Spreading properties of periodic clock changes application to interleavers

ARTICLE IN PRESS

Signal Processing 88 (2008) 221–235 www.elsevier.com/locate/sigpro

Spreading properties of periodic clock changes application to interleavers W. Chauveta,, B. Lacazea, D. Rovirasa, A. Duverdierb a

IRIT/ENSEEIHT, 2 rue Camichel, BP 7122 - 31071 Toulouse Cedex 7, France b CNES, 18 Avenue E. Belin, 31401 Toulouse Cedex 4, France

Received 12 October 2005; received in revised form 20 March 2007; accepted 7 June 2007 Available online 1 July 2007

Abstract This paper deals with the study of the digital periodic clock changes (PCC) with applications to digital communications. These PCC turn out to be part of the linear periodic time varying (LPTV) filters set. Relying on an LPTV framework, two issues are adressed. First, given a PCC, emphasis is laid on the condition for this PCC to be invertible. The second point deals with the effects of PCC on signal spectrum. Although PCC study remains a quite new research topic, equivalence between PCC and interleavers is stressed in this paper. Therefore, this paper finally proposes an original point of view on interleavers based on an LPTV framework. Particularly, new results concerning frequency effects of interleavers are presented. r 2007 Published by Elsevier B.V. Keywords: Linear periodic time varying filter; Digital periodic clock change; Interleaver; Spread spectrum; Cyclostationarity

1. Introduction Analog or digital linear periodical time varying (LPTV) filters are widely studied in telecommunications and signal processing area. [1] study the cyclostationarity properties of a stationary signal filtered by an analog LPTV filter, [2] derives the relationships between different digital representations of an LPTV filter, [3] points out the relationships between digital LPTV filters and filter banks while [4] focuses on the effect of a digital LPTV filter on a stochastic signal. LPTV applications are various. Some applications explicitly use LPTV Corresponding author. Tel.: +33 56 15 88 012; fax: +33 56 15 88 306. E-mail address: [email protected] (W. Chauvet).

0165-1684/$ - see front matter r 2007 Published by Elsevier B.V. doi:10.1016/j.sigpro.2007.06.006

filters: spread spectrum [5,6], transmultiplexing [7], blind channel estimation [8,9] and spectral scrambling [10]. Other ones implicitly use LPTV filters, like time domain scrambling (LPTV properties of periodical interleavers), or applications relying on multirate digital processing such as filter banks [11] (because of the LPTV nature of interpolation and decimation [12]). In [13], authors have introduced a special case of LPTV filters: the periodic clock changes (PCC) in analog form, used for example in [14]. The purpose of this paper is to study properties of the digital PCC using an LPTV framework. Two problems will be addressed. The first issue deals with the sufficient condition for a PCC to be invertible. This question is motivated by the context of digital communications. Assuming a PCC is included in a transmission

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system, the simplest approach is to ensure the invertibility of this PCC. Indeed, no additional extra process is required except the inverse process. The second point is related to the effect of a PCC on the input power spectral density (PSD). It is known that LPTV filters frequentially spread any oversampled input signal. Hence, in a previous paper [5], we have presented a spread spectrum multiuser system relying on a PCC. However, in [5] no theoretical expression was derived on the exact spectral effect of PCC. Although PCC remain quite marginal tools in digital processing domain, the interest of the paper is that all the obtained results can be applied to interleavers. Indeed, equivalence between PCC and interleavers will be pointed out. Thus, this paper proposes an original LPTV approach for interleavers. Furthermore, results about frequency effects of interleavers are new since the latter are usually considered for their temporal properties as in turbo code applications [15]. In Section 2, a review of LPTV filters is proposed. Relations between LPTV filters, PCC and interleavers are also described. Section 3 deals with the invertibility of the PCC. A necessary and sufficient condition (NSC) for the invertibility of any PCC is proposed. It is illustrated how the knowledge of this condition is helpful in designing an invertible PCC. The last part of the paper is devoted to the spectral effects of PCC. Interest of this result is illustrated using the example of a matrix interleaver. 2. Relationships between LPTV, PCC and interleavers

Fig. 1. Modulator representation of an N periodic LPTV filter.

Many representations are possible for an N periodic LPTV filter. McLernon [2] has listed the various equivalent structures in an instructive way. In the following, we use the modulator representation defined by N linear time invariant (LTI) filters with z-transform fT p ðzÞg0pppN1 . In such a representation, output and input z-transforms Y ðzÞ and X ðzÞ are related together according to the following relation where W kN stands for the Nth roots of unity, namely W kN ¼ expð2ipk=NÞ: Y ðzÞ ¼

N 1 X

(2)

p¼0

LTI filters fT p ðzÞg0pppN1 are called modulator filters and can be expressed [2] by T p ðzÞ ¼

In this section, LPTV filters, PCC and interleavers are presented. Emphasis is laid on their interrelationships.

T p ðzW pN ÞX ðzW pN Þ.

1 1 X X 1N W pn zk hðn; kÞ. N N n¼0 k¼1

(3)

Fig. 1 presents an LPTV filter modulator representation.

2.1. LPTV filters 2.1.1. Presentation of LPTV filters An N periodic discrete LPTV filter has an N periodic time varying impulse response hðn; kÞ and the output yðnÞ is expressed by the following relation according to the input signal xðnÞ: yðnÞ ¼

1 X

hðn; kÞxðn  kÞ,

k¼1

hðn þ N; kÞ ¼ hðn; kÞ.

