On spectral theory of cyclostationary signals in multirate systems

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 7, JULY 2005 2421

On Spectral Theory of CyclostationarySignals in Multirate Systems

Jiandong Wang, Tongwen Chen, and Biao Huang

Abstract—This paper studies two problems in the spectral theoryof discrete-time cyclostationary signals: the cyclospectrum repre-sentation and the cyclospectrum transformation by linear multi-rate systems. Four types of cyclospectra are presented, and theirinterrelationships are explored. In the literature, the problem ofcyclospectrum transformation by linear systems was investigatedonly for some specific configurations and was usually developedwith inordinate complexities due to lack of a systematic approach.A general multirate system that encompasses most common sys-tems—linear time-invariant systems and linear periodically time-varying systems—is proposed as the unifying framework; more im-portantly, it also includes many configurations that have not beeninvestigated before, e.g., fractional sample-rate changers with cy-clostationary inputs. The blocking technique provides a systematicsolution as it associates a multirate system with an equivalent lineartime-invariant system and cyclostationary signals with stationarysignals; thus, the original problem is elegantly converted into a rel-atively simple one, which is solved in the form of matrix multipli-cation.

Index Terms—Bispectrum, blocking, cyclic spectrum, cyclosta-tionarity, multirate systems, time-frequency representation, two-dimensional spectrum.

I. INTRODUCTION

Adiscrete-time signal is said to be cyclostationary, orstrictly speaking cyclo-wide-sense-stationary, if its mean

and/or autocorrelation are periodically time-varying sequences[16], [17], [44]. Discrete-time cyclostationary signals oftenarise due to the time-varying nature of physical phenomena,e.g., the weather [26], and certain man-made operations, e.g.,the amplitude modulation, fractional sampling, and multiratesystem filtering [14], [16]. The spectral theory of cyclosta-tionary signals has applications in different areas, e.g., blindchannel identification and equalization by fractional samplingreceived signals [41], [42], filterbank optimization by mini-mizing averaged variances of reconstruction errors [28], [32],

Manuscript received December 1, 2003; revised July 13, 2004. This workwas supported by the Natural Sciences and Engineering Research Council ofCanada. The first author acknowledges the support from Alberta Ingenuity Fundin the form of the Alberta Ingenuity Ph.D. Studentship, from Informatics Circleof Research Excellence in the form of the iCore Graduate Student Scholarship,and from University of Alberta in the form of the Izaak Walton Killam MemorialScholarship. The associate editor coordinating the review of this manuscript andapproving it for publication was Prof. Fredrik Gustafsson.

J. Wang and T. Chen are with the Department of Electrical and ComputerEngineering, University of Alberta, Edmonton, AB, Canada T6G 2V4 (e-mail:[email protected]; [email protected]).

B. Huang is with the Department of Chemical and Materials Engi-neering, University of Alberta, Edmonton, AB, Canada T6G 2G6 (e-mail:[email protected]).

Digital Object Identifier 10.1109/TSP.2005.849192

and system identification by introducing cyclostationary ex-ternal excitation [13], [15] and by fast sampling system outputs[36], [43].

The spectral theory of discrete-time cyclostationary signalsmainly consists of two parts, namely, the cyclospectrum rep-resentation and the cyclospectrum transformation by linearsystems. Here, the cyclospectrum is the counterpart of thepower spectrum defined for discrete-time stationary or strictlyspeaking wide-sense-stationary signals. The theory was firstdeveloped by Gladyshev [17]; a complex function that iscurrently referred to as the cyclic spectrum was defined as thespectrum of a -periodically correlated1 sequence; the spec-tral relationship between the original sequence and a higherdimensional sequence that is actually the blocked signal wasdiscussed. Motivated by the sampling operation, the cyclicspectrum of discrete-time cyclostationary signals was defined,but only a very limited study has been given in Gardner’sbooks [11], [12]; as a complement, linear time-invariant (LTI)and linear periodically time-varying (LPTV) filtering of cy-clostationary signals was discussed briefly in [14]. Using theGardner’s notation (e.g., that in [12]), Ohno and Sakai [28]derived the output cyclic spectrum of a filterbank (an LPTVsystem) mostly from definitions and used it in the optimalfilterbank design. To avoid the cumbersome derivation in [28],Sakai and Ohno [32] studied the cyclic spectrum relationshipsamong the original, the modulated, and the blocked signals, andobtained the same expression of the cyclic spectrum in [28] viathese relationships. In an excellent overview [16], Giannakispresented some results in terms of the cyclic spectrum on theLPTV filtering, fractional sampling, and multirate processing.Besides the cyclic spectrum, there are some other cyclospectra,namely, the time-frequency representation (TFR), the bispec-trum, and the two-dimensional (2-D) spectrum. After giving anobservation that the cyclic spectrum is not “very illustrative”(a character actually caused by derivation without a systematicapproach), Lall et al. [21] analyzed the output of a filterbank interms of the TFR. Akkarakaran and Vaidyanathan [1] used thebispectrum as a tool to generalize most results in [33] (studyingeffects of multirate blocks on scalar cyclostationary signals)into the vector case; they also gave the bispectrum of the outputof a single-input and single-output (SISO) LPTV system andfound the conditions under which a SISO LPTV system wouldproduce stationary outputs for all stationary inputs. The 2-Dspectrum, indeed a coordinate transform of the bispectrum,was proposed in the context of periodic random processes in

