)

where the integers 6 q correspond to vector pairs relative to theindex i N 2=2 and the identity

w N 2

6 q N w

N 2

N w6 q N w

6 q N

is employed. The case of q 0 corresponds to i N 2=2 andyields the real vector e N 22

1

ffiffiffiffi N p 1; 1; 1; ; 1; 1T (18)

which has the same value for each element with alternating signsfrom element to element. For q 6 0 the terms w6 q N are complexconjugates according to the proof of Corollary 7 such thate N 2=2 q and e N 2=2 q are complex conjugate pairs. There are( N 2)/2 such pairs corresponding to q f1; 2; ; N 2=2ginEq. (17). Finally, the case of i1 always yields the real vector

e1 1

ffiffiffiffi N p 1; 1; 1; ; 1; 1T (19)

for even and odd N . A similar formulation for odd N shows that e1is real and distinct and the remaining ( N 1)/2 vectors appear incomplex conjugate pairs. This is shown by example in Sec. 3.3 inthe context of vibration modes for a cyclic structure with a singleDOF per sector.

A key feature of the Fourier matrix is that it is unitary. This isessentially a statement of orthogonality of each column of E N andis captured by the following lemmas.

LEMMA 1 (Finite Geometric Series Identity). Let N 2Z and q 2C . Then

Xs N 1

r sqr

qs1 q N 1 q

for any s 2Z and q 6 1. Proof. Consider the nite geometric series

Xs N 1

r sqr qs qs1 qs2 qs N 1

qs1 q q2 q N 1Multiplying from the left by q yields

q Xs N 1

r sqr qsq q2 q3 q N

Subtraction of the second equation from the rst results in

1 qXs N 1

r sqr qs1 q N



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from which the proof is established by division of the term (1 q)because q 6 1 by restriction. Lemma 1 is used to establish the following result, which isrequired to show that the Fourier matrix is unitary. The orthogon-ality condition is fundamental to the diagonalization of circulantsin Secs. 2.5.3 2.5.5 , and the relationship between the DFT andIDFT in Sec. 2.7 .

LEMMA 2. Let w N be the primitive Nth root of unity with N 2Z . Then

Xs N 1

r sw

r

i k

N N ; i k

mN

0; otherwise

for i; k 2Z and any s ; m 2Z . Proof. Let q w

i k N e j 2p N i k and note that q N 1. If i k mN , then q e j 2pm 1 for any integer m, and it follows that

Xs N 1

r swr i k N X

s N 1

r sqr

1s 1s1 1s N 1 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl N terms N For the case of i k

6 mN it follows from Lemma 1 that

X N 1

r 0wr i k N X

N 1

r 0qr

1 q N

1 q

1 11 q

0

(Lem. 1)

which completes the proof. Example 11 . Consider the orthogonality condition given by

Lemma 2 for s 0. Then if i k N 5,

X5 1

r 0wr 55 X

4

r 0e j 2p5

5r

X4

r 0e j

2p5 5r

e0 e j 2p e j 4p e j 6p e j 8p 1 1 1 1 1 5

which is numerically equal to N, as expected. If instead we seti k 5 but N 4, then

X4 1

r

0

wr 54

X3

r

0

e j 2p4

5r

X3

r 0e j

2p4 5r

X3

r 0e j

5p2 r

e5p2 0 e

5p2 1 e

5p2 2 e

5p2 3

e0 e5p2 e5p e

15p2

1 j 1 j 0

sums to zero.

Lemma 2 allows for representations of the N N identity, ip,and cyclic forward shift matrices in terms of certain conditions ontheir indices relative to N .

COROLLARY 8. For i ; k 1; ; N and any integer m, the (i, k)elements of the N N identity, ip, and cyclic forward shift matri-ces can be represented by the summations

dik I N ik 1 N

X

N 1

r 0wr i k N

1; i k mN 0; otherwise(

j N ik 1 N X

N 1

r 0wr ik 2 N

1; i k 2 mN 0; otherwise(

r N ik 1 N X

N 1

r 0wr i k 1 N

1; i k 1 mN 0; otherwise(

where dik is the Kronecker delta . The reader can verify Corollary 8 for the special case of N 3by inspection of the arrays in Fig. 3.We are now ready to state the key result required to diagonalize

a general circulant matrix. Corollaries 7 and 8 are used to showthat the Fourier matrix is unitary.

THEOREM 6 (Unitary Fourier Matrix). The Fourier matrix E N isunitary.

Proof. For 1 i; k N , the (i, k ) entry of EH N E N is given by

EH N E N ik X N

r 1EH N ir E N rk

X N

r 11

ffiffiffiffi N p wi 1r 1 N

1

ffiffiffiffi N p wr 1k 1 N

1 N X N

r 1wr 1i k N

1 N X

N 1

r 0wr i k N

(Eq. 15 and Cor. 7)

I N ik (Cor. 8)

from which it follows that EH N E N I N . Example 12 . Consider the matrix E 4 from Example 9. Becausethe Fourier matrix is unitary, it follows that

Fig. 3 Arrays showing the ( i , k ) elements of the ( a ) identity, ( b )ip, and ( c ) cyclic forward shift matrices of dimension N 5 3 fori, k 5 1, 2, 3



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I N I M I NM (Thm. 6)

where I N and I NM are identity matrices of dimension N and NM ,respectively.

COROLLARY 12. If ei denotes the ith column of the Fourier matrixE N , I M is the identity matrix of dimension M, and dik is the Kro-necker delta, then the NM M matrices ei I M are such that

ei

I M

Hek

I M

dik I M

for i; k 1; 2; ; N . Proof. Consider the matrix product

ei I M Hek I M eHi IH M ek I M (Eq. 6b) eHi ek IH M I M (Eq. 4) dik I M (Cor. 10) dik I M

which completes the proof. Corollary 12 can also be obtained directly from Corollary 11 by

writing

EH N I M

eH1eH2

..

.

eH N

266666664377777775

I M

eH1 I M eH2 I M

..