ð1Þ

2.1.2. Frequency spreading of LPTV filters An LPTV filter exhibits the property of spreading oversampled input signals. This point will be discussed in Section 4 and deeply studied for the particular case of PCC. The aim of this subsection is just to shed light on this frequency spreading property. For that purpose, we use relation (2) with z ¼ e2ipf where f stands for the normalized frequency. We obtain the following relation between the input

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223

and output Fourier transforms: Y ðe2jpf Þ ¼

N 1 X

T p ðe2ipðf p=NÞ ÞX ðe2ipðf p=NÞ Þ.

(4)

p¼0

From this frequency relation, it is obvious that an LPTV filter will spread any oversampled signal. For a more precise understanding of this LPTV property, interested reader can refer to [4] where the bifrequential representation of LPTV filters is presented. Example. We consider an LPTV filter with period N ¼ 2 and modulator filters T 0 ðzÞ and T 1 ðzÞ given by Fig. 2. The input signal Fourier transform X ðe2ipf Þ is given by Fig. 2. For more convenience of representation, we make the assumption that these three Fourier transforms X ðe2ipf Þ, T 0 ðe2ipf Þ and T 1 ðe2ipf Þ are real and positive. According to relation (4), Y ðe2ipf Þ is given by Y ðe

2jpf

Þ ¼ T 0 ðe

2ipf

þ T 1 ðe

ÞX ðe

2ipf

2ipðf 0:5Þ

Þ ÞX ðe

Fig. 3. Output Fourier transform.

1 PCC function f(n) 0

2

3

4

5

6

7

x(4)

x(5)

x(6)

x(7)

8

n

-1 2ipðf 0:5Þ

Þ.

Using (5), it is easy to compute Y ðe Output spectrum is clearly spread.

2jpf

ð5Þ Þ (Fig. 3).

x(p) x(0)

An N PCC is a transformation defined by an N periodic function f ðnÞ, from Z into Z, such that output signal yðnÞ is given by the following relation according to input signal xðnÞ: f ðn þ NÞ ¼ f ðnÞ.

x(1)

x(2) x(3)

f(0)

2.2. Periodic clock changes

yðnÞ ¼ xðn  f ðnÞÞ;

1

0

y(n)

n= 0 x(1)

f(3)

f(1) n= 1 n=2 x(0)

PCC

x(8) input

x(2)

f(4) n= 3 n = 4

n = 5 n= 6 n=7

x(4)

x(5) x(7)

x(3)

x(6)

n= 8 PCC x(8) output

Fig. 4. An example of PCC with period 3.

(6)

Function f ðnÞ is called the PCC function. In the following, we will denote PCCf as the PCC with function f ðnÞ.

A PCC introduces periodic delays f ðnÞ. Fig. 4 illustrates this process with a period 3 PCC. On this representation, periodicity appears with the periodicity of the arrays pattern. For following developments, it is convenient to introduce notations concerning Euclidian division of n by N where dneN and bncN , respectively, stand for the quotient and the modulus of the division of n by N: N

n ¼ Ndne þ bncN

with 0pbncN pN  1.

(7)

Given a PCCf with period N, we introduce now other useful functions with capital letters: F ðnÞ, F q ðnÞ, F m ðnÞ. Indices q and m, respectively, stand for quotient and modulus. These functions are related to PCC function f ðnÞ by F ðnÞ ¼ n  f ðnÞ; Fig. 2. Input Fourier transform and modulator filters.

F m ðnÞ ¼ bF ðnÞcN .

F q ðnÞ ¼ dF ðnÞeN , ð8Þ

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N periodicity of f ðnÞ implies following properties for functions F ðnÞ, F q ðnÞ and F m ðnÞ: 9 F ðn þ NÞ ¼ F ðnÞ þ N; > = F q ðn þ NÞ ¼ F q ðnÞ þ 1; . (9) > ; F m ðn þ NÞ ¼ F m ðnÞ

P 140 x(0) x(Q) x(2Q)

x(Q(P-1)) 100

x(1)

rea ding

Q

A PCC is entirely defined by the knowledge of either f ðnÞ, F ðnÞ or the couple ½F q ðnÞ; F m ðnÞ on an N length interval. In addition, it is easy to show that a PCC with function f ðnÞ is an LPTV filter with periodic impulse response hðn; kÞ and modulator filters T p ðzÞ given by hðn; kÞ ¼ dðk  f ðnÞÞ 1 T p ðzÞ ¼ N

N 1 X

and

60

x(2) f (n)

224

writi ng

20 0 -20 -60 -100

x(Q-1)

x(PQ-1) -140 0

20 40 60 80 100 120 140 160 180 n

Fig. 5. (a) Principle of the matrix interleaver, (b) PCC function with ðQ; PÞ ¼ ð8; 21Þ.

f ðnÞ W pn . Nz

ð10Þ

n¼0

for a ðQ; PÞ matrix interleaver have the following theoretical expression:

2.3. Interleavers An N periodic interleaver is defined by a function pðnÞ from Z into Z such that input xðnÞ and output yðnÞ are related according to yðnÞ ¼ xðpðnÞÞ

with pðnÞ ¼ pðn þ NÞ þ N.

(11)

Identifying (6) and (11), we can see that an N periodic interleaver is an N periodic PCC with following relation between PCC function f ðnÞ and interleaver function pðnÞ: pðnÞ ¼ F ðnÞ ¼ n  f ðnÞ.

