Multi-channel filter banks associated with linear canonical transform

Signal Processing 93 (2013) 695–705

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Multi-channel filter banks associated with linear canonical transform$ Juan Zhao, Ran Tao n, Yue Wang Fundamental Science on Multiple Information Systems Laboratory, Department of Electronic Engineering, Beijing Institute of Technology, No. 5, South ZhongGuanCun Street, Haidian district, Beijing 100081, China

a r t i c l e i n f o

abstract

Article history: Received 11 January 2012 Received in revised form 31 July 2012 Accepted 14 September 2012 Available online 26 September 2012

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. This paper investigates multi-channel filter banks associated with the LCT. First, the perfect reconstruction (PR) conditions are analyzed and design method of PR filter banks for the LCT is proposed, which demonstrates that the LCT based filter banks can inherit conventional design methods of filter banks in the Fourier domain. Then polyphase decompositions in the LCT domain are defined and polyphase realization of the LCT based filter banks is derived in terms of polyphase matrices. Furthermore, multi-channel cyclic filter banks associated with the LCT are proposed by defining circular convolution in the LCT domain. The PR design method and polyphase representation of cyclic filter banks for the LCT are derived similarly. Finally, simulations validate the proposed design methods of the LCT based filter banks and also demonstrate potential application of the LCT based cyclic filter banks in image subband decomposition. & 2012 Elsevier B.V. All rights reserved.

Keywords: Linear canonical transform Filter banks Perfect reconstruction Polyphase decomposition

The linear canonical transform (LCT) has recently attracted much attention in the area of signal processing and optics [1–3], which is an integral transform with four parameters a,b,c,d and many transforms such as the Fourier transform (FT), the fractional Fourier transform (FrFT) and the Fresnel transform (FRT) are its special cases. The LCT of a signal xðtÞ with parameter M ¼ ða,b,c,dÞ is defined as [2] X M ðuÞ ¼ Lða,b,c,dÞ ðxðtÞÞðuÞ 8 qffiffiffiffiffiffiffi d 2 R a t 2 j1ut þ1 j2b 1 j2bu < e e b dt, 1 xðtÞe j2pb ¼ pffiffiffi cd 2 : dej 2 u xðduÞ,

ba0 ð1Þ b¼0

where a,b,c,d 2 R and adbc ¼ 1. When ða,b,c,dÞ ¼ ðcos a, sin a, sin a, cos aÞ, the LCT reduces to the FrFT, i.e., pffiffiffiffiffiffiffiffiffi Lðcos a, sin a, sin a, cos aÞ ðxðtÞÞðuÞ ¼ eja X a ðuÞ ð2Þ

$ This work was partially supported by the National Natural Science Foundation of China (Grant 60890070, 60901058 and 61001199), the National Key Basic Research Program Founded by MOST (Grant 2009CB724003 and 2010CB731902) and Program for Changjiang Scholars and Innovative Research Team in University (IRT1005). n Corresponding author. E-mail address: [email protected] (R. Tao).

0165-1684/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2012.09.008

where X a ðuÞ is the FrFT of the signal xðtÞ with order a. For further details about the definition and properties of the LCT, [1–3] can be referred. Since the LCT has three free parameters, it is more flexible and has been found many applications in radar system analysis, filter design, phase retrieval, pattern recognition, encryption and many other areas [1–6]. As a generalization of the FT and FrFT, the basic theories of the LCT have been developed including uncertainty principles [7–11], convolution theorem [12,13], Hilbert transform [14,15], sampling theory [16–21], discretization [22–24] and so on, which can enrich the theoretical framework of the LCT and advance the application of the LCT. The theory of filter banks can be applied in almost any domain, which can separate a signal into a set of subband signals or combine many such subband signals into a single composite signal, and realize multi-resolution analysis of signals in the corresponding domain. Since the LCT has shown to be a powerful signal processing tool, it is necessary to study the filter banks theory associated with the LCT. Recently some researchers have studied the filter banks in the FrFT and LCT domains in order to realize subband decomposition of those signals that may be band limited in a certain FrFT or LCT

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J. Zhao et al. / Signal Processing 93 (2013) 695–705

domain. Meng et al. [25] has given the analysis of cyclic filter banks in the FrFT domain. Shinde has proposed twochannel paraunitary filter banks based on the LCT [26]. However multi-channel filter banks associated with the LCT has never been presented before. In this paper we investigate L-channel filter banks associated with the LCT including the design method of perfect reconstruction (PR) filter banks and polyphase decomposition in the LCT domain. Furthermore, cyclic filter banks associated with the LCT are also proposed by defining circular convolution in the LCT domain, which are appropriate for dealing with finite length signals such as images. The simulations verify the effectiveness of the proposed design methods of filter banks in the LCT domain and also discuss potential application of the LCT based cyclic filter banks in image subband decomposition. The rest of this paper is organized as follows. In Section 1 we consider L-channel filter banks associated with the LCT. The conditions for PR reconstruction are analyzed and the design method of filter banks for the LCT is proposed. Moreover, polyphase decompositions in the LCT domain are given. In Section 2 L-channel circular filter banks associated with the LCT are proposed. The PR design method and polyphase representation of cyclic filter banks for the LCT are also presented. In Section 3 some simulated examples are given to verify the achieved results. Finally we make conclusions in Section 4. 1. L-channel filter banks associated with the LCT

Let xðnÞ be a discrete time signal sampled from the continuous signal xc ðtÞ with sampling interval Dt, i.e., xðnÞ ¼ xc ðnDtÞ. The discrete time LCT (DTLCT) of xðnÞ with parameter M ¼ ða,b,c,dÞ is defined as [20] 2 Xn ¼ þ 1 2 2 X~ M ðoÞ ¼ ejðd=2bÞðbo=DtÞ xðnÞejða=2bÞn Dt ejnosgnðbÞ n ¼ 1 ð3Þ where o is the digital frequency in the LCT domain and sgnðbÞ is sign of b. Without loss of generality, we assume that b 4 0 in this paper. Expander and decimator are two basic operations in multirate digital signal processing. Let yðnÞ ¼ xðnÞmL denotes expander by an integer L, the DTLCT of yðnÞ with parameter M ¼ ða,b,c,dÞ is [20] ð4Þ

