the interlaced chirp z transform

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Signal Processing 86 (2006) 2221–2232

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The interlaced chirp Z transform

Indranil Sarkar�, Adly T. Fam

Department of Electrical Engineering, The State University of New York at Buffalo, NY 14260, USA

Received 2 April 2005; received in revised form 6 August 2005; accepted 14 October 2005

Available online 28 November 2005

Abstract

In this paper we introduce the interlaced chirp Z transform (Interlaced CZT). It is based on the computation of several

carefully staggered CZT that are progressively interlaced to result in a spectrum that has denser frequency samples where

needed. This simple modification of the CZT is shown to result in significant computational savings over the regular CZT,

as well as the zoom CZT (ZCZT). The CZT computes uniformly spaced frequency samples over a desired range with an

efficiency similar to that of the FFT algorithm. In practice, several CZT’s over increasingly smaller ranges are required to

obtain denser frequency samples where needed. This process is referred to as the ZCZT. The interlaced CZT improves on

the ZCZT by interlacing successive CZT’s such that, in each step, the previous samples are included with the new ones,

resulting in dense frequency sampling with increased computational efficiency. The regular CZT is also compared to the

ZCZT based on a typical regimen and the conditions under which either is superior are elucidated.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Chirp Z transform; Discrete Fourier transform; Interlaced chirp Z transform; Warped discrete Fourier transform; Zoom chirp

Z transform

1. Introduction

The discrete Fourier transform (DFT) computesuniformly spaced samples in the frequency domain.The fast Fourier transform (FFT) is a computa-tionally efficient implementation of the DFT. Thechirp Z transform (CZT) allows us to efficientlycompute frequency samples that are uniformlyspaced over any desired arc of the unit circle withan efficiency similar to that of the FFT. In variousreal time applications, fast hardware FFT imple-mentation for a particular limited-length data is

e front matter r 2005 Elsevier B.V. All rights reserved

pro.2005.10.004

ing author. Tel.: +1716 645 2422x2504;

3656.

sses: [email protected] (I. Sarkar),

alo.edu (A.T. Fam).

available. The CZT is then repetitively used toobtain a higher resolution by zooming onto thedesired part of the spectrum. This method is widelyused in the industry especially in localized high-resolution spectrum analysis. It is to be noted thatsome technical manuals refer to this process simplyas the CZT or the zoom FFT [1,2]. In practice, it isoften required to carry out a series of CZT’s toachieve progressively higher zooming. We will usethe term ZCZT throughout this paper to distinguishthis process from a single CZT. Applications of theZCZT include ultrasonic blood flow analysis, RFcommunications, mechanical stress analysis, Dop-pler radar, side band analysis, and modulationanalysis.

The importance of unequally spaced frequencysamples has been discussed in the literature from as

.

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ARTICLE IN PRESSI. Sarkar, A.T. Fam / Signal Processing 86 (2006) 2221–22322222

early as 1971 [3]. The simplest way to calculateunequally spaced frequency samples would be tocalculate each and every sample individually. Suchbrute force evaluation is computationally intensive,requiring OðNÞ complex multiplications and addi-tions per frequency sample.

In this paper, we introduce the interlaced CZT toobtain unequally spaced frequency samples. It isbased on the ZCZT which computes a series ofCZT’s over decreasing ranges to produce a denserfrequency grid. However, the ZCZT does notincorporate previously computed samples and henceinvolves some redundant computational cost. Theinterlaced CZT improves on the ZCZT by carefullystaggering the successive CZT’s such that the newlycomputed frequency samples are interlaced with thepreviously calculated ones. This simple modificationof the ZCZT efficiently produces unequally spacedfrequency samples which are dense where neededand sparse elsewhere.

We show that the interlaced CZT results in asignificant reduction in computational complexityas compared to the regular CZT as well as theZCZT. In addition to that, it inherits the desirableproperties and applications of the CZT. The firstexample of such an application is converting anexisting FFT algorithm that requires a power of 2length to an efficient algorithm for any DFT length.In such cases, the desired DFT is first converted to aconvolution via the CZT [4]. The convolution isthen computed via a fast algorithm. Since paddingwith zeros is possible in this case, a power of 2length could be achieved. The above-mentionedFFT algorithm that requires a power of 2 lengthcould then be used to compute this convolution withcomparable computational efficiency. The secondapplication is also based on the CZT converting theDFT into a convolution. This allows charge-coupled devices (CCD) that implement the dis-crete-time FIR filters to be used to implement theDFT. This application of CCD is of particularinterest in very high-frequency systems where thesignals are discrete in time but not digitized.