(12)

As a consequence, any interleaver can be considered as a PCC and conversely. 2.4. An illustration: the matrix interleaver The ðQ; PÞ matrix interleaver consists in writing the input xðnÞ in a Q  P matrix column by column according to Fig. 5(a). Output yðnÞ is then obtained by reading the matrix in the inverse way (namely row by row). It can be shown that the PCC function of this N ¼ PQ periodic interleaver has the following expression: f ðnÞ ¼ ðQ  1ÞcP ðnÞ þ ðP  1ÞbP;Q ðnÞ cP ðnÞ ¼ bncP ;

P

bP;Q ðnÞ ¼ bdne cQ .

with ð13Þ

One period of this N periodic function is given in Fig. 5(b) in the case ðP; QÞ ¼ ð21; 8Þ. Because of the PCC/interleaver equivalence and the LPTV nature of PCC, an interleaver can be defined by its modulator filters T p ðzÞ defined by (10). Using (13), we can show that modulators filters

T p ðf Þ ¼

expðjpp=NÞ sinðpfPðQ  1Þ  ðp=QÞpÞ N sinðpf ðQ  1Þ  ðp=NÞpÞ sinðpfQðP  1ÞÞ . ð14Þ  sinðpf ðP  1Þ þ ðp=QÞpÞ

Fig. 6 illustrates some modulator filters (T 0 ðf Þ, T 1 ðf Þ, T 6 ðf Þ and T 7 ðf Þ) for a ðQ; PÞ ¼ ð3; 4Þ matrix interleaver. In this section, the LPTV nature of interleavers has been pointed out and any interleaver appears to be a possible tool in order to design a spreading system. For such an application, it is interesting to compute the theoretical effect of an interleaver on the input spectrum. This issue will be adressed in Section 4.

3. Invertibility of periodical clock changes In a digital communication context, transmitted digital information has to be retrieved at the receiver side. It is thus interesting to study the invertibility of PCC. If the signal is processed by a PCC at the emitter side, the inverse processing is needed at the receiver side. Here, it can be noted that, in a digital transmission context, perfect invertibility can be avoided provided that it is possible to estimate the transmitted symbols at the receiver side. Before studying such noninvertible PCC, this paper focuses on invertible PCC study as a first necessary step. The purpose of this section is to present a NSC for the PCC invertibility.

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Fig. 6. Frequency response of modulator filters for the matrix interleaver ðQ; PÞ ¼ ð3; 4Þ.

x(n)

y(n) PCC f

w(n) PCCg

x(n)

w(n) PCC f

This result is obtained by successively applying relation (6) to outputs yðnÞ and wðnÞ. In addition, it is easy to check that the lowest common multiple (LCM) of ðP; QÞ is a period of hðnÞ. According to this proposition, the set of PCC with period N is closed under the cascade operation. It is also easy to check that this operation is associative and that PCCe with eðnÞ ¼ 0, for n 2 Z, is the unit element for this operation. However, this cascade operation is not commutative.

PCCg = PCCh

3.2. Invertibility of a PCC Fig. 7. The cascade operation.

3.1. The cascade operation Let us consider PCCf and PCCg two PCC with respective periods P and Q. The cascade of these two PCC is also a PCC. Cascade operation will be denoted as PCCh ¼ PCCf  PCCg according to Fig. 7. Resulting PCC is characterized by function hðnÞ that is related to gðnÞ and f ðnÞ according to hðnÞ ¼ gðnÞ þ f ðn  gðnÞÞ.

(15)

Invertibility of LPTV filters has already been studied [16,17]. Invertibility condition is related to the invertibility of LPTV matrices. Because these last matrices are complicated, it is untractable to relate matrices invertibility to PCC function f ðnÞ. Originality of this paper is to relate invertibility to very simple properties of PCC function f ðnÞ.

3.2.1. Characterization of the PCC invertibility First of all, we define the concept of PCC invertibility. Given an N periodic PCCf , this PCC is invertible if there exists an N periodic PCCf 0 such

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W. Chauvet et al. / Signal Processing 88 (2008) 221–235

that PCCf  PCCf 0 ¼ PCCe , namely f 0 ðnÞ must fulfill 8n 2 Z

f 0 ðnÞ þ f ðn  f 0 ðnÞÞ ¼ 0.

(16)

According to this definition and to Eq. (15), a PCC is invertible if there exists an other PCC that cancels its effects. Invertibility characterization of a PCC can be related to function F m by the following proposition where ½0 : N  1 stands for the set of integers between 0 and N  1:

Fm(n) 2 1 0 0 f(n)

1

2

3

4

5

p= 0

p= 1

p= 2

p= 3

p= 4

p= 5

x(0)

x(1)

x(2)

x(3)

x(4)

x(5)

n= 0 y(0)

n= 1 y(1)

n =2 y(2)

n=3 y(3)

n= 4 y(4)

n= 5 y(5)

0 -1 -2

x(p)

A PCCf with period N is invertible

PCC input

() F m ðnÞ is bijective on ½0: N  1. Proof of this NSC can be found in Appendix A.1. A consequence of this proposition is that PCC invertibility is independent of F q ðnÞ and invertibility study can be reduced to the study of F m ðnÞ on an N length interval.

3.2.2. Illustration of PCC invertibility We choose an N periodic linear PCCf defined by f ðnÞ ¼ KbncN , where K is a positive integer and n 2 Z. According to (8), F m ðnÞ is given by F m ðnÞ ¼ bn  f ðnÞcN ¼ bð1 þ KÞncN .

