polynomial-based interpolation filters for dsp … · 1 tampere university of technology...

1

Tampere University of Technology

POLYNOMIAL-BASED INTERPOLATION FILTERS FOR DSP APPLICATIONS

DESIGN, IMPLEMENTATION, AND APPLICATIONS

Djordje Babic, Jussi Vesma, Tapio Saramäki, Vesa Lehtinen, Markku Renfors

Tampere University of Technology, Finland

• This pile of lecture notes is based on the research work done by Dr. Tech. Jussi Vesma, Prof. Tapio Saramäki, Dr. Tech. Djordje Babic, and Prof. Markku Renfors.

• The lecture notes have been prepared by Prof. Saramäki and Dr. Vesma, and updated by Dr. Babic and M.Sc. Vesa Lehtinen.

2


POLYNOMIAL-BASED INTERPOLATION FILTERS FOR DSP APPLICATIONS

DESIGN, IMPLEMENTATION, AND APPLICATIONS

Speaker: Vesa Lehtinen Tampere University of Technology, Finland

e-mail: [email protected]

Contents

1. Interpolation Filters 2. Fractional-Delay Filters 3. Lagrange Interpolation 4. Analog Model for Interpolation Filter 5. Polynomial-Based Interpolation Filters 6. Design 7. Applications

3


INTERPOLATION FILTERS

• In many DSP applications there is a need to know the value of a signal also between the existing discrete-time samples x(n).

• Special interpolation filters can be used to compute new sample values y(l)=ya(tl) at arbitrary points t l=(nl+µl)Tin.

• Here ya(t) approximates the original continuous-time signal xa(t).

• The output sample time is thus determined by the fractional interval (or delay) µl∈ [0,1) and the integer index nl.

• Terms "interpolator" and "fractional delay (FD) filter" are also used in the literature.

nlTin

= x(n)= y(l)

µ lTin

ya(t)

tltl-1 tl+1

µ l+1Tin

nl-1Tin nl+1Tin

µ l-1Tin

Time

Fig. 1. Interpolation in the time domain.

4


Terminology pitfalls: Mathematical interpolation vs. interpolation in DSP vs. decimation

• In mathematics, interpolation stands for constructing a smooth curve that travels through all given points. o Constructing a curve that is as close as possible

but does not include all the given points is referred to as approximation. Cf., e.g., least-squares approximation.

• In DSP, upsampling + antiimage filtering commonly referred to as “interpolation” (vs. decimation). o Not all DSP “interpolators” interpolate in the

mathematical sense. o Filters that interpolate in the math sense are also

known as Nyquist filters (recall the zero intersymbol interference property).

o It is the antiimage filter that performs the interpolation, not the upsampler.

• In decimation by fractional (nonintegral) ratios, output samples are generated between the input samples. => Fractional decimation incorporates interpolation / approximation for the purpose of antialiasing.

5


Applications for Interpolation Filters

• Timing adjustment in all-digital receivers (symbol synchronization)

• Time delay estimation • Conversion between arbitrary sample rates • Conversion between uniform and nonuniform sample

grids • Echo cancellation • Phased array antenna systems • Speech coding and synthesis • Computer simulation of continuous-time systems • Image processing (image resizing and rotation)

6


Statement of the general interpolation problem for uniformly sampled input

Interpolation filterh(k,µl)

y(l)x(n)

nl µl

Fig. 2. Simplified block diagram for interpolation filter.

• The input parameters nl and lµ are used to determine the time instant tl for the l th output sample y(l)=ya(tl) as t l=(nl+µl)Tin.

• Given tl, these parameters are determined by

inll Ttn /= and inlinll TtTt // −=µ (1)

• After knowing nl and µl, the interpolation filter calculates y(l) according to the convolution

/ 2 1

/ 2( ) ( ) ( , ).

N

l lk N

y l x n k h k µ−

=−

= −∑ (2)

where N is the length and h(k, µl) is the (discrete-time) impulse response of the interpolation filter.

7


Interpretation of the convolution sum

• First, the location of the existing sample preceding (or occurring at) the new sampling instant tl is determined and denoted by nl.

• Based on the location of this sample, the existing samples located at n=nl−N/2+1, nl−N/2+2,···, nl+N/2 are used. This means that there are N/2 existing discrete-time samples before and after the desired new time instant tl.

• Second, the distance between tl and nlTin is measured as a fraction of Tin, giving the fractional interval µl.

• In the above convolution, µl determines the coefficient values h(k,µl). The purpose is to determine them in such a way that y(l) is found according to some criterion to be discussed next.

8


Requirements for the impulse-response coefficients h((((k,,,,µµµµl))))

1. h(k, µl) should be optimized such that y(l)≈xa((nl+µl)Tin) for all values of µl ∈ [0,1) and l.

2. The fundamental idea in the optimization is that ya(t) approximates the original continuous-time signal xa(t) according to some time-domain or frequency-domain criterion.

3. Frequency-domain criteria are preferred for DSP applications.

4. The interpolation filter should have an efficient digital implementation structure.

9


Filter Synthesis and Analysis Problem

• The problem is now to design a filter which calculates the interpolated samples y(l) for a given value of µl based on the above discussion.

• And to analyze the performance of the filter (what is the error between ya(t) and xa(t)).

• Note that the direct analysis of the above interpolation filter is difficult, because the system is time-varying.

Three approaches to find a solution

1. The frequency response of the ideal fractional delay is e−jωµ, find a FIR filter to approximate this response (FD filter approach) for each value of µ.

2. Use some classical interpolation method to calculate y(l), e.g., Lagrange or B-spline interpolation (time-domain approach).

3. Utilize the continuous-time (analog) model for the interpolation filter (frequency-domain approach).

10


FD FILTERS

• There are several synthesis methods [1] to design FIR filters having a phase delay of Dint+µ, where Dint=N/2−1 is the integer delay determined by the length of the filter N and µ∈ [0,1) is the fractional delay.

• The amplitude response of this FIR filter should approximate unity on the given frequency band.

• Good solution when µ is fixed. Otherwise, several FD FIR filters are designed and coefficients are stored in a lookup table. ⇒ Need for a large memory.

• See Matlab demo #1.

11


Example: FD Filter

• The length of the filter N=8, Ωp=[0, 0.75π], six different filters having µ=0, 0.1, 0.2, 0.3, 0.4, and 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Angular Frequency ω / π

Ampl

itude

Res

pons

e

Fig. 3. Amplitude response for the example FD filter.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

3

3.1

3.2

3.3

3.4

3.5

Angular Frequency ω / π

Phas

e D

elay

Res

pons

e

Fig. 4. Phase delay response.