Let yðnÞ ¼ xðnÞkN denotes decimator by an integer N, the DTLCT of yðnÞ with parameter M ¼ ða,b,c,dÞ is [20]   1 XN1 ~ o2pk j2pkbdðopkÞ=ðNDtÞ2 e ð5Þ X Y~ M ðoÞ ¼ k¼0 M N N The convolution is a fundamental operator in filter banks. The convolution of xðnÞ and hðnÞ for the LCT with parameter M ¼ ða,b,c,dÞ is defined as [12,26] 2

2

2

2

yðnÞ ¼ xðnÞYhðnÞ ¼ ejða=2bÞn Dt ½xðnÞejða=2bÞn Dt Xk ¼ þ 1 2 2 nhðnÞejða=2bÞn Dt  ¼ hðkÞxðnkÞ k ¼ 1 ejða=bÞDt

2

kðnkÞ

~ M ðoÞejðdb=2Dt2 Þo2 Y~ M ðoÞ ¼ X~ M ðoÞH

ð7Þ

The time delay operator is relevant to polyphase decomposition and PR of filter banks. The time delay operator for the LCT with parameter M ¼ ða,b,c,dÞ on a signal xðnÞ is defined as [26] Dk ½xðnÞ ¼ xðnkÞejða=2bÞDt

2

2

ð2nk þ k Þ

ð8Þ

Note that the delay operator is relevant with the sampling interval Dt. The delay operator D½U has the following properties [26]: Dl ½Dk ½xðnÞ ¼ Dl þ k ½xðnÞ

If yðnÞ ¼ Dk ½xðnÞ,

If yðnÞ ¼ xðnÞYhðnÞ,

then

then

ð9Þ

Y~ M ðoÞ ¼ ejko X~ M ðoÞ

ð10Þ

Dl ½xðnÞYDk ½hðnÞ ¼ Dl þ k ½yðnÞ ð11Þ

1.2. Design of L-channel PR filter banks associated with the LCT At first we show that the time delay operator given by Eq. (8) can be written as the convolution of xðnÞ and dðnkÞ in the LCT domain.

1.1. Preliminaries

Y~ M ðoÞ ¼ X~ M ðLoÞ

where n denotes the conventional convolution. The DTLCT of convolution yðnÞ is

ð6Þ

Lemma 1. The time delay operator D½U on a signal xðnÞ can be written as 2

Dk ½xðnÞ ¼ ðxðnÞYdðnkÞÞejða=2bÞk

Dt 2

ð12Þ

Proof. Using Eq. (6), we have xðnÞYdðnkÞ ¼ xðnkÞ 2 ejða=bÞDt kðnkÞ . Thus, it is obvious that Eq. (12) holds.& Lemma 1 means that the time delay operator D½U is consistent with that defined by the convolution with d function in [25]. We take the time delay operator D½U in this paper because it has the property given by Eq. (9). The structure of L-channel filter banks associated with the LCT is shown in Fig. 1 with convolution as defined in ~ l,M ðoÞ and G~ l,M ðoÞ are the DTLCT of hl,M ðnÞ Eq. (7), where H and g l,M ðnÞ, respectively, l ¼ 0,1,:::,L1. The subfilters fhl,M ðnÞ, l ¼ 0, 1,. . .,L1g are known as the analysis filters which split the original signal xðnÞ into a number of subband signals in the LCT domain, and the subfilters fg l,M ðnÞ, l ¼ 0,1,. . .,L1g are known as the synthesis filter banks which combine the subband signals to reconstruct the original signal. The reconstructed signal yðnÞ should be as close to the original input signal as possible in a certain well defined sense, i.e., Y~ M ðoÞ ¼ cejn0 o X~ M ðoÞ or yðnÞ ¼ cDn0 ½xðnÞ, where c,n0 are constants and the corresponding LCT based filter banks are PR. First we analyze the input–output relation of the filter banks in the LCT domain as shown in Fig. 1. Using Eqs. (4), (5) and (7), we obtain the DTLCT of the output of the filter

J. Zhao et al. / Signal Processing 93 (2013) 695–705

Proof. Using Eq. (6), the output of L-channel filter banks associated with the LCT as shown in Fig. 1 is equivalent to the output of the structure shown in Fig. 2, where the convolution is the conventional convolution associated with the FT and  denotes multiplication opera2 2 2 2 tor, hi ðnÞ ¼ hi,M ðnÞejða=2bÞn Dt , g i ðnÞ ¼ g i,M ðnÞejða=2bÞn Dt , i ¼ 0,1,. . .,L1. It should be pointed that the outputs fx~ l ðnÞ, l ¼ 0, 1,. . .,L1g of analysis filters of Fig. 2 are not equal to the outputs fxl ðnÞ, l ¼ 0, 1,. . .,L1g of analysis filters in Fig. 1 and the relationships between them are 2 2 xi ðnÞ ¼ x~ i ðnÞejða=2bÞn Dt , i ¼ 0,1,. . .,L1.

Fig. 1. The structure of filter banks associated with the LCT.

banks as     1 XL1 ~ 2pl XL1 ~ 2pl Y~ M ðoÞ ¼ X M o H i,M o l ¼ 0 i ¼ 0 L L L

According to the assumption that fhl ðnÞ, g l ðnÞ, l ¼ 0, 1, . . .,L1g are PR filter banks in the FT domain, it is obvious that

2 2 G~ i,M ðoÞejðdb=Dt Þðoð2pl=LÞÞ

1~ 2 2 X M ðoÞB0 ðoÞejðdb=Dt Þo L   2 1 XL1 ~ 2pl 2 Bl ðoÞejðdb=Dt Þðoð2pl=LÞÞ X þ o  M l¼1 L L ð13Þ  PL1 ~  where Bl ðoÞ ¼ i ¼ 0 H i,M o 2Lpl G~ i,M ðoÞ, 0 r l r L1. The second term in the above equation is precisely due to the aliasing caused by sampling rate alteration. If the analysis and synthesis filters are chosen to satisfy the following condition ¼

