On Spectral Theory of Cyclostationary Signals in Multirate Systems
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On Spectral Theory of Cyclostationary Signals in Multirate Systems Jiandong Wang, Tongwen Chen, and Biao Huang
Abstract—This paper studies two problems in the spectral theory of discrete-time cyclostationary signals: the cyclospectrum representation and the cyclospectrum transformation by linear multirate systems. Four types of cyclospectra are presented, and their interrelationships are explored. In the literature, the problem of cyclospectrum transformation by linear systems was investigated only for some specific configurations and was usually developed with inordinate complexities due to lack of a systematic approach. A general multirate system that encompasses most common systems—linear time-invariant systems and linear periodically timevarying systems—is proposed as the unifying framework; more importantly, it also includes many configurations that have not been investigated before, e.g., fractional sample-rate changers with cyclostationary inputs. The blocking technique provides a systematic solution as it associates a multirate system with an equivalent linear time-invariant system and cyclostationary signals with stationary signals; thus, the original problem is elegantly converted into a relatively simple one, which is solved in the form of matrix multiplication. Index Terms—Bispectrum, blocking, cyclic spectrum, cyclostationarity, multirate systems, time-frequency representation, twodimensional spectrum.
I. INTRODUCTION
A
discrete-time signal is said to be cyclostationary, or strictly speaking cyclo-wide-sense-stationary, if its mean and/or autocorrelation are periodically time-varying sequences [16], [17], [44]. Discrete-time cyclostationary signals often arise 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 multirate system filtering [14], [16]. The spectral theory of cyclostationary signals has applications in different areas, e.g., blind channel identification and equalization by fractional sampling received signals [41], [42], filterbank optimization by minimizing averaged variances of reconstruction errors [28], [32],
Manuscript received December 1, 2003; revised July 13, 2004. This work was supported by the Natural Sciences and Engineering Research Council of Canada. The first author acknowledges the support from Alberta Ingenuity Fund in the form of the Alberta Ingenuity Ph.D. Studentship, from Informatics Circle of Research Excellence in the form of the iCore Graduate Student Scholarship, and from University of Alberta in the form of the Izaak Walton Killam Memorial Scholarship. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Fredrik Gustafsson. J. Wang and T. Chen are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada T6G 2V4 (e-mail: [email protected]; [email protected]). B. Huang is with the Department of Chemical and Materials Engineering, 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 external excitation [13], [15] and by fast sampling system outputs [36], [43]. The spectral theory of discrete-time cyclostationary signals mainly consists of two parts, namely, the cyclospectrum representation and the cyclospectrum transformation by linear systems. Here, the cyclospectrum is the counterpart of the power spectrum defined for discrete-time stationary or strictly speaking wide-sense-stationary signals. The theory was first developed by Gladyshev [17]; a complex function that is currently referred to as the cyclic spectrum was defined as the spectrum of a -periodically correlated1 sequence; the spectral relationship between the original sequence and a higher dimensional sequence that is actually the blocked signal was discussed. Motivated by the sampling operation, the cyclic spectrum of discrete-time cyclostationary signals was defined, but only a very limited study has been given in Gardner’s books [11], [12]; as a complement, linear time-invariant (LTI) and linear periodically time-varying (LPTV) filtering of cyclostationary signals was discussed briefly in [14]. Using the Gardner’s notation (e.g., that in [12]), Ohno and Sakai [28] derived the output cyclic spectrum of a filterbank (an LPTV system) mostly from definitions and used it in the optimal filterbank design. To avoid the cumbersome derivation in [28], Sakai and Ohno [32] studied the cyclic spectrum relationships among the original, the modulated, and the blocked signals, and obtained the same expression of the cyclic spectrum in [28] via these relationships. In an excellent overview [16], Giannakis presented some results in terms of the cyclic spectrum on the LPTV filtering, fractional sampling, and multirate processing. Besides the cyclic spectrum, there are some other cyclospectra, namely, the time-frequency representation (TFR), the bispectrum, and the two-dimensional (2-D) spectrum. After giving an observation that the cyclic spectrum is not “very illustrative” (a character actually caused by derivation without a systematic approach), Lall et al. [21] analyzed the output of a filterbank in terms of the TFR. Akkarakaran and Vaidyanathan [1] used the bispectrum as a tool to generalize most results in [33] (studying effects of multirate blocks on scalar cyclostationary signals) into the vector case; they also gave the bispectrum of the output of a single-input and single-output (SISO) LPTV system and found the conditions under which a SISO LPTV system would produce stationary outputs for all stationary inputs. The 2-D spectrum, 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 in the mathematical field [8].
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[38]–[40], where it was related to the cyclic spectrum and the TFR. These four types of cyclospectra should have some interrelationships, since they all describe second-order statistical properties of cyclostationary signals. The first contribution of this paper, which is also of some tutorial value, is to summarize these cyclospectra 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 to others in one specific context or other. The kernel problem of the spectral theory is the cyclospectrum transformation by linear systems, i.e., to represent the cyclospectrum of the system output in terms of that of the input. For this problem, there exist two limitations in the above cited literature: First, only some specific configurations have been investigated, e.g., the input of an -band filterbank has to be either stationary [28], [32] or cyclostationary with the same period as [21] ; second, most of existing results, e.g., the cyclic spectrum of the output of an LPTV system ([16, Eq. (17.44)]), are developed via definitions, and hence, their derivations are so overwhelming that generalization to more complex systems, e.g., multirate systems, becomes almost impossible, unless a systematic approach is adopted [see the two proofs of (21) in Example 4 and in the Appendix]. The second contribution of this paper is to remove these limitations: The problem of the cyclospectrum transformation is attacked in the framework of multirate systems using the blocking technique. A discrete-time linear system can always be represented by a as Green’s function
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 signals only, which can be readily solved using some well-known results. More specifically, the kernel problem is separated into the following two subquestions. in Fig. 1 • Given a linear SISO multirate system and that the input is cyclo-wide-sense-stationary with period and abbreviated as CWSS , is the output stationary or cyclostationary? If is cyclostationary, what is its period? • What is the cyclospectrum transformation in Fig. 1, i.e., how do we represent the cyclospectrum of in terms of that of ? The rest of the paper is organized as follows. Section II aims at summarizing the different cyclospectra and exploring their interrelationships. Section III studies the effects of the blocking operation on statistical properties of cyclostationary signals. Section IV answers the two subquestions and presents some examples as illustration. Finally, Section V provides concluding remarks. II. CYCLOSPECTRUM
(1) where , which is the set of integers [6]. A linear SISO has the so-called -shift invariance multirate system2 property ( and are integers) if shifting the input by samples results in shifting the output by samples [4]. In terms of -shift invariance is characterized by Green’s function, (2) Fig. 1 depicts such a linear SISO multirate system in which ” denotes that is -shift inthe notation “ variant. Such a multirate system covers many familiar sys, the tems as special cases, e.g., the LTI system , and the cascade of upsampler, LTI LPTV system system, and downsampler ( and are coprime) depicted in Fig. 2. Blocking in signal processing [27], [45] or lifting in control [3], [20] has been shown to be a powerful technique in dealing with multirate systems and cyclostationary signals; by the blocking technique, one can associate the multirate system with an equivalent multi-input and multi-output LTI system [20], [27]; blocking the cyclostationary signal can result in a higher dimensional stationary signal [17], [32], [33]. Therefore, our main idea is to block multirate systems and cyclostationary signals properly and convert the original 2SISO
multirate systems are also called dual-rate systems [4].