.

eH N I M

266666664377777775

and

E N I M e1; e2; ; e N I M e1 I M ; e2 I M ; ; e N I M

expanding these matrices similarly to Eq. (20), and setting theresult equal to I NM

diag

I M ; I M ; ; I M

.

Next we derive a relationship between the Fourier and ipmatrices.

THEOREM 7. Let E N and j N denote the N N Fourier and ipmatrices, respectively. Then

E 2 N j N EH N 2: Proof. We rst shown that E2 N E N E N j N . For any inte-ger m and i; k 1; 2; ; N , the (i, k ) entry of E N E N is givenby

E N E N ik

X N

r

1E N ir E N rk

X N

r 11

ffiffiffiffi N p wi 1r 1 N

1

ffiffiffiffi N p wr 1k 1 N (Eq. 15)

1 N X

N 1

r 0wr ik 2 N

j N ik (Cor. 8)

from which it follows that E2 N j N . The result j N EH N 2follows from complex conjugation and transposition of j N E N E N , and by invoking the properties j H N j N andE 2 N H EH N 2.

A number of properties follow directly from Theorem 7.COROLLARY 13. Let E N and j N be the N N Fourier and ip

matrices. Then

(a) E N j N j N E N ;(b) j 2 N I N or j N ffiffiffiffiffiI N p ; and(c) E4 N I N or E N ffiffiffiffiffiI N 4p where I N is the N N identity matrix . Property ( a) of Corollary 13 says that the ip and Fourier matri-

ces commute or, since E N is unitary, that j N is invariant under a

unitary transformation with respect to E N . Thus, j N is not diago-nalizable by E N . Properties ( b) and (c) give alternative denitionsof the ip and Fourier matrices. Moreover, because the power of adiagonal matrix is obtained by raising each diagonal element tothe power in question, if follows that the eigenvalues of j N are6 1 and those of E N are 6 1 and 6 j , each with the appropriatemultiplicities.

2.5.3 Diagonalization of the Cyclic Forward Shift Matrix. Inlight of Corollary 5, diagonalization of a general circulant or block circulant matrix begins by diagonalizing the cyclic forwardshift matrix.

THEOREM 8. Let E N be the N N Fourier matrix and r N be the N N cyclic forward shift matrix. Then

EH N r N E N X N is a diagonal matrix, where X N is given by Denition 16 .

Proof. For i; k 1; 2; ; N , the (i, k ) entry of E N X N EH N is givenby

E N X N EH N ik X N

r 1 X N

p1E N ipX N pr EH N rk

X N

r 1 X N

p11

ffiffiffiffi N p wi 1 p 1 N d pr w

r 1 N

1

ffiffiffiffi N p w r 1k 1 N (Eq. 15)

1 N X

N

r 1wi 1r 1 N w

r 1 N w r 1k 1

1 N X

N

r 1wr 1i k 1 N

1 N X

N 1

r 0wr i k 1 N

r N ik (Cor. 8)from which it follows that E N X N EH N r N . The desired result fol-lows by multiplying from the left by E

H N , multiplying from the

right by E N , and invoking Theorem 6. Theorem 8 implies that r N is unitarily similar to a diagonal ma-

trix whose diagonal elements are the distinct N th root of unity(i.e., Denition 16). Because the eigenvalues of a matrix are pre-served under such a transformation (this is guaranteed by Theo-rem 1), it follows that

ar N aX N f1; w N ; w2 N ; ; w N 1 N gwhere a denotes the matrix spectrum. The eigenvectors of thecirculant matrix r N are the linearly independent columns of E N e1; e2; ; e N , which are given by Denition 18. In fact,all circulant matrices contained in C N share the same eigenvectors



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ei, which is shown in Sec. 2.8 . In light of Corollary 2, we have thefollowing results.

COROLLARY 14. Let E N and r N be the N N Fourier and cyclic forward shift matrices. Then for any n 2Z ,

EH N r n N E N X n N where X N given by Denition 16 .

COROLLARY 15. Let ei be the ith column of the Fourier matrixE N and r N be cyclic forward shift matrix. Then for any n 2Z and i ; k 1; 2; N,

eHi r n N ek wi 1 N ndik w

ni 1 N dik

where dik is the Kronecker delta . Corollary 15 shows that the columns of the Fourier matrix are

orthogonal with respect to the cyclic forward shift matrix and itspowers. It is shown in Sec. 2.5.4 that they are in fact orthogonalwith respect to any circulant matrix.

2.5.4 Diagonalization of a Circulant.Corollary 16. Let the matrix X N be populated with the distinct

Nth roots of unity according to Denition 16 and s be an arbitrarysquare matrix. Then

qX N ; s diagi1; ; N qwi 1 N ;s

where q is given by Denition 13 . Proof. Consider the representationqX N ; s qdiag1; w N ; w2 N ; ; w N 1 N ; s

X N

k 1diagw

0 k 1 N ;w1 k 1 N ;; w

N 1k 1 N s

X N

k 1diagsw

0 k 1 N ;sw1 k 1 N ; ; sw

N 1k 1 N

diagi1;; N X N

k 1swi 1k 1 N !

diagi1;; N qwi 1 N ;s

which is diagonal when s is a scalar and block diagonal when s isa matrix.

THEOREM 9 (Diagonalization of a Circulant). Let C 2C N havegenerating elements c 1; c2; ; c N . Then if E N is the N N Fourier matrix ,

EH N CE N

k1 0

k2

. ..

0 k N

2666664

3777775

is a diagonal matrix. For i 1; 2; ; N, the diagonal elementsareki qwi 1 N ;ck X

N

k 1ck wk 1i 1 N (21)

where w N is the primitive Nth root of unity and the function q isgiven by Denition 13 .

Proof. Consider the representation

C qr N ; ck (Cor. 5)

qE N X N EH N ; ck (Thm. 8)

E N qX N ; ck EH N (Thms. 2 and 6)

where the last step follows directly from the proof of Theorem 2and the polynomial term

qX N ; ck diagi1;; N X N

k 1ck wk 1i 1 N !

diagi1;; N qwi 1 N ;ck

follows from Corollary 16 and Denition 13. The desired result isobtained by multiplying from the left by EH N , multiplying from theright by E N , and invoking Theorem 6.