In Section 2, we assess the comparative computa-tional performance of the regular dense grid CZTversus the ZCZT using a representative zoomingregimen. In Section 3, the interlaced CZT isintroduced where each successive CZT is chosensuch that its data samples are interlaced with theprevious ones which are not discarded. In Section 4,the regular CZT and the interlaced CZT arecompared in terms of their computational complex-

ities. In Section 5, we compare the interlaced CZTand ZCZT in terms of their zooming ability andcomplexities. The warped discrete Fourier transform(WDFT) [5,6] is a recently introduced method toproduce unequally spaced frequency samples. InSection 6, we compare the computational complexityof our method with that of the WDFT. In Section 7,we show the effective usage of the interlaced CZTthrough an example of spectral analysis. Finally, weconclude after a brief discussion on potentialapplications and further research scope.

2. Quantitative analysis of the zoom chirp Z

transform

Even though the ZCZT is widely used, there doesnot seem to be a comparative quantitative assess-ment of its performance vis-a-vis the regular CZT.In this section we compare the two methods toelucidate the conditions under which it is moreefficient to use the ZCZT.

At this point, it is in order to formally define theCZT and the parameters that govern it. The CZTwas introduced in 1969 by Rabiner et al. [7]. If x½n�

is a N point sequence and its Fourier transform isgiven by X ðejoÞ, the CZT computes M equi-spacedsamples of X ðejoÞ at frequencies

ok ¼ o0 þ kDo, (1)

where k ¼ 0; 1; . . . ;M � 1. The starting frequencyo0 and the frequency increment Do can be chosenarbitrarily. The Fourier transform corresponding tothis general set of frequency samples is then given by

X ðejok Þ ¼XN�1n¼0

x½n�e�jokn. (2)

Defining W ¼ ejDo, and using (1), (2) reduces to

X ðejok Þ ¼XN�1n¼0

x½n�e�jo0nW nk. (3)

It can easily be shown that (3) can be converted to aconvolution using the simple yet ingenious substitu-tion due to Bluestein [8]:

nk ¼n2 þ k2

� ðk � nÞ2

2. (4)

We refer the reader to [9] for a complete analysis ofthe CZT.

For our present purposes, we see that the CZTcan be calculated over any specified arc on the unitcircle. This can be done by choosing o0 and Do

ARTICLE IN PRESSI. Sarkar, A.T. Fam / Signal Processing 86 (2006) 2221–2232 2223

appropriately. In case of ZCZT, the CZT is carriedout repetitively, thereby zooming into region(s)where the energy is found to be concentrated.

As a representative analysis of the ZCZT, weconsider the case where the number of frequencysamples remains constant each time, but the arcover which the CZT is calculated, is diminished by afactor of a from one step to the next. Here, each steprepresents computing a new CZT over a diminishedrange of frequencies.

Let Dy0 ¼ Angle subtended at the center by theinitial arc.

Dyi ¼ Angle subtended at the center by the arcobtained in the ith step.

We have

Dy04Dy14Dy24 � � � , (5)

where

Dyiþ1 ¼ aðDyiÞ (6)

which results in

Dyi ¼ aiðDy0Þ. (7)

a is defined to be the zooming parameter such that

0oao1. (8)

This way we parameterize the ZCZT. In practice theZCZT can actually be parameterized in a variety ofways. Here, we consider a representative regimenfor the sake of comparison with the regular CZT aswell as the interlaced CZT. We assume the numberof frequency points (L) in each step to be constant.The following analysis elucidates how the computa-tional complexity of the ZCZT depends on thezooming parameter a. By the term complexity wemean the total number of complex multiplications.In case of the CZT this is also approximately equalto the number of complex additions.

Let N be the number of samples in the datavector, L the number of frequency points consideredin each step of the repetitive CZT, n the number ofzooming steps.

The single equivalent CZT should have the samedense grid as that of the last step of the repetitiveprocedure. This high density is to be maintainedover the entire initial arc.

