Sampling rate conversion for linear canonical transform

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Contents lists available at ScienceDirect

Signal Processing

Signal Processing 88 (2008) 2825– 2832

0165-16

doi:10.1

� Cor

E-m

journal homepage: www.elsevier.com/locate/sigpro

Sampling rate conversion for linear canonical transform

Juan Zhao, Ran Tao �, Yue Wang

Department of Electronic Engineering, Beijing Institute of Technology, Haidian District, Beijing 100081, China

a r t i c l e i n f o

Article history:

Received 10 January 2008

Received in revised form

24 March 2008

Accepted 10 June 2008Available online 13 June 2008

Keywords:

Linear canonical transform

Interpolation

Decimation

Sampling rate conversion

84/$ - see front matter & 2008 Elsevier B.V. A

016/j.sigpro.2008.06.008

responding author.

ail address: [email protected] (R. Tao).

a b s t r a c t

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and

signal processing. This paper investigates the sampling rate conversion problem in the

LCT domain. Firstly, the discrete-time LCT is introduced and the formulas of

interpolation and decimation in the LCT domain are derived. Then, based on the

sampling theorem expansion in the LCT domain, the formulas of sampling rate

conversion by real factors for the LCT in time domain are proposed. The spectral analysis

of sampling rate conversion by real factors in the LCT domain is also illustrated. The

sampling rate conversion theories in the Fourier domain and the fractional Fourier

domain are shown to be special cases of the achieved results. The simulations verify the

effectiveness of the obtained results.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

The linear canonical transform (LCT) [1–5] is anintegral transform with three free parameters. It wasintroduced in the 1970s and many transforms such as theFourier transform (FT), the fractional Fourier transform(FrFT), the Fresnel transform and scaling operations arespecial cases of the LCT [4–6]. Recently, along withapplications of the FrFT in the signal processing commu-nity, the role of the LCT for signal processing has also beennoticed. It has found many applications in radar systemanalysis, filter design, phase retrieval, pattern recognition,and many other areas [3–8].

The sampling process is central in almost any domainand it explains how to sample continuous signals withoutaliasing. The sampling theorem expansions for the LCThave been derived in [9–11], which provides the linkbetween the continuous signals and the discrete signals,and can be used to reconstruct the original signal fromtheir samples satisfying the Nyquist rate of that domain.Moreover, with the development of digital signal proces-sing, computational amount and storage load have

ll rights reserved.

gradually increased. To reduce the computational load aswell as saving the storage space, different sampling ratesand the conversion between them are typically required inmany applications such as communications, image pro-cessing, digital audio and multimedia. The samplingrates conversion theory is the basis of multirate signalprocessing and it helps to convert the sampling rate of adiscrete-time signal to obtain another discrete-timeversion without significantly destroying the signal com-ponents of interest. Since the LCT has shown to be apowerful signal processing tool, it is necessary to studythe sampling rates conversion theory in the LCT domain.The conventional sampling rate conversion by rationalfactors and irrational factors in the FT domain has beenstudied in [12–18], respectively. Recently, the samplingrate conversion by rational factors associated with theFrFT has been analyzed by Tao [19] and Meng [20].However, the two papers focused on the spectral analysisof interpolation and decimation in the FrFT domain anddid not give closed form formulas of interpolation anddecimation in time domain. In addition, their methodscould not deal with the sampling rate conversion problemby irrational factors. To the best of our knowledge, thesampling rate conversion associated with the LCT hasnever been presented before. It is therefore worthwhile tostudy the sampling theory in the LCT domain, which can

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not only generalize the sampling theories of the FT[12–18] and the FrFT [19,20], but also provide a theoreticalsupport for reducing the sampling rate and advanceapplications such as filter banks theorem for the LCT.

To overcome the sampling rate conversion problem inthe LCT domain, we need to define discrete-time LCT(DTLCT) to analyze the spectrum of sampled signal in theLCT domain. In this paper, we firstly give the definition ofDTLCT by introducing digital frequency in that domain,which is a generalization of the classical DTFT. Then theformulas for interpolation and decimation associated withthe DTLCT are derived using the results in the FT domain.Moreover, based on the sampling theorem expansion inthe LCT domain, the formulas of sampling rate conversionby real factors for the LCT in time domain are deduced.The LCT domain analysis of sampling rate conversion byreal factors is also given. The sampling rate conversiontheories in the FT domain and the FrFT domain are specialcases of the achieved results. The simulations verify theeffectiveness of the obtained results.