1“Periodically correlated” is a synonym of “cyclostationary” mainly used inthe mathematical field [8].

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2422 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 7, JULY 2005

[38]–[40], where it was related to the cyclic spectrum and theTFR.

These four types of cyclospectra should have some interrela-tionships, since they all describe second-order statistical prop-erties of cyclostationary signals. The first contribution of thispaper, which is also of some tutorial value, is to summarize thesecyclospectra and find their interrelationships. As shown later,they are indeed related to each other and mutually convertible,even though each has its own features and may be superior toothers in one specific context or other.

The kernel problem of the spectral theory is the cyclospec-trum transformation by linear systems, i.e., to represent the cy-clospectrum of the system output in terms of that of the input.For this problem, there exist two limitations in the above citedliterature: First, only some specific configurations have been in-vestigated, e.g., the input of an -band filterbank has to be eitherstationary [28], [32] or cyclostationary with the same period as[21] ; second, most of existing results, e.g., the cyclic spectrumof the output of an LPTV system ([16, Eq. (17.44)]), are devel-oped via definitions, and hence, their derivations are so over-whelming that generalization to more complex systems, e.g.,multirate systems, becomes almost impossible, unless a system-atic approach is adopted [see the two proofs of (21) in Example4 and in the Appendix]. The second contribution of this paperis to remove these limitations: The problem of the cyclospec-trum transformation is attacked in the framework of multiratesystems using the blocking technique.

A discrete-time linear system can always be represented by aGreen’s function as

(1)

where , which is the set of integers [6]. A linear SISOmultirate system2 has the so-called -shift invarianceproperty ( and are integers) if shifting the input by sam-ples results in shifting the output by samples [4]. In terms ofGreen’s function, -shift invariance is characterized by

(2)

Fig. 1 depicts such a linear SISO multirate system in whichthe notation “ ” denotes that is -shift in-variant. Such a multirate system covers many familiar sys-tems as special cases, e.g., the LTI system , theLPTV system , and the cascade of upsampler, LTIsystem, and downsampler ( and are coprime) depicted inFig. 2.

Blocking in signal processing [27], [45] or lifting in control[3], [20] has been shown to be a powerful technique in dealingwith multirate systems and cyclostationary signals; by theblocking technique, one can associate the multirate system withan equivalent multi-input and multi-output LTI system [20],[27]; blocking the cyclostationary signal can result in a higherdimensional stationary signal [17], [32], [33].

Therefore, our main idea is to block multirate systemsand cyclostationary signals properly and convert the original

2SISO multirate systems are also called dual-rate systems [4].

Fig. 1. Linear SISO multirate system.

Fig. 2. Cascade of upsampler (" m), LTI system (H), and downsampler (# n).

problem into one involving LTI systems and stationary signalsonly, which can be readily solved using some well-knownresults. More specifically, the kernel problem is separated intothe following two subquestions.

• Given a linear SISO multirate system in Fig. 1and that the input is cyclo-wide-sense-stationary withperiod and abbreviated as CWSS , is the outputstationary or cyclostationary? If is cyclostationary, whatis its period?

• What is the cyclospectrum transformation in Fig. 1, i.e.,how do we represent the cyclospectrum of in terms ofthat of ?