We first review some basic concepts of stationary signals and introduce cyclostationarity and four types of cyclospectra. Next, these cyclospectra are shown to be related to each other. After presenting the cyclospectrum transformation by LTI systems, we choose the cyclic spectrum as the representation of the cyclospectrum in the rest of the sections. A. Stationary Signals A discrete-time signal , perhaps vector-valued, is said to be stationary 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 the conjugate transpose. The power spectrum of is defined as the discrete-time Fourier transform (DTFT) of the autocorrelation (5) It is well known that when a stationary signal with power is passed through an LTI system with transfer spectrum , the output is also a stationary signal with power function spectrum [23], [29] (6)
WANG et al.: SPECTRAL THEORY OF CYCLOSTATIONARY SIGNALS IN MULTIRATE SYSTEMS
B. Cyclostationary Signals A discrete-time signal is called cyclostationary or cyclowide-sense-stationary with period ( being a positive integer) [16], [17], [44] if its mean is periodic (7)
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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 also known as Rihaczek spectrum [31] or the time-varying spectrum [34]. is defined as the Bispectrum: The bispectrum 2-D DTFT of the autocorrelation [1], [29]
is pe-
and/or its autocorrelation riodic such that
(8) If , (7) and (8) imply that (3) and (4) hold, i.e., stationary signals can be regarded as cyclostationary signals with period 1. There are mainly four types of cyclospectra, namely, the cyclic spectrum, the time frequency representation, the bispectrum, and the 2-D spectrum. , which Cyclic Spectrum: We know from (8) that is an equivalence of , is a periodic sequence of with has the following period for a fixed . Therefore, discrete Fourier expansion: (9) where the discrete Fourier series coefficient
(14) Like the TFR, the bispectrum is also a very general concept that can describe the second-order statistical property of nonstationary signals. Specifically, the bispectrum of a cyclostationary plane [1], signal lies on some parallel lines in the (15) where . It is shown later that the bispectrum component on the th line is exactly the th cyclic spectrum defined in (11). Note that the terminology “bispectrum” has a different meaning in the literature—the 2-D DTFT of the third-order moment [37]. Two-Dimensional Spectrum: The 2-D spectrum is defined as [38]–[40] the 2-D Fourier transform of
is (16) (10)
The DTFT of [11], [17]
is defined as the cyclic spectrum of
(11) where the subscript stands for “cyclic power spectrum.” It follows from (8) and (10) that is periodic in with period (12) The set thus forms a full description of the cyclospectrum of a CWSS signal . The cyclic spectrum can be considered as a genis eralization of the power spectrum in (5); that is, if stationary, , and for . can be conTime-Frequency Representation: sidered also as a sequence of for a fixed . The time-frequency taking representation is defined as the DTFT of as the changing variable [21], (13) It follows from (8) that
is periodic in with period
The 2-D spectrum and bispectrum are very similar [see (18) is conlater]; however, for a cyclostationary signal , in (14) is contintinuous in and discrete in , but uous both in and . The 2-D spectrum is also referred to as the dual-frequency spectrum [34], which is defined for nonstationary signals. These cyclospectra are related to each other. First, it follows from (10), (11), and (13) that the cyclic spectrum and the TFR are a discrete Fourier transform pair [21] (17) Second, it follows easily from (14) and (16) that the bispectrum and the 2-D spectrum are a coordinate transform of each other with a scaling factor (18) Third, the th cyclic spectrum is exactly the bispectrum component 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-D spectrum [39]
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Fig. 4. Blocking a linear SISO multirate system.
Fig. 3.
cyclic spectrum in the sequel, and the subscript without confusion.
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 sufficient to use only one of them to represent the cyclospectrum in the rest of the development. Before making a choice, we introduce the cyclospectrum transformation by LTI systems to further capture the characteristics of these cyclospectra. Let a CWSS signal be the input of a discrete-time LTI system with transfer function . As LTI systems preserve the 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 4 and the Appendix using two different approaches. The TFR and bispectrum 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 the cyclostationary processes, in terms of cyclic spectral density3, which, although a means of characterization, is not very illustrative, particularly in the context of filterbank analysis” [21]; however, (22) reveals that the TFR is not compact. The bispectrum is originally defined for nonstationary signals so that it has been cautioned to be unwieldy in mathematics [30] or too general and inefficient for indiscriminate use [1]. The 2-D spectrum is simply a coordinate transformation of the bispectrum, and thus, they share the same problems. The cyclic spectrum is defined in the way of incorporating the spectral information along with periodicity, and thus, it displays directly the fundamental characteristic of cyclostationary signals; even though (21) is not as compact as (23) and (24), the cyclic spectrum is very convenient once all the th cyclic spectra are enclosed in the so-called cyclic spectrum matrix [17], [32]. Therefore, we will use the 3The
cyclic spectral density is synonymous with the cyclic spectrum.
is dropped
III. BLOCKING OPERATOR In Section I, we have briefly discussed the idea of blocking multirate systems and cyclostationary signals properly to form a relatively simple problem. This section devotes to studying the effects of the blocking operator on multirate systems and cyclostationary signals. First, we review the definition of the blocking operator. Let be a discrete-time signal defined on , which is the set of nonnegative integers. The -fold discrete blocking operator is defined as the mapping from to , where underlining denotes blocking [20], [27]:
.. .
(25)
.. .
If the underlying sampling frequency of is , that of the . Meanwhile, the signal’s dimension inblocked signal is creases by a factor of . It can be shown from both the time and frequency domains that no information is lost in the blocking is defined as operation. The inverse of the blocking operator the reverse operation of (25); thus, and , where denotes the identity system. One of the advantages of the blocking operator is that it can associate multirate systems that are essentially time-varying with some equivalent LTI systems to which many existing LTI techniques can be applied. For the multirate system in and , Fig. 1, blocking the input and the output by , which respectively, yields a blocked system has inputs and outputs. The blocking procedure is depicted -shift invariant (see (2)), is LTI [27] in Fig. 4. As is transfer matrix [4] and has an
.. .
.. .
whose entries relate to the Green’s function of
.. .
(26)
[see (1)] as (27)
Second, we focus on the effects of the blocking operator on cyclostationary signals. The original and the blocked signals have the following statistical relationships. Lemma 1: A scalar signal is CWSS iff its -fold blocked version is CWSS .
WANG et al.: SPECTRAL THEORY OF CYCLOSTATIONARY SIGNALS IN MULTIRATE SYSTEMS
is used to change
Fig. 5. Decompose L
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by a new variable
into L and L .
Proof: The case of was proved from definitions of stationarity and cyclostationarity in [32] and [33], independently. The general case was given via bispectrum in [1]. Theorem 1: An -dimensional vector signal is CWSS iff its -fold blocked version is CWSS . Proof: It can certainly be proved from definitions; however, here, we take a shortcut. Followed from Lemma 1, a iff its -fold blocked version scalar signal is CWSS is CWSS . Equation (25) implies that can be decomposed into a series of and , as depicted in Fig. 5. , Lemma 1 and (25) give that , which is the output of -dimensional vector signal. Therefore, is is a CWSS CWSS iff its -fold blocked signal is CWSS . Gladyshev [17] first proposed the relationship between a cyclic spectrum matrix of the original signal and the power spectrum of the blocked signal, which was proved by Sakai and Ohno [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 terms of the cyclic spectrum for vector signals. of a Theorem 2: The cyclic spectrum matrix CWSS -dimensional vector signal is connected with the of its -fold blocked version as power spectrum
where the second equality is reached by replacing with a variable , and the last equality follows from the cyclostationarity property described in (8). From (9) and (11)
(31) Next, we show that (31) is equivalent to
(28) is an matrix whose th where component is determined by the cyclic spectrum of
block (29)
is an and component is
unitary matrix whose
th
block
(30) Here, , , , and is a identity matrix. Proof: Theorem 1 says that the blocked signal is sta. From (5) and tionary; thus, it has a power spectrum (25), the th component of is
(32) Comparing (31) and (32), their difference for a certain is as in the equation shown at the bottom of the next page, where the second equality uses the periodicity of the cyclic spectrum in (12), and the third equality is obtained by changing and in the last two summing variables terms, respectively. Finally, (28) is obtained from (32). Remark: There are two important differences between Theorem 2 and its counterpart in [32]: first, the proof in [32] takes a modulation representation of cyclostationary signals as an intermediate step, whereas we attack the problem more directly, and thus, the proof is much simpler; second, the result in [32] holds under a condition (33) which is actually introduced by the modulation representation, whereas our proof shows that (33) is superfluous. Removing the limiting condition (33) is extremely important, because the valid range of using the blocking technique will become too small to be meaningful, if (33) has to be satisfied. IV. CYCLOSTATIONARY SIGNALS IN MULTIRATE SYSTEMS
An identity is an integer otherwise
We are ready to attack the two subquestions proposed in Section I using the blocking technique. The first subquestion is answered in Theorem 3; the second is solved in the form
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Fig. 6. Blocked SISO multirate system.