12


LAGRANGE INTERPOLATION

• In Lagrange interpolation, the interpolated signal ya(t) is a piecewise polynomial of an odd degree M.

ya((n+µ)Tin)=ŷn((n+µ)Tin)

I.e., the polynomial ŷn(t) is used for t∈ [nTin, (n+1)Tin). • The polynomial pieces ŷn(t) are defined such that ŷn(kTin)=x(kTin) for N=M+1 consecutive sample points k=n−N/2+1,…,n+N/2. Notice that these points extend beyond the range in which ŷn(t) is applied.

• Disadvantage: Does not utilize the frequency-domain information of the input signal x(n).

• Lagrange interpolation is discussed in more detail later on.

• See Matlab demo #2.

13


Example: Lagrange Interpolation

• How to obtain y(l)?

= x(n)= y(l)

µlTin

ZERO-ORDER HOLD

LINEAR INTERPOLATION

CUBIC LAGRANGE INTERPOLATION

nl (nl+µl)Tin nl+1

µlTin

nl (nl+µl)Tin nl+1

= x(n)= y(l)

= x(n)= y(l)

µlTin

nl (nl+µl)Tin nl+1

ya(t)

ya(t)

ya(t)

Fig. 5. Lagrange interpolation.

14


ANALOG MODEL FOR INTERPOLATION FILTER

y(l)x(n)

DAC ha(t)

Sample at tl =(n l+µ l)Tin

ya(t) xs(t)

Fig. 6. Analog model for interpolation filter.

• Interpolation is a reconstruction problem where the approximating signal ya(t) is reconstructed based on the existing discrete-time samples x(n).

• Therefore, one useful way to model discrete-time interpolation filters is to use the analog system shown in Fig. 6.

• In this system, x(n) is first converted to the sequence of weighted and shifted Dirac impulses xs(t).

• xs(t) is then filtered using an analog, linear time-invariant (LTI) reconstruction filter ha(t) resulting in the following convolution:

.)()()()()( ∑∫

∞

−∞=

∞

∞−

−=−=k

inaasa kTthkxdthxty τττ (3)

15


Why to Use the Analog Model?

• Interpolation is generally considered as a time-domain polynomial fitting problem which is not a very practical approach for DSP applications. This is because the time-domain characteristics of the signal are not known usually. What is known is the frequency band of the signal.

• The use of the analog model turns interpolation from a time-domain problem to a frequency-domain filtering problem.

• Various interpolation methods can, indeed, be expressed as convolution when the input samples are uniformly spaced.

16


Role of ha(t) in the Frequency Domain

• How to design ha(t) in the frequency domain? • Such that it preserves the signal components in the

range [0,Fin /2] and attenuates them elsewhere.

Xa( f )

f

Fin/2

Fig. 7. The spectrum of the original continuous-time signal.

X(e j2πf ////Fin)

f

Ha( j2πf )

Fin/2 Fin 2Fin

Fig. 8. The spectrum of the input signal x(n) and the frequency response of the reconstruction filter ha(t).

17


Farrow structure – traditional approach

• The main idea of this part is the understanding of the special implementation form of polynomial-based interpolation filters.

• y(l) is then obtained by sampling ya(t) at tl giving

.))(()()()(

12/

2/∑

−

−=

+−==N

Nkinlalla Tkhknxtyly µ

(4)

• It is assumed that ha(t) is zero outside the interval −NTin /2≤ t≤NTin /2.

• By comparing (2) and (4), it can be seen that the impulse responses of the analog and discrete-time filters are related as follows:

))((),( inlal Tkhkh µµ += (5)

for k=−N/2, −N/2+1,···, N/2−1. • In the causal case, ha(t) is delayed by NTin/2 and the

output is given by

.))2/(()2/( )(

1

0∑

−

=

−++−=N

kinlal TNkhNknxly µ

(6)

18


• For a polynomial-based interpolation filter, the impulse response ha(t) is expressed in each interval of length Tin by means of a polynomial as follows:

m

l

M

mminla kcTkh µµ ∑

=

=+0

)(ˆ))(( (7)

for k=−N/2,···, N/2−1. Here the )(ˆ kcm ’s are the coefficients and M is the degree of the polynomials.

• By using the analog model, the digital implementation structure for polynomial-based interpolation filters can be derived by substituting (7) into (6) giving

∑∑ ∑=

−

= =

=−+−=M

m

mllm

N

k

ml

M

mml nvNkcNknxly

0

1

0 0

)()2/(ˆ)2/( )( µµ (8)

where

∑−

=

−+−=1

0

)2/(ˆ)2/()(N

kmllm NkcNknxnv

(9)

are the output samples of the M+1 FIR filters having the transfer functions

.)2/(ˆ)(ˆ

1

0∑

−

=

−−=N

k

kmm zNkczC

(10)

19


Interpretation of the Above Equation

• According to Equation (9), the l th output sample y(l) at the time instant t = (nl+µl)Tin can be generated based on the N existing samples x(n) being located at n=nl−N/2+1, nl−N/2+2,···, nl+N/2 in the following three steps:

• First, these samples are filtered using M+1 FIR filters having the transfer functions of the form given in (10)

• Second, the outputs of these filters, denoted by vm(nl), are multiplied with constants m

lµ .

• Third, the multiplication results are added, leading to the so-called original Farrow structure shown in Fig. 10.

• In this figure, the input to both structures is denoted by x(nl + N/2) to emphasize the fact that this is the last existing sample value when evaluating y(l).

• If desired, the transfer functions )(ˆ zCm can share the delay elements.

20


x ( n l )

C M ( z ) C 0 ( z ) C 1( z ) C 2( z )

v 0 ( n l ) v 1( n l) v 2 ( n l ) v M ( n l )

µµµµ l

y ( l )

Hold & sample

^ ^ ^ ^

Fig. 9. The Farrow structure for interpolation.

y(l)

x(nl + N/2)

−−−− 1 Z

c M (N −−−− 1)

c M (1)

c M (0)

−−−− 1 Z

−−−−1Z

−−−−1Z

c1(N −−−− 1)

c1(1)

c1(0)

−−−−1Z

−−−−1Z

c 0 (N −−−− 1)

c 0 (1)

c 0 (0)

v1(nl) v 0 (nl)

x(nl + N/2)

µµµµl

Foutv M (nl)

Fin

Fig. 10. The Farrow structure.