(17)

Using the previous NSC, invertibility of this PCC only relies on F m ðnÞ. With this NSC and the NSC for the group structure of Z=nZ [18], we have the following result: a linear PCC with PCC function f ðnÞ ¼ KbncN is invertible if and only if K þ 1 and N are coprime. The two following examples illustrate this proposition. Example 1 (K þ 1 and N are coprime). We consider a linear PCC function f 1 ðnÞ ¼ K 1 bncN with N ¼ 3 and K 1 ¼ 1. Using the above invertibility condition, this PCC is invertible since K 1 þ 1 ¼ 2 and N ¼ 3 are coprime. Fig. 8 gives a plot of F m ðnÞ. It is easy to check that this function is bijective on f0; 1; 2g. Invertibility of the PCC appears on the figure by a bijective correspondance between input and output samples. Example 2 (K þ 1 and N are not coprime). Here we choose f 2 ðnÞ ¼ K 2 bncN with N ¼ 3 and K 2 ¼ 2. This PCC is not invertible since K 1 þ 1 ¼ 3 and N ¼ 3 are not coprime. Looking at F m ðnÞ in Fig. 9, we see clearly that this function is not bijective on the set f0; 1; 2g. Some input samples (with warning label) are not present in the output signal proving the noninvertibility of the PCC process.

y(n)

PCC output

Fig. 8. An invertible linear PCC (K 1 ¼ 1, N ¼ 3).

Fm(n) 2 1 0

0 f(n)

1

2

3

4

5

0 -2 -4

n

x(p)

p=0 x(0)

p= 1 x(1)

p= 2 x(2)

p= 3 x(3)

p =4 x(4)

p= 5 x(5)

PCC input

y(n)

n= 0 y(0)

n=1 y(1)

n= 2 y(2)

n= 3 y(3)

n= 4 y(4)

n= 5 y(5)

PCC output

Fig. 9. A noninvertible linear PCC (K 2 ¼ 1, N ¼ 3).

4. Spreading properties of PCC In this section, we study frequency effects of a PCC (or an interleaver). Although relation (4) was helpful to understand spreading properties of LPTV filters, the derivation of the output theoretical PSD relying on this relation appears untractable. Hence we propose here an alternative method for computing the theoretical PSD of the output PCC. This method relies on the transformation by an LPTV filter (and so a PCC) [13] of any wide sense stationary (WSS) process into a wide sense cyclostationary (WSC) process. After exctracting the stationary part of the PCC output, a theoretical expression of the output PSD will be obtained.

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4.1. PSD of WSC processes

227 PCC input

x(F(p+r))

x(F(r))

x(n)

Here, we review the expression of WSC process bispectrum. This bispectrum consists of a stationary part and a cyclic part. We are here interested in the stationary part.

distance=| (r,p)|

PCCf

4.1.1. Bispectrum of WSC processes The autocorrelation function of an L-WSC process yðnÞ is defined as Ry ðm; nÞ ¼ E½yðmÞy ðnÞ. Wide sense L cyclostationarity implies that autocorrelation function is periodic in n and m with period L, namely Ry ðm þ L; n þ LÞ ¼ Ry ðm; nÞ. The bispectrum S y ðf 0 ; f Þ (where f 0 and f are normalized bifrequencies) of such a WSC process is defined by the two dimensional Fourier transform of Ry ðm; nÞ leading to the following expression [4]: 1  X q d f f0 þ S y ðf 0 ; f Þ ¼ L q¼1 " # L1 X 1 1X j2pf 0 m j2pfr  . Ry ðm; rÞe e L r¼0 m¼1 ð18Þ Expression (18) contains a stationary part and a cyclic part. Stationary part corresponds to the main diagonal, namely for bqcL ¼ 0. In addition, given a WSC process, it is always possible to stationarize this process by introducing a random delay [13] (delay uniformly distributed on the interval ½0 : L  1). PSD of this stationarized process is then equal to the stationary part of the previous bispectrum. 4.1.2. Stationary part of a WSC process We consider an L-WSC process with a cyclic autocorrelation Ry ðm; nÞ and an associated bispectrum Sy ðf 0 ; f Þ (18). Stationary part Ssta y ðf Þ of the bispectrum is given by S sta y ðf Þ ¼ S y ðf ; f Þ. After some derivations we have the following expression for the stationary part of autocorrelation function: Rsta y ðpÞ ¼

L1 1X Ry ðp þ r; rÞ. L r¼0

(19)

4.2. Stationary part of a PCC output We consider a PCC with function f ðnÞ. A WSS input process xðnÞ (with autocorrelation function Rx ðpÞ) is processed by this N periodic PCCf . As a particular case of LPTV filter [13], this N periodic

distance = |p|

y(n)

y(p+r)

y(r) PCC output

Fig. 10. Function Lðr; pÞ.

PCC turns this WSS process xðnÞ into an N-WSC process yðnÞ with autocorrelation function Ry ðm; nÞ. Using (19), the stationary part of the output autocorrelation Rsta y ðpÞ can be obtained (Appendix A.2): Rsta y ðpÞ ¼

1 X 1N Rx ðLðr; pÞÞ N r¼0

(20)

with Lðr; pÞ ¼ p þ f ðrÞ  f ðp þ rÞ ¼ F ðp þ rÞ  F ðrÞ. (21) Eq. (20) highlights how function Lðr; pÞ is of importance for the theoretical determination of the PCC output autocorrelation function. Given two samples yðrÞ and yðp þ rÞ in the output signal, jLðr; pÞj is the distance between these two samples before PCC process. Fig. 10 illustrates the meaning of Lðr; pÞ.