The rest of this paper is organized as follows. In Section2, the definition of the LCT and the sampling theoremexpansions in the LCT domain are reviewed, then thedefinition of DTLCT is given. In Section 3, the formulas ofinterpolation and decimation in the LCT domain arederived. Moreover, the formulas of the sampling rateconversion by real factors for the LCT in time domain arededuced and the LCT domain analysis of sampling rateconversion by real factors is also given. In Section 4, someexamples are given to verify the achieved results. Finally,we make conclusions in Section 5.

2. Discrete-time LCT

2.1. The LCT

The LCT of a signal x(t) with parameter (a, b, c, d) isdefined as [1,3]

Xða;b;c;dÞðuÞ ¼ Lða;b;c;dÞðxðtÞÞðuÞ

¼

ffiffiffiffiffiffiffiffiffiffi1

j2pb

sejðd=2bÞu2 Rþ1

�1xðtÞejða=2bÞt2

e�jð1=bÞut dt; ba0;

ffiffiffidp

ejðcd=2Þu2xðduÞ; b ¼ 0;

8>>><>>>:

(1)

where ad�bc ¼ 1. In general, parameter a, b, c, d are realnumbers and we only consider the case of b6¼0, since theLCT is just a chirp multiplication operation if b ¼ 0. When(a, b, c, d) ¼ (cosa, sina, �sina, cosa), the LCT reduces tothe FrFT, i.e.,

Lðcos a;sin a;� sin a;cos aÞðxðtÞÞðuÞ ¼ffiffiffiffiffiffiffiffiffie�ja

pFaðxðtÞÞðuÞ,

where Fa(x(t))(u) is the FrFT of the signal x(t). The LCT hasthe reversible property

Lðd;�b;�c;aÞLða;b;c;dÞðxðtÞÞ ¼ xðtÞ.

For further details about the definition and properties ofthe LCT, [1–5] can be referred.

2.2. Sampling theorem for the LCT

The uniform sampled signal of the continuous signalx(t) with sampling period Dt is

xsðtÞ ¼ xðtÞXm¼þ1

m¼�1

dðt �mDtÞ.

The LCT of the sampled signal xs(t) is [10]

Xsða;b;c;dÞðuÞ ¼ Lða;b;c;dÞXm¼þ1

m¼�1

xðmDtÞdðt �mDtÞ

!ðuÞ

¼


j2pb

sejðd=2bÞu2

Xm¼þ1

m¼�1

xðmDtÞejða=2bÞm2Dt2

e�jð1=bÞumDt

¼1

Dtejðd=2bÞu2

Xm¼þ1

m¼�1

Xða;b;c;dÞ u�m2pb

Dt

� �e�jðd=2bÞðu�mð2pb=DtÞÞ2 ,

(2)

where X(a, b, c, d)(u) is the LCT of x(t). Eq. (2) shows that theLCT of the sampled signal xs(t) replicates X(a, b, c, d)(u) witha period of 2p|b|/Dt along with linear phase modulation.

If the continuous signal x(t) is band-limited in the LCTdomain, i.e., X(a, b, c, d)(u) ¼ 0 for |u|4uc, then from Eq. (2)we know that no overlapping occurs in Xs(a, b, c, d)(u) ifDtpp|b|/uc. Thus, the original band-limited continuoussignal x(t) can be reconstructed by the sampled signal,that is to say, the following sampling theorem expansionin the LCT domain holds:

xðtÞ ¼ e�jða=2bÞt2Xk¼þ1

k¼�1

xðkDtÞejða=2bÞk2Dt2

sin ct

Dt� k

� �, (3)

where Dtpp|b|/uc (Eq. (3) can be derived using the similarmethod in [11, Theorem 1]).