21


Characteristics of the Farrow Structure

• The number of FIR subfilters is M+1. • The length of these subfilters is N. • Filter coefficients are directly determined by the

polynomial coefficients of the impulse response ha(t). • The main advantage of the Farrow structure is that all

the filter coefficients are fixed. • The only changeable parameters are the fractional

interval µl as well as nl that depend on the l th output sampling instant.

• During computation, the control of µl is easier than in the implementation based on fixed FD filters.

• The resolution of µl is limited only by the precision of the arithmetic, not by the size of the coefficient memory.

• These characteristics of the Farrow structure make it a very attractive structure to be implemented using a VLSI circuit or a signal processor.

• In fact, the Farrow structure is a general-purpose adjustable FIR filter whose taps are polynomials in a control variable. Any filter property may be varied.

22


POLYNOMIAL-BASED INTERPOLATION FILTERS

General approach: How to construct the impulse response ha(t)

• ha(t) nonzero for 0 ≤t<NT and zero elsewhere. • In each subinterval nT ≤t<(n+1)T for n=0, 1, …, N−1,

ha(t) is expressible as ( )∑ == Mm mma tTnfncth 0 ,,)()( that is,

ha(t) is a piecewise-polynomial for 0 ≤t<NT and is of degree M in each subinterval.

• ha(t) is symmetric around t = NT/2, that is, ha(NT−t) = ha(t), possibly excluding the segment edges (time instants t = nT for n=0, 1, …, N/2−1 and n= N/2+1, N/2+2, …, N).

• Lagrange interpolation filters have the same properties (plus some more).

• Mathematical representation:

( )tTnfncth m

N

n

M

mma ,,)()(

1

0 0∑ ∑

−

= == (11)

( )

+≤

−

−=

,otherwise 0

)1(<for 12),,( TntnT

TnTt m

tTnfm (12)

23


• The symmetry property ha(−t)=ha(t) is achieved by

)1()1()( −−−= ncnc mm

m (13)

for m=0, 1,···, M and n=0, 1,···, N/2−1. This condition halves the number of unknowns. • ha(t) can be now constructed as follows

∑ ∑−

= =

=12/

0 0

),,()()(N

n

M

mma tmngncth

(14)

where cm(n)’s are unknown coefficients and g(n,m,t)’s are the basis functions given by

( ). )2/,,1(

)2/,,(1),,(NTtTnNf

NTtTnftTng

m

mm

m

−−−+−−= (15)

−2 −1 0 1 2−1

0

1

Time in Tin

Am

plitu

de

Fig. 11. The basis function g(n,m, t) for n=1 and m=3.

24


Example how to construct ha(t) for N=8 and M=3.

−0.20

0.7

−0.6

0

0.6

−0.14

0

0.17

−0.1

0

0.1

−8 −6 −4 −2 0 2 4 6 8−0.2

0

1 Overall impulse response ha ( t )

t/T

(a)

(b)

(d)

(c)

(e)

Fig. 12. Construction of the overall impulse response ha(t) for N=8 and M=3. (a)−(d) The weighted basis functions cm(n)g(n,m, t) for

m=n=0, 1 , 2, and 3. (e) The resulting impulse response ha(t).

• T, the length of polynomial segments can be chosen as T = βTin or T = βTout, where β is unity, an integer, one divided by an integer.

• The polynomial based filters are efficiently implemented by using Farrow structure and its modifications.

25


“MODIFIED” FARROW STRUCTURE

• In this case, the desired ha(t) is obtained from (2) by selecting T to be input sampling period in Fig. 1, that is, T = Tin, yielding

( )tTnfncth inm

N

n

M

mma ,,)()(

1

0 0∑ ∑

−

= == (16)

where the basis functions fm(n, Tin, t) are given by

( )

+≤

−

−=

. otherwise 0

)1(<for 12

),,( ininin

in

inmTntnT

TnTt m

tTnf (17)

• By substituting the resulting ha(t) into (6), the lth output sample can be expressed, after some manipulations, in the following form:

∑ ∑−

= =−+−=

1

0 0)12)(()2/()(

N

k

M

m

mlml kcNknxly µ , (18)

where linoutlinoutl nTlTTlTn −== µ and . (19)

26


• Compared to the ‘regular’ case, here, instead of using µl we use 2µl−1 as a basis polynomial to construct ha(t) as follows:

∑=

−=+M

m

mlminla ncTnh

0

)12)(())(( µµ (20)

for n=−N/2, −N/2+1,···, N/2−1. Here the cm(n)’s are the unknown polynomial coefficients.

0 1−1

1

Am

plitu

de

0 1−1

1

0 1−1

1

0 1−1

1

Time in Tin

Fig. 13. Polynomials (2µl −1)m for m=0, 1, 2, and 3.

• Alternatively, y(l) can be rewritten as

( )∑=

−=M

m

mllm nvly

012)( )( µ , (21)

where

)()2/()(1

0kcNknxnv m

N

kllm +−= ∑

−

= (22)

27


“Modified” Farrow structure

• Note that the polynomial coefficients cm(n) are symmetrical according to (13).

• And 2µl−1 is used instead of µl . • Therefore, interpolation filters based on this approach

can be implemented using the Modified Farrow structure

• This modified structure has two differences compared to the original structure.

1. The output samples vm(nl) of the FIR filters are multiplied with 2µl−1 instead of µl.

2. FIR filters having the transfer functions

∑−

=

− =−=1

0

,,1 ,0for )2/()(N

k

kmm MmzNkczC !

(23)

possess the properties given by (13). • When exploiting the above symmetries, the number of

coefficients to be implemented can be reduced from (M+1)N to (M+1)N/2 for N even.

28


x ( n l )

C M ( z ) C 0 ( z ) C 1( z ) C 2( z )

v 0 ( n l ) v 1( n l) v 2 ( n l ) v M ( n l )

2222 µµµµ l -1

y ( l )

Hold & sample

Fig. 14. The Modified Farrow structure for interpolation.

y(l)

x(nl + N/2)

−−−− 1 Z

c M (N −−−− 1)

c M (1)

c M (0)

−−−− 1 Z

−−−−1Z

−−−−1Z

c1(N −−−− 1)

c1(1)

c1(0)

−−−−1Z

−−−−1Z

c 0 (N −−−− 1)

c 0 (1)

c 0 (0)

v1(nl) v 0 (nl)

x(nl + N/2)

2222µµµµl −1−1−1−1

Foutv M (nl)

Fin

Fig. 15. The Modified Farrow structure, subfilter structure (unoptimized) shown.