4.3. Application: matrix interleaver for spread spectrum applications In a previous paper [5], we have presented a method for building a set of invertible and orthogonal LPTV filters. The choice of a matrix interleaver as a generator LPTV filter results in a spread spectrum multiuser system with very good performance in terms of multiuser interference. However, no attention has been paid (in [5]) on the spreading effects of the system. In the following, we present all different steps for output PSD computation in the case of the ðQ; PÞ matrix interleaver. Influence of parameters P and Q on the spreading effects will also be discussed.

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4.3.1. Theoretical expression of Lðr; pÞ for the ðQ; PÞ matrix interleaver We consider a ðQ; PÞ matrix interleaver with period equal to N ¼ PQ. We derive theoretical expression of function Lðr; pÞ for 0prpN  1 and p a positive integer (see Appendix A.3). Only positive values of p are considered because of Hermitian symmetry. Theoretical values are given by the four following relations depending on the values of cP ðrÞ and bP;Q ðrÞ previously defined by (13): If 0pcP ðrÞpP  1  cP ðpÞ:

Lðr; pÞ ¼ p  f ðpÞ þ QðP  1Þ for Q  bP ðpÞpbP;Q ðrÞpQ  1. If P  cP ðpÞpcP ðrÞpP  1: Lðr; pÞ ¼ p  f ðpÞ þ 1  N for 0pbP;Q ðrÞpQ  2  bP ðpÞ, for Q  bP ðpÞ  1pbP;Q ðrÞpQ  1.

ð22Þ

4.3.2. Stationary autocorrelation We propose now to express the stationary autocorrelation function of the matrix interleaver output. We choose an input NRZ process with oversampling factor L as input process. After a review of NRZ process, we will express the output stationarized autocorelation function if a NRZ process is interleaved by a matrix interleaver. An NRZ signal with oversampling factor L (L samples per NRZ symbol) is an L-WSC process.

0

Λ=11

3

30

1

2 Λ=-15

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

Λ (r,7)

20

10

0

-10

21 -20 4

8

12

16

r

20

ð25Þ

Given p a positive integer, for all r 2 ½0 : N  1, Lðr; pÞ can take only four different values. These four different zones can be summarized by the matrix representation of Fig. 11, where the matrix is filled row by row with increasing values of r. Function Lðr; pÞ is of prime importance when computing output stationary autocorrelation function (20). We can remark that this function depends only on the PCC (or interleaver) and not on the input process. Fig. 12 illustrates function Lðr; pÞ for a ðQ; PÞ ¼ ð9; 3Þ interleaver and for p ¼ 7.

Fig. 11. Values of Lðr; pÞ according to r for a given value of p.

0

ð24Þ

Lðr; pÞ ¼ p  f ðpÞ þ 1  Q

Lðr; pÞ ¼ p  f ðpÞ for 0pbP;Q ðrÞpQ  1  bP ðpÞ,

ð23Þ

24

28

24

Λ=29

20 Λ=3

22

23

25

26

p=7 Fig. 12. Function Lðr; pÞ for p ¼ 7, Q ¼ 9 and P ¼ 3.

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Fig. 13. Computation of Rsta y ð1Þ.

In the following, input NRZ process will be chosen with oversampling factor L equal to four. Fig. 13 illustrates how to compute one value of Rsta using (20) in the particular case y ðpÞ (Q,PÞ ¼ ð9; 3Þ. For example, let assume we want to compute Rsta y ð1Þ. We have to evaluate the input autocorrelation Rx for the N values Lðr; 1Þ ð0prpN  1Þ in (20). We pointed out in relation (22)–(25) that this function took no more than four values. According to Fig. 13, possible values of Lðr; 1Þ are only ð9; 17; 1Þ. The output autocorrelation will be given by a linear combination of the three values Rx ð9Þ, Rx ð17Þ and Rx ð1Þ. Coefficients are equal to the occurence of 9; 17 and 1 for Lðr; 1Þ. Using (20), Rsta y ð1Þ equals: 1 Rsta y ð1Þ ¼ 27ð18Rx ð9Þ þ 8Rx ð17Þ þ Rx ð1ÞÞ.

(27)

With Rx ð9Þ ¼ 0, Rx ð17Þ ¼ 0 and Rx ð1Þ ¼ 0:75, 0:75 we have Rsta y ð1Þ ¼ 27 ¼ 0:03. This example shows that many values of Rsta y ðpÞ will be zero. In general, provided that Q4L, it can

Simulated output autocorrelation Theoretical output autocorrelation

1.1 0.9 0.7

Rstay(p)

If f stands for the normalized frequency, the autocorrelation of the stationarized NRZ process Rx ðmÞ and the PSD Sx ðf Þ are given by 8 > < Rx ðmÞ ¼ 1  jmj if m 2 ½L : L L and > : else Rx ðmÞ ¼ 0   1 sinðpfLÞ 2 S x ðf Þ ¼ . ð26Þ L sinðpf Þ

0.5 0.3 0.1

p -0.1 0 Zone 1

4

8

12

Zone 2

16

20

24

28

Zone 3

Fig. 14. Comparaison of theoretical and simulated autocorrelation function.

be demonstrated (Appendix A.4) that the output autocorrelation function has only 3L  2 nonnull values. These values are given by following relations: Zone 1: for 0pjpL  1,   j sta Ry ðjPÞ ¼ Rx ðjÞ 1  . Q Zone 2: for 0pjoL  2, jþ1 . Rsta y ðjP þ 1Þ ¼ Rx ðj þ 1Þ N Zone 3: for 1pjoL  1,

ð28Þ

ð29Þ

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Fig. 15. Input and output signal, autocorrelation and PSD.