Bl ðoÞ ¼ 0,

1 r l rL1,

ð14Þ

then Eq. (13) can be expressed as 1 2 2 Y~ M ðoÞ ¼ X~ M ðoÞB0 ðoÞejðdb=Dt Þo L where B0 ðoÞ is called the distortion transfer function and it is possible to lead to PR by choosing the analysis and synthesis filters satisfying XL1 ~ i,M ðoÞG~ i,M ðoÞ ¼ cejn0 o ejðdb=Dt2 Þo2 B 0 ð oÞ ¼ ð15Þ H i¼0 Thus, the PR conditions of Eqs. (14) and (15) can be written in the following matrix form 2 6 6 6 6 6 4

~ 0,M ðoÞ H   ~ H 0,M o 2Lp ^   ~ 0,M o 2pðL1Þ H L

~ 1,M ðoÞ H   ~ H 1,M o 2Lp ^   ~ 1,M o 2pðL1Þ H L

3 2 ~ 3 2 2 2 G 0,M ðoÞ cejn0 o ejðdb=Dt Þo 7 6 ~ 7 6 7 6 G 1,M ðoÞ 7 6 0 7 6 7¼6 6 7 6 7 ^ ^ 5 4 5 4 G~ L1,M ðoÞ 0

  & 

697

3 ~ L1,M ðoÞ H   7 ~ L1,M o 2p 7 H L 7 7 ^  7 5 2 p ðL1Þ ~ L1,M o H L

ð16Þ

Eq. (16) just gives the PR conditions that the LCT based filter banks should satisfy in the LCT domain, which is too complex to design the filter banks in practice. In the following we discuss the design method of a class of such filter banks. Theorem 1. Consider L-channel filter banks associated with the LCT as shown in Fig. 1. The sampling interval of input signal xðnÞ is Dt. Let fhl ðnÞ, g l ðnÞ, l ¼ 0, 1,. . .,L1g be the PR filter banks in the FT domain, then fhl,M ðnÞ ¼ hl ðnÞ 2 2 2 2 ejða=2bÞn Dt , g l,M ðnÞ ¼ g l ðnÞejða=2bÞn Dt , l ¼ 0, 1,. . .,L1g are the PR filter banks associate with the LCT.

2

jða=2bÞðnn0 Þ ~ ~ yðnÞ ¼ cxðnn 0 Þ ¼ cxðnn0 Þe

Dt 2

Thus, we have jða=2bÞn ~ yðnÞ ¼ yðnÞe

2

Dt 2

2

¼ cxðnn0 Þejða=2bÞðnn0 Þ

¼ cxðnn0 Þejða=2bÞDt

2

ð2nn0 þ n0 2 Þ

Dt2 jða=2bÞn2 Dt 2

e

¼ cDn0 ½xðnÞ

The theorem is proved.& Theorem 1 shows that the LCT based filter banks can be designed by using conventional filter design in the Fourier domain. The result of two-channel paraunitary filter banks based on the LCT [26] can be regard as a special case of Theorem 1. Now we consider the complexity of the LCT based filter banks shown in Fig. 1. Assume the length of input signal xðnÞ is N and according to the relationship between Fig. 1 and Fig. 2, it is obvious that the complexity of the LCT based filter banks in time domain is comparable to that of conventional filter banks and it just increases ðLþ 1ÞN multiplications to implement the analysis filter banks when compared with the complexity of conventional analysis filter banks. Thus, the complexity of the LCT based filter banks in time domain totally increases 2ðLþ 1ÞN multiplications when compared with that of conventional filter banks. 1.3. Polyphase decomposition of filter banks in the LCT domain The polyphase decomposition of a sequence xðnÞ is important in the framework of multirate signal processing theory, which can realize filter banks more efficiently. We define the polyphase component of the signal xðnÞ for the LCT as xl ðnÞ ¼ xðLn þlÞejða=2bÞDt

2

lð2Ln þ lÞ

,

l ¼ 0,1,. . .,L1

Fig. 2. The equivalent structure of the LCT based filter banks.

ð17Þ

698

J. Zhao et al. / Signal Processing 93 (2013) 695–705

which can also be written as Dl ½xðnÞkL by using Eq. (8). The sampling interval of xl ðnÞ is LDt, the DTLCT of the polyphase component xl ðnÞ is 2 Xn ¼ þ 1 2 l 2 E~ M ðoÞ ¼ ejðd=2bÞðbo=LDtÞ xl ðnÞejða=2bÞn ðLDtÞ ejno n ¼ 1 2 Xn ¼ þ 1 2 ¼ ejðd=2bÞðbo=LDtÞ xðLn þlÞejða=2bÞDt lð2Ln þ lÞ n ¼ 1 ejða=2bÞn

2

ðLDtÞ2 jno

jðd=2bÞðbo=LDtÞ2

¼e

e Xn ¼

jða=2bÞDt 2 ðLn þ lÞ2

e

þ1

n ¼ 1 jno

xðLn þlÞ

e

ð18Þ

Substituting n ¼ Lk þl, l ¼ 0,1,. . .,L1 into Eq. (3), we have 2 Xn ¼ þ 1 2 2 xðnÞejða=2bÞn Dt ejno X~ M ðoÞ ¼ ejðd=2bÞðbo=DtÞ n ¼ 1 XL1 2 ¼ ejlo ejðdb=2Þðo=DtÞ l¼0 Xk ¼ þ 1 2 2  xðLkþ lÞejða=2bÞDt ðLk þ lÞ ejLko k ¼ 1

Fig. 3. Realization of filter banks based on polyphase decomposition.