Theorem 9 shows that the Fourier matrix E N diagonalizes any N N circulant matrix. As discussed in Sec. 2.8, the columnse1; e2; ; e N of E N are the eigenvectors of C. The scalars ki inTheorem 9 are the eigenvalues of C . Unlike the eigenvectors, theydepend on the elements (i.e., generating elements) of C. In fact,Eq. (21) has the same form as the DFT of a discrete time series,which is clear by comparing it to Denition 20 of Sec. 2.7 . In thiscase, ck k 1; 2; ; N and kii 1; 2; ; N are analogous to adiscrete signal and its DFT, and are related by

k1

k2

..

.

k N

2666664

3777775

ffiffiffiffi N p E N

c1

c2

..

.

c N

2666664

3777775

(22)

which has the same form as the matrixvector representation of the DFT dened by Eq. (27) of Sec. 2.7 . Thus, the eigenvalues kican be calculated directly using Eq. (21) or Eq. (22) , which areequivalent.

If the circulant matrix in Theorem 9 is also symmetric, theeigenvalues are real-valued and certain ones are repeated, asshown in Corollary 17.

COROLLARY 17. If C 2SC N is a symmetric circulant, Eq. (21)reduces to

ki

c1

2

X N =2

k 2ck cos

2pk 1i 1 N 1i 1c N 22 ; N even

c1 2 X N 1=2

k 2ck cos

2pk 1i 1 N ; N odd

8>>>>>>>>>>>>>>>>>:where the generating elements are given by Eq. (10) .

Proof. If N is even, the generating elements of C are given by

c1; c2; ; c N 2; c N 22 ; c N 2 ; ; c3; c2

and Eq. (21) reduces to



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ki X N

k 1ck wk 1i 1 N

c1w0 i 1 N c2w

1 i 1 N c3w2 i 1 N

c N 2 w N 2 1 i 1 N c N 22 w

N 22 1 i 1 N

c N 2 w N 42 1 i 1 N

c3w N 1 1i 1

N c2w N 1i 1

N

c1 c2 wi 1 N w

N 1i 1 N c3 w

2i 1 N w N 2i 1 N

c N 2 w N 2

2 i 1 N w

N N 22 i 1 N c N 22 w N 2i 1 N c1 2X

N =2

k 2ck cos

2pk 1i 1 N c N 22 1i 1

where the identity

wk i 1 N

w N k i 1 N

2 cos 2pk

N i 1

is employed. If N is odd, the generating elements of C are given byc1; c2; ; c N 1

2; c N 12 ; c N 12 ; c N 12 ; ; c3; c2

and Eq. (21) reduces similarly to the case for even N , which is leftas an exercise for the reader.

The cosinusoidal nature of k i in Corollary 17 implies that k1 isdistinct for a symmetric circulant matrix C 2SC N , but theremaining elements ki k N 2 i appear in repeated pairs. How-ever, k N 2=2 is also distinct if N is even. Example 15 . Let C circ4; 1; 0; 1 2SC 4 be a symmetriccirculant matrix. Then the product

EH4 CE 4 1

ffiffiffi4p 1 1 1 1

1 j 1 j 1 1 1 11 j 1 j 26666664

37777775

4

1 0

1

1 4 1 00 1 4 11 0 1 4

2666666437777775

1

ffiffiffi4p 1 1 1 1

1 j 1 j 1 1 1 11 j 1 j

2666666437777775

14

1 1 1 1

1 j 1 j 1 1 1 11 j 1 j 26666664 37777775

2 4 6 4

2 4 j 6 4 j 2 4 6 42 4 j 6 4 j 26666664 37777775

2 0 0 0

0 4 0 0

0 0 6 0

0 0 0 4

2666666437777775

diag2; 4; 6; 4

is a diagonal matrix. The diagonal elements can be computeddirectly using Eq. (21) for N 4. For example,

k3 X4

k 1ck wk 13 14

4w1 13 14 1w

2 13 14

0 w3 13 14 1w

4 13 14

41 e j 2p4 2

0 e j 2p4 6

4 e j p 0 e j 3p

4 10 1 6which is recognized to be the third diagonal element of the matrixproduct EH4 CE 4 . Because C is a symmetric circulant matrix, Corol-lary 17 can also be used. Observing that N 4 is even, it follows that

k3 c1 2X4=2

k 2ck cos

2pk 13 14 13 1c422

4 2 1cos 2p2 13 1

4 12 0 4 2cos p 0 4 2 10 6

as before. The eigenvalues and eigenvectors of C are discussed inExample 24 of Sec. 2.8 .

Example 16 . Let C circ4; 1; 0; 1 2C 4 be a nonsymmetriccirculant matrix. Then the product

EH4 CE 4 1

ffiffiffi4p 1 1 1 1

1 j 1 j 1 1 1 11 j 1 j

2666664

3777775

4 1 0 11 4 1 00 1 4 11 0 1 4

2666664

3777775 1 ffiffiffi4p

1 1 1 11 j 1 j 1 1 1 11 j 1 j 2666664

3777775

14

1 1 1 1

1 j 1 j 1 1 1 11 j 1 j

26666643777775

4 4 2 j 4 4 2 j 4 2 4 j 4 2 4 j 4 4 2 j 4 4 2 j 4 2 4 j 4 2 4 j

26666643777775

4 0 0 0

0 4 2 j 0 0

0 0 4 0

0 0 0 4 2 j

2666664

3777775

is a diagonal matrix. The diagonal elements are the eigenvalues of C,which is stated explicitly for general C in the following corollary. Theymay be computed directly using Eq. (21), as it is done in Example 15,butCorollary 17 cannot be used because C is not symmetric.