The rest of the paper is organized as follows. Section II aimsat summarizing the different cyclospectra and exploring theirinterrelationships. Section III studies the effects of the blockingoperation on statistical properties of cyclostationary signals.Section IV answers the two subquestions and presents someexamples as illustration. Finally, Section V provides concludingremarks.

II. CYCLOSPECTRUM

We first review some basic concepts of stationary signals andintroduce cyclostationarity and four types of cyclospectra. Next,these cyclospectra are shown to be related to each other. Afterpresenting the cyclospectrum transformation by LTI systems,we choose the cyclic spectrum as the representation of the cyclo-spectrum in the rest of the sections.

A. Stationary Signals

A discrete-time signal , perhaps vector-valued, is said to bestationary or wide-sense-stationary if its mean is constant

(3)

and its autocorrelation depends only on the time difference

(4)

for all integers and [29]. Here, superscript denotes theconjugate transpose. The power spectrum of is defined as thediscrete-time Fourier transform (DTFT) of the autocorrelation

(5)

It is well known that when a stationary signal with powerspectrum is passed through an LTI system with transferfunction , the output is also a stationary signal with powerspectrum [23], [29]

(6)

WANG et al.: SPECTRAL THEORY OF CYCLOSTATIONARY SIGNALS IN MULTIRATE SYSTEMS 2423

B. Cyclostationary Signals

A discrete-time signal is called cyclostationary or cyclo-wide-sense-stationary with period ( being a positive integer)[16], [17], [44] if its mean is periodic

(7)

and/or its autocorrelation is pe-riodic such that

(8)

If , (7) and (8) imply that (3) and (4) hold, i.e., stationarysignals can be regarded as cyclostationary signals with period 1.

There are mainly four types of cyclospectra, namely, thecyclic spectrum, the time frequency representation, the bispec-trum, and the 2-D spectrum.

Cyclic Spectrum: We know from (8) that , whichis an equivalence of , is a periodic sequence of withperiod for a fixed . Therefore, has the followingdiscrete Fourier expansion:

(9)

where the discrete Fourier series coefficient is

(10)

The DTFT of is defined as the cyclic spectrum of[11], [17]

(11)

where the subscript stands for “cyclic power spectrum.”It follows from (8) and (10) that is periodic inwith period

(12)

The set thus

forms a full description of the cyclospectrum of a CWSSsignal . The cyclic spectrum can be considered as a gen-eralization of the power spectrum in (5); that is, if isstationary, , and for

.Time-Frequency Representation: can be con-

sidered also as a sequence of for a fixed . The time-frequencyrepresentation is defined as the DTFT of takingas the changing variable [21],

(13)

It follows from (8) that is periodic in with period

If , i.e., is stationary, , for all. The TFR is a broad concept that characterizes nonstationary

signals over a jointly time-frequency domain [2], [18]. It is alsoknown as Rihaczek spectrum [31] or the time-varying spectrum[34].

Bispectrum: The bispectrum is defined as the2-D DTFT of the autocorrelation [1], [29]

(14)Like the TFR, the bispectrum is also a very general conceptthat can describe the second-order statistical property of nonsta-tionary signals. Specifically, the bispectrum of a cyclostationarysignal lies on some parallel lines in the plane [1],

(15)

where . It is shown later that the bispectrum componenton the th line is exactly the th cyclic spectrum defined in (11).Note that the terminology “bispectrum” has a different meaningin the literature—the 2-D DTFT of the third-order moment [37].

Two-Dimensional Spectrum: The 2-D spectrum is defined asthe 2-D Fourier transform of [38]–[40]

(16)

The 2-D spectrum and bispectrum are very similar [see (18)later]; however, for a cyclostationary signal , is con-tinuous in and discrete in , but in (14) is contin-uous both in and . The 2-D spectrum is also referred to asthe dual-frequency spectrum [34], which is defined for nonsta-tionary signals.

These cyclospectra are related to each other. First, it followsfrom (10), (11), and (13) that the cyclic spectrum and the TFRare a discrete Fourier transform pair [21]

(17)

Second, it follows easily from (14) and (16) that the bispectrumand the 2-D spectrum are a coordinate transform of each otherwith a scaling factor

(18)

Third, the th cyclic spectrum is exactly the bispectrum compo-nent that lies on the th line described in (15), i.e.,

(19)where is an integer, and denotes the Dirac delta function[16]. Certainly, the th cyclic spectrum is also related to the 2-Dspectrum [39]


Fig. 3. Interrelationships among the four cyclospectra.