Combining Eq. (18) and the above equation, Type I polyphase decomposition of the signal xðnÞ in the LCT domain is XL1 l X~ M ðoÞ ¼ ejlo E~ M ðLoÞ ð19Þ l¼0 l

where E~ M ðoÞ can be called the polyphase component of X~ M ðoÞ. It is obvious that Eq. (19) reduces to the result in [26] when L ¼ 2. Similarly, we can obtain the following Type II and Type III polyphase decomposition, respectively, XL1 l Type II : X~ M ðoÞ ¼ ejðL1lÞo R~ M ðLoÞ ð20Þ l¼0 l L1l ðoÞ, 0 r l rL1, where R~ M ðoÞ ¼ E~ M XL1 l ejlo Q~ M ðLoÞ Type III : X~ M ðoÞ ¼ l¼0

ð21Þ

l where Q~ M ðoÞ is the DTLCT of the polyphase component 2 xl ðnÞ ¼ xðLnlÞejða=2bÞDt lð2LnlÞ , 0 rl r L1. Now we consider the polyphase representations of filter banks associated with the LCT. Using Type I and II polyphase decomposition, respectively, the analysis and the synthesis filters can be represented as 0 ~ 1 0 1 H 0,M ðoÞ 1 B ~ C j o B C B H 1,M ðoÞ C e C C ¼ EM ðLoÞB HM ðoÞ ¼ B ð22Þ B C B C @ A ^ ^ @ A ~ L1,M ðoÞ ejðL1Þo H

and

h i GM ðoÞ ¼ G~ 0,M ðoÞ G~ 1,M ðoÞ    G~ L1,M ðoÞ

¼ ejðL1Þo ejðL2Þo    1 RM ðLoÞ

ð23Þ

where EM ðoÞ and RM ðoÞ are the analysis and synthesis polyphase matrices, which have the polyphase compok i nents E~ i,M ðoÞ and R~ k,M ðoÞ in the i-th row k-th column k k elements, respectively. And E~ i,M ðoÞ,R~ i,M ðoÞ denote the ~ i,M ðoÞ and G~ i,M ðoÞ, k-th polyphase component of H respectively. Furthermore, by invoking the cascade equivalences of interpolation and decimation in the LCT domain [Fig. 4 and Fig. 6, 21], we arrive at an equivalent realization of Lchannel filter banks in terms of polyphase matrix shown in Fig. 3. Note that Fig. 3 gives the polyphase realization of

Fig. 4. The structure of cyclic filter banks associated with the LCT.

the LCT based filter banks in the LCT domain. Using Eq. (6), Eq. (8) and the similar analysis in Theorem 1, we can obtain the equivalent structure of Fig. 3 and find that the complexity of the LCT based filter banks based on polyphase decomposition is comparable to that of conventional filter banks in the polyphase decomposition form. Assume that the length of input signal xðnÞ is N, it is similar to the results in Section 1.2 that the complexity of the LCT based filter banks based on polyphase decomposition totally increases 2ðLþ 1ÞN multiplications when compared with that of conventional filter banks in polyphase decomposition form.

2. L-channel cyclic filter banks associated with the LCT In practice, we often deal with finite discrete signal using circular convolution and DFT. The cyclic multirate systems were proposed by Vaidyanathan et al. [27–30], which are generalization of the classical multirate signal processing and widely used in image subband coding [31,32]. In this section, we investigate the cyclic filter banks based on the LCT, which will advance the applications of filter banks in the LCT domain on the finite signal field, such as digital image processing.

2.1. The design method of L-channel PR cyclic filter banks associated with the LCT Assume xðnÞ be a discrete time signal with sampling interval Dt and length N, the discrete LCT (DLCT) and the inverse DLCT algorithm are given by [20,22], 2 Xn ¼ N1 2 2 2 X~ M ðmÞ ¼ ejðd=2bÞm ðDuÞ xðnÞejða=2bÞn Dt ejð2pnm=NÞ n¼0 ð24aÞ

J. Zhao et al. / Signal Processing 93 (2013) 695–705

xðnÞ ¼

2 1 jða=2bÞn2 Dt2 Xm ¼ N1 ~ 2 e X M ðmÞejðd=2bÞm ðDuÞ ejð2pnm=NÞ m¼0 N

699

Lemma 2. The cyclic time delay operator DN ½U has the following properties:

ð24bÞ where Du ¼ 2pb=N Dt, m ¼ 0,1,. . .,N1. The cyclic convolution is very important in cyclic filter banks. Assume xðnÞ and hðnÞ be two discrete time signals with sampling interval Dt and length N, we define the cyclic convolution of xðnÞ and hðnÞ for the LCT with parameter M ¼ ða,b,c,dÞ as follows N

yðnÞ ¼ xðnÞYhðnÞ ¼ ejða=2bÞn ½xðnÞejða=2bÞn

2

2

DlN ½DkN ½xðnÞ ¼ DlNþ k ½xðnÞ If yðnÞ ¼ DkN ½xðnÞ,

ð28Þ Y~ M ðmÞ ¼ ejð2p=NÞkm X~ M ðmÞ

then

N

N

If yðnÞ ¼ xðnÞYhðnÞ,then DlN ½xðnÞYDkN ½hðnÞ ¼ DlNþ k ½yðnÞ ð30Þ

Dt 2

Dt 2 N

2 2 nhðnÞejða=2bÞn Dt 

ð25Þ

N

where n denotes the conventional N-length cyclic convolution. The DLCT of the cyclic convolution yðnÞ is

Proof. 1) DlN ½DkN ½xðnÞ ¼ ejða=2bÞn ¼ ejða=2bÞn

~ M ðmÞejðd=2bÞm2 Du2 Y~ M ðmÞ ¼ X~ M ðmÞH

2

Dt

2

2

Dt2

½xðnÞejða=2bÞn

The above theorem extends the cyclic filter banks in the LCT domain. The result in the FrFT domain [25] can be regard as a special case of Theorem 2. Consider the complexity of the cyclic filter banks associated with the LCT, it is similar to the analysis of Section 1.2 that the complexity of the LCT based cyclic filter banks in time domain just increases 2ðLþ 1ÞN multiplications when compared with that of conventional cyclic filter banks.