The eigenvalue magnitudes of the nonsymmetric matrix C inExample 16 are observed to exhibit the same multiplicity (andsymmetry) as the eigenvalues in Example 15 for a symmetric cir-culant. It is shown in Sec. 2.8.3 that the eigenvalues of any circu-lant matrix C 2C N with real-valued generating elements exhibit



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a certain symmetry about the so-called Nyquist component,which is analogous to the DFT of a real-valued sequence. This isbecause Eq. (22) represents the DFT of the generating elementsc1; c2; ; c N .

COROLLARY 18. Let ei be the ith column of the Fourier matrixE N and C be a N N circulant matrix. Then for i ; k 1; 2; ; N,

eHi Cek kidik where dik is the Kronecker delta and k i is dened by Eq. (21) for C

2C N or Corollary 17 if C

2SC N .

Thus, the columns of the Fourier matrix are mutually orthogo-nal (Corollary 10) and orthogonal with respect to any circulantmatrix (Corollary 18), not just the cyclic forward shift matrix andits integer powers (Corollary 15).

Example 17 . Consider the circulant C circ4; 1; 0; 1from Example 15. Then

eH3 Ce 3 1

ffiffiffi4p 1 1 1 1 4 1 0 11 4 1 0

0 1 4 11 0 1 4

2666437775

1

ffiffiffi4p 1

1

1

1

2666437775

1

4 1

1 1

1

6

6

66

26664

3777514 24 6

which is recognized to be the third diagonal element k3 of the ma-trix EH4 CE 4 in Example 15. However, the scalar

eH3 Ce 1 1

ffiffiffi4p 1 1 1 1 4 1 0 11 4 1 0

0 1 4 11 0 1 4

26666643777775

1

ffiffiffi4p 1

1

1

1

26666643777775

14

1 1 1 1

2

2

2

2

26666643777775

14

0 0vanishes, as expected, because i 6 k in Corollary 18 such thatdik 0.

2.5.5 Block Diagonalization of a Block Circulant. Theorem 9is generalized to handle block circulants using the Fourier andidentity matrices together with the Kronecker product. The choiceof diagonalizing matrix EH N I M is discussed in Sec. 2.6, wheregeneralizations of Theorem 10 are considered.THEOREM 10 (Block Diagonalization of a Block Circulant). Let C 2BC M ; N and denote its M M generating matrices byC1; C 2; ; C N . Then if E N is the N N Fourier matrix and I M isthe identity matrix of dimension M,

EH N I M CE N I M

K 1 0K 2

. ..

0 K N

266664377775

is a NM NM block diagonal matrix. For i 1; 2; ; N, the M M diagonal blocks are

K i qwi 1 N ;C k X N

k 1C k wk 1i 1 N (23)

where w N is the primitive Nth root of unity and the function q isgiven by Denition 13. Proof. Consider the representation

C X N

k 1r k 1 N C k (Cor. 5)

X N

k 1 E N X

k 1 N EH N C k (Cor. 14)

X N

k 1E N I M X k 1 N C k EH N I M (Eq. 4)

E N I M qX N ; Ck EH N I M (Def. 13)where

qX N ; C k diagi1;; N X N

k 1C k wk 1i 1 N !

diagi1;; N qwi 1 N ;C k

follows from Corollary 16 and Denition 13. The desired resultfollows by multiplying from the left by

EH N I M E N I M Hmultiplying from the right by E N I M , and invoking Corollary11.

Thus, the unitary matrix E N I M reduces any NM NM blockcirculant matrix with M M blocks to a block diagonal matrixwith M M diagonal blocks.

Example 18 . Consider C circA; B; 0; B 2BC 2;4 fromExample 6. It can be block diagonalized via the transformationEH4 I2CE 4 I2. That is,

1

ffiffiffi4p

1 0 1 0 1 0 1 0

0 1 0 1 0 1 0 11 0 j 0 1 0 j 00 1 0 j 0 1 0 j 1 0 1 0 1 0 1 00 1 0 1 0 1 0 11 0 j 0 1 0 j 00 1 0 j 0 1 0 j

377777777777777777775

C

266666666666666666664

1

ffiffiffi4p

1 0 1 0 1 0 1 0

0 1 0 1 0 1 0 1

1 0 j 0 1 0 j 00 1 0 j 0 1 0 j 1 0 1 0 1 0 1 00 1 0 1 0 1 0 11 0 j 0 1 0 j 00 1 0 j 0 1 0 j

377777777777777777775

266666666666666666664

diag0 11 0" #; 2 11 2" #; 4 11 4" #; 2 11 2" # !



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which is a block diagonal matrix with 2 2 diagonal blocks. Theeigenvalues and eigenvectors of C are discussed in Example 25 of Sec. 2.8 .

COROLLARY 19. Let C 2BC M ; N have M M generating matri-ces C 1; C 2; ; C N and K i be dened by Eq. (23). Then if each C iis symmetric, it follows that K i is symmetric for i 1; 2; ; N . Proof. If each Ck is symmetric for k 1; 2; ; N , then so tooare the matrices Ck wk 1i 1 N for each i because wk 1i 1 N is ascalar. Moreover, the sum and difference of two symmetric matri-ces is again symmetric. If follows that

K i X N

k 1Ck wk 1i 1 N

is symmetric for i 1; 2; ; N . COROLLARY 20. Let C 2BC M ; N be a NM NM block circulant matrix. Then for i ; k 1; 2; ; N ,

eHi I M Cek I M K idik where K i is dened by Eq. (23).

Example 19 . Consider C circA; B; 0; B 2BC 2;4 fromExamples 6 and 18. Then

eH3 I2Ce3 I2 1

ffiffiffi4p 1 1 1 1 1 0

0 1 C 1

ffiffiffi4p 1

1

1

1

2666437775

1 0

0 1 0BBB@1CCCA

4 1

1 4 is the third 2 2 block of the block diagonal matrix obtained inExample 18.