2.3. Discrete-time LCT

Consider the discrete-time signal x(n) sampled fromthe continuous signal x(t) with sampling period Dt, fromEq. (2) the DTLCT of x(n) can be defined as follows:

~Xða;b;c;dÞðuÞ ¼


j2pb

sejðd=2bÞu2

Xn¼þ1n¼�1

xðnÞejða=2bÞn2Dt2

e�jð1=bÞunDt . (4)

As in the classical FT domain, we define

o ¼ 1

jbjuDt (5)

as the digital frequency in the LCT domain, and note that itis related to |b|. Substituting Eq. (5) into Eq. (4), we have

~Xða;b;c;dÞðoÞ ¼


j2pb

sejðd=2bÞðbo=DtÞ2

Xn¼þ1n¼�1


e�jno sgnðbÞ,

where sgnðbÞ ¼1; b40

�1; bo0

(. The corresponding inverse

transform is

xðnÞ ¼

ffiffiffiffiffiffijb

2p

re�jða=2bÞn2Dt2

Z p

�p~Xða;b;c;dÞðoÞe�jðd=2bÞðbo=DtÞ2

ejno sgnðbÞ do.

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J. Zhao et al. / Signal Processing 88 (2008) 2825–2832 2827

Since omitting some constants does not affect the DTLCT,we can obtain the following DTLCT related to digitalfrequency in the LCT domain and its inverse transform

~Xða;b;c;dÞðoÞ ¼ ejðd=2bÞðbo=DtÞ2Xn¼þ1

n¼�1


e�jnosgnðbÞ, (6a)

xðnÞ ¼1

2pe�jða=2bÞn2Dt2

Z p

�p~Xða;b;c;dÞðoÞe�jðd=2bÞðbo=DtÞ2

ejno sgnðbÞ do. (6b)

When (a, b, c, d) ¼ (0, 1, �1, 0), Eqs. (6a and b) reduce tothe following DTFT and DTIFT:

~XðoÞ ¼Xn¼þ1

n¼�1

xðnÞe�jno, (7a)

xðnÞ ¼1

2p

Z p

�p~XðoÞejno do. (7b)

If x(n) ¼ 0 for |n|4N, let Do ¼ 2p/(2M+1), mA[�M, M],MXN. Substituting o ¼ m �Do into Eq. (6a), we have

~Xða;b;c;dÞðmÞ ¼ ejðd=2bÞm2ðb2p=Dtð2Mþ1ÞÞ2Xn¼N

n¼�N


e�jð2pnm=2Mþ1ÞsgnðbÞ. (8a)

According to the DIFT, its inverse transform is

xðnÞ ¼1

2M þ 1e�jða=2bÞn2Dt2

Xm¼M

m¼�M

~Xða;b;c;dÞðmÞe�jðd=2bÞm2ðb2p=Dtð2Mþ1ÞÞ2

ejð2pnm=2Mþ1ÞsgnðbÞ (8b)

Eqs. (8a and b) are just the DAFT algorithm proposed in[21].

3. Sampling rate conversion in the LCT domain

In this section, we assume that the original discrete-time signal x(n) is sampled with sampling period Dtx fromcontinuous signal x(t) band-limited in the LCT domain, i.e.,X(a, b, c, d)(u) ¼ 0 for |u|4uc, where Dtxpp|b|/uc.

3.1. The LCT domain analysis of interpolation and

decimation

Interpolation and decimation are two basic operationsin multirate digital signal processing. In this subsection,we analyze the spectrum in the LCT domain of sampledsignal after interpolation and decimation by integer,which are given by the following theorems.

Theorem 1. To increase the sampling rate by an integer

factor L, it can be achieved by passing the original signal x(n)through an L-fold expander and the output sequence y(n) is

yðnÞ ¼xðn=LÞ n ¼ Lk; k 2 z;

0 otherwise:

((9)

The sampling period of y(n) is Dty ¼ Dtx/L. Then the DTLCT

of y(n) is

~Y ða;b;c;dÞðoÞ ¼ ~Xða;b;c;dÞðLoÞ. (10)

Proof. According to Eqs. (6a) and (9), we have

~Y ða;b;c;dÞðoÞ ¼ ejðd=2bÞðbo=DtyÞ2 Xn¼þ1

n¼�1

yðnÞejða=2bÞn2Dt2y e�jno sigðbÞ

¼ ejðd=2bÞðbLo=DtxÞ2 Xk¼þ1

k¼�1

xðkÞejða=2bÞðDtxkÞ2

e�jLko sigðbÞ

¼ ~Xða;b;c;dÞðLoÞ.