Finally, the TFR and the 2-D spectrum are a DTFT pair [38],

(20)

These interrelationships are shown in Fig. 3.Since these cyclospectra are mutually convertible, it is suffi-

cient to use only one of them to represent the cyclospectrum inthe rest of the development. Before making a choice, we intro-duce the cyclospectrum transformation by LTI systems to fur-ther capture the characteristics of these cyclospectra.

Let a CWSS signal be the input of a discrete-time LTIsystem with transfer function . As LTI systems preservethe cyclostationarity [1], [21], [33], the output is CWSS .The cyclospectrum of is associated with that of as follows.In terms of the cyclic spectrum

(21)

where . Similar observations to (21) were noticed in[14] and [42] without proofs, which are provided in Example 4and the Appendix using two different approaches. The TFR andbispectrum of were given in [21] and [1], respectively

(22)

(23)

Here, is the impulse response of . Equations (18) and (23)give the 2-D spectrum of

(24)

The TFR is introduced because “the representation of thecyclostationary processes, in terms of cyclic spectral density3,which, although a means of characterization, is not very illus-trative, particularly in the context of filterbank analysis” [21];however, (22) reveals that the TFR is not compact. The bispec-trum is originally defined for nonstationary signals so that it hasbeen cautioned to be unwieldy in mathematics [30] or too gen-eral and inefficient for indiscriminate use [1]. The 2-D spectrumis simply a coordinate transformation of the bispectrum, andthus, they share the same problems. The cyclic spectrum is de-fined in the way of incorporating the spectral information alongwith periodicity, and thus, it displays directly the fundamentalcharacteristic of cyclostationary signals; even though (21) is notas compact as (23) and (24), the cyclic spectrum is very conve-nient once all the th cyclic spectra are enclosed in the so-calledcyclic spectrum matrix [17], [32]. Therefore, we will use the

3The cyclic spectral density is synonymous with the cyclic spectrum.

Fig. 4. Blocking a linear SISO multirate system.

cyclic spectrum in the sequel, and the subscript is droppedwithout confusion.

III. BLOCKING OPERATOR

In Section I, we have briefly discussed the idea of blockingmultirate systems and cyclostationary signals properly to forma relatively simple problem. This section devotes to studyingthe effects of the blocking operator on multirate systems andcyclostationary signals.

First, we review the definition of the blocking operator. Letbe a discrete-time signal defined on , which is the set of non-negative integers. The -fold discrete blocking operator isdefined as the mapping from to , where underlining denotesblocking [20], [27]:

......

(25)

If the underlying sampling frequency of is , that of theblocked signal is . Meanwhile, the signal’s dimension in-creases by a factor of . It can be shown from both the time andfrequency domains that no information is lost in the blockingoperation. The inverse of the blocking operator is defined asthe reverse operation of (25); thus, and ,where denotes the identity system.

One of the advantages of the blocking operator is that it canassociate multirate systems that are essentially time-varyingwith some equivalent LTI systems to which many existingLTI techniques can be applied. For the multirate system inFig. 1, blocking the input and the output by and ,respectively, yields a blocked system , whichhas inputs and outputs. The blocking procedure is depictedin Fig. 4. As is -shift invariant (see (2)), is LTI [27]and has an transfer matrix [4]

......

...(26)

whose entries relate to the Green’s function of [see (1)] as

(27)

Second, we focus on the effects of the blocking operator oncyclostationary signals. The original and the blocked signalshave the following statistical relationships.

Lemma 1: A scalar signal is CWSS iff its -fold blockedversion is CWSS .


Fig. 5. Decompose L into L and L .

Proof: The case of was proved from definitionsof stationarity and cyclostationarity in [32] and [33], indepen-dently. The general case was given via bispectrum in [1].

Theorem 1: An -dimensional vector signal is CWSS iffits -fold blocked version is CWSS .

Proof: It can certainly be proved from definitions; how-ever, here, we take a shortcut. Followed from Lemma 1, ascalar signal is CWSS iff its -fold blocked version

is CWSS . Equation (25) implies that can be de-composed into a series of and , as depicted in Fig. 5.Lemma 1 and (25) give that , which is the output of ,is a CWSS -dimensional vector signal. Therefore, isCWSS iff its -fold blocked signal is CWSS .