2.6 Generalizations. Let denote an arbitrary operationthat takes a square matrix as its argument and returns another square matrix with the same dimension. For example, the opera-tion could denote a matrix inverse such that 1 . Let #be another arbitrary matrix operation with the same restrictions(i.e., returns another square matrix with the same dimension).Then if C 2BC M ; N has generating matrices C 1; C 2; ; C N , it fol-lows that

A B# CA B

A B#

X N

k

1

r k 1 N Ck !A B (Cor. 5)X

N

k 1A r k 1 N B# Ck A B (Eq. 4)

X N

k 1A r k 1 N A B# C k B (Eq. 4)

for any matrices A 2C N N and B 2C M M . The importance of this result is that C can be decomposed into a summation of directproducts of two separate equivalence transformations, one thatoperates on the cyclic forward shift matrix and the other on thegenerating matrices of C . This decomposition justies the diago-nalizing matrix used in Sec. 2.5, motivates some generalizations

of Theorem 10, and aids in proving orthogonality relationships for the cyclic eigenvalue problems described in Sec. 2.8.

In light of Corollary 14, it is clear that the choice of A E N and H accomplishes block diagonalization of a matrixC 2BC M ; N . Then if B I M , the appropriate diagonalizing matrixto block decouple C without operating on its generating matricesis E N I M (see Theorem 10). However, if B and # are keptgeneral, we have the following result.

THEOREM 11. Let C 2BC M ; N have M M generating matricesC1; C 2; ; C N and E N be the N N Fourier matrix. Then for anarbitrary matrix B 2C

M M and operator

#,

EH N B# CE N B

W 1 0W 2

. ..

0 W N

2666437775

is a block diagonal matrix, where

W i qwi 1 N ;B# C k B X N

k 1B# Ck Bwk 1i 1 N (24)

is the ith M M diagonal block for i 1; 2; ; N . COROLLARY 21. Let C 2

BC M ; N be a NM NM block circulant

matrix, B 2C M M be an arbitrary square matrix, and # denotean arbitrary operation that takes a square matrix as its argument and returns another square matrix with the same dimension. Then for i; k 1; 2; ; N ,

eHi B# Cek B W idik where W i is dened by Eq. (24) .

Theorem 11 is useful if there exists an equivalence transforma-tion B# Ck B that simplies each of the generating matrices. For example, if each Ck is a circulant of type M , then the additionalchoice of B E M and

#

H fully diagonalizes a block cir-culant matrix C 2BC N ; M with circulant blocks.COROLLARY 22. Let C 2BC M ; N have generating matricesC1; C 2; ; C N 2

C M and denote the generating elements of eachC i by c1i ;c

2i ; ; c M i . Then

EH N EH M CE N E M diagi1;; N

k1i 0k2i

. ..

0 k M i

2666666437777775

is a NM NM diagonal matrix, where

k pi X

N

k 1 X M

l1clk w

l 1 p 1 M wk 1i 1 N (25)

is the pth diagonal element of the ith M M block for i 1; 2; ; N and p 1; 2; ; M .

Corollary 22 shows that the NM -dimensional eigenvectors q piof C are the columns of E N E M . This is in contrast to rotation-ally periodic structures, where q is partitioned into N M -vectorscorresponding to each sector and decomposed into a set of N reduced-order eigenvalue problems described in Sec. 2.8.

Example 20 . Consider C circA; B; 0; B 2BC 2;4 fromExamples 6, 18, and 19. Because each of its generating matrices isa circulant, that is, A; B; 0 2C 2 , the block circulant C is diagon-alized via the transformation



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EH4 EH2 CE 4 E 2 1

ffiffiffi4p 1 1 1 1

1 j 1 j 1 1 1 11 j 1 j

26666643777775

1 1

1 j " #0BBBBB@1CCCCCA

C

1

ffiffiffi4p

1 1 1 1

1 j 1 j 1

1 1

1

1 j 1 j

2666664

3777775

1 1

1

j " #0B

BBBB@1CCCCCA

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 3 0 0 0 0

0 0 0 0 3 0 0 0

0 0 0 0 0 5 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 3

3777777777777777775

2666666666666666664from which is follows that aC f1; 1; 1; 3; 3; 5; 1; 3g. Observ-ing that

c11 2; c12 1; c

13 0; c14 1

c21 1; c22 0; c

23 0; c24 0

9=;are the generating elements of C, the NM 4 2 8 diagonalelements can be calculated directly using Eq. (25). For example,the second diagonal element ( p2) of the third 4 4 diagonalblock ( i 3) is given by

k23 c11 w

1 12 12 w1 13 14 c

21 w2 12 12 w

1 13 14

c1

2 w1 1

2 1

2 w2 1

3 1

4 c2

2 w2 1

2 1

2 w2 1

3 1

4

c13 w

1 12 12 w3 13 14 c

23 w2 12 12 w

3 13 14

c14 w

1 12 12 w4 13 14 c

24 w2 12 12 w

4 13 14

211 111 111 011011 011 111 011

5The reader can compare this matrix decomposition to the results

in Example 25 of Sec. 2.8.1 . Finally, the eigenvectors q p

i are thecolumns of E4 E 2 and are stated explicitly in Example 25.

2.7 Relationship to the Discrete Fourier Transform.Before turning to the circulant eigenvalue problem, we consider asomewhat tangential but relevant subject on the DFT and itsinverse, which are central to the analysis of experimental data in awide range of elds, including mechanical vibrations. In thisdetour we show that computation of the DFT is, in fact, a multipli-cation of an N -vector of discrete signal samples by the Fourier matrix and a constant c f . The IDFT is similarly dened using theHermitian of the Fourier matrix and a constant ci, wherec f ci 1= N . More importantly, it is shown that the DFT

computation is exactly analogous to the determination of theeigenvalues of a circulant matrix given its generating elements.We begin by dening the DFT sinusoids, which provide a conven-ient means of representing the DFT of a discretized signal, andthen develop the relationships of interest for the DFT and theIDFT. We present only the basic results as they relate to thetheory and mathematics of circulants. The reader can nd a vastliterature on related topics [ 89 91,102 110 ].