of matrix multiplication. Both are followed by some specific configurations as illustration. A. Cyclostationarity of the Output There are basically three approaches to answer the first subquestion: i) to explicitly write out some statistics of the system output, e.g., the autocorrelation or the cyclic spectrum, as what was done in [1], [21]; ii) to reduce the multirate system into simpler building blocks such a upsamplers, LTI systems, and downsamplers and study cyclostationary properties of each block, as in [1], [33]; iii) to use the blocking technique. The last approach is the simplest for most systems and hence is adopted here. Theorem 3: Given a linear SISO multirate system in Fig. 1 and a CWSS input , the output is CWSS , where means the greatest common divisor of and . Proof: The use of the blocking operator needs to comply with two principles; the blocked system needs to be LTI, and the blocked signal is stationary. Thus, the fold of the blocking operator at the input side must be an integer multiple of as is LTI (see Fig. 4) and, at the same time, be an integer and by , multiple of (see Theorem 1). Blocking by as depicted in Fig. 6, will satisfy the two principles, where (34) In Fig. 6, the blocked system is LTI, and the blocked input is stationary. Therefore, the blocked output is stationary, which, according to Lemma 1, implies that is CWSS . Example 1: The multirate system is indeed an LTI system , under which Theorem 3 says that if the input is if CWSS , then the output is CWSS too. In other words, LTI systems preserve the cyclostationarity, which is consistent with the conclusions in [1], [16], [21], and [33].
, the multirate system Example 2: If reduces to an LPTV system that appears frequently in signal processing and control, such as multirate filterbanks [45] and LPTV controllers [10], [20]. More specifically, if the input is stationary or CWSS , Theorem 3 gives that the output of an ) is CWSS . LPTV system with period (i.e., This conclusion is consistent with those in [1], [16], [33]. . For , the mulExamples 1 and 2 are both with tirate system is also named the fractional sample-rate changer, which has two configurations as follows. First, if and are is equivalent to the cascade coprime, the multirate system of upsampler, LTI system and downsampler, depicted in Fig. 2, which has been studied extensively [9], [24], [33], [35], [45]. and have some nontrivial common factor, Second, if is not equivalent to the cascade system in Fig. 2 [4], [35]; it stands for a more general building block that finds applications in the nonuniform filterbanks [4] and the multichannel nonuniform transmultiplexers [22]. Example 3: For the cascade system in Fig. 2 ( and are coprime), if the input is stationary , Theorem 3 says that the output is CWSS , which is consistent with that in [33], which is obtained by analyzing the cascade in Fig. 2. As a comparison, the other two approaches mentioned at the beginning of this subsection are explored for the same configuration. Clearly, the first approach has difficulties as the explicit statistical expression of the system output has not been given in the literature. The second approach proceeds as follows. The upsampler and downsampler have the properties: If the input of a -fold [21]; if upsampler is CWSS , the output will be CWSS the input of a -fold downsampler is CWSS , the output is CWSS [33]. Applying the two properties to the cascade 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’s method [19], Martin–Kedem’s method [25], Dandawaté–Giannakis’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 blocked as that in Fig. 6 (see the proof of Theorem 3). Blocking the CWSS input by for the in (34) implies that , where is an integer. A general solution of the second subquestion consists of two cases.
WANG et al.: SPECTRAL THEORY OF CYCLOSTATIONARY SIGNALS IN MULTIRATE SYSTEMS
Fig. 7.
Case 1—
Blocked SISO multirate system: pr
: From Theorem 2, we have (see Fig. 6) (35) (36)
where . Since LTI, (6) gives
is stationary and
is
(37) is represented by the Green’s function of in (26). where Therefore, the cyclic spectrum of is associated with that of via (35)–(37)
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= qn.
is associated with that of , which is shown in (41) at the bottom of the page. Remark: For a fixed frequency , either (38) or (41) can be numerically computed as the matrix multiplication.4 The next example is on the cyclospectrum transformation by LTI systems. The purpose of the example is three-fold: i) to illustrate 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) (38) : With , (36) and (37) still Case 2— is decomposed hold. However, one more step is needed: into a series of and , which makes Fig. 6 equivalent to Fig. 7. Theorem 2 gives
The unitary matrix
is [see (30)] (43)
An LTI system is fully characterized by its impulse response , i.e.,
(39) (44)
Since is stationary, its cyclic spectrum matrix is block diagonal. Taking it as a special case of Theorem 2 results in
Comparing (1) and (44) gives the connection between the Green’s function and the impulse response (45) diag
.. .
(40)
where , and diag denotes a diagonal matrix taking the elements of the operand vector as the diagonal entries. From (36), (37), (39), and (40), the cyclic spectrum of
diag
The blocked system (26)]
has a transfer matrix [see
(46) 4See
an example at http://www.ece.ualberta.ca/~jwang/research.htm.
.. . (41)
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where (27) and (45) result
Fig. 8. Maximally decimated filterbank.
Thus, (46) becomes5 (47) where and are the well-known type-1 polyphase components of [45],
Fig. 9. Polyphase representation of a filterbank.
(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 the last equality follows from (48). Note that the zeros on the off-diagonal 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 the bottom of the page, which also proves (21). 5The
generalization of (47) is given in [3, Th. 8.2.1].
Multirate filterbanks are typical examples of LPTV systems and have been of particular interest in signal processing [45]. Unlike the approaches in [1], [21], [28], [32], and [33], the statistical properties (in terms of cyclic spectrum) of the output or the reconstructed signal of the filterbank are elegantly found by the blocking technique in the next example. Example 5: Fig. 8 depicts a maximally decimated filterbank [45], where and ( ) are analysis filters and synthesis filters, respectively. Let the input be CWSS . With the type-1 polyphase representation of [45], the type-3 polyphase representation of [9], and noble identities [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)
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(50)
and in Fig. 9 are exactly the blocked versions of and , respectively. Thus, the original filterbank in Fig. 8 is represented in terms of the . Here, both blocked signals and with an LTI system and are systems; the th elements of their transfer matrices are
where and
and of and
in Theorem 3, and the cyclospectrum of the output is associated with 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
relate to the impulse responses , respectively, as 7
Finally, the cyclic spectrum of the output can be associated with that of the input via the following three equations [see (38)]
Let be the impulse response of the LTI system volution of and gives
. The con-
V. CONCLUSION In this paper, we have studied the spectral theory of discretetime cyclostationary signals: the cyclospectrum representation and the cyclospectrum transformation by linear multirate systems. The four types of cyclospectra, namely, the cyclic spectrum, the time frequency representation, the bispectrum, and the 2-D spectrum are shown to be closely related and mutually convertible (see Fig. 3). The cyclospectrum transformation by linear systems are solved in a systematic manner by using multirate systems as the unifying framework and the blocking technique as the main tool. The effects of the blocking operator on cyclostationary signals are investigated in Theorems 1 and 2. The cyclostationarity of the output of the multirate system is studied 7Superscripts (
) and (
sentations, respectively.
) mean the type-1 and type-3 polyphase repre-
Changing by a new variable yields (50), shown at the top of this page, where the second and the last equalities follow from (9) and (11), respectively. By an identity is an integer otherwise we get
, and thus, (50) results in (21). REFERENCES
[1] S. Akkarakaran and P. P. Vaidyanathan, “Bifrequency and bispectrum maps: A new look at multirate systems with stochastic inputs,” IEEE Trans. Signal Process., vol. 48, no. 3, pp. 723–736, Mar. 2000.