2.1. ZCZT

The computational complexity for each stage isgiven by [10,11]:

C ¼ 3ðN þ L� 1Þ logðN þ L� 1Þ. (9)

Therefore, the total complexity for the n stageswould be:

CZ ¼ nC ¼ 3nðN þ L� 1Þ logðN þ L� 1Þ. (10)

Hence the complexity in this method is independentof the zooming parameter a.

2.2. Regular CZT

In order to compute the complexity of the singleequivalent regular CZT, we calculate the number offrequency points required to achieve the sameresolution as that of the last step of the ZCZT.The angle range after n steps in the ZCZT can befound using (7).

Since the number of frequency points within thisarc ¼ L, the total number of points required tomaintain the same resolution over the entire initialarc of angular range is given by:

M ¼L

an. (11)

The total complexity for the regular CZT is there-fore:

CR ¼ 3ðN þM � 1Þ logðN þM � 1Þ. (12)

Since M is dependent on a, then so is the complexityCR. The comparative complexities for two differentL values, L ¼ 1000 and 100; 000 are shown in Figs.1(a) and (b), respectively. The number of points N

in the data vector and the number of steps n areidentical in both cases.

It can be seen from each of the above figures thatthe variation in L affects the position of theintersection point of the two curves for the ZCZTand the regular CZT. The complexities of both theZCZT and regular CZT increase monotonicallywith L. From (11) and (12) we can deduce that avariation in N would affect the curves similarly to L.

Although the variation in cost of the ZCZT isindependent of the zooming parameter a, it isdependent on L, the number of points on thefrequency spectrum that are considered, as shown inFig. 2.

The comparative study of the ZCZT and thesingle high-resolution regular CZT in Figs. 1(a) and(b) shows that the former is superior to the latter interms of complexity only when the zoomingparameter a is less than a certain value. This islogical since a high a implies lesser zooming,requiring more repetitions to reach a desired final

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N=1000L=100000r=10

104

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107

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109

1010

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1012

1013

(a)

(b)

Fig. 1. Comparative complexity of the regular CZT and ZCZT.

(a) L ¼ 1000, (b) L ¼ 100; 000.

0 1 2 3 4 5 6 7 8 9 10

x 104

105

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107

108

No of points on the frequency spectrum

Co

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Fig. 2. Cost curve for the ZCZT.

I. Sarkar, A.T. Fam / Signal Processing 86 (2006) 2221–22322224

arc. Depending on N, L and n, the cross-over pointvaries.

3. The interlaced chirp Z transform

The interlaced CZT is similar to the ZCZT in itssuccessive zooming behavior. However, unlike theZCZT, it combines the new samples in each stepwith those from previous ones resulting in a denserspectrum. In addition, the starting and endingpoints in each step are judiciously chosen to reflectthe sector where increased resolution is desired. InFig. 3, the positions of the first and the last samplesfrom the ith step are denoted by yi;f and yi;l,respectively. Let this sector subtend an angle of ci.Let the inter-sample angle in the ith step be dyi. Thepositions of the samples in the ði þ 1Þth step arechosen such that g inter-sample intervals from theith step are left vacant at each end of the sector.Each new sample is inserted at the midpoint of theremaining inter-sample intervals from the previoussteps. Hence the position of the new sample at oneend in the ði þ 1Þth step is

yiþ1;1 ¼ yi;f � ðgþ 0:5Þdyi, (13)

where i ¼ 1; 2; . . . ; n.The new sample at the other end in this step is

placed at

yiþ1;Tiþ1¼ yi;l þ ðgþ 0:5Þdyi. (14)

Fig. 3. Position of the frequency samples after 3 steps of

interlaced CZT. The circles, the crosses and the diamonds

represent the samples obtained in the first, second and third

steps, respectively.

ARTICLE IN PRESS

Samples from the CZT of the first step

New samples obtained through the CZT of the second step

Interlaced old and new samples of the above two CZT’s

The homogeneous part of the above interlaced samples. This is chosen as the starting point for the next step. The whole procedure shown till here then repeats itself.

Interlaced old and new samples after the third step. The new samples obtained in the third step are denoted by diamonds.

(a)

(b)

(c)

(d)

Fig. 4. Example elucidating the interlacing scheme.