The theorem is proved. &

Theorem 1 shows that the expander creates an imagingeffect, which is similar to the result in the FT domain [13].

Theorem 2. To reduce the sampling rate by an integer factor

M, it can be achieved by passing the original signal x(n)through an M-fold decimator and the output sequence y(n) is

yðnÞ ¼ xðMnÞ. (11)

The sampling period of y(n) is Dty ¼ MDtx. Then theDTLCT of y(n) is

~Y ða;b;c;dÞðoÞ ¼1

M

XM�1

k¼0

~Xða;b;c;dÞo� sgnðbÞ2pk

M

� �ej2pkbdðsgnðbÞo�pkÞ=ðMDtxÞ

2

(12)

Proof. Let x1ðnÞ ¼ xðnÞejða=2bÞn2Dt2x and y1ðnÞ ¼ x1ðMnÞ ¼

yðnÞejða=2bÞn2Dt2y . From Eqs. (6a) and (7a), it is obvious that

~Xða;b;c;dÞðoÞ ¼ ejðd=2bÞðbo=DtxÞ2 ~X1ðsgnðbÞoÞ,

~Y ða;b;c;dÞðoÞ ¼ ejðd=2bÞðbo=DtyÞ2 ~Y1ðsgnðbÞoÞ,

where ~X1ðoÞ, ~Y1ðoÞ are the DTFT of the signals x1(n), y1(n),respectively. Using the result in the FT domain [13], wehave

~Y1ðoÞ ¼1

M

XM�1

k¼0

~X1o� 2pk

M

� �.

Combining the above equations, the DTLCT of y(n) is

~Y ða;b;c;dÞðoÞ ¼ ejðd=2bÞðbo=DtyÞ2 ~Y1ðsgnðbÞoÞ

¼ ejðd=2bÞðbo=MDtxÞ2 1

M

XM�1

k¼0

~X1

sgnðbÞo� sgnðbÞ2pk

M

� �

¼ ejðd=2bÞðbo=MDtxÞ2 1

M

XM�1

k¼0

~Xða;b;c;dÞ

o� sgnðbÞ2pk

M

� �e�jðd=2bÞðbðo�sgnðbÞ2pkÞ=MDtxÞ

2

¼1

M

XM�1

k¼0

~Xða;b;c;dÞo� sgnðbÞ2pk

M

� �

ej2pkbdðsgnðbÞo�pkÞ=ðMDtxÞ2

.


If the original discrete-time signal x(n) is band-limitedin the LCT domain with cutoff frequency oc, i.e.,~Xða;b;c;dÞðoÞ ¼ 0 for |o|4oc, it is obvious that decimationwill create aliasing when M4p/oc according to Eq. (12).

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3.2. Sampling rate conversion by real factors

Now we consider sampling rate conversion problemby a real factor r40. From the results of Section 3.1,we know that interpolation and decimation will lead toimage and aliasing, respectively. To remove unwantedimages and avoid aliasing, a lowpass filter is needed.Moreover, to achieve the sampling rate conversion by arational factor, a common method is combination ofinterpolation, filtering, and decimation to remove un-wanted images and avoid aliasing. However, this generalmethod cannot deal with sampling rate conversionproblem by an irrational factor. In the following, we willuse the sampling theorem expansion in the LCT domain toderive the formulas of sampling rate conversion by a realfactor for the LCT, which is given by the followingtheorem.

Theorem 3. To change the sampling rate conversion by a

real factor r40, it can be achieved by the following formulas

xrðnÞ ¼ e�jða=2bÞn2ðDtx=rÞ2Xk¼þ1

k¼�1

xðkÞejða=2bÞk2Dt2x

sin cn

r� k

� �for r41, (13)

xrðnÞ ¼ re�jða=2bÞn2ðDtx=rÞ2Xk¼þ1

k¼�1


sin cðn� rkÞ for ro1. (14)