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[2] B. Boashash, “Time-frequency signal analysis: Past, present and future trends,” in Control and Dynamic Systems, C. T. Leondes, Ed. New York: Academic, 1996, vol. 78, pp. 1–69. [3] T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems. London, U.K.: Springer, 1995. [4] T. Chen, L. Qiu, and E. W. Bai, “General multirate building structures with application to nonuniform filterbanks,” IEEE Trans. Circuits Syst. II, vol. 45, no. 8, pp. 948–958, Aug. 1998. [5] T. Chen, “Nonuniform multirate filterbanks: Analysis and design with performance measure,” IEEE Trans. Signal Process., vol. 45, an no. 3, pp. 572–582, Mar. 1997. [6] R. E. Crochiere and L. R. Rabiner, Multirate Digital Signal Processing. Engelwood Cliffs, NJ: Prentice-Hall, 1983. [7] A. V. Dandawaté and G. B. Giannakis, “Statistical tests for presence of cyclostationarity,” IEEE Trans. Signal Process., vol. 42, no. 9, pp. 2355–2369, Sep. 1994. [8] D. Dehay and H. L. Hurd, “Representation and estimation for periodically and almost periodically correlated random processes,” in Cyclostationarity in Communications and Signal Processing, W. A. Gardner, Ed. Piscataway, NJ: IEEE Press, 1994. [9] N. J. Fliege, Multirate Digital Signal Processing. New York: Wiley, 1994. [10] B. A. Francis and T. T. Georgiou, “Stability theory for linear time-invariant plants with periodic digital controllers,” IEEE Trans. Autom. Control, vol. 33, no. 9, pp. 820–832, Sep. 1988. [11] W. A. Gardner, Statistical Spectral Analysis: A Nonprobabilistic Theory. Engelwood Cliffs, NJ: Prentice-Hill, 1988. , Introduction to Random Processes With Application to Signals and [12] Systems, Second ed. New York: McGraw-Hill, 1990. , “Identification of systems with cyclostationary input and corre[13] lated input/output measurement noise,” IEEE Trans. Autom. Control, vol. 35, no. 4, pp. 449–452, Apr. 1990. , “An introduction to cyclostationary signals,” in Cyclostationarity [14] in Communications and Signal Processing, W. A. Gardner, Ed. Piscataway, NJ: IEEE Press, 1994. [15] G. B. Giannakis, “Polyspectral and cyclostationary approaches for identification of closed-loop systems,” IEEE Trans. Autom. Control, vol. 40, no. 5, pp. 882–885, May 1995. , “Cyclostationary signal analysis,” in Digital Signal Processing [16] Handbook. Boca Raton, FL: CRC, 1999. [17] E. G. Gladyshev, “Periodically correlated random sequence,” Soviet Math., vol. 2, pp. 385–388, 1961. [18] F. Hlawatsch and G. F. Boudreaux-Bartels, “Linear and quadratic timefrequency signal representations,” IEEE Signal Process. Mag., vol. 9, no. 2, pp. 21–67, Apr. 1992. [19] H. L. Hurd and N. L. Gerr, “Graphical methods for determining the presence of periodic correlation,” J. Times Ser. Anal., vol. 12, no. 4, pp. 337–350, 1991. [20] P. P. Khargonekar, K. Poolla, and A. Tannenbaum, “Robust control of linear time-invariant plants using periodic compensation,” IEEE Trans. Autom. Control, vol. AC–30, no. 11, pp. 1088–1096, Nov. 1985. [21] B. Lall, S. D. Joshi, and R. K. P. Bhatt, “Second-order statistical characterization of the filterbank and its elements,” IEEE Trans. Signal Process., vol. 47, no. 6, pp. 1745–1749, Jun. 1999. [22] T. Liu and T. Chen, “Design of multichannel nonuniform transmultiplexers using general building blocks,” IEEE Trans. Signal Process., vol. 49, no. 1, pp. 91–99, Jan. 2001. [23] L. Ljung, System Identification: Theory for the User, Second ed. Englewood Cliffs, NJ: Prentice Hall, 1999. [24] C. M. Loeffler and C. S. Burrus, “Optimal design of periodically time-varying and multirate digital filters,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-32, no. 5, pp. 991–997, Oct. 1984. [25] D. E. K. Martin and B. Kedem, “Estimation of the period of periodically correlated sequences,” J. Times Ser. Anal., vol. 14, no. 2, pp. 193–205, 1993. [26] D. E. K. Martin, “Detection of periodic autocorrelation in time series data via zero-crossing,” J. Times Ser. Anal., vol. 20, no. 4, pp. 435–452, 1999. [27] R. A. Meyer and C. S. Burrus, “A unified analysis of multirate and periodically time-varying digital filters,” IEEE Trans. Circuits Syst., vol. CAS-22, no. 3, pp. 162–168, Mar. 1975. [28] S. Ohno and H. Sakai, “Optimization of filterbanks using cyclostationary spectral analysis,” IEEE Trans. Signal Process., vol. 44, no. 11, pp. 2718–2725, Nov. 1996.
H
[29] A. Papoulis, Probability, Random Variables and Stochastic Processes. New York: McGraw-Hill, 1965. [30] U. Petersohn, H. Unger, and N. J. Fliege, “Exact deterministic and stochastic analysis of multirate systems with application to fractional sampling rate alteration,” in Proc. IEEE ISCAS, 1994, pp. 177–180. [31] A. W. Rihaczek, “Signal energy distribution in time and frequency,” IEEE Trans. Inf. Theory, vol. 14, no. 3, pp. 369–374, May 1968. [32] H. Sakai and S. Ohno, “Theory of cyclostationary processes and its application,” in Statistical Methods in Control and Signal Processing. New York: Marcel Dekker, 1997, pp. 327–354. [33] V. P. Sathe and P. P. Vaidyanathan, “Effects of multirate systems on the statistical properties of random signals,” IEEE Trans. Signal Process., vol. 41, no. 1, pp. 131–146, Jan. 1993. [34] L. L. Scharf and B. Friedlander, “Toeplitz and hankel kernels for estimating time-varying spectra of discrete-time random processes,” IEEE Trans. Signal Process., vol. 49, no. 1, pp. 179–189, Jan. 2001. [35] R. G. Shenoy, “Multirate specifications via alias-component matrices,” IEEE Trans. Circuits Syst. II, vol. 45, no. 3, pp. 314–320, Mar. 1998. [36] L. Sun, H. Ohmori, and A. Sano, “Frequency domain approach to closedloop identification based on output inter-sampling scheme,” in Proc. Amer. Control Conf., vol. 3, Jun. 2000, pp. 1802–1806. [37] C. W. Therrien, “Overview of statistical signal processing,” in Digital Signal Processing Handbook. Boca Raton, FL: CRC, 1999. , “Issues in multirate statistical signal processing,” in Proc. [38] Thirty-Fifth Asilomar Conf. Signals, Syst. Comput., vol. 1, Nov. 2001, pp. 573–576. [39] C. W. Therrien and R. Cristi, “Two-dimensional spectral representation of periodic, cyclostationary, and more general random processes,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 4, May 2002, pp. 3561–3563. [40] C. W. Therrien, “Some considerations for statistical characterization of nonstationary random processes,” in Proc. Thirty-Sixth Asilomar Conf. Signals, Syst., Comput., vol. 2, Nov. 2002, pp. 1554–1558. [41] L. Tong, G. Xu, and T. Kailath, “Blind identification and equalization based on second-order statistics: A time domain approach,” IEEE Trans. Inf. Theory, vol. 40, no. 2, pp. 340–349, Mar. 1994. [42] L. Tong, G. Xu, B. Hassibi, and T. Kailath, “Blind identification and equalization based on second-order statistics: A frequency-domain approach,” IEEE Trans. Inf. Theory, vol. 41, no. 1, pp. 329–334, Jan. 1995. [43] J. Wang, T. Chen, and B. Huang, “Closed-loop identification via output fast sampling,” J. Process Control, vol. 14, no. 5, pp. 555–570, 2004. , “Cyclo-period estimation for discrete-time cyclostationary sig[44] nals,” IEEE Trans. Signal Process., to be published. [45] P. P. Vaidyanathan, Multirate Systems and Filterbanks. Englewood Cliffs, NJ: Prentice-Hall, 1993.
Jiandong Wang received the B.E. degree in automatic control from Beijing University of Chemical Technology, Beijing, China, in 1997 and the M.Sc. degree in electrical and computer engineering from the University of Alberta, Edmonton, AB, Canada, in 2003. Currently, he is pursuing the Ph.D. degree at the University of Alberta. From 1997 to 2001, he was a control engineer with the Beijing Tsinghua Energy Simulation Company. His current research work involves system identification, cyclostationary signal processing, and multirate systems. Mr. Wang received the Izaak Walton Killam Memorial Scholarship, the Alberta Ingenuity Ph.D. Studentship, and the iCore Graduate Student Scholarship at the University of Alberta, all since May of 2004.