DEFINITION 19 (DFT Sinusoids). Let wk N denote the distinct Nthroots of unity, where w N e j

2p N is the primitive root. Then the

DFT sinusoids are

Sk r wk N r wkr N e j 2p N kr

for k ; r 0; 1; ; N 1. COROLLARY 23. The DFT sinusoids are orthogonal . Proof. Consider the DFT sinusoids

Sir e j 2p N ir

Sk r e j 2p N kr ); i; k ; r 0; 1; ; N 1

The inner product of Si(r ) and Sk (r ) is given by

hSi

r

; Sk

r

i X N 1

r 0Si

r

Sk

r

X

N 1

r 0wir N w

kr N (Def. 19)

X N 1

r 0wir N w

kr N (Cor. 7)

X N 1

r 0wr i k N

N ; i k 0; otherwise( (Lem. 2)

which shows that the DFT sinusoids are orthogonal. The Fourier matrix in Denition 17 can be written element

wise as

E N ik 1

ffiffiffiffi N p wk 1i 1 N

1

ffiffiffiffi N p e j 2p N k 1i 1

1

ffiffiffiffi N p Si 1k 1 1

ffiffiffiffi N p Sk 1i 1for each i; k 1; 2; ; N . It follows that

E N 1

ffiffiffiffi N p S00 S01 S0 N 1S10 S11 S1 N 1S20 S21 S2 N 1

..

. ... . .

. ...

S N 10 S N 11 S N 1 N 1

2666666666437777777775

1

ffiffiffiffi N p W N (26)is a representation of the Fourier matrix in terms of the DFT sinu-soids, where W N is the DFT sinusoid matrix. The DFT is formally



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dened next and subsequently reformulated in terms of a matrixmultiplication involving W N .

DEFINITION 20 (DFT). Let x(i) denote a nite sequence with indi-ces i 1; 2; ; N. Then for k 1; 2; ; N, the DFT of x(i) is thesequence

X k X N

i1 xie j

2p N i 1k 1

X N

i1 x

i

wi 1k 1 N

X N

i1 xiSi 1k 1

where w N is the primitive Nth root of unity and S ik wik N is a DFT sinusoid . The DFT preserves the units of x(i). That is, if x(i) has engineer-

ing units EU , then the units of X (k ) are also EU . This is clear fromDenition 20, where the exponential function is dimensionless.

Expanding each X (k ) in Denition 20 yields

X 1 x1S00 x N S0 N 1 X 2 x1S10 x N S1 N 1

..

.

X k x1Si 10 x N Si 1 N 1...

X N x1S N 10 x N S N 1 N 1

9>>>>>>>>>>>>=>>>>>>>>>>>>;If the discrete samples x(i) and corresponding sequence of DFTs

X (k ) are assembled into the conguration vectors

x N x1; x2; ; x N TX N X 1; X 2; ; X N T)

then the DFT of x N can be represented in the matrixvector form

X N W N x N (27)where W N ffiffiffiffi N p E N follows from Eq. (26). Thus, the DFT com-putation is simply a multiplication of the conguration vector x N with the DFT sinusoid matrix W N and constant c f 1. Equation(27) has exactly the same form as Eq. (22), where the generatingelements of a circulant matrix are analogous to the sequence of signals x(i) and the resulting eigenvalues are analogous to thesequence of DFTs X (k ).

Example 21 . Consider the conguration vector of samplesx4 0; 1; 2; 3T . The DFT of x4 follows from Eq. (27) with N 4 and is given by

X4 W 4x4

S00 S01 S02 S03S10 S11 S12 S13S20 S21 S22 S23S30 S31 S32 S33

266664377775

x1 x2 x3 x4

266664377775

1 1 1 1

1 j 1 j 1 1 1 11 j 1 j

266664377775

0

1

2

3

266664377775

6; 2 2 j ; 2; 2 2 j T

where W4 ffiffiffi4p E4 follows from Example 9 or by elementwisedirect computation according to Denition 19. Alternatively, eachelement X (k ) of X4 can be computed using the summation givenin Denition 20. If k 3, for example,

X 3 X4

i1 xie j

2p4 i 13 1

X4

i1 x

i

e j pi 1

x1e j p 0 x2e j p 1 x3e j p 2 x4e j p 3

01 11 21 31 0 1 2 3 2

which is the third element of X4 , as expected. Results for k 1, 2,4 follow similarly.If X N is known, the N -vector x N is recovered from Eq. (27) by

multiplying from the left by EH N and invoking Theorem 6. Then

EH N X N EH N W N x N (Eq. 27) EH N ffiffiffiffi N p E N x N (Eq. 26) ffiffiffiffi N p EH N E N x N ffiffiffiffi N p I N x N (Thm. 6) ffiffiffiffi N p x N and it follows that

x N 1

ffiffiffiffi N p EH N X N 1 N

WH N X N (28)

is a matrixvector representation of the IDFT of X N . That is, the

IDFT computation is simply a matrix multiplication of X N withthe Hermitian of W N and constant ci 1= N such that c f ci 1= N .In light of Corollary 7, Eq. (28) can be written as

x1 x2

..

.

x N

2666666437777775

1 N

S00 S01 S0 N 1S10 S11 S1 N 1

..

. ... . .

. ...

S N 10 S N 11 S N 1 N 1

266666664377777775

X 1 X 2

..

.

X N

2666666437777775

Expanding each row yields

x

1

1

N X

1

S0

0

X

N

S0

N

1

x2

1 N X 1S10 X N S1 N 1

..

.

xi 1 N X 1Sk 10 X N Sk 1 N 1

..

.

x N 1 N X 1S N 10 X N S N 1 N 1

9>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>;which provides an alternative representation of the IDFT.



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DEFINITION 21 (IDFT). Let the sequence X(k) be dened accord-ing to Denition 20. Then for each i 1; 2; ; N, the IDFT of X(k) is the sequence

xi 1 N X

N

k 1 X k Sk 1i 1

1 N X

N

k 1 X k w

i 1k 1 N

1 N X

N

k 1 X k e j

2p N i 1k 1

where w N is the primitive Nth root of unity and S ik wik N is a DFT sinusoid . The IDFT preserves the units of X (k ). If X (k ) has engineering

units EU , then the units of x(i) are also EU . This is clear from Def-inition 21, where the exponential function is dimensionless, as isthe number N .