Proof. Since the original discrete-time signal x(n) issampled with sampling period Dtx from continuous signalx(t) band-limited in the LCT domain, i.e., X(a, b, c, d)(u) ¼ 0for |u|4uc, we have the following Eq. (15) according toEq. (3):

xðtÞ ¼ e�jða=2bÞt2Xk¼þ1

k¼�1


sin ct

Dtx� k

� �, (15)

where Dtxpp|b|/uc. When the sampling rate is changed bya real factor r, the sampling period of the sampled signalxr(n) after sampling rate conversion will be Dtxr ¼ Dtx=r.For sampling rate r41, the sampled signal xr(n) can beobtained by sampling directly from the original contin-uous signal x(t) with sampling period Dtxr . So fromEq. (15), we have

xrðnÞ ¼ xðnDtxr Þ ¼ e�jða=2bÞn2ðDtx=rÞ2Xk¼þ1

k¼�1

xðkÞ

ejða=2bÞk2Dt2x sin c

n

r� k

� �.

For sampling rate ro1, from Eq. (2) we know thatsampling directly from the original continuous signal x(t)with sampling period Dtxr may lead to aliasing ifuc4pjbj=Dtxr . Therefore, it need to prefilter x(t) with alowpass filter to avoid aliasing effect, where the transferfunction of the lowpass filter in the LCT domain with

parameter (a, b, c, d) can be written as

Hða;b;c;dÞðuÞ ¼1; jujp

pjbjDtxr

0; otherwise

8><>: . (16)

The signal filtered by the lowpass filter given by Eq. (16) is

xLPðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

�j2pb

se�jða=2bÞt2

Z þ1�1

Xða;b;c;dÞðuÞHða;b;c;dÞðuÞ

e�jðd=2bÞu2

ejð1=bÞut du

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

�j2pb

se�jða=2bÞt2

Z pjbj=Dtxr

�pjbj=Dtxr

Xða;b;c;dÞðuÞ

e�jðd=2bÞu2

ejð1=bÞut du.

Substituting Eqs. (1) and (15) into the above equation andnoting that sin c is even function give

xLPðtÞ ¼1

2pjbj e�jða=2bÞt2

Z þ1�1

xðt0Þejða=2bÞt02

Z pjbj=Dtxr

�pjbj=Dtxr

ejð1=bÞuðt�t0 Þ du

!dt0

¼1

Dtxre�jða=2bÞt2

Z þ1�1

xðt0Þejða=2bÞt02

sin csgnðbÞðt � t0Þ

Dtxr

� �dt0

¼1

Dtxre�jða=2bÞt2

Xk¼þ1k¼�1


sin ct

Dtx� k

� �sin c

t

Dtxr

� �

¼ re�jða=2bÞt2 Xk¼þ1k¼�1


sin ct � kDtx

Dtxr

� �. (17)

So for sampling rate ro1, using Eq. (17) we can obtain

xrðnÞ ¼ xLPðnDtxr Þ ¼ re�jða=2bÞn2ðDtx=rÞ2Xk¼þ1

k¼�1


sin cðn� rkÞ.


Theorem 3 can be seen as a generalization of the samplingrate conversion in the FT domain [18], and Eqs. (13) and (14)reduce to the result in [18] when (a, b, c, d) ¼ (0,1,�1,0).Simultaneously, we can obtain the following corollary.

Corollary 1. When (a, b, c, d) ¼ (cosa, sina, �sina, cosa),the formulas of the sampling rate conversion by a real factor

r40 for the FrFT in time domain are as follows:

xrðnÞ ¼ e�jðcot a=2Þn2ðDtx=rÞ2Xk¼þ1

k¼�1

xðkÞejðcot a=2Þk2Dt2x

sin cn

r� k

� �for r41, (18)

xrðnÞ ¼ re�jðcot a=2Þn2ðDtx=rÞ2Xk¼þ1

k¼�1

xðkÞejðcot a=2Þk2Dt2x

sin cðn� rkÞ for ro1. (19)

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Next, we will analyze the spectrum in the LCT domain ofthe sampling rate conversion by a real factor r, which isdescribed by the theorem given below.