2.2. Polyphase representations of cyclic filter banks associated with the LCT

2

Dt 2 N

ndðnkÞ

½xðnÞ

2

2

2

Since yðnÞejða=2bÞn Dt ¼ ½xðnÞejða=2bÞn Eq. (29) can be obtained. 3)

N

DlN ½xðnÞYDkN ½hðnÞ ¼ ejða=2bÞn N

N

2

2

Dt 2

N Dt 2 n dðnkÞ,

½xðnÞejða=2bÞn

2

ð27Þ

then

Dt 2

2N

ndðnkÞnhðnÞejða=2bÞn Dt ndðnlÞ

¼ ejða=2bÞn

2

Dt 2

2

½xðnÞejða=2bÞn

Dt 2 N

2 2 nhðnÞejða=2bÞn Dt

DlNþ k ½yðnÞ

The lemma is proved. &

Consider the signal xðnÞ with sampling interval Dt and length N ¼ LK, we define the polyphase component of xðnÞ for the LCT as ( 2 xðLi þ lÞejða=2bÞDt lð2Li þ lÞ , n ¼ Li, i ¼ 0,1,. . .,K1 l x ðnÞ ¼ 0, otherwise l ¼ 0,1,. . .,L1

ð31Þ

The sampling interval of xl ðnÞ is Dt, the DLCT of the polyphase component xl ðnÞ is 2 Xn ¼ N1 2 l 2 E~ M ðmÞ ¼ ejðd=2bÞðmDuÞ xl ðnÞejða=2bÞn ðDtÞ ejð2p=NÞnm n¼0 2 Xi ¼ K1 2 2 ¼ ejðd=2bÞðmDuÞ xðLi þlÞejða=2bÞDt ðLi þ lÞ i¼0 ejð2p=KÞim

The time delay operator is relevant to polyphase representation of filter banks. Let the sampling interval of a signal xðnÞ with length N be Dt. We define the cyclic time delay operator for the LCT on the signal xðnÞ with parameter M ¼ ða,b,c,dÞ as ½xðnÞejða=2bÞn

lþk

ndðnklÞ ¼ DN

where X~ 1 ðmÞ is the DFT of the signals x1 ðnÞ. Using the property of DFT, we have 2 2 2 2 Y~ M ðmÞejðd=2bÞm Du ¼ ejð2p=NÞkm X~ M ðmÞejðd=2bÞm Du

ndðnklÞ ¼

Dt 2

Dt

N

ndðnkÞndðnlÞ

2N

2) Let x1 ðnÞ ¼ xðnÞejða=2bÞn Dt , it is obvious that 2 X~ M ðmÞ ¼ ejðd=2bÞðmDuÞ X~ 1 ðmÞ,

N

2

2

2

2

Theorem 2. Consider L-channel cyclic filter banks associated with the LCT as shown in Fig. 4. The sampling interval of input signal xðnÞ is Dt. Let fhl ðnÞ, g l ðnÞ, l ¼ 0, 1,. . .,L1g be the PR cyclic filter banks in the FT domain, 2 2 2 2 then fhl,M ðnÞ ¼ hl ðnÞejða=2bÞn Dt , g l,M ðnÞ ¼ g l ðnÞejða=2bÞn Dt , l ¼ 0, 1,. . .,L1g are the PR filter banks in the LCT domain.

Dt 2 N

2

½xðnÞejða=2bÞn

ð26Þ

The structure of L-channel cyclic filter banks associated with the LCT is shown in Fig. 4 with convolution as defined in Eq. (26), where mL denotes L-fold cyclic inter~ l,M ðmÞ and polation, kL denotes cyclic decimation by L. H ~ G l,M ðmÞ are the DLCT of hl,M ðnÞand g l,M ðnÞ, respectively, l ¼ 0, 1,. . .,L1. fhl,M ðnÞ, l ¼ 0, 1,. . .,L1g and fg l,M ðnÞ, l ¼ 0, 1,. . .,L1g are the cyclic analysis and synthesis filters with sampling interval Dt and length N, respectively. Assume the input xðnÞ with sampling interval Dt and length N, the reconstructed signal yðnÞ should be as close to the original input signal as possible, i.e., yðnÞ ¼ xðnÞ. Similarly, we can obtain the following design method for PR cyclic filter banks.

DkN ½xðnÞ ¼ ejða=2bÞn

ð29Þ

ð32Þ

Substituting n ¼ Lk þl, l ¼ 0,1,. . .,L1, k ¼ 0,1,. . ., K1 into Eq. (24a), we have 2 Xn ¼ N1 2 2 X~ M ðmÞ ¼ ejðd=2bÞðmDuÞ xðnÞejða=2bÞn Dt ejð2p=NÞnm n¼0 XL1 2 ¼ ejð2p=NÞlm ejðd=2bÞðmDuÞ l¼0 Xk ¼ K1 2 2  xðLk þ lÞejða=2bÞDt ðLk þ lÞ ejð2p=KÞkm k¼0

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Combing Eq. (32) and the above equation, Type I polyphase decomposition of the signal xðnÞ in the LCT domain is XL1 l X~ M ðmÞ ¼ ejð2p=NÞlm E~ M ðmÞ ð33Þ l¼0 l

where E~ M ðmÞ can be called the polyphase component of X~ M ðmÞ. Similarly, we can obtain the following Type II polyphase decomposition, XL1 l Type II : X~ M ðmÞ ¼ ejð2p=NÞlm R~ M ðmÞ ð34Þ l¼0 l where R~ M ðmÞ is the DLCT of the polyphase component ( 2 i ¼ 0,1,. . .,K1 xðLilÞejða=2bÞDt lð2LilÞ , n ¼ Li, xl ðnÞ ¼ 0, otherwise

0 r l r L1

g ki ðnÞ ¼ g i ðLnkÞejða=2bÞDt k ¼ 0,1,. . .,L1,

2

kð2LnkÞ

,

n ¼ 0,1,. . .,K1

ð37Þ

Assume the length of input signal xðnÞ is N. Consider the complexity of the cyclic filter banks associated with the LCT based on polyphase decomposition as shown in Fig. 9, it is similar to the analysis of Section 1.3 that the complexity of the LCT based cyclic filter banks in polyphase decomposition form just increases 2ðL þ1ÞN multiplications when compared with that of conventional cyclic filter banks in polyphase decomposition form.

ð35Þ

Now we give the following cascade equivalences of cyclic decimation and interpolation in the LCT domain shown in ~ N,M ðmÞ and H~ K,M ðmÞ Fig. 5 and Fig. 6, respectively, where H are the DLCT of hN ðnÞ and hK ðnÞ, respectively. hN ðnÞ is derived from hK ðnÞ by L-fold cyclic interpolation and N ¼ LK. Fig. 5 and Fig. 6 can be derived by using the similar method, here we only give the derivation of Fig. 5. Using Eq. (25), the output of the left system in Fig. 5 is equivalent to the output of the structure shown by Fig. 7, where the convolution is the conventional convolution in the Fourier domain and h~ ðnÞ ¼ h ðnÞ N

2

2

Fig. 7. The equivalent structure of the left system in Fig. 5.