Gladyshev [17] first proposed the relationship between acyclic spectrum matrix of the original signal and the powerspectrum of the blocked signal, which was proved by Sakai andOhno [32] for scalar signals. Akkarakaran and Vaidyanathan[1] also noticed this kind of relationship between bispectra.Here, we offer a theorem describing this relationship in termsof the cyclic spectrum for vector signals.

Theorem 2: The cyclic spectrum matrix of aCWSS -dimensional vector signal is connected with the

power spectrum of its -fold blocked version as

(28)

where is an matrix whose th blockcomponent is determined by the cyclic spectrum of

(29)

and is an unitary matrix whose th blockcomponent is

(30)

Here, , , , andis a identity matrix.

Proof: Theorem 1 says that the blocked signal is sta-tionary; thus, it has a power spectrum . From (5) and(25), the th component of is

An identity

is an integerotherwise

is used to change by a new variable

where the second equality is reached by replacing with a vari-able , and the last equality follows from thecyclostationarity property described in (8). From (9) and (11)

(31)

Next, we show that (31) is equivalent to

(32)

Comparing (31) and (32), their difference for a certainis as in the equation shown at the bottom of the next

page, where the second equality uses the periodicity of the cyclicspectrum in (12), and the third equality is obtained by changingvariables and in the last two summingterms, respectively. Finally, (28) is obtained from (32).

Remark: There are two important differences betweenTheorem 2 and its counterpart in [32]: first, the proof in [32]takes a modulation representation of cyclostationary signalsas an intermediate step, whereas we attack the problem moredirectly, and thus, the proof is much simpler; second, the resultin [32] holds under a condition

(33)

which is actually introduced by the modulation representation,whereas our proof shows that (33) is superfluous. Removing thelimiting condition (33) is extremely important, because the validrange of using the blocking technique will become too small tobe meaningful, if (33) has to be satisfied.

IV. CYCLOSTATIONARY SIGNALS IN MULTIRATE SYSTEMS

We are ready to attack the two subquestions proposed inSection I using the blocking technique. The first subquestionis answered in Theorem 3; the second is solved in the form


Fig. 6. Blocked SISO multirate system.

of matrix multiplication. Both are followed by some specificconfigurations as illustration.

A. Cyclostationarity of the Output

There are basically three approaches to answer the first sub-question: i) to explicitly write out some statistics of the systemoutput, e.g., the autocorrelation or the cyclic spectrum, as whatwas done in [1], [21]; ii) to reduce the multirate system into sim-pler building blocks such a upsamplers, LTI systems, and down-samplers and study cyclostationary properties of each block, asin [1], [33]; iii) to use the blocking technique. The last approachis the simplest for most systems and hence is adopted here.

Theorem 3: Given a linear SISO multirate systemin Fig. 1 and a CWSS input , the output

is CWSS , where means the greatestcommon divisor of and .

Proof: The use of the blocking operator needs to complywith two principles; the blocked system needs to be LTI, andthe blocked signal is stationary. Thus, the fold of the blockingoperator at the input side must be an integer multiple of as

is LTI (see Fig. 4) and, at the same time, be an integermultiple of (see Theorem 1). Blocking by and by ,as depicted in Fig. 6, will satisfy the two principles, where

(34)

In Fig. 6, the blocked system is LTI, and theblocked input is stationary. Therefore, the blocked outputis stationary, which, according to Lemma 1, implies that isCWSS .

Example 1: The multirate system is indeed an LTI systemif , under which Theorem 3 says that if the input isCWSS , then the output is CWSS too. In other words, LTI

systems preserve the cyclostationarity, which is consistent withthe conclusions in [1], [16], [21], and [33].

Example 2: If , the multirate systemreduces to an LPTV system that appears frequently in signalprocessing and control, such as multirate filterbanks [45] andLPTV controllers [10], [20]. More specifically, if the input isstationary or CWSS , Theorem 3 gives that the output of anLPTV system with period (i.e., ) is CWSS .This conclusion is consistent with those in [1], [16], [33].

Examples 1 and 2 are both with . For , the mul-tirate system is also named the fractional sample-rate changer,which has two configurations as follows. First, if and arecoprime, the multirate system is equivalent to the cascadeof upsampler, LTI system and downsampler, depicted in Fig. 2,which has been studied extensively [9], [24], [33], [35], [45].Second, if and have some nontrivial common factor,is not equivalent to the cascade system in Fig. 2 [4], [35]; itstands for a more general building block that finds applicationsin the nonuniform filterbanks [4] and the multichannel nonuni-form transmultiplexers [22].