Theorem 4. For the sampled signal xr(n) after the sampling

rate conversion by a real factor r, the DTLCT of the signal xr(n)

is

~Xr

ða;b;c;dÞðoÞ ¼Xk¼þ1

k¼�1

r ~Xða;b;c;dÞðrðo� 2pkÞÞ

ej2pkbdr2ðo�pk=Dt2x ÞHrðo� 2pkÞ (20)

where

HrðoÞ ¼1 jojpp=r

0 otherwise

(if r41, (21)

HrðoÞ ¼1 jojpp0 otherwise

if ro1. (22)

Proof. Firstly, we consider the case where r41.Let x1ðnÞ ¼ xðnÞejða=2bÞn2Dt2

x , x2ðnÞ ¼ xrðnÞejða=2bÞn2ðDtx=rÞ2 , fromEq. (13) we have

x2ðnÞ ¼Xk¼þ1

k¼�1

x1ðkÞ sin cn

r� k

� �.

Since sin c½ðn=rÞ � k� ¼ r=2pR p=r�p=r ejðn�rkÞo0 do0, the DTFT of

x2(n) is given below:

~X2ðoÞ ¼Xn¼þ1

n¼�1

Xk¼þ1k¼�1

x1ðkÞ sin cn

r� k

� �" #e�jon

¼Xn¼þ1

n¼�1

r

2p

Z p=r

�p=r

~X1ðro0Þejno0do0" #

e�jon

¼ rXk¼þ1

k¼�1

Z p=r

�p=r

~X1ðro0Þdðo0 �oþ 2pkÞdo0:

Using Eq. (21) the above equation becomes

~X2ðoÞ ¼ rXk¼þ1

k¼�1

Z þ1�1

~X1ðro0ÞHrðo0Þdðo0 �oþ 2pkÞdo0

¼ rXk¼þ1

k¼�1

~X1ðrðo� 2pkÞÞHrðo� 2pkÞ.

At the same time, from Eq. (6a), we can obtain

~Xða;b;c;dÞðoÞ ¼ ejðd=2bÞðbo=DtxÞ2 ~X1ðsgnðbÞoÞ,

~Xr

ða;b;c;dÞðoÞ ¼ ejðd=2bÞðbro=DtxÞ2 ~X2ðsgnðbÞoÞ.

Therefore, combining the above equations provides

~Xr

ða;b;c;dÞðsgnðbÞoÞ ¼ rejðd=2bÞðbro=DtxÞ2

Xk¼þ1k¼�1

~Xða;b;c;dÞðsgnðbÞrðo� 2pkÞÞ

e�jðd=2bÞ½brðo�2pkÞ=Dtx�2

Hrðo� 2pkÞ.

Substituting o ¼ sgn(b)o0 into the above equation andsimplifying it, we can obtain Eq. (20).

Secondly, we consider the case where ro1. Noting

sin cðn� rkÞ ¼ 1=2pR p�p ejðn�rkÞo0 do0, the DTLCT of xr(n)

can be given by Eqs. (20) and (22) using the similar

method. The theorem is proved. &

Theorem 4 shows that the formulas for sampling rateconversion by a real factor for the LCT given by Eqs. (13)and (14) can remove unwanted images and avoid aliasingsimultaneously. Especially, when k ¼ 0 in Eq. (20) (i.e.,oA[�p, p]),

~Xr

ða;b;c;dÞðoÞ ¼ r ~Xða;b;c;dÞðroÞHrðoÞ. (23)

4. Simulations

Some examples are given in this section to verify theachieved results. There are some literatures discussingnumerical algorithms for the LCT such as [21–24]. Here,we use the DAFT algorithm proposed in [21]. In thefollowing, we assume that the parameter of the LCT (a, b,c, d) ¼ (1, 5, 1/5, 2). The original discrete signal x(n) issampled from continuous signal x(t) band-limited in theLCT domain with cutoff frequency uc ¼ 9.3779. Thesampling period Dtx ¼ 1 and the sampling interval is[�100 s, 100 s].

At first, we discuss interpolation by integer L in the LCTdomain. Fig. 1(a) demonstrates the DTLCT of the originalsignal x(n), which is band-limited with cutoff frequencyoc ¼ 1.8756. Fig. 1(b) shows the DTLCT of the signal afterzero-valued interpolation with L ¼ 3 which is a three-foldcompressed version of the DTLCT of the original signal.Fig. 1(c) shows the DTLCT of the signal after interpolationgiven by Eq. (13) when r ¼ 3 and the unwanted images areremoved.