N

ejða=2bÞn Dt . Let h~ K ðnÞ ¼ hK ðnÞejða=2bÞn ðLDtÞ , we can obtain that h~ N ðnÞ is L-fold cyclic interpolation of h~ K ðnÞ. Then using the identity for cyclic decimation in the FT domain [28], the system shown in Fig. 7 becomes the structure shown in Fig. 8, which is obviously equivalent to the right system in Fig. 5. Using Type I and Type II polyphase decomposition, and the above cascade equivalences of cyclic decimation and interpolation in the LCT domain as shown in Fig. 5 and Fig. 6, we arrive at the polyphase realization of L-channel cyclic filter banks as shown in Fig. 9, where EM ðmÞ and RM ðmÞ are the analysis and synthesis polyphase matrices, which have the polyphase components k i E~ i,M ðmÞ and R~ k,M ðmÞ in the i-th row k-th column ele2

k and R~ i,M ðmÞ is the DLCT of

2

Fig. 8. The equivalent structure of the left system in Fig. 7.

k ments, respectively. And E~ i,M ðmÞ is the DLCT of k

hi ðnÞ ¼ hi ðLn þkÞejða=2bÞDt n ¼ 0,1,. . .,K1

2

kð2Ln þ kÞ

,

k ¼ 0,1,. . .,L1, ð36Þ

Fig. 9. Realization of cyclic filter banks based on polyphase decomposition.

Fig. 5. Identical relation for cyclic decimation in the LCT domain.

Fig. 6. Identical relation for cyclic interpolation in the LCT domain.

J. Zhao et al. / Signal Processing 93 (2013) 695–705

3. Simulations In this section we give simulations to verify the design method of the filter banks associated with the LCT by utilizing some PR filter banks in the Fourier domain and discuss potential application of the LCT based filter banks in image subband decomposition. There are some literatures discussing numerical algorithms for the LCT such as [22–24]. Here we use the DAFT algorithm proposed in [22].

701

shows the magnitudes of the DTLCT of analysis filter banks. Fig. 12 gives the magnitudes of the DTLCT of the outputs of analysis filter banks before decimation and slight aliasing occurs after decimation as shown in Fig. 13, from which we can see that the LCT spectrum of the original signal is decomposed into three subbands in the LCT domain. Fig. 14(b) plots the magnitude of the DTLCT of the reconstructed signal. Fig. 10(c) and (d) give the reconstructed signal in time domain. The simulations show that the filter banks associated with the LCT are PR and can reconstruct the original signal well.

3.1. Simulation for PR filter banks in the LCT domain In this subsection we consider a one-dimensional discrete signal xðnÞ which is band-limited in the LCT domain with the parameter M ¼ ða,b,c,dÞ ¼ ð1, 5, 1=5, 2Þ and Fig. 14 (a) demonstrates the DTLCT X~ M ðoÞ of the original signal xðnÞ, which is band-limited with cutoff digital frequency oc ¼ 1.8756. The original signal xðnÞ is produced by performing inverse DLCT of X~ M ðmÞ, which is the discrete form of X~ M ðoÞ. Fig. 10(a) and (b) give the real part and imaginary part of the signal xðnÞ, respectively. The sampling interval Dt ¼ 1 and the sampling time scope is [  100 s, 100 s]. Here we consider three-channel filter banks to show subband decomposition of the LCT spectrum of the signal xðnÞ. The prototype filters of three-channel filter banks in Fourier domain are quadrature mirror filter banks whose impulse response coefficients of analysis bank filters are from the table II in [33]. According to Theorem 1 we can obtain the analysis and synthesis filters bank associated with the LCT, whose lengths are 15. The synthesis bank filters are chosen to guarantee c ¼ 1 and n0 ¼ 0. Fig. 11

Fig. 11. The magnitudes of the DTLCT of analysis filter banks.

Fig. 10. The original signal and reconstructed signal. (a) The real part of the original signal x(n). (b) The imaginary part of the original signal x(n). (c) The real part of the reconstructed signal y(n). (d) The imaginary part of the reconstructed signal y(n).

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Fig. 12. The magnitudes of the DTLCT of the outputs of analysis filter banks before decimation.

Fig. 13. The magnitudes of the DTLCT of the outputs of analysis filter banks.

a

size 512512 is shown in Fig. 16(a). The PR cyclic filter banks with DFT modulations in the Fourier domain are utilized to design the PR filter banks in the LCT domain. Note that the LCT based filter banks are just related to parameter a=b according to Eq. (25). Let the length of cyclic filter banks be N ¼ 512, sampling interval be Dt ¼ 1=N and a=b ¼ 1=5. The prototype filters of fourchannel cyclic filter banks with DFT demodulation are derived from the design method in [30]. The prototype filters of the four-channel cyclic filter banks associated with the LCT can be obtained from Theorem 2. The original image is processed in the horizontal and vertical directions separately since the proposed fourchannel cyclic filter banks are one-dimensional filter banks. Fig. 15 gives the magnitudes of the outputs of cyclic analysis bank filters, from which we can find that the LCT based cyclic filter banks realize the subband decomposition of images. Compared with the traditional cyclic filter banks in the FT domain, the LCT based cyclic filter banks has one free parameter a=b and we can obtain different subband decomposition of images with different a=b. Fig. 16(b) shows the reconstructed image. Simulation demonstrates that the PSNR between the original image and reconstructed image is 309.8740 dB, which verifies effectiveness of the proposed PR LCT based cyclic filter banks. We can also use the proposed LCT based cyclic filter banks to analysis and reconstruct RGB color images. The original color image Parrot with size 256256 is shown in Fig. 17(a). Let the length of cyclic filters be N ¼ 256 and the prototype filters of four-channel cyclic filter banks with DFT demodulation are derived from the design method in [30]. The prototype filters of the four-channel cyclic filter banks associated with the LCT can be obtained from Theorem 2 and sampling interval be Dt ¼ 1=N and a=b ¼ 1=5. Fig. 17(b) shows the reconstructed image. The PSNRs between the original image and reconstructed image are PSNR(R) ¼309.8361 dB, PSNR(G) ¼310.1079 dB, PSNR(B)¼311.8829 dB and the average PSNR is 310.6090 dB, which verifies effectiveness of the proposed LCT based cyclic filter banks. 3.3. Simulation of the LCT based cyclic filter banks in image subband decomposition

b

Fig. 14. The magnitudes of the DTLCT of the original signal and reconstructed signal. (a) The original signal. (b) The reconstructed signal.