WANG et al.: SPECTRAL THEORY OF CYCLOSTATIONARY SIGNALS IN MULTIRATE SYSTEMS
Tongwen Chen received the B.Sc. degree from Tsinghua University, Beijing, China, in 1984 and the M.A.Sc. and Ph.D. degrees from the University of Toronto, Toronto, ON, Canada, in 1988 and 1991, respectively, all in electrical engineering. From 1991 to 1997, he was with the faculty of the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada. Since 1997, he has been with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, where he is presently a Professor of electrical engineering. He held visiting positions at the Hong Kong University of Science and Technology and Kumamoto University, Kumamoto, Japan. His current research interests include process control, multirate systems, robust control, network-based control, digital signal processing, and their applications to industrial problems. He co-authored, with B.A. Francis, the book Optimal Sampled-Data Control Systems (New York: Springer, 1995). Currently, he is an Associate Editor for Automatica, Systems and Control Letters, and DCDIS Series B. Dr. Chen received a University of Alberta McCalla Professorship for 2000/2001 and a Fellowship from the Japan Society for the Promotion of Science for 2004. He was an Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL from 1998 to 2000. He is a registered Professional Engineer in Alberta.
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Biao Huang received the B.Sc. and M.Sc. degrees from Beijing University of Aeronautics and Astronautics, Beijing, China, in 1983 and 1986, respectively, and the Ph.D. degree from University of Alberta, Edmonton, AB, Canada, in 1997. He is currently a professor with the Department of Chemical and Materials Engineering, University of Alberta. His research interest is in the areas of system identification, process control, and control monitoring.
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On Spectral Theory of Cyclostationary Signals in Multirate Systems Jiandong Wang, Tongwen Chen, and Biao Huang
Abstract—This paper studies two problems in the spectral theory of discrete-time cyclostationary signals: the cyclospectrum representation and the cyclospectrum transformation by linear multirate systems. Four types of cyclospectra are presented, and their interrelationships are explored. In the literature, the problem of cyclospectrum transformation by linear systems was investigated only for some specific configurations and was usually developed with inordinate complexities due to lack of a systematic approach. A general multirate system that encompasses most common systems—linear time-invariant systems and linear periodically timevarying systems—is proposed as the unifying framework; more importantly, it also includes many configurations that have not been investigated before, e.g., fractional sample-rate changers with cyclostationary inputs. The blocking technique provides a systematic solution as it associates a multirate system with an equivalent linear time-invariant system and cyclostationary signals with stationary signals; thus, the original problem is elegantly converted into a relatively simple one, which is solved in the form of matrix multiplication. Index Terms—Bispectrum, blocking, cyclic spectrum, cyclostationarity, multirate systems, time-frequency representation, twodimensional spectrum.
I. INTRODUCTION
A
discrete-time signal is said to be cyclostationary, or strictly speaking cyclo-wide-sense-stationary, if its mean and/or autocorrelation are periodically time-varying sequences [16], [17], [44]. Discrete-time cyclostationary signals often arise 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 multirate system filtering [14], [16]. The spectral theory of cyclostationary signals has applications in different areas, e.g., blind channel identification and equalization by fractional sampling received signals [41], [42], filterbank optimization by minimizing averaged variances of reconstruction errors [28], [32],
Manuscript received December 1, 2003; revised July 13, 2004. This work was supported by the Natural Sciences and Engineering Research Council of Canada. The first author acknowledges the support from Alberta Ingenuity Fund in the form of the Alberta Ingenuity Ph.D. Studentship, from Informatics Circle of Research Excellence in the form of the iCore Graduate Student Scholarship, and from University of Alberta in the form of the Izaak Walton Killam Memorial Scholarship. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Fredrik Gustafsson. J. Wang and T. Chen are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada T6G 2V4 (e-mail: [email protected]; [email protected]). B. Huang is with the Department of Chemical and Materials Engineering, 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 external excitation [13], [15] and by fast sampling system outputs [36], [43]. The spectral theory of discrete-time cyclostationary signals mainly consists of two parts, namely, the cyclospectrum representation and the cyclospectrum transformation by linear systems. Here, the cyclospectrum is the counterpart of the power spectrum defined for discrete-time stationary or strictly speaking wide-sense-stationary signals. The theory was first developed by Gladyshev [17]; a complex function that is currently referred to as the cyclic spectrum was defined as the spectrum of a -periodically correlated1 sequence; the spectral relationship between the original sequence and a higher dimensional sequence that is actually the blocked signal was discussed. Motivated by the sampling operation, the cyclic spectrum of discrete-time cyclostationary signals was defined, but only a very limited study has been given in Gardner’s books [11], [12]; as a complement, linear time-invariant (LTI) and linear periodically time-varying (LPTV) filtering of cyclostationary signals was discussed briefly in [14]. Using the Gardner’s notation (e.g., that in [12]), Ohno and Sakai [28] derived the output cyclic spectrum of a filterbank (an LPTV system) mostly from definitions and used it in the optimal filterbank design. To avoid the cumbersome derivation in [28], Sakai and Ohno [32] studied the cyclic spectrum relationships among the original, the modulated, and the blocked signals, and obtained the same expression of the cyclic spectrum in [28] via these relationships. In an excellent overview [16], Giannakis presented some results in terms of the cyclic spectrum on the LPTV filtering, fractional sampling, and multirate processing. Besides the cyclic spectrum, there are some other cyclospectra, namely, the time-frequency representation (TFR), the bispectrum, and the two-dimensional (2-D) spectrum. After giving an observation that the cyclic spectrum is not “very illustrative” (a character actually caused by derivation without a systematic approach), Lall et al. [21] analyzed the output of a filterbank in terms of the TFR. Akkarakaran and Vaidyanathan [1] used the bispectrum as a tool to generalize most results in [33] (studying effects of multirate blocks on scalar cyclostationary signals) into the vector case; they also gave the bispectrum of the output of a single-input and single-output (SISO) LPTV system and found the conditions under which a SISO LPTV system would produce stationary outputs for all stationary inputs. The 2-D spectrum, 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 in the mathematical field [8].
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[38]–[40], where it was related to the cyclic spectrum and the TFR. These four types of cyclospectra should have some interrelationships, since they all describe second-order statistical properties of cyclostationary signals. The first contribution of this paper, which is also of some tutorial value, is to summarize these cyclospectra 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 to others in one specific context or other. The kernel problem of the spectral theory is the cyclospectrum transformation by linear systems, i.e., to represent the cyclospectrum of the system output in terms of that of the input. For this problem, there exist two limitations in the above cited literature: First, only some specific configurations have been investigated, e.g., the input of an -band filterbank has to be either stationary [28], [32] or cyclostationary with the same period as [21] ; second, most of existing results, e.g., the cyclic spectrum of the output of an LPTV system ([16, Eq. (17.44)]), are developed via definitions, and hence, their derivations are so overwhelming that generalization to more complex systems, e.g., multirate systems, becomes almost impossible, unless a systematic approach is adopted [see the two proofs of (21) in Example 4 and in the Appendix]. The second contribution of this paper is to remove these limitations: The problem of the cyclospectrum transformation is attacked in the framework of multirate systems using the blocking technique. A discrete-time linear system can always be represented by a as Green’s function
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 signals only, which can be readily solved using some well-known results. More specifically, the kernel problem is separated into the following two subquestions. in Fig. 1 • Given a linear SISO multirate system and that the input is cyclo-wide-sense-stationary with period and abbreviated as CWSS , is the output stationary or cyclostationary? If is cyclostationary, what is its period? • What is the cyclospectrum transformation in Fig. 1, i.e., how do we represent the cyclospectrum of in terms of that of ? The rest of the paper is organized as follows. Section II aims at summarizing the different cyclospectra and exploring their interrelationships. Section III studies the effects of the blocking operation on statistical properties of cyclostationary signals. Section IV answers the two subquestions and presents some examples as illustration. Finally, Section V provides concluding remarks. II. CYCLOSPECTRUM
(1) where , which is the set of integers [6]. A linear SISO has the so-called -shift invariance multirate system2 property ( and are integers) if shifting the input by samples results in shifting the output by samples [4]. In terms of -shift invariance is characterized by Green’s function, (2) Fig. 1 depicts such a linear SISO multirate system in which ” denotes that is -shift inthe notation “ variant. Such a multirate system covers many familiar sys, the tems as special cases, e.g., the LTI system , and the cascade of upsampler, LTI LPTV system system, and downsampler ( and are coprime) depicted in Fig. 2. Blocking in signal processing [27], [45] or lifting in control [3], [20] has been shown to be a powerful technique in dealing with multirate systems and cyclostationary signals; by the blocking technique, one can associate the multirate system with an equivalent multi-input and multi-output LTI system [20], [27]; blocking the cyclostationary signal can result in a higher dimensional stationary signal [17], [32], [33]. Therefore, our main idea is to block multirate systems and cyclostationary signals properly and convert the original 2SISO
multirate systems are also called dual-rate systems [4].