I. Sarkar, A.T. Fam / Signal Processing 86 (2006) 2221–2232 2225

The old samples immediately following the newsample at yiþ1;1 and immediately preceding the oneat yiþ1;Tiþ1

, along with all the samples in between,form a series of frequency points separated by acloser angular spacing dyiþ1. We define the sectorformed by these samples as ciþ1. Hence thepositions of the first and last samples of this sectorare given by the equations:

yiþ1;f ¼ yiþ1;1 þ dyiþ1, (15)

yiþ1;l ¼ yiþ1;Tiþ1� dyiþ1. (16)

We define an equivalent zooming parameter for theinterlaced CZT as

ai ¼Dyi

Dyi�1, (17)

where i ¼ 2; 3; . . . ; n and a2 is taken as the initialzooming parameter. In Fig. 3, we show a represen-tative case where centralized zooming is achievedand 2 gaps are left vacant on each side in each step.In this example, only 3 steps are shown. More stepsare added as required. In actual applications, thedistribution of the vacant intervals can be adap-tively changed from one step to the next dependingon the region(s) where peaks are detected.

Let P be the number of frequency points in thefirst step, g the number of vacant intervals in eachstep at each end. Symmetric zooming is assumedwithout loss in generality, Pi the number of newfrequency points obtained in the ith step.

In the interlaced CZT, the number of newsamples obtained in the second step is given by

P2 ¼ K ¼ P� ð2gþ 1Þ. (18)

The number of new samples obtained in the ith stepis given by

Pi ¼ 2ðPi�1 � gÞ, (19)

where i ¼ 3; 4; . . . ; n.We observe from (18) that depending on the

values of P and g, the number of frequency samplesobtained in the second step could be either even orodd. However, from (19), the number of newfrequency samples obtained from the third steponwards will always be even. The number of pointsN in the data vector is constant for all steps.Neglecting the insignificant complexity due to pre-and post-multipliers, the complexity for the ith stepis given by [10]

Ci ¼ 3ðN þ Pi � 1Þ logðN þ Pi � 1Þ. (20)

The number of samples after n steps, as shown inAppendix A.1, is given by

Sn ¼ Pþ 2gðn� 1Þ þ ðP� 4g� 1Þð2n�1 � 1Þ. (21)

If it is desired for the interlacing to be continued foras many steps as required, the term ðP� 4g� 1Þmust be positive. This results in the condition:

gpP� 1

4

� �. (22)

There could be several ways in which the newlyobtained samples are interlaced with the old ones.We further elucidate the process used in this paperwith another example as shown in Fig. 4. For thepurpose of illustration, we show the samplesobtained in different steps depicted on a line insteadof the unit circle. The initial number of samples inthis example has been taken as P ¼ 13. These arerepresented by the circles as shown in Fig. 4a. Thenumber of gaps left vacant in each step is g ¼ 2. Thenewly obtained samples in the second step aredenoted by crosses as shown in Fig. 4a. Theinterlacing for the second step is explicitly shownin Fig. 4b. We observe that the number of newsamples obtained in the second step is 8, which isconsistent with (18). In Fig. 4c, we show how weselect the homogeneous (or equally spaced) part ofthe samples obtained after the second step. This istaken as the starting point of the next step and thesamples for the third step are calculated by leaving g

intervals on each side of this set of samples. The newsamples obtained in the third step are denoted bydiamonds. The interlaced samples after the thirdstep is shown in Fig. 4d. The number of newsamples obtained in the third step is 12 as expected

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105

106

107

108

Co

mp

lexi

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105

106

107

108

Interlaced CZTRegular CZT

(a)

(b)

Fig. 5. Comparative complexities of the interlaced CZT and the

regular CZT. (a) g ¼ 200, N ¼ 1000, P ¼ 1000, (b) g ¼ 249,

N ¼ 1000, P ¼ 1000.


from (19). If more steps are carried out, the numberof new samples in every step will be governedby (19).