29


TRANSPOSED FARROW STRUCTURE

• In interpolation (increasing the sample rate), the same Farrow interpolator can be used for various rate conversion factors. o The antiimage filtering effect & requirements are

independent of the output sample rate! • In decimation, the direct Farrow structure often

performs poorly, and the coefficients need to be optimized separately for each decimation factor, i.e., the flexibility is lost.

=> Is there something for decimation that would correspond to Farrow in interpolation?? • Yes! It is known as the transposed Farrow structure. • Entire structure & subblocks transposed. Hold &

sample (H&S) becomes its dual, integrate & dump (I&D).

• When generating the transposed Modified Farrow structure, the desired ha(t) is obtained from (2) by selecting the length T of the polynomial segments to equal the output sample period, that is, T = Tout, giving

( )tTnfncth outm

N

n

M

mma ,,)()(

1

0 0∑ ∑

−

= == , (24)

30


where ( )( )

+≤−−=

.otherwise 0)1(<for 12),,( outoutoutout

outmTntTnTnTt m

tTnf (25)

By substituting the resulting ha(t) into (3), the lth output sample can be expressed, after some manipulations,

,),()()(0

1

0∑ ∑=

−

==

M

m

N

nmm lnvncly (26)

where

( ) 12)(),(),(

),(∑

=−=

lnk

lnkk

mkm

up

low

kxlnv µ (27)

with inoutlow TTnNllnk )12/(),( −−+= , (28)

( ) −

=otherwise, ),(

integeran is ),( if 1 ),(,

lnslns lns

lnkup (29)

inout TTnNllns /)2/(),( −+= , (30) and . // outinoutink TkTTkT −=µ (31)

• Here, the fractional interval µk is the distance between the current input sample x(k) and the previous output sample position y(l−1) divided by Tout.

• This structure is used only for decimation.

31


• In single-rate FD filtering, H&S and I&D vanish, hence the direct & transposed Farrow structures are as good.

• Notice that in FD filtering, if µ changes relatively seldom, Farrow is inefficient – it is better to only calculate all the N filter taps whenever µ changes and to use those values until µ changes again.

=> N multiplies per sample instead of (M+1)N/2.

2222µµµµk −−−− 1111

y(l)

−−−−1Z

c M (N −−−− 1)

cM (1)

c M (0)

c 0 (N −−−− 1)

c 0 (1)

c 0 (0)

c1(N −−−− 1)

c1(1)

c1(0)

v M (n,l) v1(n,l)v 0 (n,l)

−−−− 1 Z

−−−− 1 Z

−−−−1Z

I&D

Fin

Fout

x(k)

ov(l)

−−−−1Z

Fig. 16. Transposed Modified Farrow structure.

32


TIME-SCALED FARROW VARIANTS *

• The impulse response of a polynomial interpolator can be scaled in the time domain.

o Stretching in time domain = shrinking in frequency domain

o Shrinking in time domain = stretching in frequency domain

• Time-shrinking: J polynomial segments per input sample interval

o Lower polynomial degree o Longer subfilters, i.e., more coefficients o Subfilter output only needs to be computed for

those segments for which output is requested. o Referred to as the “generalized Farrow structure”

by some authors. • Time-stretching: Each polynomial segment covers J

input sample intervals. o Narrower passband with the same subfilter

coefficients. o Requires J sets of basis multipliers. o Known as the prolonged Farrow structure.

33


STRUCTURE CONSISTING OF A CASCADE OF A LINEAR-PHASE FIR INTERPOLATOR AND A MODIFIED

FARROW STRUCTURE

• The proposed structure consists of two basic building blocks. The first block is a linear-phase FIR filter transfer function of the following form:

∑=

−=IK

k

kII zkhzH

0)()( , (32)

where the impulse-response of HI (z) is symmetric, that is,

. ,,1,0for )()( IIII KkkhkKh …==− (33)

The second building block involves generating the desired lth output sample of Fig. 17 according to

∑ ∑−

= =−+−=

1

0 0)12)(()2/()(

N

k

M

m

mlml kcNknzly µ (34)

where

( ) ( ) . and l linoutinoutl nLTlTLTlTn −== µ (35)

FIR filterHI(z)

Farrowstructure↑↑↑↑L

x(n) z(j) y(l)Fin LFin LFin Fout

Fig. 17. Linear-phase FIR interpolation filter in cascade with Modified Farrow structure for interpolation.

34


• Alternatively, the resulting overall system corresponds to that of Fig. 6, where ha(t) is replaced by

∑=

−=IN

linaIa LlTthlhtg

0)./()()( (36)

• The resulting impulse response ga(t) is characterized by the following properties:

1)ga(t) is formed as a shifted and weighted sum of the start-up piecewise polynomial ha(t) with the length of polynomial segments being equal to Tin/L.

2)The resulting ga(t) is nonzero for 0 ≤ t<(KI+N)Tin/L and of order M in each of the subintervals nTin/L

≤ t<(n+1) Tin/L for n=0, 1, …, KI+N −1. • The main advantage of using the cascaded structure of

Fig. 17, instead of the direct Modified Farrow structure of Fig. 14, lies in the fact that when jointly optimizing the two building blocks the computational complexity to generate practically the same filtering performance is drastically decreased. This is due to the following facts: 1) The implementation of a fixed linear-phase FIR

interpolator is not very costly. 2) The requirements for implementing the Modified

Farrow structure become significantly milder.

35


TRANSPOSED CASCADED STRUCTURE

• The previous interpolation structure has its dual implementation form. The main properties are the same.

FIR filter

HD(z)Transposed

Farrow ↓↓↓↓Lx(n) z(j) y(l)Fin LFout LFout Fout

Fig. 18. Transposed Farrow structure in cascade with a linear-phase FIR decimation filter for decimation.

36


What Do We Have Up to Now:

1. The fundamental idea is that the approximating signal ya(t) approximates the original continuous-time signal xa(t) according to some condition, and then the output sample is y(l)=ya((nl+µl)Tin).

2. By using the analog model, this approximation problem can be converted to the problem of designing the reconstruction filter ha(t).

3. If ha(t) is a piecewise polynomial, then the Farrow structure or its modifications can be used for implementation.

• Next we will discuss how to obtain the coefficients )(ˆ kcm for the polynomial-based impulse response

ha(t).

37


DESIGN

• We consider only polynomial-based interpolation filters because they can be efficiently implemented using the Farrow structure or its modifications.