We first review some basic concepts of stationary signals and introduce cyclostationarity and four types of cyclospectra. Next, these cyclospectra are shown to be related to each other. After presenting the cyclospectrum transformation by LTI systems, we choose the cyclic spectrum as the representation of the cyclospectrum in the rest of the sections. A. Stationary Signals A discrete-time signal , perhaps vector-valued, is said to be stationary 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 the conjugate transpose. The power spectrum of is defined as the discrete-time Fourier transform (DTFT) of the autocorrelation (5) It is well known that when a stationary signal with power is passed through an LTI system with transfer spectrum , the output is also a stationary signal with power function spectrum [23], [29] (6)
WANG et al.: SPECTRAL THEORY OF CYCLOSTATIONARY SIGNALS IN MULTIRATE SYSTEMS
B. Cyclostationary Signals A discrete-time signal is called cyclostationary or cyclowide-sense-stationary with period ( being a positive integer) [16], [17], [44] if its mean is periodic (7)
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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 also known as Rihaczek spectrum [31] or the time-varying spectrum [34]. is defined as the Bispectrum: The bispectrum 2-D DTFT of the autocorrelation [1], [29]
is pe-
and/or its autocorrelation riodic such that
(8) If , (7) and (8) imply that (3) and (4) hold, i.e., stationary signals can be regarded as cyclostationary signals with period 1. There are mainly four types of cyclospectra, namely, the cyclic spectrum, the time frequency representation, the bispectrum, and the 2-D spectrum. , which Cyclic Spectrum: We know from (8) that is an equivalence of , is a periodic sequence of with has the following period for a fixed . Therefore, discrete Fourier expansion: (9) where the discrete Fourier series coefficient
(14) Like the TFR, the bispectrum is also a very general concept that can describe the second-order statistical property of nonstationary signals. Specifically, the bispectrum of a cyclostationary plane [1], signal lies on some parallel lines in the (15) where . It is shown later that the bispectrum component on the th line is exactly the th cyclic spectrum defined in (11). Note that the terminology “bispectrum” has a different meaning in the literature—the 2-D DTFT of the third-order moment [37]. Two-Dimensional Spectrum: The 2-D spectrum is defined as [38]–[40] the 2-D Fourier transform of
is (16) (10)
The DTFT of [11], [17]
is defined as the cyclic spectrum of
(11) where the subscript stands for “cyclic power spectrum.” It follows from (8) and (10) that is periodic in with period (12) The set thus forms a full description of the cyclospectrum of a CWSS signal . The cyclic spectrum can be considered as a genis eralization of the power spectrum in (5); that is, if stationary, , and for . can be conTime-Frequency Representation: sidered also as a sequence of for a fixed . The time-frequency taking representation is defined as the DTFT of as the changing variable [21], (13) It follows from (8) that
is periodic in with period
The 2-D spectrum and bispectrum are very similar [see (18) is conlater]; however, for a cyclostationary signal , in (14) is contintinuous in and discrete in , but uous both in and . The 2-D spectrum is also referred to as the dual-frequency spectrum [34], which is defined for nonstationary signals. These cyclospectra are related to each other. First, it follows from (10), (11), and (13) that the cyclic spectrum and the TFR are a discrete Fourier transform pair [21] (17) Second, it follows easily from (14) and (16) that the bispectrum and the 2-D spectrum are a coordinate transform of each other with a scaling factor (18) Third, the th cyclic spectrum is exactly the bispectrum component 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-D spectrum [39]
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Fig. 4. Blocking a linear SISO multirate system.
Fig. 3.
cyclic spectrum in the sequel, and the subscript without confusion.
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 sufficient to use only one of them to represent the cyclospectrum in the rest of the development. Before making a choice, we introduce the cyclospectrum transformation by LTI systems to further capture the characteristics of these cyclospectra. Let a CWSS signal be the input of a discrete-time LTI system with transfer function . As LTI systems preserve the 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 4 and the Appendix using two different approaches. The TFR and bispectrum 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 the cyclostationary processes, in terms of cyclic spectral density3, which, although a means of characterization, is not very illustrative, particularly in the context of filterbank analysis” [21]; however, (22) reveals that the TFR is not compact. The bispectrum is originally defined for nonstationary signals so that it has been cautioned to be unwieldy in mathematics [30] or too general and inefficient for indiscriminate use [1]. The 2-D spectrum is simply a coordinate transformation of the bispectrum, and thus, they share the same problems. The cyclic spectrum is defined in the way of incorporating the spectral information along with periodicity, and thus, it displays directly the fundamental characteristic of cyclostationary signals; even though (21) is not as compact as (23) and (24), the cyclic spectrum is very convenient once all the th cyclic spectra are enclosed in the so-called cyclic spectrum matrix [17], [32]. Therefore, we will use the 3The
cyclic spectral density is synonymous with the cyclic spectrum.
is dropped
III. BLOCKING OPERATOR In Section I, we have briefly discussed the idea of blocking multirate systems and cyclostationary signals properly to form a relatively simple problem. This section devotes to studying the effects of the blocking operator on multirate systems and cyclostationary signals. First, we review the definition of the blocking operator. Let be a discrete-time signal defined on , which is the set of nonnegative integers. The -fold discrete blocking operator is defined as the mapping from to , where underlining denotes blocking [20], [27]:
.. .
(25)
.. .
If the underlying sampling frequency of is , that of the . Meanwhile, the signal’s dimension inblocked signal is creases by a factor of . It can be shown from both the time and frequency domains that no information is lost in the blocking is defined as operation. The inverse of the blocking operator the reverse operation of (25); thus, and , where denotes the identity system. One of the advantages of the blocking operator is that it can associate multirate systems that are essentially time-varying with some equivalent LTI systems to which many existing LTI techniques can be applied. For the multirate system in and , Fig. 1, blocking the input and the output by , which respectively, yields a blocked system has inputs and outputs. The blocking procedure is depicted -shift invariant (see (2)), is LTI [27] in Fig. 4. As is transfer matrix [4] and has an
.. .
.. .
whose entries relate to the Green’s function of
.. .
(26)
[see (1)] as (27)
Second, we focus on the effects of the blocking operator on cyclostationary signals. The original and the blocked signals have the following statistical relationships. Lemma 1: A scalar signal is CWSS iff its -fold blocked version is CWSS .
WANG et al.: SPECTRAL THEORY OF CYCLOSTATIONARY SIGNALS IN MULTIRATE SYSTEMS
is used to change
Fig. 5. Decompose L
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by a new variable
into L and L .