While the number of intervals that are left vacantin each step remains constant, the equivalentzooming parameter ai varies as a function of i, asinvestigated next. From (17), we find the zoomingparameter for the ith step. Let Ti�1 denote thenumber of frequency samples in the sector ci�1

obtained in the ði � 1Þth step. If dyi is the anglesubtended at the center of the unit circle by twoconsecutive samples, then the total angle rangecovered in the ði � 1Þth step is given by

Dyi�1 ¼ dyi�1ðTi�1 � 1Þ. (23)

For the ith step, the first and last g inter-sampleintervals from the ði � 1Þth step are left vacant andthe new samples are placed at the midpoints of theremaining intervals. The total angle range coveredin the ith step is therefore given by

Dyi ¼ dyi�1fðTi�1 � 1Þ � ð2gþ 1Þg. (24)

Substituting values from (23) and (24) in (17), weobtain

ai ¼ 1�2gþ 1

Ti�1 � 1, (25)

using (21) to find out Ti�1 as

Ti�1 ¼ Si�1 � Si�2

and substituting in (25) the expression for ai reducesto

aiðgÞ ¼ 1�2gþ 1

2gþ ðP� 4g� 1Þ2i�3 � 1, (26)

where ai is explicitly indicated as a function of g.From (26), it is seen that when g remains

constant, the zooming parameter varies with i.The interlaced CZT also allows us to vary thezooming parameter by adaptively varying g in everystep. In the next section, we compare the computa-tional performance of the interlaced CZT with thatof the regular CZT.

4. Comparison of The interlaced CZT with the

regular CZT

In this section, we compare the interlaced CZTwith a dense grid regular CZT calculated over theentire range of interest. The highest densityobtained in the last step of the interlaced CZT isused to define the single dense grid regular CZT

over the entire arc. The complexity of the regularCZT is calculated based on [10].

As shown in Appendix A.2, the number ofsamples in the equivalent regular CZT that replacesthe n steps of the interlaced CZT is given by

S ¼ Sn þ gð2n � n� 1Þ. (27)

The comparative complexities are shown in Fig. 5for two different values of g.

It is seen that as g increases, the performance ofthe interlaced CZT becomes increasingly better thanthe regular CZT. The equivalent zooming para-meter in the interlaced CZT is a monotonedecreasing function of g. Therefore, the averageequivalent zooming parameter also decreases mono-tonically with g. Hence the comparative perfor-mance of the interlaced CZT improves as theaverage equivalent zooming parameter is decreased.

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16

18

com

ple

xity

average zooming parameter

x 106

Interlaced CZTZCZT

Fig. 6. Comparative complexities of the interlaced CZT and

ZCZT.


This can be compared to our analysis in Section 2where we showed that the performance of theregular CZT becomes increasingly poorer, ascompared to the ZCZT, as a decreases.

It is observed that for a given choice of P, thechoice of g is constrained by (22). In the represen-tative case that has been considered, for P ¼ 1000,the maximum value of g that can be used is 249.This value of g also gives the best comparativeperformance of the interlaced CZT. Hence for agiven value of P, the optimum g, that achieves thebest comparative performance over the regularCZT, is governed by

g ¼P� 1

4

� �. (28)

It maybe argued that the regular CZT gives densefrequency samples over the entire range, whereasthe interlaced CZT gives the same high density onlyover a part of that range. However, the fairness ofthe comparison lies in the fact that we areconsidering situations where we need the densesamples only over certain parts of the spectrum andnot over the entire range.

5. Comparison of the interlaced CZT with the ZCZT

The comparison is based on the following:

(a)
Both begin with the same initial sector angle andend with the same highest zoom angle;
(b)
Both have the same number of steps; and (c) Both achieve the same high density frequency
grid in the final step.

For the interlaced CZT, we select N and g, both ofwhich remain constant in every step. This results ina varying equivalent zooming parameter. However,to validate the comparison, we calculate the averagezooming parameter of all steps. For the ZCZT, thisis the same as the actual zooming parameter becausethe zooming parameter remains constant in the caseof the ZCZT. The complexities of the interlacedCZT and the ZCZT are shown in Fig. 6 as functionsof this average zooming parameter. It should benoted that in this figure, ZCZT appears to be afunction of a even though we mentioned in Section2 that it is not. This happens due to the fact thatwhen comparing with the interlaced CZT, adetermines the number of steps n required toachieve the same dense frequency grid as obtainedthrough the interlaced CZT. Since the complexity of

the ZCZT is a function of n as shown in (10), it alsoappears to be a function of a in this particularcomparison.