3.2. Simulation for PR cyclic filter banks in the LCT domain In this subsection we use cyclic filter banks to analysis and reconstruction image. The original image Lena with

In practical application, some subbands of image will be discarded after subband decomposition in order to reduce the output bit rate. Using incomplete subbands to reconstruct image will lead to a degradation of the image quality. Since the proposed LCT based filter banks are complex-valued, it is appropriate to compare its performance with the corresponding complex-valued filter banks in the FT domain such as the four-channel cyclic filter banks in [30]. In this subsection we will test the sensitiveness of the four-channel LCT based cyclic filter banks to subbands discarding and make a comparison with the conventional cyclic filter banks in [30]. The conventional four-channel cyclic filter banks in [30] are complex-valued except the first and third channels, thus it will produce 16 subband images, among which there are 12 subband images with complex values.

J. Zhao et al. / Signal Processing 93 (2013) 695–705

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Fig. 15. The magnitudes of the outputs of analysis cyclic bank filters.

Fig. 16. The original image Lena and the reconstructed image. (a) The original image. (b) The reconstructed image.

Fig. 17. The original image Parrot and the reconstructed image. (a) The original image. (b) The reconstructed image. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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versus free parameter a=b when using different incomplete subbands to reconstruct image, from which we can obtain that the optimal parameters for these four cases are same, i.e., a=b ¼ 13. Table 1 gives the experiment results of the two kinds of cyclic filter banks with different number of subbands. When the number of subbands for reconstruction increases, both PSNR value and subjective quality of image will be improved. It can be observed that the quality of reconstructed images with incomplete subbands by the LCT based cyclic filter banks with appropriate parameters is better than that by the corresponding cyclic filter banks in the FT domain. Similarly the RGB color image Parrot 256256 shown in Fig. 17(a) is used to compare the performances of the two types of four-channel cyclic filter banks with the length N ¼ 256 and sampling interval is Dt ¼ 1=N. Let the step size of parameter a=b is 0.5 and Fig. 19 plots the curves of average PSNR versus free parameter a=b when using different incomplete subbands, from which we can also obtain the optimal parameter a=b ¼ 13. Table 2 gives the experiment results of the two cyclic filter banks with different number of subbands. It is shown that we can obtain better quality of reconstructed images with incomplete subbands by using the LCT based cyclic filter banks.

While the four-channel LCT based filter banks will produce 16 subband images with complex values. In order to compare these two filter banks fairly, the imaginary parts of all the subband images of the original image will be discarded after subband decomposition by these two filter banks. In the following we compare these two filter banks in reconstruction of image with incomplete subbands. The i-th row k-th column subband image obtained by using four-channel filter banks is denoted by ði,kÞ. We consider reconstruct the image with the following four cases of incomplete subbands: Case 1: use real parts of subbands C1 ¼ fð1,1Þ,ð1,2Þ,ð2,1Þ,ð2,2Þg. Case 2: use real parts of subbands C2 ¼ fC1,ð1,3Þ,ð1,4Þ,ð2,3Þ,ð2,4Þg. Case 3: use real parts of subbands C3 ¼ fC2,ð3,1Þ,ð3,2Þ,ð4,1Þ,ð4,2Þg. Case 4: use real parts of subbands C4 ¼ fC3,ð3,3Þ,ð3,4Þ,ð4,3Þ,ð4,4Þg.

In general the parameter of the LCT based cyclic filter banks a=b 2 ½1,1, here we only consider a=b 2 ½0,20 to optimize the quality of reconstructed images with given subbands, i.e., the optimal a=b is determined to maximize the PSNR between the original image and reconstructed image with given subbands. First we still use the standard gray-level image Lena 512512 shown in Fig. 16(a) as the input image. The length of the four-channel cyclic filter banks is N ¼ 512 and sampling interval is Dt ¼ 1=N. Let the step size of parameter a=b is 0.5 and Fig. 18 plots the curves of PSNR

4. Conclusions In this paper, we develop L-channel filter banks associated with the LCT, which can be useful in subband decomposition of those signals that are band limited in an LCT domain. We show that the design method of PR filter

Fig. 18. The PSNR versus the parameter a/b when using different subbands to reconstruct the gray level image Lena.

Fig. 19. The average PSNR versus the parameter a/b when using different subbands to reconstruct the RGB color image Parrot.

Table 1 PSNR(dB) of reconstructed image using two cyclic filter banks. Subbands

Case 1

Case 2

Case 3

Case 4

The LCT based cyclic filter banks with a/b ¼13 The conventional cyclic filter banks in [30]

31.1387 31.1090

31.5677 31.5346

32.4570 32.4482

32.6541 32.6441

J. Zhao et al. / Signal Processing 93 (2013) 695–705

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Table 2 subbands

Case 1

Case 3

Case 4

(a) PSNR(dB) of reconstructed image using the LCT based cyclic filter banks with a/b¼ 13. PSNR(R) (dB) 26.9463 27.5901 PSNR(G) (dB) 27.0991 27.7595 PSNR(B) (dB) 26.3821 27.1067 Average PSNR (dB) 26.8092 27.4854

28.0858 28.2188 27.6200 27.9748

28.2650 28.4086 27.7865 28.1534

(b) PSNR(dB) of reconstructed image using the conventional cyclic filter banks in [30]. PSNR(R) (dB) 26.8894 27.5390 PSNR(G) (dB) 27.0699 27.7212 PSNR(B) (dB) 26.3165 27.0336 Average PSNR (dB) 26.7586 27.4313