Proof: The case of was proved from definitions of stationarity and cyclostationarity in [32] and [33], independently. The general case was given via bispectrum in [1]. Theorem 1: An -dimensional vector signal is CWSS iff its -fold blocked version is CWSS . Proof: It can certainly be proved from definitions; however, here, we take a shortcut. Followed from Lemma 1, a iff its -fold blocked version scalar signal is CWSS is CWSS . Equation (25) implies that can be decomposed into a series of and , as depicted in Fig. 5. , Lemma 1 and (25) give that , which is the output of -dimensional vector signal. Therefore, is is a CWSS CWSS iff its -fold blocked signal is CWSS . Gladyshev [17] first proposed the relationship between a cyclic spectrum matrix of the original signal and the power spectrum of the blocked signal, which was proved by Sakai and Ohno [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 terms of the cyclic spectrum for vector signals. of a Theorem 2: The cyclic spectrum matrix CWSS -dimensional vector signal is connected with the of its -fold blocked version as power spectrum
where the second equality is reached by replacing with a variable , and the last equality follows from the cyclostationarity property described in (8). From (9) and (11)
(31) Next, we show that (31) is equivalent to
(28) is an matrix whose th where component is determined by the cyclic spectrum of
block (29)
is an and component is
unitary matrix whose
th
block
(30) Here, , , , and is a identity matrix. Proof: Theorem 1 says that the blocked signal is sta. From (5) and tionary; thus, it has a power spectrum (25), the th component of is
(32) Comparing (31) and (32), their difference for a certain is as in the equation shown at the bottom of the next page, where the second equality uses the periodicity of the cyclic spectrum in (12), and the third equality is obtained by changing and in the last two summing variables terms, respectively. Finally, (28) is obtained from (32). Remark: There are two important differences between Theorem 2 and its counterpart in [32]: first, the proof in [32] takes a modulation representation of cyclostationary signals as an intermediate step, whereas we attack the problem more directly, and thus, the proof is much simpler; second, the result in [32] holds under a condition (33) which is actually introduced by the modulation representation, whereas our proof shows that (33) is superfluous. Removing the limiting condition (33) is extremely important, because the valid range of using the blocking technique will become too small to be meaningful, if (33) has to be satisfied. IV. CYCLOSTATIONARY SIGNALS IN MULTIRATE SYSTEMS
An identity is an integer otherwise
We are ready to attack the two subquestions proposed in Section I using the blocking technique. The first subquestion is answered in Theorem 3; the second is solved in the form
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Fig. 6. Blocked SISO multirate system.
of matrix multiplication. Both are followed by some specific configurations as illustration. A. Cyclostationarity of the Output There are basically three approaches to answer the first subquestion: i) to explicitly write out some statistics of the system output, e.g., the autocorrelation or the cyclic spectrum, as what was done in [1], [21]; ii) to reduce the multirate system into simpler building blocks such a upsamplers, LTI systems, and downsamplers and study cyclostationary properties of each block, as in [1], [33]; iii) to use the blocking technique. The last approach is the simplest for most systems and hence is adopted here. Theorem 3: Given a linear SISO multirate system in Fig. 1 and a CWSS input , the output is CWSS , where means the greatest common divisor of and . Proof: The use of the blocking operator needs to comply with two principles; the blocked system needs to be LTI, and the blocked signal is stationary. Thus, the fold of the blocking operator at the input side must be an integer multiple of as is LTI (see Fig. 4) and, at the same time, be an integer and by , multiple of (see Theorem 1). Blocking by as depicted in Fig. 6, will satisfy the two principles, where (34) In Fig. 6, the blocked system is LTI, and the blocked input is stationary. Therefore, the blocked output is stationary, which, according to Lemma 1, implies that is CWSS . Example 1: The multirate system is indeed an LTI system , under which Theorem 3 says that if the input is if CWSS , then the output is CWSS too. In other words, LTI systems preserve the cyclostationarity, which is consistent with the conclusions in [1], [16], [21], and [33].
, the multirate system Example 2: If reduces to an LPTV system that appears frequently in signal processing and control, such as multirate filterbanks [45] and LPTV controllers [10], [20]. More specifically, if the input is stationary or CWSS , Theorem 3 gives that the output of an ) is CWSS . LPTV system with period (i.e., This conclusion is consistent with those in [1], [16], [33]. . For , the mulExamples 1 and 2 are both with tirate system is also named the fractional sample-rate changer, which has two configurations as follows. First, if and are is equivalent to the cascade coprime, the multirate system of upsampler, LTI system and downsampler, depicted in Fig. 2, which has been studied extensively [9], [24], [33], [35], [45]. and have some nontrivial common factor, Second, if is not equivalent to the cascade system in Fig. 2 [4], [35]; it stands for a more general building block that finds applications in the nonuniform filterbanks [4] and the multichannel nonuniform transmultiplexers [22]. Example 3: For the cascade system in Fig. 2 ( and are coprime), if the input is stationary , Theorem 3 says that the output is CWSS , which is consistent with that in [33], which is obtained by analyzing the cascade in Fig. 2. As a comparison, the other two approaches mentioned at the beginning of this subsection are explored for the same configuration. Clearly, the first approach has difficulties as the explicit statistical expression of the system output has not been given in the literature. The second approach proceeds as follows. The upsampler and downsampler have the properties: If the input of a -fold [21]; if upsampler is CWSS , the output will be CWSS the input of a -fold downsampler is CWSS , the output is CWSS [33]. Applying the two properties to the cascade 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’s method [19], Martin–Kedem’s method [25], Dandawaté–Giannakis’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 blocked as that in Fig. 6 (see the proof of Theorem 3). Blocking the CWSS input by for the in (34) implies that , where is an integer. A general solution of the second subquestion consists of two cases.
WANG et al.: SPECTRAL THEORY OF CYCLOSTATIONARY SIGNALS IN MULTIRATE SYSTEMS
Fig. 7.
Case 1—
Blocked SISO multirate system: pr
: From Theorem 2, we have (see Fig. 6) (35) (36)
where . Since LTI, (6) gives
is stationary and
is
(37) is represented by the Green’s function of in (26). where Therefore, the cyclic spectrum of is associated with that of via (35)–(37)
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= qn.
is associated with that of , which is shown in (41) at the bottom of the page. Remark: For a fixed frequency , either (38) or (41) can be numerically computed as the matrix multiplication.4 The next example is on the cyclospectrum transformation by LTI systems. The purpose of the example is three-fold: i) to illustrate 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) (38) : With , (36) and (37) still Case 2— is decomposed hold. However, one more step is needed: into a series of and , which makes Fig. 6 equivalent to Fig. 7. Theorem 2 gives
The unitary matrix
is [see (30)] (43)
An LTI system is fully characterized by its impulse response , i.e.,
(39) (44)
Since is stationary, its cyclic spectrum matrix is block diagonal. Taking it as a special case of Theorem 2 results in
Comparing (1) and (44) gives the connection between the Green’s function and the impulse response (45) diag
.. .
(40)
where , and diag denotes a diagonal matrix taking the elements of the operand vector as the diagonal entries. From (36), (37), (39), and (40), the cyclic spectrum of
diag
The blocked system (26)]
has a transfer matrix [see
(46) 4See
an example at http://www.ece.ualberta.ca/~jwang/research.htm.
.. . (41)
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where (27) and (45) result
Fig. 8. Maximally decimated filterbank.
Thus, (46) becomes5 (47) where and are the well-known type-1 polyphase components of [45],
Fig. 9. Polyphase representation of a filterbank.
(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 the last equality follows from (48). Note that the zeros on the off-diagonal 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 the bottom of the page, which also proves (21). 5The
generalization of (47) is given in [3, Th. 8.2.1].
Multirate filterbanks are typical examples of LPTV systems and have been of particular interest in signal processing [45]. Unlike the approaches in [1], [21], [28], [32], and [33], the statistical properties (in terms of cyclic spectrum) of the output or the reconstructed signal of the filterbank are elegantly found by the blocking technique in the next example. Example 5: Fig. 8 depicts a maximally decimated filterbank [45], where and ( ) are analysis filters and synthesis filters, respectively. Let the input be CWSS . With the type-1 polyphase representation of [45], the type-3 polyphase representation of [9], and noble identities [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)
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(50)
and in Fig. 9 are exactly the blocked versions of and , respectively. Thus, the original filterbank in Fig. 8 is represented in terms of the . Here, both blocked signals and with an LTI system and are systems; the th elements of their transfer matrices are
where and
and of and
in Theorem 3, and the cyclospectrum of the output is associated with 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
relate to the impulse responses , respectively, as 7
Finally, the cyclic spectrum of the output can be associated with that of the input via the following three equations [see (38)]
Let be the impulse response of the LTI system volution of and gives
. The con-
V. CONCLUSION In this paper, we have studied the spectral theory of discretetime cyclostationary signals: the cyclospectrum representation and the cyclospectrum transformation by linear multirate systems. The four types of cyclospectra, namely, the cyclic spectrum, the time frequency representation, the bispectrum, and the 2-D spectrum are shown to be closely related and mutually convertible (see Fig. 3). The cyclospectrum transformation by linear systems are solved in a systematic manner by using multirate systems as the unifying framework and the blocking technique as the main tool. The effects of the blocking operator on cyclostationary signals are investigated in Theorems 1 and 2. The cyclostationarity of the output of the multirate system is studied 7Superscripts (
) and (
sentations, respectively.