In spite of the higher complexity of the ZCZT, thenumber of frequency samples obtained is less thanthat of the interlaced CZT. We also study thebehavior of the equivalent zooming parameter ofthe interlaced CZT and compare it to the constantzooming parameter of the ZCZT. To obtain ameaningful comparison between the two methods,we define the following ratio in the case of interlacedCZT:

req ¼Dyn

Dy1, (29)

where n is the total number of steps involved.Next, we calculate the constant zooming para-

meter that is required in the case of the ZCZT toachieve the same ratio in as many steps n. This isplotted along with the varying ai of the interlacedCZT in Fig. 7. As shown earlier, the equivalentzooming parameter in the interlaced CZT isdependent on the number of intervals g that areleft vacant in each step on each side. Hence thecomparison has been done for different g values, asshown in Fig. 7. The number of data points N andthe initial number of frequency points P are bothtaken as 1000. As a result, the maximum value of g

that can be used is 249 as calculated using (28).Since g should always be an integer, the minimumusable value of g is 1. However, such a low value ofg will produce a very low amount of zoomingrendering the process computationally inefficient.

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aInterlaced CZTZCZT

Interlaced CZTZCZT

Interlaced CZTZCZT

Interlaced CZTZCZT

g=225g=200

g=245 g=249

Fig. 7. Comparison of the zooming parameter of the interlaced CZT and the ZCZT.


The range of a is constrained by these extremevalues of g. In practice, a sufficiently high value of g

should be chosen to achieve effective zooming. Ageneral trend is observed in each of the cases with adifferent g. The zooming parameter dips in thesecond step of the interlaced CZT to a low value. Itremains lower than the equivalent constant value ofthe ZCZT for the first few steps. Hence, the mosteffective zooming in the interlaced CZT is achievedin the first few steps of interlacing.

6. Comparison of the interlaced CZT with the

WDFT

The WDFT [5,6], has been recently introduced asa method of computing non-uniformly spacedfrequency samples. It has been proposed as anefficient way to calculate the more general non-uniform discrete Fourier transform (NDFT) [12]which calculates frequency samples at N arbitrarybut distinct points in the z-plane. The WDFTcomputes a non-uniformly spaced set of frequencysamples similar to that obtained by the interlaced

CZT. In this section, we compare the interlacedCZT with the WDFT in terms of the computationalcomplexity required to calculate the same numberof frequency samples.

First, we briefly review the concept of both theNDFT and the WDFT. A detailed analysis of thesecan be found in [12,5], respectively. The NDFTrepresents the most general form of DFT and the N

point NDFT of a length-N sequence x½n� is given by

XNDFT½k� ¼XN�1n¼0

x½n�z�nk , (30)

where 0pkpN � 1 denote N distinct frequencypoints in the z-plane. This process is computationallyburdensome requiring OðN2Þ complex multiplica-tions. The WDFT has been proposed to calculatenon-uniformly spaced frequency samples with re-duced complexity. The key idea of WDFT is to do anall-pass transformation on the frequency axis. Thiswarps the frequency axis in such a way thatuniformly-spaced samples in this warped axis becomenon-uniformly spaced in the original frequencydomain. The WDFT mapping was illustrated with

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x 104

105

106

107

108

No of points in the frequency spectrum

Co

mp

lexi

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Interlaced CZTWDFT

Fig. 10. Comparative complexities of the interlaced CZT and

WDFT.


a simple example in [5]. In this example, N equallyspaced samples in the modified or warped z-domain(z) is related to the original z-domain by a simplefirst-order all-pass function given by

z�1 ¼�aþ z

1� az�1, (31)

where jajo1 and it is called the warping parameter.The frequency mapping between the modified andoriginal z-domain is shown to be

tano2

� �¼

1þ a

1� a

� �tan

o2

� �. (32)

The frequency mapping and its nature of dependenceon a is shown in Fig 8. The positions of the frequencysamples on the unit circle are shown for both DFTand WDFT in Fig 9. For illustrative purposes, thenumber of frequency samples is taken as 16 with the

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

Original frequency

War

ped

fre

qu

ency

a=0.5a=0a=0.5

Fig. 8. Frequency mapping using the first-order all-pass trans-

formation as shown in [5].

π 0 π 0

DFT WDFTN=16 N=16, a=0.5

Fig. 9. Position of the frequency samples on the unit circle for

DFT and WDFT.

last sample overlapping with the first one. It caneasily be noted from Figs. 3 and 9 that the WDFTcalculates non-uniformly spaced frequency sampleslike the interlaced CZT. We now compare thecomputational complexities of these two methods.