Example 3: For the cascade system in Fig. 2 ( and are co-prime), if the input is stationary , Theorem 3 says thatthe output is CWSS , which is consistent with that in [33],which is obtained by analyzing the cascade in Fig. 2. As a com-parison, the other two approaches mentioned at the beginning ofthis subsection are explored for the same configuration. Clearly,the first approach has difficulties as the explicit statistical ex-pression of the system output has not been given in the litera-ture. The second approach proceeds as follows. The upsamplerand downsampler have the properties: If the input of a -foldupsampler is CWSS , the output will be CWSS [21]; ifthe input of a -fold downsampler is CWSS , the output isCWSS [33]. Applying the two properties to the cas-

cade system in Fig. 2 gives the same result.Remark: Theorem 3 can be verified by estimating the period

of the output via some numerical methods, e.g., Hurd–Gerr’smethod [19], Martin–Kedem’s method [25], Dandawaté–Gian-nakis’s method [7], and the variability method [44].

B. Cyclospectrum of the Output

We follow the same idea used in Section IV-A. Specifically,the multirate system and cyclostationary signals are blockedas that in Fig. 6 (see the proof of Theorem 3). Blocking theCWSS input by for the in (34) implies that ,

where is an integer. A general solution of the second subques-tion consists of two cases.


Fig. 7. Blocked SISO multirate system: pr = qn.

Case 1— : From Theorem 2, we have (see Fig. 6)

(35)

(36)

where . Since is stationary and isLTI, (6) gives

(37)

where is represented by the Green’s function of in (26).Therefore, the cyclic spectrum of is associated with that ofvia (35)–(37)

(38)

Case 2— : With , (36) and (37) stillhold. However, one more step is needed: is decomposedinto a series of and , which makes Fig. 6 equivalent toFig. 7. Theorem 2 gives

(39)

Since is stationary, its cyclic spectrum matrix is block diag-onal. Taking it as a special case of Theorem 2 results in

diag ...(40)

where , and diag denotes a diagonal ma-trix taking the elements of the operand vector as the diagonalentries. From (36), (37), (39), and (40), the cyclic spectrum of

is associated with that of , which is shown in (41) at the bottomof the page.

Remark: For a fixed frequency , either (38) or (41) can benumerically computed as the matrix multiplication.4

The next example is on the cyclospectrum transformation byLTI systems. The purpose of the example is three-fold: i) toillustrate what happens beyond the matrix multiplication in (38)and (41); ii) to give a concrete example showing a realization of(26); and iii) to provide an alternative proof of (21).

Example 4: Let be LTI and be CWSS , i.e.,and in Fig. 6. Equation (38) becomes

(42)

The unitary matrix is [see (30)]

(43)

An LTI system is fully characterized by its impulse response, i.e.,

(44)

Comparing (1) and (44) gives the connection between theGreen’s function and the impulse response

(45)

The blocked system has a transfer matrix [see(26)]

(46)

4See an example at http://www.ece.ualberta.ca/~jwang/research.htm.

diag ...

(41)


where (27) and (45) result

Thus, (46) becomes5

(47)

where and are the well-known type-1 polyphasecomponents of [45],

(48)

From (43) and (47), the product of the first three matrices in(42) is as in (49), shown at the bottom of the page, where thelast equality follows from (48). Note that the zeros on the off-di-agonal entries imply an exact spectrum cancelation. From (42),(49), and the cyclic spectrum matrix of [see (29)]

is obtained, as shown in the second equation at thebottom of the page, which also proves (21).

5The generalization of (47) is given in [3, Th. 8.2.1].

Fig. 8. Maximally decimated filterbank.

Fig. 9. Polyphase representation of a filterbank.

Multirate filterbanks are typical examples of LPTV systemsand have been of particular interest in signal processing [45].Unlike the approaches in [1], [21], [28], [32], and [33], the sta-tistical properties (in terms of cyclic spectrum) of the output orthe reconstructed signal of the filterbank are elegantly found bythe blocking technique in the next example.

Example 5: Fig. 8 depicts a maximally decimated filterbank[45], where and ( ) are analysisfilters and synthesis filters, respectively. Let the input beCWSS . With the type-1 polyphase representation of

[45], the type-3 polyphase representation of [9], and nobleidentities [45], Fig. 8 is equivalent to Fig. 9, where superscript

denotes the transpose.6 It is easy to see from (25) that

6A similar observation was also noticed in [5].