• Design methods for polynomial-based interpolation filters to be considered

1. Conventional time domain design: Lagrange interpolation.

2. Frequency domain design based on the analog model: Minimax and least-mean-square optimization of the interpolation filters.

• The design methods will be compared by using the impulse response and frequency response of the corresponding reconstruction filter ha(t).

38


Lagrange Interpolation

• Lagrange interpolation was originally used in mathematics, not in signal processing (discovered by Joseph-Louis Lagrange 1736-1813). • Does not offer a good filtering characteristics.

• Filter design is done in the time domain and the filter coefficients can be given in a closed form. ⇒ No need for optimization.

1. Choose the degree M of interpolation, the length of the filter is then N=M+1.

2. Based on the time-domain conditions for ya(t) in (3), the polynomial coefficients for ha(t) are determined by the following equation:

∏∑

≠+−== +−

−=2/

12/0

)(ˆN

kjNj

M

m

mm jk

xjxkc (37)

for k=−N/2, −N/2+1,···, N/2−1.

39


Example

−2 −1 0 1 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time in Tin

Impu

lse

resp

onse

ha(t

)

Fig. 19. Impulse response ha(t) for the cubic (solid line) and linear (dashed line) Lagrange interpolation filters

0 0.5 1 1.5 2 2.5 3 3.5 4−60

−50

−40

−30

−20

−10

0

Mag

nitu

de in

dB

Frequency in Fin

Fig. 20. Magnitude responses for the cubic (solid line) and linear (dashed line) Lagrange interpolation filters

40


Disadvantages of the Lagrange interpolation filters

• The frequency response can not be changed: stopband attenuation, passband ripple as well as edge frequencies are fixed.

• If the degree of the interpolation is M, then the length of the filter is always N=M+1.

• The stopband attenuation can not be improved easily by increasing the degree of the interpolation (e.g., 5 or 7), but the number of filter coefficients increases fast.

• Do not have characteristics of a good lowpass (reconstruction) filter, at least if the input signal contains frequency components close to half the sampling frequency.

• Solution: make the input signal narrowband by increasing the sampling rate before the Farrow structure.

• For instance, the third order Lagrange interpolation filter attenuates the image frequencies at least by 60dB if the bandwidth of the input signal is 0.1Fin as shown in Fig. 21.

41


0 0.5 1 1.5 2 2.5 3 3.5 4−90

−80

−70

−60

−50

−40

−30

−20

−10

0

Frequency in Fin

Mag

nitu

de in

dB M=1

M=9

Fig. 21. The magnitude responses |Ha( f)| of the Lagrange interpolation filters with degree M=1,3, 5, 7, and 9 (solid lines) and the spectrum Xs( j2πf) of the input signal with the bandwidth of 0.1Fin (dashed-line).

42


The frequency response of polynomial–based filter ha(t) in general

• The zero-phase frequency response of polynomial-based filter in general is given by:

∑ ∑−

= =

=12/

0 0

),,()()(N

n

M

mma fmnGncfH

(38)

• Since g(n,m, t) is symmetrical around t=0, G(n,m, f) is real and is given by

Φ+−

Φ−+

=+ odd, for ),())(sin(!)1(2

even for 2/

)2/sin(+),(!)1())(cos(2

),,(

212/)1(

2/21

mfmnm

m

fmmn

mnGm

m

ω

ωωω

ω (39)

with ω=2πfT and

.)12()cos()sin(

)!2()1()(),(

2/)1(

0

2∑−

=

−

+

−−=Φm

k

kmk

kTf

TfTf

kTffm π

πππ (40)

43


General Frequency-Domain Synthesis Technique for Interpolation Filters

• It is desired to design in the analog model of Fig. 6 the reconstruction filter with impulse response ha(t) only subject to the restriction that the overall system can be implemented using the Farrow structure or its modifications.

• In this case, M and N can be selected arbitrarily. • The main goal to is to optimize the coefficients of the

Farrow structure and the corresponding impulse response ha(t) in such a manner that the overall system provides the desired frequency-domain behavior that depends on the application.

• The purpose is to determine the coefficients in such a manner that the amplitude response of the reconstruction filter approximates an arbitrary amplitude response either in the least-squares (L2)

sense or in the minimax (L∞) sense. • It is also desired to allow the use of some time-

domain conditions for the overall impulse response ha(t).

44


Optimization problem: find the unknowns cm(n) to minimize

( ) ( ) ( )max aW f H f D ff X

δ∞ = −∈

(41)

where X is the approximation region, D( f) is the desired function, and W( f) is the weighting function.

• These interpolation filters can also be optimized in L2 sense. In this case, the function to be minimized is

[ ] dffDfHfWX a

2

2 )()()(∫ −=δ (42)

• It is also possible to set some time-domain conditions for ha(t), e.g., ha(t)=1 for t=0 and ha(kTin)=0 for k=±1, ±2,…, ±N/2.

• For minimax problem, linear programming can be used to optimize the filter coefficients.

• Design parameters for the optimization are − edge frequencies for passband(s) and stopband(s) − desired amplitude and weight for every band − the length of the filter and − the polynomial degree

45


Example I (minimax)

• Design parameters are: − M = 7, N = 24, fp=0.4Fin, fs=0.6Fin, weights:

Wp=0.002, and Ws=1

0 0.5 1 1.5 2 2.5 3 3.5 4

−100

−80

−60

−40

−20

0

Mag

nitu

de in

dB

Frequency in Fin

−12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12−0.2

0

0.2

0.4

0.6

0.8

1

Time in Tin

Impu

lse

resp

onse

ha(t

)

46


Example II (L2-norm)

• Design parameters are: − M = 5, N = 8, fp=0.35Fin, fs=0.65Fin, weights:

Wp=0.02, and Ws=1

0 0.5 1 1.5 2 2.5 3 3.5 4−90

−80

−70

−60

−50

−40

−30

−20

−10

0

Mag

nitu

de in

dB

Frequency in Fin

−4 −3 −2 −1 0 1 2 3 4−0.2

0

0.2

0.4

0.6

0.8

1

Time in Tin

Impu

lse

resp

onse

ha(t

)

47


APPLICATIONS

Conversion Between Arbitrary Sample Rates

Criteria for Uniform Sampling: Interpolation and Decimation

• If y(l) is generated at the time instants outl lTt = , then

∑∞

∞−=

−=k

outaoutFfj kFfjYFeY out ))(2()( /2 ππ , (43)

where Fout = 1/Tout is the sampling rate of the output signal y(l) and the baseband of interest is [0, Fout/2]. • The case 1/ >= inout FFβ corresponds to interpolation. • The case 1/ <= inout FFβ corresponds to decimation. • In both cases, the ideal response for Ha( j2πf)

avoiding both imaging and aliasing is given by

( ) 1/ for 0 /2 0 for /2,

in C

C

F f FD f

f F≤ ≤

= > (44)

where ( )outinC FFF ,min= . (45)

48


• Note that in the interpolation case, it is enough to attenuate the images of X(ej2πf ////Fin).