From [5] the total number of real multiplicationsrequired for a N point WDFT is given by

CWDFT ¼ NðN þ 2 log2 N þ 4Þ. (33)

Assuming 4 real multiplications to be equivalent toone complex multiplication, and neglecting the realadditions required for the process, we compare thecomputational complexity of WDFT with that ofthe interlaced CZT as a function of the number ofobtained frequency samples. The plot for thecomparison is shown in Fig. 10.

The WDFT is a very efficient and elegant methodfor obtaining unequally spaced frequency sampleswhen the number of frequency samples are rela-tively low. However, it can be observed from Fig. 10that when the number of frequency samples in thespectrum is large, the interlaced CZT proves to becomputationally more efficient than the WDFT.Another distinct advantage of the interlaced CZT isthat the distribution of g can be adaptively variedand more than one region in the spectrum can bezoomed into simultaneously.

7. Illustrative example

Here, we give an example of spectral estimationto show the effective use of the interlaced CZT. Wehave chosen three very closely spaced sinusoids at


2000, 2010 and 2025Hz, respectively. The samplingfrequency is taken as 8000Hz. High varianceadditive white Gaussian noise is added to the mixedsignal. Figs. 11a–c show how the interlaced CZT

0 500 1000 1500 2000 2500 3000 3500 40000

500

1000

1500

2000

2500

3000

3500

4000

Frequency

Mag

nit

ud

e

0 500 1000 1500 2000 2500 3000 3500 40000

500

1000

1500

2000

2500

3000

3500

4000

4500

Frequency

Mag

nit

ud

e

0 500 1000 1500 2000 2500 3000 3500 40000

500

1000

1500

2000

2500

3000

3500

4000

4500

Frequency

Mag

nit

ud

e

(a)

(b)

(c)

Fig. 11. Interlaced CZT of a signal with three sinusoids. (a) 3

steps of interlaced CZT: one speak is detected, (b) 4 steps of

interlaced CZT: two speaks are detected, (c) 5 steps of interlaced

CZT: all three peaks are detected.

produces progressively denser frequency grids insuccessive steps.

From Fig. 11(a), we observe that three steps ofinterlacing detects only one of the components. Wedo the fourth step of interlacing such that we get adenser grid near the detected peak. This fourth stepdetects the second component as shown in Fig.11(b). In order to examine the spectrum even moreclosely near the peaks, we do a fifth step ofinterlacing, which reveals all the three componentsin the spectrum as shown in Fig. 11(c). Furtherinterlacing steps could be carried out to examine ifmore peaks exist. It can be observed from Fig. 11that the frequency samples are unevenly distributedand become denser in the region where peaks aredetected. The ZCZT is widely used to zoom into aparticular region of the frequency spectrum, but asshown in Section 5, the interlaced CZT does thismore efficiently. The interlaced CZT is similar to theZCZT in the sense that every step tells us about theregion where further zooming is required, butimproves over the ZCZT, incorporating previoussamples via interlacing and thereby increasingefficiency.

8. Applications and future scope

In many applications, we need a denser frequencygrid only in a part of the spectrum. The importanceof unequally spaced frequency samples have beendiscussed in the literature at least since 1971 [3]. Adirect application of such an idea can be found in[13]. As discussed in Section 6, the WDFT is one ofthe latest major contributions in this area. The mainidea of WDFT is to warp the frequency axis usingan all-pass transformation, such that equally spacedpoints on the warped axis becomes equivalent tounequally spaced points on the original axis. Thewarping can be controlled by a warping parameterthat conglomerates the frequency samples at somedesired part of the spectrum thus giving a densergrid in that range. The WDFT has been shown toperform excellently for short data records. For alarge number of frequency samples and long datarecord length, the interlaced CZT outperforms theWDFT in terms of computational complexity.

The WDFT has been effectively used in manyapplications. In [14], it has been used as apsychoacoustic model. It has also been used forspectral analysis of monitored power systems [15]. Itwould be of interest to investigate the performanceof the interlaced CZT in these applications

ARTICLE IN PRESSI. Sarkar, A.T. Fam / Signal Processing 86 (2006) 2221–2232 2231

especially the ones described in [13,14]. Anotherpotential application of the interlaced CZT is ininterference cancellation in OFDM systems. AnFFT implementation of the same is proposed in[16]. ZCZT reduces the computational complexityof the system. The complexity could further bedecreased if the interlaced CZT is used. Theinterlaced could also be potentially used in the areaof parameter estimation. Li and Stoica [17] haveproposed the RELAX algorithm for estimatingsinusoidal parameters in the presence of colorednoise. The method is based on the calculation ofFFT. The system could be made more efficient byusing the interlaced CZT.