ARTICLE IN PRESS W. Chauvet et al. / Signal Processing 88 (2008) 221–235

ðP  1Þ j N

231

else Rsta y ¼ 0.

process and the output corresponding process. Input autocorrelation is also plotted as well as the input and output PSD.

Expressions (28)–(30) hold for any input process with autocorrelation function Rx ðmÞ fulfilling the property: jRx ðmÞj ¼ 0 for me½ðL  1Þ : L  1. In order to validate theoretical expressions (28)–(30), we compare, in Fig. 14, theoretical results and simulated autocorrelation function of the stationarized output for ðL; P; QÞ ¼ ð4; 3; 9Þ. Fig. 15 proposes a synthetic illustration of the effect of the matrix interleaver on an oversampled NRZ process. Parameters are ðL; P; QÞ ¼ ð4; 3; 9Þ. This figure presents one realisation of the input

4.3.3. Influence of parameters P and Q Thanks to expressions (28)–(30) of the output autocorrelation function, it is easy to conclude that for any value of P, when Q increases, absolute values of Rsta y ðpÞ in zone 2 and 3 tend to 0. Therefore, output autocorrelation function tends to function RðpÞ given by ( Rx ðp=PÞ if bpcP ¼ 0; sta Ry ðpÞ ! RðpÞ ¼ (31) 0 if bpcP a0: large Q

Rsta y ðN  1  PjÞ ¼ Rx ðjÞ

ð30Þ

Fig. 16. Comparaison of theoretical PSD for increasing values of Q and limit PSD.

ARTICLE IN PRESS W. Chauvet et al. / Signal Processing 88 (2008) 221–235

232

Using Fourier transform, and some derivations, we have in the frequency domain: Ssta y ðf Þ ! Sðf Þ ¼ S x ðPf Þ. large Q

(32)

As a consequence, PSD of the output process for large Q is obtained from input PSD S x ðf Þ by a periodization of Sx ðPf Þ at frequencies multiple of 1=P. For a large value of Q, output PSD only depends on parameter P. In the particular case of an NRZ input process, output PSD is given by   1 sinðPpLf Þ 2 Ssta ðf Þ ! Sðf Þ ¼ . (33) y increasing Q L sinðPpf Þ Fig. 16 illustrates the theoretical PSD (FFT of the theoretical autocorrelation function given by (28)–(30) for increasing values of Q: ðL; P; QÞ ¼ ð5; 6; 5Þ, ð5; 6; 10Þ, ð5; 6; 60Þ. This figure also displays the PSD theoretical limit Sðf Þ (for Q ! 1) (33). According to Fig. 16, theoretical expression of PSD meets limit expression of (32) for large values of Q. Furthermore, for large values of Q, output PSD only depends on P. The number of peaks in the output PSD only depends on P. 5. Conclusion Analog PCC have been studied in the past. In this paper digital PCC have been investigated. LPTV nature of PCC has been stressed as well as the equivalence between PCC and interleavers. Both properties lead to the original conclusion that an interleaver can be studied through an LPTV approach. This paper has focused on two issues. The first one was to find conditions of PCC invertibility based on simple results concerning the PCC function f ðnÞ. The second issue was to compute theoretically the spectral effects of PCC and interleavers on input signals. Concerning PCC invertibility, a very simple NSC has been demonstrated. Invertibility relies on the bijectivity of function F m ðnÞ on the PCC period. F m ðnÞ is related to PCC function in a very simple way. Frequency effects of PCC and interleavers have been studied. After computing the bispectrum of the PCC output (cyclostationary process), the PSD of the stationarized process has been obtained. Particularly, the stationarized autocorrelation function can be related to a linear combination of input autocorrelation process at given delays. These

delays are related to the PCC function. This result is original and can be applied to interleavers in order to compute their spreading effects. Furthermore, if a spreading system has forbidden frequency subbands, these theoretical results are helpful for designing a suitable interleaver. Such a constraint appears, for example in positioning systems where the two systems (Galileo and GPS) must not interfere. Concerning interleavers, most of the studies have focused on their temporal effects. Frequency LPTV approach offers a new formalism for studying interleavers effects.

Appendix A A.1. NSC for PCC invertibility Let us make the assumption that PCCf is invertible. We have to show that F m ðnÞ is bijective on ½0 : N  12 . We consider an integer n 2 ½0 : N  1. Since PCCf is invertible, there is a function f 0 ðnÞ such that (16) is satisfied. Using associated functions (8) for PCCf and (16), we have the following result: n ¼ NF q ðn  f 0 ðnÞÞ þ F m ðbn  f 0 ðnÞcN Þ.

(34)

But n 2 ½0 : N  1 and because of unicity of Euclidian Division (7), relation (34) turns into n ¼ F m ðbn  f 0 ðnÞcN Þ.

(35)

Thus, function F m is surjective on ½0 : N  1 since for any integer n 2 ½0 : N  1, there is an integer p 2 ½0 : N  1 such that n ¼ F m ðpÞ. p is given by p ¼ bn  f 0 ðnÞcN . Since function F m is defined on a finite set, surjectivity implies bijectivity and we have the expected result. Conversely, we make the assumption that F m ðnÞ is bijective on ½0 : N  12 . We have to show that there exists a function f 0 ðnÞ such that (16) is satisfied. By assumption, F m ðnÞ is bijective on ½0 : N  12 and we denote H 2 ðnÞ as the reciprocal function, i.e. H 2 ðnÞ ¼ F 1 m ðnÞ. We define the following function H: HðnÞ ¼ dneN  F q ðH 2 ðnÞÞ þ H 2 ðnÞ.