• In the decimation case, X(ej2πf ////Fin) should be band-limited into the range [0, Fout/2], that is, the region [Fout/2, Fin/2] should be attenuated in order to avoid aliasing.

Y(e j2πf ////Fout

f

Ha( j2πf )

Fout/2 Fout 2Fout

Fig. 22. The spectrum of the output signal y(l) and the frequency response of the filter ha(t).

49


Practical Criteria

• Like for conventional digital interpolators and decimators, the criteria can be stated as

[ ]ppainp fffjHF ,0for 1)2(1 ∈+≤≤− δπδ (46)

ssain ffjHF Ω∈≤ for )2( δπ , (47)

where 2/ Cp Ff < and

[ )[ )

[ ]

+−

∞−∞

=Ω∞

=

C. Typefor ,

B Typefor ,A Typefor ,2/

1∪k

pCpC

pC

C

s

fkFfkF

fFF

(48)

• For Type A, no aliasing or imaging is allowed. • For Type C decimation case, aliasing is allowed into

the transition band [fp, Fout/2]. For Type B, aliasing into this band is allowed only from band [ Fout/2, Fout−fp].

• In the interpolation case, Types B and C are useful if most of the energy of the incoming signal is in the range [0, fp].

50


• The Farrow structure can be directly used for providing the conversion between an arbitrary input sampling rate Fin and an arbitrary output sampling rate Fout.

• It is desired that Ha( f) approximates unity for 0 ≤ f ≤ 0.4Fout with tolerance of 0.01 and zero for f ≥ 0.5Fout with tolerance of 0.001 (60-dB attenuation). The sample rate is reduced (decimation).

• When using the minimax optimization, and direct implementation of the transposed Modified Farrow the given criteria are met by N ==== 28 and M ==== 4. This implementation requires 5 fixed branch filters of length 28.

• The implementation can be simplified using fixed linear-phase FIR interpolators before the Farrow structure, i.e., cascaded structure.

• N ==== 4 and M ==== 3 are required if the sampling rate is decreased by a factor of four by using a two-stage fixed decimator with decimation factors of two and two and FIR filters of order 3 and 52, respectively.

• The same requirements are met by the prolonged Modified Farrow structure by N = 14, M = 6, and J = 2. This implementation requires 7 fixed branch filters of length 14.

51


Table 1. Requirements for the structures under consideration

Multiplications/s No. of multipliers

No. of I&Ds

TF 4Fin+70Fout 74 5 PTF 12Fin+49Fout 61 13 TF+FIR 3Fin+59Fout 38 4 TF+FIR1+FIR2 2Fin+55Fout 37 3 TF is the transposed Farrow structure, PTF is the prolonged transposed Farrow structure, TF+FIR is the transposed Farrow structure in cascade with a one-stage FIR decimator, and TF+FIR1+FIR2 is the transposed Farrow structure in cascade with a two stage FIR decimator.

52


Direct design

0 1 2 3 4 5 6 7 8−100

−80

−60

−40

−20

0

Mag

nitu

de in

dB

Frequency relative to Fout

TFPTF

Fig. 23. Magnitude responses for the transposed Farrow structure (TF) and the prolonged transposed Farrow structure (PTF).

53


Design with fixed two stage decimator after the transposed Farrow structure

0 1 2 3 4 5 6 7 8−120

−100

−80

−60

−40

−20

0

Frequency in Fout

Mag

nitu

de in

dB

FIR 1FIR 2FarrowOverall

Fig. 24. The magnitude responses for both FIR filters in the two-stage decimator, the Farrow structure, and the overall system.

54


Response in rational sample rate conversion

• When Fout / Fin = K / L > 1 is rational, µl becomes periodic, with the periodicity of K. The set of µ values contains K members and can be written as

µ(0)+ k/K | k=0…K-1, 0≤ µ(0)<1/K (49)

• Rational SRC can be modelled as a cascade of an upsampler, a discrete-time LTI filter, and a downsampler. This filter is a sampled version of the filter ha(t) of the analog model.

• Takes into account the sampler of the analog model!

• The formula horrors of page 42 are mostly avoided. => Simpler formulas, simpler programs.

• BUT: A filter designed using the discrete-time model may not be good for other conversion factors!

FIR filter

Hd(z) ↑↑↑↑K x(n) z(j) Fin KFin KFin

y(l) Fout=KFin/L↓↓↓↓L

Fig. 25. Model for rational sample rate conversion.

55


Emulating Continuous-Time Signal Processing

• The Farrow structure can be easily generalized to process digitally the reconstructed signal ya(t).

• These applications include, among others, interpolating the derivative or integral of ya(t).

• Derivative is widely utilized, for example, in finding the location of a local maximum or minimum of a signal.

• Integral can be used to calculate the energy of a signal over a given time interval.

• We concentrate on determining the derivative of ya(t).

Farrow Structure for Determining the Derivative of ya(t)

• In the intervals nTi n≤ t<(n+1)Ti n for n=0, 1, 2,··· ya(t) can be expressed as

[ ]∑

=+= −==

M

m

mmTnta nvnpty

in0

)( 12)(),()( µµµ (50)

where the vm(n)’s are the output samples of the FIR branch filters in the Modified Farrow structure.

• The derivative of ya(t) in the intervals is thus given by

56


[ ] .122)(),()(

0

1)( ∑

=

−+= −==

M

m

mmTnt

a mnvdnpd

tdtyd

inµ

µµ

µ (51)

• The derivative of ya(t) at t = (n+µ)Ti n can be determined by multiplying the vm(n)’s by 2m(2µ−1)m−1, instead of (2µ −1) m in the Modified Farrow structure.

• If the frequency response of the interpolation filter is Ha(ω), then the corresponding response for the differentiator is jωHa(ω).

• The location of an extremum can be found by setting (51) to zero and solving µ.

ω

1

|Ha(ω)|

ω

1

|jωHa(ω)|

Fig. 26. Ideal frequency responses for interpolator and differentiator.

57


Example on the Derivative Approximation

• Design parameters for the differentiator are: L=5, N=8, ωp ∈ [ 0, 0.35fs] ,ωs=0.65fs, Wp=0.035,and Ws=1.