Further research is needed to assess the effective-ness of the interlaced CZT in various otherapplications.

9. Conclusions

We have proposed a simple modification of theCZT in which successive CZT’s are interlaced toresult in a progressively denser grid spectrum. Inspite of its simplicity, the interlaced CZT achievessignificant computational savings. An algorithm ispresented, which ensures that no two frequencysamples overlap, even when the density of thesamples becomes very high. This way, frequencysamples are neither calculated more than once nordiscarded, thereby removing the redundancy of theZCZT. The computational complexity of the inter-laced CZT is shown to be superior to both theregular CZT, ZCZT and WDFT.

The interlaced CZT is ideally suited for applica-tions where a denser grid is desired in only a fewlocal regions of the frequency spectrum. It has beenmentioned in Section 1 how the CZT is amenable toimplementation on CCD’s. As the interlaced CZT isbased on the CZT algorithm, it inherits thisdesirable property of the CZT. A comparison hasalso been done between the regular CZT and theZCZT, based on a typical regimen, and theconditions under which either is superior have beenestablished.

Acknowledgments

We wish to thank the anonymous reviewers fortheir helpful comments and suggestions, whichresulted in a significant improvement of the paper.

Appendix A

A.1. Derivation of Eq. (21)

In the first step of the interlaced CZT, P equi-spaced frequency samples are chosen. We proceedto find the total number of samples obtained in thesucceeding ðn� 1Þ steps. Let the number of pointsobtained in the second step be denoted by

T 01 ¼ K . (34)

The number of points obtained in the third step isgiven by

T 02 ¼ K þ ðK � 2gÞ. (35)

Proceeding similarly, the general term is given by

T 0m ¼ K þ fðK � 2gÞ þ 2ðK � 2gÞ

þ � � � þ 2m�2ðK � 2gÞg. ð36Þ

The total number of points obtained in these ðn� 1Þsteps is obtained by the sum:

S0n ¼Xn�1m¼1

T 0m

¼Xn�1m¼1

fK þ ðK � 2gÞð1þ 2þ 4þ � � � þ 2m�2Þg

¼ Kðn� 1Þ þ ðK � 2gÞXn�1m¼1

ð2m�1 � 1Þ

¼ 2gðn� 1Þ þXn�1m¼1

ðK � 2gÞ2m�1.

Therefore, we have:

S0n ¼ 2gðn� 1Þ þ ðK � 2gÞð2n�1 � 1Þ. (37)

Adding this to the P samples chosen in the first step,and substituting for K using (18), the expression forthe total number of samples obtained after n stepsreduces to:

Sn ¼ Pþ 2gðn� 1Þ þ ðP� 4g� 1Þð2n�1 � 1Þ. (38)

A.2. Derivation of Eq. (27)

To calculate the equivalent uniform grid over theentire arc or range of interest, we must fill all thevacant intervals on both sides of the innermost arcobtained through the interlaced CZT. This inner-most arc corresponds to the last (nth) step of theinterlaced CZT and has the highest densityfrequency grid. As evident in Fig. 3 the size of the


vacant intervals would gradually increase as wemove outward from the innermost arc. The extrasamples required on each side to achieve uniformspacing over the entire range are determined asfollows.

The 1st series of g intervals will require 1 extrasample each.

The 2nd series of g intervals will require 3 extrasamples each.

Proceeding in the same way, the mth series of g

intervals will require ð2m � 1Þ extra samples each.There will be ðn� 1Þ such series of g intervals.

P0 ¼Xn�1m¼1

gð2m � 1Þ

¼ 2gð2n�1 � 1Þ � gðn� 1Þ. ð39Þ

The total number of samples required in the regularCZT would be the sum of the Sn samples obtainedthrough the n steps of the interlaced CZT and theseextra samples. This yields:

S ¼ Sn þ gð2n � n� 1Þ. (40)

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the interlaced chirp z transform

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