Example 22 . Reconsider Example 21, where it is shown that theDFT of x4 0; 1; 2; 3T is the four-vector X4 6; 2 2 j ;

2; 2 2 j T . The IDFT of X4 follows from Eq. (28) with N 4and is given byx4 1 N WH4 X4

1 N

S00 S01 S02 S03S10 S11 S12 S13S20 S21 S22 S23S30 S31 S32 S33

2666666437777775

X 1 X 2 X 3 X 4

2666666437777775

14

1 1 1 1

1 j 1 j 1 1 1 11 j 1 j

26666664

37777775

6

2 2 j 2

2 2 j

26666664

37777775 14 0; 4; 8; 12T

0; 1; 2; 3T

which is the same four-vector from Example 21. Alternatively,each element x(i) of x4 can be computed using the summationgiven in Denition 21. If i 3, for example,

x3 14X

4

k 1 X k e j

2p4 3 1k 1

14 X 1e j p 0 X 2e j p 1 X 3e j p 2 X 4e j p 3 14 612 2 j 1212 2 j 1 2

which is the third element of x4 , as expected. Results for k 1, 2,4 follow similarly.There are other suitable denitions of the DFT/IDFT pair. For

example, the signs of the exponents are irrelevant as long as theyare opposite in the DFT and IDFT. To see this, suppose that thesign of the exponential in Denition 20 is negative instead of

positive. Then each entry Sk (r ) in W N is replaced with Sk r toproduce W N (see Corollary 7), and the DFT is instead given byX N W N x N (29)

The corresponding IDFT takes the form

x N 1 N

WH N X N (30)

which follows in the same way as Eq. (28) by replacing E N withE N and invoking Corollary 9. Similarly, the multiplicative con-stants that dene the DFT/IDFT pair are arbitrary as long as theproduct of the constants is equal to 1/ N . This is important for applications involving a transformation of data to the frequencydomain for analysis and then back to the time domain for results.However, the multiplicative constant is irrelevant if the analysisgoal is only to identify periodicities in a data set (e.g., frequenciesthat correspond to amplication of a structural response).

Thus, Denitions 20 and 21 form a representation of the DFT/ IDFT pair, where Eqs. (27) and (28) are the correspondingmatrixvector forms. The DFT and IDFT pair can be written inthe general form

X k c f X N

i1 xie

6

j 2p N i 1k 1 (31a)

xi ciX N

k 1 X k e j

2p N i 1k 1 (31b)

where ci and c f are such that c f ci 1= N but are otherwise arbi-trary. Common choices are fc f ; cig f1= ffiffiffi N p ;1= ffiffi N p g or fc f ; cig f1; 1= N g, where the multiplicative constants in thematrixvector formulation (i.e., Eqs. (27) and (28) or Eqs. (29)and (30)) are adjusted accordingly.

If the generally complex sequences x(i) and X (k ) are dissectedinto their real and imaginary parts, the DFT/IDFT pair is repre-sented by

xi x Ri jx I i (32a) X k X Rk jX I k (32b)

where the subscripts R and I denote the real and imaginary partsand each sequence pair x Ri; x I i and X Rk ; X I k is real for i; k 1; 2; ; N . An alternative representation of the DFT andIDFT is obtained by substituting Eq. (32) in Denition 20, writingthe complex exponential in terms of sines and cosines, and col-lecting the real and imaginary terms. The result is

X Rk X N

i1 x Ricos

2pi 1k 1 N

x I isin 2pi 1k 1 N (33a) X I k X

N

i1 x Risin

2pi 1k 1 N

x I icos 2pi 1k 1

N (33b)which together form the DFT of x(i) according to Eq. (32b). Asimilar relationship for the IDFT is obtained by substitutingEq. (32) into Denition 21 and collecting the real and imaginaryterms. Then



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x Ri 1 N X

N

i1 X Rk cos

2pi 1k 1 N

X I k sin 2pi 1k 1

N (34a) x I i

1 N X

N

i1 X Rk sin

2pi 1k 1 N

X I k cos 2pi 1k 1

N

(34b)

together form the IDFT of X (k ) according to Eq. (32a). The DFT/ IDFT pair dened by Eqs. (32) (34) are equivalent to Denitions20 and 21.

Example 23 . Reconsider Example 21, where it is shown that theDFT of x4 0; 1; 2; 3T is the four-vector X4 6; 2 2 j ;2; 2 2 j T . Because x4 is real, Eq. (33) can be used to com-pute each X k X Rk jX I k for k 1, 2, 3, 4. If k 2, for example, the real and imaginary DFT components are given by

X R2X4

i1 x Ricos

2pi 12 14

0 cos

2p1 114

1 cos 2p2 11

4 2 cos 2p3 114 3 cos 2p4 114 0 cos

0p2 1 cos 1p2 2 cos 2p2 3 cos 3p2

01102130 2

X I 2 X4

i1 x Risin

2pi 12 14

0 sin 0p

2

1 sin

1p2

2 sin

2p2

3 sin

3p2

00 11 20 31 2such that X 2 2 2 j , as expected. Results for k 1, 3, 4 fol-low similarly.

2.8 The Circulant Eigenvalue Problem. The eigenvaluesand eigenvectors of circulants and block circulants with circulantblocks have thus far been inferred from Theorem 9 (circulant matrix)and Corollary 22 (block circulant matrix with circulant blocks),where these matrices are fully diagonalized. Determination of theeigenvalues and eigenvectors of a block circulant matrix with gener-ally noncirculant and nonsymmetric blocks is systematicallydescribed here, which reinforces the results already obtained for the

special cases of C 2C

N and C 2BC

M ; N with circulant generatingmatrices. The standard cEVP is discussed in Sec. 2.8.1 , where aneigensolution is obtained for a general matrix C 2BC M ; N . Section2.8.2 generalizes these results to handle ( M ,K) systems with systemmatrices contained in BC M ; N , which arise naturally in vibration stud-ies of rotationally periodic structures. The structure of the eigenval-ues and eigenvectors is discussed in Sec. 2.8.3 , which builds uponthe relationship to the DFT described in Sec. 2.7. Section 2.8.4 intro-duces fundamental orthogonality conditions for the special case of block circulants with symmetric generating matrices.