−4 −3 −2 −1 0 1 2 3 4−1.5

−1

−0.5

0

0.5

1

1.5Impulse response for dh(t)/dt

Time / T

Am

plitu

de

Fig. 27. Impulse response ha(t) for the differentiator.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

Frequency f / Fs

Am

plitu

de

(a)

Fig. 28. Amplitude response.

58


Symbol Synchronization in Digital Receivers

∼

Interpolatorh(k,µl)

Timingestimation

r(n)

Fin=1/ Tin

y(l)x(n)

µl

Decisionâ(l)

ADCr(t) Matched filter

hR(n)

nl

Fig. 29. Digital receiver with non-synchronized sampling.

• The sampling of the received signal is performed by a fixed sampling clock, and thus, sampling is not synchronized to the received symbols. ⇒ Timing adjustment must be done by digital

methods after sampling. • Can be done by using an interpolation filter. • Advantages of nonsynchronized sampling:

− Separates the analog and digital parts (no feedback across the A/D interface).

− Easy to change the sampling rate. − Sampling rate does not have to be a multiple of

the symbol rate (only high enough to avoid aliasing).

− No need for a complex PLL circuit.

59


DESIGN EXAMPLE

• The input signal x(n) has a raised cosine pulse shape with the excess bandwidth of α =0.5.

• The sample rate Fin is twice the symbol rate. ⇒ The highest baseband frequency component of

x(n) is (1+α )Fin /4=0.375Fin. • The interpolator is designed in the least-mean-square

sense with the following specifications: N=8, M=5, the passband edge fp=0.375Fin, and the stopband edge fs=0.625Fin.

• The following pages show the characteristics for the resulting interpolator as well as for the linear and third order Lagrange interpolator.

60


L2 design

0 0.5 1 1.5 2 2.5 3 3.5 4−80

−70

−60

−50

−40

−30

−20

−10

0

Mag

nitu

de in

dB

Frequency in Fin

Fig. 30. The magnitude response for the L2 interpolator (solid line) and the amplitude spectrum of the reconstructed signal ya(t) (dark

area).

61


Linear interpolation filter

0 0.5 1 1.5 2 2.5 3 3.5 4−80

−70

−60

−50

−40

−30

−20

−10

0

Mag

nitu

de in

dB

Frequency in Fin

Fig. 31. The magnitude response for the linear interpolation filter (solid line) and the amplitude spectrum of the reconstructed signal

ya(t) (dark area).

62


Cubic Lagrange interpolation filter

0 0.5 1 1.5 2 2.5 3 3.5 4−80

−70

−60

−50

−40

−30

−20

−10

0

Mag

nitu

de in

dB

Frequency in Fin

Fig. 32. The magnitude response for the cubic Lagrange interpolation filter (solid line) and the amplitude spectrum of the

reconstructed signal ya(t) (dark area).

63


Some remarks

• Cubic Lagrange or even linear interpolation may be sufficient for the cases where the oversampling factor is high, e.g. >>2.

• However, if there are signal components close to Fin/2, i.e., the oversampling factor is small (<2), the linear and cubic Lagrange interpolations are not good enough.

• The complexity of the Farrow structure can be reduced by increasing the sampling rate (by 2, 4, 8,…) before the Farrow structure.

• It is also possible to optimize this fixed interpolation and the Farrow structure together.

64


References [1] T. I. Laakso, V. Välimäki, M. Karjalainen, and U. K. Laine, “Splitting the unit delay,” IEEE Signal

Processing Magazine, vol. 13, pp. 30-60, Jan. 1996. [2] C. W. Farrow, “A continuously variable digital delay element,” in Proc. IEEE Int. Symp. Circuits

& Syst., Espoo, Finland, June 1988, pp. 2641-2645. [3] F. M. Gardner, “Interpolation in digital modems - Part I: Fundamentals,” IEEE Trans. Commun.,

vol. 41, pp. 501-507, Mar. 1993. [4] L. Erup, F. M. Gardner, and R. A. Harris, “Interpolation in digital modems - Part II: Implementa-

tion and performance,” IEEE Trans. Commun., vol. 41, pp. 998-1008, June 1993. [5] D. Kincaid and W. Cheney, Numerical Analysis. Pacific Grove, 1991. [6] J. Vesma and T. Saramäki, “Interpolation filters with arbitrary frequency response for all-digital

receivers,” in Proc. IEEE Int. Symp. Circuits & Syst., Atlanta, GA, May 1996, pp. 568-571. [7] J. Vesma, M. Renfors, and J. Rinne, “Comparison of efficient interpolation techniques for symbol

timing recovery,” in Proc. IEEE Globecom 96, London, UK, Nov. 1996, pp. 953-957. [8] J. Vesma and T. Saramäki, “Optimization and efficient implementation of FIR filters with

adjustable fractional delay,” in Proc. IEEE Int. Symp. Circuits & Syst., Hong Kong, June 1997, pp. 2256-2259.

[9] H. Ridha, J. Vesma, T. Saramäki, and M. Renfors, “Derivative approximations for sampled signals based on polynomial interpolation,” in Proc. 13th Int. Conf. on Digital Signal Processing, Santorini, Greece, July 1997, pp. 939-942.

[10] H. Ridha, J. Vesma, M. Renfors, and T. Saramäki, “Discrete-time simulation of continuous-time systems using generalized interpolation techniques,” in Proc. 1997 Summer Computer Simulation Conference, Arlington, Virginia, USA, July 1997, pp. 914-919.

[11] V. Tuukkanen, J. Vesma, and M. Renfors, “Combined interpolation and maximum likelihood symbol timing recovery in digital receivers,” to be presented in 1997 IEEE Int. Conference on Universal Personal Communications, San Diego, CA, USA, Oct. 1997.

[12] T. Saramäki and M. Ritoniemi, "An efficient approach for conversion between arbitrary sampling frequencies," in Proc. IEEE Int. Symp. Circuits & Syst., Atlanta, GA, May 1996, pp. 285-288.

[13] J. Vesma, R. Hamila, T. Saramäki, and M. Renfors, “Design of polynomial interpolation filters based on Taylor series,” in Proc. IX European Signal Processing Conf., Rhodes, Greece, Sep. 1998, pp. 283-286.

[14] J. Vesma, R. Hamila, M. Renfors, and T. Saramäki, “Continuous-time signal processing based on polynomial approximation,” in Proc. IEEE Int. Symp. on Circuits and Systems, Monterey, CA, USA, May 1998, vol. 5, pp. 61-65.