(36)

Then it is possible to check that f 0 ðnÞ ¼ n  HðnÞ fulfills relation (16) proving invertibility of PCCf .

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A.2. Stationarized autocorrelation function at the PCC output We denote xðnÞ as a WSS process. This process is filtered by PCCf with a period N. As a particular LPTV filter, PCCf turns the input WSS process xðnÞ into an N-WSC process yðnÞ. We denote Rx ðpÞ and Ry ðp; nÞ as the autocorrelation functions of, respectively, xðnÞ and yðnÞ. According to (19), the stationarized autocorrelation Rsta y ðpÞ of the N-WSC process yðnÞ is given by Rsta y ðpÞ ¼

1 X 1N E½yðp þ rÞy ðrÞ. N r¼0

(37)

Since yðnÞ is the output of the PCC, yðnÞ is given by the relation (6) in terms of f ðnÞ and xðnÞ. Therefore, relation (37) can be reformulated as Rsta y ðpÞ ¼

1 X 1N E½xðp þ r  f ðp þ rÞÞx ðr  f ðrÞÞ. N r¼0

(38) But, xðnÞ is a WSS process and its autocorrelation function is defined by Rx ðpÞ ¼ E½xðnÞx ðn  pÞ. Thus, the stationarized autocorrelation function (38) can be expressed by the following relation according to the input autocorrelation function Rx ðpÞ: Rsta y ðpÞ ¼

1 X 1N Rx ðp þ f ðrÞ  f ðr þ pÞÞ. N r¼0

(39)

Finally, we get the expected result if we define the function Lðr; pÞ by Lðr; pÞ ¼ p þ f ðrÞ  f ðp þ rÞ ¼ F ðp þ rÞ  F ðrÞ.

AN ðnÞ ¼ dneN .

233

ð44Þ

In addition, for the ðQ; PÞ matrix interleaver, we have the expression (13) for the PCC function. According to this expression (13) of f ðnÞ and since F ðnÞ ¼ n  f ðnÞ (by definition (8)), it is easy to derive the following expression for F ðnÞ: F ðnÞ ¼ NAN ðnÞ þ QcP ðnÞ þ bP;Q ðnÞ.

(45)

Furthermore, the function Lðr; pÞ is defined by relation (21). Then, using the previous expression (45) for F ðnÞ, we have for Lðr; pÞ: Lðr; pÞ ¼ NAN ðp þ rÞ þ Q½cP ðp þ rÞ  cP ðrÞ þ bP;Q ðp þ rÞ  bP;Q ðrÞ.

ð46Þ

Now, we substitute cP ðp þ rÞ and cP ðrÞ thanks to relation (41) for p þ r and r, and we get the following expression for Lðr; pÞ under the assumption r 2 ½0 : N  1: Lðr; pÞ ¼ NAN ðp þ rÞð1  QÞ þ ½bP;Q ðp þ rÞ  bP;Q ðrÞ½1  N þ Qp.

ð47Þ

Given an integer p 2 N, four cases have to be discussed for the expression of Lðr; pÞ depending on the value of r 2 ½0 : N  1. With the assumption r 2 ½0; N  1 and according to the definitions (42) and (43) of cP ðrÞ and bP;Q ðrÞ, we have cP ðrÞ 2 ½0 : P  1 and bP;Q ðrÞ 2 ½0 : Q  1. We propose to explain the method for the first case. Other cases would be discussed in a similar way. Case 1: ( 0pcP ðrÞpP  1  cP ðpÞ; (48) 0pbP;Q ðrÞpQ  1  bP ðpÞ:

ð40Þ Under the assumptions (48), we have

A.3. Theoretical expression of Lðr; pÞ for the ðQ; PÞ matrix interleaver According to Euclidian decomposition (13), it is possible to decompose any integer n 2 N according to the following unique decomposition with N ¼ PQ:

bP;Q ðp þ rÞ ¼ bP;Q ðpÞ þ bP;Q ðrÞ,

ð49Þ

AN ðp þ rÞ ¼ AN ðpÞ.

ð50Þ

As a consequence, relation (47) for Lðr; pÞ turns into Lðr; pÞ ¼ NAN ðpÞð1  QÞ þ bP;Q ðpÞ½1  N þ Qp.

(41)

(51)

Coefficient AN ðnÞ, bP;Q ðnÞ and cP ðnÞ are unique and are defined by the three following relations: ð42Þ

Then, in this expression (51), we substitute p by the decomposition (41), and after simplification, we get

ð43Þ

Lðr; pÞ ¼ NAN ðpÞ þ QcP ðpÞ þ bP;Q ðpÞ.

n ¼ AN ðnÞN þ PbP;Q ðnÞ þ cP ðnÞ.

cP ðnÞ ¼ bncP , P

bP;Q ðnÞ ¼ bdne cQ ,

(52)

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Finally, by identification of (52) and (45), we conclude for the final expression of Lðr; pÞ: Lðr; pÞ ¼ F ðpÞ ¼ p  f ðpÞ.