2.8.1 Standard Circulant Eigenvalue Problem. Consider theblock circulant matrix C 2BC M ; N with arbitrary generating mat-rices C1; C2; ; C N . The standard cEVP involves determination

of the scalar eigenvalues k and NM 1 eigenvectors q of C suchthat

0 NM C kI NM q (35)where 0 NM is a NM 1 vector of zeros and I NM is the identity ma-trix of dimension NM . The problem can be simplied consider-ably by exploiting the block circulant nature of C. To this end,partition q q1; q2; ; q N T into M 1 vectors q i i 1; ; N and introduce the change of coordinates

q

E

N I M

u (36)

where I M is the M M identity matrix (same dimension as the gener-ating matrices of C) and u u1; u2; ; u N T is composed of N M -vectors u i. Substituting Eq. (36) into Eq. (35) and multiplying fromthe left by the unitary matrix E N I M H EH N I M yields0 NM EH N I M C kI NM E N I M u

EH N I M CE N I M kEH N I M I NM E N I M u

K 1 0

K 2

. ..

0 K N

26666664

37777775

k

I M 0

I M

. ..

0 I M

26666664

37777775

0BBBBBB@

1CCCCCCA

u1

u2

..

.

u N

26666664

37777775

which follows from Theorem 10, and from

EH N I M I NM E N I M E N I M HE N I M I NM diag I M ; I M ; ; I M |fflfflfflffl N terms

in light of Corollary 11. Thus, the single eigenvalue problemdened by Eq. (35) is decomposed into the N reduced-order stand-ard EVPs

0 M K i kI M u i; i 1; 2; ; N (37)where 0 M is a M 1 vector of zeros and each K i is dened by Eq.(23) in terms of the generating matrices of C. Because the transforma-tion dened by Eq. (36) is unitary, it preserves the eigenvalues of C.Thus, the NM eigenvalues of C are the eigenvalues of the N M M matrices K i, which follow from the characteristic polynomials

detK i kI M 0; i 1; 2; ; N (38)If k k

pi denotes the pth eigenvalue of the ith matrix K i for p 1; 2; ; M , then the attendant M 1 eigenvector u

pi isobtained from Eq. (37) . The corresponding NM 1 eigenvector of C follows from Eq. (36) and is given by

q pi E N I M 0 M ; ; u pi ; ; 0 M

T

e1; ; ei; ; e N I M 0 M ; ; u pi ; ; 0 M T

e1 I M ; ; ei I M ; e N I M 0 M ; ; u pi ;; 0 M T

e i I M u pi

ei u pi

(39)



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for i 1; 2; ; N and p 1; 2; ; M . If the scalars a pi are such

that each

~u pi a pi u pi (40)

is orthonormal with respect to K i , then the corresponding ortho-normal eigenvectors of C are given by

~q pi ei ~u pi

ei

a pi u pi

a pi ei u pi a

pi q pi

(41)

for i 1; 2; ; N and p 1; 2; ; M .If C is an ordinary (not block) circulant with M 1, Eq. (41)reduces to~q1i q

1i ei 1 ei; M 1 (42)which conrms that all circulants contained in C N share the same N 1 eigenvectors e1; e2; ; e N , and each e i is orthonormal withrespect to C . The eigenvalues of a matrix C 2C N with generatingelements c1; c2; ; c N are given by Eq. (21), or equivalently byEq. (22). If C 2

SC N , the eigenvalues follow from Corollary 17. Example 24 . Consider C circ4; 1; 0; 1 2C 4 fromExamples 15 and 17, where it is shown thatEH4 CE 4 diag2; 4; 6; 4

Thus, the eigenvalues of C are 2, 4, 6, and 4. Because N 4 iseven, k1 2 and k N 2=2 k3 6 are distinct and k2 k4 4are repeated. The reason for this eigenvalue symmetry is dis-cussed in Sec. 2.8.3 . The eigenvalues can be veried using

ki 4 2cosp2 i 1; i 1; 2; 3; 4

which follows from Corollary 17 for the case of even N because Cis also contained in SC 4 . Because C is an ordinary circulant, its

orthonormal eigenvectors are simply the columns of the Fourier matrix E 4 , and are given by Denition 18 according to Eq. (42).For example,

e2 1

ffiffiffi4p 1; w14; w

24; w

34 T

12

1; e j 2p4 1; e j

2p4 2; e j

2p4 3 T

12

1; e j p2; e j p ; e j

3p2 T

12 1; j ; 1; j

T

which can be visualized in Fig. 2 for the case of N 4. The com-plete set of eigenvectors is given by

e1 12

1

1

1

1

266664377775

; e2 12

1

j

1

j

266664377775

; e3 12

1

1

1

1

266664377775

; e4 12

1

j

1

j

266664377775

The eigenvectors e1 and e3 are real and correspond to the distincteigenvalues k1 2 and k3 6. The eigenvectors e2 and e4 arecomplex conjugates according to Eq. (17) and correspond to therepeated eigenvalue k2 k4 4. Example 29 of Sec. 2.8.4 dis-cusses orthogonality of the orthonormal eigenvectors 1 =2 ei withrespect to the matrix C.

The same basic results also hold for the nonsymmetric circulantC circ4; 1; 0; 1 2C 4 in Example 16, where it is shown thatEH4 CE 4 diag4; 4 2 j ; 4; 4 2 j . In this case, the eigenvaluesare given by

aC f4; 4 2 j ; 4; 4 2 j gand the corresponding eigenvectors are the same as those inExample 24 because all circulants

amr_066_04_040803

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