(49)


(50)

and in Fig. 9are exactly the blocked versions of and , respectively. Thus,the original filterbank in Fig. 8 is represented in terms of theblocked signals and with an LTI system . Here, both

and are systems; the th elements of their transfermatrices are

where and 7 relate to the impulse responsesand of and , respectively, as

Finally, the cyclic spectrum of the output can be associatedwith that of the input via the following three equations [see(38)]

V. CONCLUSION

In this paper, we have studied the spectral theory of discrete-time cyclostationary signals: the cyclospectrum representationand the cyclospectrum transformation by linear multirate sys-tems. The four types of cyclospectra, namely, the cyclic spec-trum, the time frequency representation, the bispectrum, and the2-D spectrum are shown to be closely related and mutually con-vertible (see Fig. 3). The cyclospectrum transformation by linearsystems are solved in a systematic manner by using multiratesystems as the unifying framework and the blocking techniqueas the main tool. The effects of the blocking operator on cy-clostationary signals are investigated in Theorems 1 and 2. Thecyclostationarity of the output of the multirate system is studied

7Superscripts ( ) and ( ) mean the type-1 and type-3 polyphase repre-sentations, respectively.

in Theorem 3, and the cyclospectrum of the output is associatedwith that of the input in the form of matrix multiplication in (38)and (41).

APPENDIX

This Appendix serves to prove (21) via definitions. From (11)and (10), we have

Let be the impulse response of the LTI system . The con-volution of and gives

Changing by a new variable yields (50), shownat the top of this page, where the second and the last equalitiesfollow from (9) and (11), respectively. By an identity

is an integerotherwise

we get , and thus, (50) results in (21).

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Jiandong Wang received the B.E. degree in auto-matic control from Beijing University of ChemicalTechnology, Beijing, China, in 1997 and the M.Sc.degree in electrical and computer engineering fromthe University of Alberta, Edmonton, AB, Canada,in 2003. Currently, he is pursuing the Ph.D. degree atthe University of Alberta.

From 1997 to 2001, he was a control engineerwith the Beijing Tsinghua Energy Simulation Com-pany. His current research work involves systemidentification, cyclostationary signal processing, and

multirate systems.Mr. Wang received the Izaak Walton Killam Memorial Scholarship, the Al-

berta Ingenuity Ph.D. Studentship, and the iCore Graduate Student Scholarshipat the University of Alberta, all since May of 2004.


Tongwen Chen received the B.Sc. degree from Ts-inghua University, Beijing, China, in 1984 and theM.A.Sc. and Ph.D. degrees from the University ofToronto, Toronto, ON, Canada, in 1988 and 1991, re-spectively, all in electrical engineering.

From 1991 to 1997, he was with the faculty of theDepartment of Electrical and Computer Engineering,University of Calgary, Calgary, AB, Canada. Since1997, he has been with the Department of Electricaland Computer Engineering, University of Alberta,Edmonton, AB, where he is presently a Professor of

electrical engineering. He held visiting positions at the Hong Kong Universityof Science and Technology and Kumamoto University, Kumamoto, Japan. Hiscurrent research interests include process control, multirate systems, robustcontrol, network-based control, digital signal processing, and their applicationsto industrial problems. He co-authored, with B.A. Francis, the book OptimalSampled-Data Control Systems (New York: Springer, 1995). Currently, he isan Associate Editor for Automatica, Systems and Control Letters, and DCDISSeries B.

Dr. Chen received a University of Alberta McCalla Professorship for2000/2001 and a Fellowship from the Japan Society for the Promotion ofScience for 2004. He was an Associate Editor for the IEEE TRANSACTIONS

ON AUTOMATIC CONTROL from 1998 to 2000. He is a registered ProfessionalEngineer in Alberta.

Biao Huang received the B.Sc. and M.Sc. degreesfrom Beijing University of Aeronautics and Astro-nautics, Beijing, China, in 1983 and 1986, respec-tively, and the Ph.D. degree from University of Al-berta, Edmonton, AB, Canada, in 1997.

He is currently a professor with the Departmentof Chemical and Materials Engineering, Universityof Alberta. His research interest is in the areas ofsystem identification, process control, and controlmonitoring.

On spectral theory of cyclostationary signals in multirate systems

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