[15] D. Fu and A. N. Willson, Jr., “Interpolation in timing recovery using a trigonometric polynomial and its implementation,” in IEEE Globecom 1998 Communications Mini Conference Record, Sydney, Australia, Nov. 1998, pp. 173−178.

[16] f. harris, “Performance and design considerations of Farrow filter used for arbitrary resampling,” in Proc. 13th Int. Conf. on Digital Signal Processing, Santorini, Greece, July 1997, pp. 595−599.

[17] G. Oetken, “A new approach for the design of digital interpolation filters,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP−27, pp. 637−643, Dec. 1979.

[18] T. A. Ramstad, “Digital methods for conversion between arbitrary sampling frequencies,” IEEE Trans. Acoust. Speech, Signal Processing, vol. ASSP−32, pp. 577−591, June 1984.

[19] T. A. Ramstad, “Fractional rate decimator and interpolator design,” in Proc. IX European Signal Processing Conf., Rhodes, Greece, Sep. 1998, pp. 1949−1952.

65


[20] R. W. Schafer and L. R. Rabiner, “A digital signal processing approach to interpolation,” Proc. IEEE, vol. 61, pp. 692−702, June 1973.

[21] M. Unser, A. Aldroubi, and M. Eden, “Fast B-spline transforms for continuous image representation and interpolation,” Trans. Pat. Anal., Mach. Int., vol. 13, pp. 277−285, Mar. 1991.

[22] M. Unser, A. Aldroubi, and M. Eden, “Polynomial spline signal approximations: Filter design and asymptotic equivalence with Shannon’s sampling theorem,” IEEE Trans. Information Theory, vol. 38, pp. 95−103, Jan. 1992.

[23] J. Vesma, Timing Adjustment in Digital Receivers Using Interpolation. M.Sc. Thesis, Tampere, Finland: Tampere University of Tech., Department of Information Technology, Nov. 1995.

[24] V. Välimäki, Discrete-Time Modeling of Acoustic Tubes Using Fractional Delay Filters. Doctoral thesis, Espoo, Finland: Helsinki University of Technology, Dec. 1995.

[25] S. R. Dooley and A. K. Nandi, “On explicit time delay estimation using the Farrow structure,” Signal Processing, vol. 72, pp. 53−57, Jan. 1999.

[26] J. Vesma, “A frequency-domain approach to polynomial-based interpolation and the Farrow structure,” to appear IEEE Trans. on Circuits and Systems II, March 2000.

[27] J. Vesma, Optimization and Applications of Polynomial-Based Interpolation Filters. Dr. Tech. Thesis, Tampere, Finland: Tampere University of Tech., Department of Information Technology, May 1999

[28] D. Babic, J. Vesma, T. Saramäki, M. Renfors, “Implementation of the transposed Farrow structure,” in Proc. 2002 IEEE Int. Symp. Circuits and Systems, Scotsdale, Arizona, USA, 2002, vol. 4, pp. 4−8.

[29] D. Babic, T. Saramäki and M. Renfors, “Conversion between arbitrary sampling frequencies using polynomial-based interpolation filters,” in Proc. Int. Workshop on Spectral Methods and Multirate Signal Processing, SMMSP02, Toulouse, France, September 2002, pp. 57−64.

TLT-5806 Receiver Architectures and Signal Processing

Addition to the lecture:Polynomial-Based Interpolation

Filters for DSP Applications

NEWTON INTERPOLATION FORFRACTIONAL-DELAY FILTERING

Vesa LehtinenDepartment of Communications Engineering

Tampere University of TechnologyP.O.Box 553, 33101 Tampere, Finland

[email protected]

TLT-5806 Receiver Architectures and Signal Processing: Newton Interpolation for Fractional-Delay Filtering

Vesa Lehtinen 2/3 29.10.2008

The Newton structure

Fig. 1: The Newton structure for fractional-delay filte ing.

• Efficient implementation o Newton’s backward difference for-mula for interpolation– Anticausal version would implement Newton’s forward

difference formula– Discrete-time counterpart for Taylor series– Sometimes incorrectly referred to as the Taylor structure

• Equivalent to Lagrange & Pascal interpolation• Proposed by various authors at different times• D : total delay in sample intervals

– Keep close to N / 2 for best response.– Cf. Farrow: D not in [0,1)N : interpolator order, i.e., number of stages

• Easy to see: y[n]=x[n–D] for integer D.• Demonstrated by newtondemo.m

1 z 1–– 1 z 1–– 1 z 1––D–

D– 1+2

------------------ D– N 1–+N

-----------------------------x n[ ]

y n[ ] Building block

TLT-5806 Receiver Architectures and Signal Processing: Newton Interpolation for Fractional-Delay Filtering

Vesa Lehtinen 3/3 29.10.2008

Advantages of the Newton structure:• Simple: multiplier-free subfilter , few multipliers• Wide delay range: far beyond one sample interval

– Order-delay-response trade-off possible• Modular structure: good for VLSI, interpolator order can be

easily changed

• O(N) complexity! (vs. Farrow’s O(N2))=> high orders feasible

Disadvantages:• Poor response (Lagrange)

– Requires an oversampled signal• High order (due to poor response)

=> Longer delay in causal realisations

References[1] L. Elden, L. Wittmeyer-Koch, and H.B. Nielsen, Introduction to Numerical Com-

putation. Studentlitteratur, Lund, 2004, pp. 107–113.[2] http://mathworld.wolfram.com/UmbralCalculus.html[3] S. Tassart and Ph. Depalle, “Fractional delays using Lagrange interpolators,“ in

Proc. Nordic Acoustic Meeting, Helsinki, Finland, 12–14 June, 1996.[4] C. Candan, “An efficient filt ing structure for Lagrange interpolation,” in IEEE

Signal Processing Letters, Vol. 14, No. 1, Jan 2007, pp. 17–19.[5] T.J. Goodman, M.F. Aburdene, “Interpolation Using the Discrete Pascal Trans-

form,” Proc. 40th Annual Conf. Information Sciences and Systems, 22–24March 2006, pp. 1079–1083.

[6] Vesa Lehtinen, Markku Renfors, "Structures for Interpolation, Decimation, and Nonuniform Sampling Based on Newton's Interpolation Formula," in Proc.

Sampling Theory and Applications (SAMPTA), Marseille, France, 18-22 May 2009. http://hal.archives-ouvertes.fr/hal-00451769/

polynomial-based interpolation filters for dsp … · 1 tampere university of technology...

Documents