We then discuss decimation by M in the LCT domain.Fig. 2(a) shows the DTLCT of the original signal. Fig. 2(b)shows the DTLCT of the signal after decimator with M ¼ 2,it is obvious that aliasing takes place since M4p/oc.Fig. 2(c) demonstrates the DTLCT of the signal afterdecimation given by Eq. (14) when r ¼ 1/2 and aliasingis avoided.

Finally, we discuss the sampling rate conversion in theLCT domain by rational and irrational factors. Fig. 3(a–c)demonstrate the DTLCTs of the original signal, the signalafter sampling rate conversion by r ¼ 3/2 given byEq. (13), and the signal after sampling rate conversionby r ¼ 2/3 given by Eq. (14), respectively. Fig. 4(a–c) showthe DTLCTs of the original signal, the signal after samplingrate conversion by r ¼ O5 given by Eq. (13), and the signalafter sampling rate conversion by r ¼ 1/p given byEq. (14), respectively. Figs. 3 and 4 illustrate that theformulas for sampling rate conversion by a real factor forthe LCT proposed in Theorem 3 can remove unwantedimages and avoid aliasing simultaneously.

5. Conclusions

In this paper, we discuss the sampling rate conversionproblem for the LCT. At first, we define DTLCT byintroducing digital frequency in that domain; then theformulas for interpolation and decimation in the LCTdomain are derived in a simple way based on the results inthe Fourier domain. Furthermore, using the samplingtheorem expansion in the LCT domain, the formulas forsampling rate conversion by real factors for the LCT in

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1

0.5

0

1

0.5

0

1

0.5

0

-4 -3 -2 -1 0 1 2 3 4ω

|X(a,b,c,d

) (�

)|~

|X(a,b,c,d

) (�

)|~

|Y(a,b,c,d

) (�

)|~

-4 -2 0 2 4 6 8 10ω

-4 -2 0 2 4 6 8 10ω

1 2

Fig. 2. The DTLCT of the discrete signal decimated with M ¼ 2.

1

0.5

0-4 -3 -2 -1 0 1 2 3 4

ω

1

0.5

0-4 -3 -2 -1 0 1 2 3 4

ω

1

0.5

0-4 -3 -2 -1 0 1 2 3 4

ω

|X(a,b,c,d

) (�

)||Y

(a,b,c,d

) (�

)||X

-3(a,b,c,d

) (�

)|~

~~

Fig. 1. The DTLCT of the discrete signal after interpolation with L ¼ 3.


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1

0.5

0

1

0.5

0

-4 -3 -2 -1 0 1 2 3 4ω

ω

|X(a,b,c,d

) (�

)|~

|X(a,b,c,d

) (�

)|~

|X(a,b,c,d

) (�

)|~

-4 -2 0 2 4 6 8 10

ω-4 -2 0 2 4 6 8 10

2

1

0

3 232

Fig. 3. The sampling rate conversion by rational factor 3/2 and 2/3.

1

0.5

0-4 -3 -2 -1 0 1 2 3 4

ω

ω

|X(a,b,c,d

) (�

)|~

|X(a,b,c,d

) (�

)|~

|X(a,b,c,d

) (�

)|~

4

2

0

0.4

0.2

0

-4 -2 0 2 4 6 8 10

ω-4 -2 0 2 4 6 8 10

5

1 �

Fig. 4. The sampling rate conversion by irrational factor O5 and 1/p.


ARTICLE IN PRESS


time domain are deduced, which can remove unwantedimages and avoid aliasing simultaneously in the LCTdomain. The simulations verify the effectiveness of theobtained results. The sampling rate conversion for the LCTproposed here generalizes the sampling rate conversiontheories in the Fourier domain and the fractional Fourierdomain. Future works will investigate multirate systemsand filter banks in the LCT domain and their applicationsin digital signal processing.

Acknowledgments

The authors would like to thank the anonymousreviewers for their valuable comments and suggestionthat improved the clarity and quality of this manuscript.We would also like to thank Dr. Bingzhao Li of BeijingInstitute of Technology for many fruitful discussions andthe proof reading of the paper.

Sponsored in part by the National Nature ScienceFoundation of China (nos. 60232010 and 60572094) andthe National Science Foundation of China for Distin-guished Young Scholars (no. 60625104).

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Sampling rate conversion for linear canonical transform