28.0839 28.2175 27.6075 27.9696

banks for the LCT can inherit conventional filter design approach in the Fourier domain. The polyphase decompositions in the LCT domain are also given. Furthermore the cyclic filter banks associated with the LCT are proposed based on the circular convolution in the LCT domain. The design method and the polyphase representation of PR cyclic filter banks are derived similarly. The simulations verify the effectiveness of the proposed design methods of filter banks in the LCT domain and discuss potential application of the LCT based cyclic filter banks in image subband decomposition. References [1] H.M. Ozaktas, M.A. Kutay, Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing, Wiley, New York, 2000. [2] S.C. Pei, J.J. Ding, Eigenfunctions of linear canonical transform, IEEE Transactions on Signal Processing 50 (1) (2002) 11–26. [3] R. Tao, L. Qi, Y. Wang, Theory and Applications of the Fractional Fourier Transform, Tsinghua University Press, Beijing, 2004. [4] B. Barshan, M.A. Kutay, H.M. Ozaktas, Optimal filters with linear canonical transformations, Optics Communication 135 (1997) 32–36. [5] K.K. Sharma, S.D. Joshi, Signal separation using linear canonical and fractional Fourier transforms, Optics Communication 265 (2006) 454–460. [6] X.M. Li, L. Dai, Reality-preserving image encryption assosiated with the chaos and the LCT, The 3rd International Congress on Image and Signal Processing 6 (2010) 2624–2627. [7] A. Stern, Uncertainty principles in linear canonical transform domains and some of their implications in optics, Journal of the Optical Society of America A: Optics, Image Science, and Vision 25 (3) (2008) 647–652. [8] K.K. Sharma, S.D. Joshi, Uncertainty principles for real signals in linear canonical transform domains, IEEE Transactions on Signal Processing 56 (7) (2008) 2677–2683. [9] J. Zhao, R. Tao, Y.L. Li, Y. Wang, Uncertainty principles for linear canonical transform, IEEE Transactions on Signal Processing 57 (7) (2009) 2856–2858. [10] G.L. Xu, X.T. Wang, X.G. Xu, Uncertainty inequalities for linear canonical transform, IET Signal Processing 3 (5) (Sep. 2009) 392–402. [11] J. Zhao, R. Tao, Y. Wang, On signal moments and uncertainty relations associated with linear canonical transform, Signal Processing 90 (9) (2010) 2686–2689. [12] B. Deng, R. Tao, Y. Wang, Convolution theorems for the linear canonical transform and their applications, Science in China Series F: Information Science 49 (5) (2006) 592–603. [13] D.Y. Wei, Q.W. Ran, Y.M. Li, et al., A convolution and product theorem for the linear canonical transform, IEEE Signal Processing Letters 16 (10) (2009) 853–856.

Case 2

28.2595 28.4032 27.7694 28.1440

[14] Y.X. Fu, L.Q. Li, Generalized analytic signal associated with linear canonical transform, Optics Communication 281 (6) (2008) 1468–1472. [15] G.L. Xu, X.T. Wang, X.G. Xu, Generalized Hilbert transform and its properties in 2D LCT domain, Signal Processing 89 (7) (2009) 1395–1402. [16] A. Stern, Sampling of linear canonical transformed signals, Signal Processing 86 (7) (2006) 1421–1425. [17] B.Z. Li, R. Tao, Y. Wang, New sampling formulae related to linear canonical transform, Signal Processing 87 (2007) 983–990. [18] Y.L. Liu, K.I. Kou, I.T. Ho, New sampling formulae for nonbandlimited signals associated with linear canonical transform and nonlinear Fourier atoms, Signal Processing 90 (3) (2010) 933–945. [19] K.K. Sharma, Vector sampling expansions and linear canonical transform, IEEE Signal Processing Letters 18 (10) (2011) 583–586. [20] J. Zhao, R. Tao, Y. Wang, Sampling rate conversion for linear canonical transform, Signal Processing 88 (11) (2008) 2825–2832. [21] G.X. Xie, B.Z. Li, Z. Wang, Identical relation of interpolation and decimation in the linear canonical transform domain, in: Ninth International Conference on Signal Processing (2008), pp. 72–75. [22] S.C. Pei, J.J. Ding, Closed-form discrete fractional and affine Fourier transforms, IEEE Transactions on Signal Processing 48 (2000) 1338–1353. [23] A. Koc, H.M. Ozaktas, et al., Digital computation of linear canonical transforms, IEEE Transactions on Signal Processing 56 (6) (2008) 2383–2394. [24] J.J. Healy, J.T. Sheridan, Sampling and discretization of the linear canonical transform, Signal Processing 89 (4) (2009) 641–648. [25] X.Y. Meng, R. Tao, Y. Wang, Fractional Fourier domain analysis of cyclic multirate signal processing, Science in China Series F: Information Science 51 (6) (2008) 803–819. [26] S. Shinde, Two channel paraunitary filter banks based on linear canonical transform, IEEE Transactions on Signal Processing 59 (2) (2011) 832–836. [27] P.P. Vaidyanathan, A. Kirac, Theory of cyclic filter banks, IEEE International Conference on Acoustics, Speech, and Signal Processing 3 (1997) 2449–2452. [28] P.P. Vaidyanathan, A. Kirac, Cyclic LTI systems in digital signal processing, IEEE Transactions on Signal Processing 47 (2) (1999) 433–447. [29] S. Sarkar, H.V. Poor, Multirate signal processing on finite field, IEE Proceedings of Vision, Image and Signal Processing 148 (8) (2001) 254–262. [30] F. Itami, E. Watanabe, A. Nishihara., A design of cyclic filter banks, Asia-Pacific Conference on Circuits and Systems 1 (2002) 65–68. [31] M.J. Smith, S.L. Eddins, Analysis/synthesis techniques for subband image coding, IEEE Transactions on Acoustic, Speech and Signal Processing 38 (8) (1990) 1446–1456. [32] L.S. Shen, W.H. Jia, P. Chen, Filter banks for image subband coding, in: Proceedings of the 34th SICE Annual Conference (1995), pp. 1201–1204. [33] P.P. Vaidyanathan, Theory and design of M channel maximally decimated quadrature mirror filters with arbitrary M, having perfect reconstruction property, IEEE Transactions on Acoustics, Speech and Signal Processing 35 (4) (1987) 476–492.