(53)

Three other cases are possible. Given an integer p 2 N, four possible values for Lðr; pÞ are possible depending on the value of r 2 ½0 : N  1. These four different values are summed up by Fig. 11 and expressed by (22)–(25). A.4. Output stationarized autocorrelation function for the ðQ; PÞ matrix interleaver We assume that the input process xðnÞ is a process with a stationarized autocorrelation function Rx ðmÞ fulfilling the property: jRx ðmÞj ¼ 0 for me½ðL 1Þ : ðL  1Þ. We consider that xðnÞ is processed by a ðQ; PÞ matrix interleaver and we denote yðnÞ as the output process. We know that this process is NWSC (where N ¼ PQÞ and that the autocorrelation function of the stationarized process Rsta y ðpÞ is given by the expression (20) where function Lðr; lÞ is given by relations (22)–(25). In addition, we make the assumption that QXL. Given a value of pNþ (value for pN will be deduced thanks to the hermitian symmetry), the function Rsta y ðpÞ is given by the sum (20). According to relations (22)–(25), Lðr; pÞ can reach only four different values. As a consequence, to determine Rsta y ðpÞ we have to denombrate how many time these four values are reached in the sum. According to Fig. 11, we have: Value of Lðr; pÞ

Occurence of this value in the sum Rsta y ðpÞ

p  f ðpÞ ¼ F ðpÞ p  f ðpÞþQðP1Þ¼F ðpÞ þ QðP  1Þ p  f ðpÞ þ 1  N ¼ FðpÞ þ 1  N p  f ðpÞ þ 1  Q ¼ F ðpÞ þ 1  Q

ðP  cP ðpÞÞðQ bP ðpÞÞ ðP  cP ðpÞÞbP ðpÞ cP ðpÞðQ  1  bP ðpÞÞ cP ðpÞðbP ðpÞ þ 1Þ

Finally, using results in this table, we have for Rsta y ðpÞ the following expression where the terms B1 , B2 , B3 and B4 are expressed by (55)–(58): 1 ½B1 ðpÞ þ B2 ðpÞ þ B3 ðpÞ þ B4 ðpÞ, N B1 ðpÞ ¼ ½P  cP ðpÞ½Q  bP ðpÞRx ðF ðpÞÞ, Rsta y ðpÞ ¼

ð54Þ ð55Þ

B2 ðpÞ ¼ ½P  cP ðpÞbP ðpÞRx ðF ðpÞ þ QðP  1ÞÞ, ð56Þ B3 ðpÞ ¼ cP ðpÞ½Q  1  bP ðpÞRx ðF ðpÞ þ 1  NÞ, ð57Þ B4 ðpÞ ¼ cP ðpÞ½bP ðpÞ þ 1Rx ðF ðpÞ þ 1  QÞ.

ð58Þ

A useful expression of F ðpÞ where AN ðnÞ, bP;Q ðnÞ and cP ðnÞ are defined by the three relations (42)–(44) is F ðpÞ ¼ NAN ðpÞ þ QcP ðpÞ þ bP;Q ðpÞ.

(59)   With the assumption QXL (with Rx ðmÞ ¼ 0 for me½ðL  1Þ : ðL  1Þ) and an integer p 2 Nþ as well as the expression (59) for F ðpÞ, it is therefore possible to show that the previous terms (55)–(58) are null in most cases. There are exactly 3L  2 þ nonnull values of Rsta y ðpÞ (for p 2 N ) that are given by (28)–(30). References [1] A. Duverdier, B. Lacaze, On the use of periodic clock changes to implement periodic time varying filters, IEEE Trans. Circuits Syst. 47 (2) (November 2000) 1152–1158. [2] D. Mc Lernon, One dimensional LPTV structures: derivations, interrelationships and properties, IEE Proc. Image Signal Process. 149 (5) (October 1999) 245–252. [3] P.P. Vaidyanathan, S.K. Mitra, Polyphase networks, block digital filtering, LPTV systems, and alias-free QMF banks: a unified approach based on pseudocirculants, IEEE Trans. Acoust. Speech Signal Process. 36 (3) (March 1988) 381–391. [4] S. Akkarakaran, P.P. Vaidyanathan, Bifrequency and bispectrum maps: a new look at multirate systems with stochastic inputs, IEEE Trans. Signal Process. 48 (3) (March 2000) 723–736. [5] W. Chauvet, B. Lacaze, D. Roviras, A. Duverdier, in: Proposition of an orthogonal LPTV filters set: application to spread spectrum multiuser system, ICASSP, Montreal, Canada, 2004. [6] D. Roviras, B. Lacaze, N. Thomas, in: Effects of discrete LPTV on stationary signals, ICASSP, Orlando, USA, May 2002, pp. 1217–1220. [7] J.S. Prater, C.M. Loeffler, Analysis and design of periodically time varying IIR filters with application to transmultiplexing, IEEE Trans. Acoust. Speech Signal Process. 40 (11) (1992) 2715–2725. [8] K. Tsatsanis, G.B. Giannakis, Transmitter induced cyclostationarity for blind channel equalization, IEEE Trans. Signal Process. 45 (7) (July 1997). [9] A.G. Orozco Lugo, D.C. McLernon, An Application of Linear Periodically Time Varying Digital Filters to Blind Equalisation, IEE, Savoy Place, London, UK, 1998. [10] R. Ishii, M. Kakishita, A design method for periodically time varying digital filter for spectrum scrambling, IEEE Trans. Acoust. Speech Signal Process. 38 (7) (1990) 1219–1222. [11] W. Chauvet, B. Lacaze, D. Roviras, A. Duverdier, Analysis banks, synthesis banks, LPTV filters: proposition of an equivalence definition and application to the design of invertible LPTV filters, in: European Signal Processing Conference, Antalya, Turkey, September 2005. [12] R. Ansari, B. Liu, Interpolators and decimators as periodically time varying filters, in: Proceedings of the IEEE International Symposium on Circuits and Syst., 1981, pp. 447–450. [13] B. Lacaze, Stationary clock changes on stationary processes, Signal Process. 55 (2) (December 1996) 191–205.

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