) mean the type-1 and type-3 polyphase repre-
Changing by a new variable yields (50), shown at the top of this page, where the second and the last equalities follow from (9) and (11), respectively. By an identity is an integer otherwise we get
, and thus, (50) results in (21). REFERENCES
[1] S. Akkarakaran and P. P. Vaidyanathan, “Bifrequency and bispectrum maps: A new look at multirate systems with stochastic inputs,” IEEE Trans. Signal Process., vol. 48, no. 3, pp. 723–736, Mar. 2000.
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[2] B. Boashash, “Time-frequency signal analysis: Past, present and future trends,” in Control and Dynamic Systems, C. T. Leondes, Ed. New York: Academic, 1996, vol. 78, pp. 1–69. [3] T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems. London, U.K.: Springer, 1995. [4] T. Chen, L. Qiu, and E. W. Bai, “General multirate building structures with application to nonuniform filterbanks,” IEEE Trans. Circuits Syst. II, vol. 45, no. 8, pp. 948–958, Aug. 1998. [5] T. Chen, “Nonuniform multirate filterbanks: Analysis and design with performance measure,” IEEE Trans. Signal Process., vol. 45, an no. 3, pp. 572–582, Mar. 1997. [6] R. E. Crochiere and L. R. Rabiner, Multirate Digital Signal Processing. Engelwood Cliffs, NJ: Prentice-Hall, 1983. [7] A. V. Dandawaté and G. B. Giannakis, “Statistical tests for presence of cyclostationarity,” IEEE Trans. Signal Process., vol. 42, no. 9, pp. 2355–2369, Sep. 1994. [8] D. Dehay and H. L. Hurd, “Representation and estimation for periodically and almost periodically correlated random processes,” in Cyclostationarity in Communications and Signal Processing, W. A. Gardner, Ed. Piscataway, NJ: IEEE Press, 1994. [9] N. J. Fliege, Multirate Digital Signal Processing. New York: Wiley, 1994. [10] B. A. Francis and T. T. Georgiou, “Stability theory for linear time-invariant plants with periodic digital controllers,” IEEE Trans. Autom. Control, vol. 33, no. 9, pp. 820–832, Sep. 1988. [11] W. A. Gardner, Statistical Spectral Analysis: A Nonprobabilistic Theory. Engelwood Cliffs, NJ: Prentice-Hill, 1988. , Introduction to Random Processes With Application to Signals and [12] Systems, Second ed. New York: McGraw-Hill, 1990. , “Identification of systems with cyclostationary input and corre[13] lated input/output measurement noise,” IEEE Trans. Autom. Control, vol. 35, no. 4, pp. 449–452, Apr. 1990. , “An introduction to cyclostationary signals,” in Cyclostationarity [14] in Communications and Signal Processing, W. A. Gardner, Ed. Piscataway, NJ: IEEE Press, 1994. [15] G. B. Giannakis, “Polyspectral and cyclostationary approaches for identification of closed-loop systems,” IEEE Trans. Autom. Control, vol. 40, no. 5, pp. 882–885, May 1995. , “Cyclostationary signal analysis,” in Digital Signal Processing [16] Handbook. Boca Raton, FL: CRC, 1999. [17] E. G. Gladyshev, “Periodically correlated random sequence,” Soviet Math., vol. 2, pp. 385–388, 1961. [18] F. Hlawatsch and G. F. Boudreaux-Bartels, “Linear and quadratic timefrequency signal representations,” IEEE Signal Process. Mag., vol. 9, no. 2, pp. 21–67, Apr. 1992. [19] H. L. Hurd and N. L. Gerr, “Graphical methods for determining the presence of periodic correlation,” J. Times Ser. Anal., vol. 12, no. 4, pp. 337–350, 1991. [20] P. P. Khargonekar, K. Poolla, and A. Tannenbaum, “Robust control of linear time-invariant plants using periodic compensation,” IEEE Trans. Autom. Control, vol. AC–30, no. 11, pp. 1088–1096, Nov. 1985. [21] B. Lall, S. D. Joshi, and R. K. P. Bhatt, “Second-order statistical characterization of the filterbank and its elements,” IEEE Trans. Signal Process., vol. 47, no. 6, pp. 1745–1749, Jun. 1999. [22] T. Liu and T. Chen, “Design of multichannel nonuniform transmultiplexers using general building blocks,” IEEE Trans. Signal Process., vol. 49, no. 1, pp. 91–99, Jan. 2001. [23] L. Ljung, System Identification: Theory for the User, Second ed. Englewood Cliffs, NJ: Prentice Hall, 1999. [24] C. M. Loeffler and C. S. Burrus, “Optimal design of periodically time-varying and multirate digital filters,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-32, no. 5, pp. 991–997, Oct. 1984. [25] D. E. K. Martin and B. Kedem, “Estimation of the period of periodically correlated sequences,” J. Times Ser. Anal., vol. 14, no. 2, pp. 193–205, 1993. [26] D. E. K. Martin, “Detection of periodic autocorrelation in time series data via zero-crossing,” J. Times Ser. Anal., vol. 20, no. 4, pp. 435–452, 1999. [27] R. A. Meyer and C. S. Burrus, “A unified analysis of multirate and periodically time-varying digital filters,” IEEE Trans. Circuits Syst., vol. CAS-22, no. 3, pp. 162–168, Mar. 1975. [28] S. Ohno and H. Sakai, “Optimization of filterbanks using cyclostationary spectral analysis,” IEEE Trans. Signal Process., vol. 44, no. 11, pp. 2718–2725, Nov. 1996.
H
[29] A. Papoulis, Probability, Random Variables and Stochastic Processes. New York: McGraw-Hill, 1965. [30] U. Petersohn, H. Unger, and N. J. Fliege, “Exact deterministic and stochastic analysis of multirate systems with application to fractional sampling rate alteration,” in Proc. IEEE ISCAS, 1994, pp. 177–180. [31] A. W. Rihaczek, “Signal energy distribution in time and frequency,” IEEE Trans. Inf. Theory, vol. 14, no. 3, pp. 369–374, May 1968. [32] H. Sakai and S. Ohno, “Theory of cyclostationary processes and its application,” in Statistical Methods in Control and Signal Processing. New York: Marcel Dekker, 1997, pp. 327–354. [33] V. P. Sathe and P. P. Vaidyanathan, “Effects of multirate systems on the statistical properties of random signals,” IEEE Trans. Signal Process., vol. 41, no. 1, pp. 131–146, Jan. 1993. [34] L. L. Scharf and B. Friedlander, “Toeplitz and hankel kernels for estimating time-varying spectra of discrete-time random processes,” IEEE Trans. Signal Process., vol. 49, no. 1, pp. 179–189, Jan. 2001. [35] R. G. Shenoy, “Multirate specifications via alias-component matrices,” IEEE Trans. Circuits Syst. II, vol. 45, no. 3, pp. 314–320, Mar. 1998. [36] L. Sun, H. Ohmori, and A. Sano, “Frequency domain approach to closedloop identification based on output inter-sampling scheme,” in Proc. Amer. Control Conf., vol. 3, Jun. 2000, pp. 1802–1806. [37] C. W. Therrien, “Overview of statistical signal processing,” in Digital Signal Processing Handbook. Boca Raton, FL: CRC, 1999. , “Issues in multirate statistical signal processing,” in Proc. [38] Thirty-Fifth Asilomar Conf. Signals, Syst. Comput., vol. 1, Nov. 2001, pp. 573–576. [39] C. W. Therrien and R. Cristi, “Two-dimensional spectral representation of periodic, cyclostationary, and more general random processes,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 4, May 2002, pp. 3561–3563. [40] C. W. Therrien, “Some considerations for statistical characterization of nonstationary random processes,” in Proc. Thirty-Sixth Asilomar Conf. Signals, Syst., Comput., vol. 2, Nov. 2002, pp. 1554–1558. [41] L. Tong, G. Xu, and T. Kailath, “Blind identification and equalization based on second-order statistics: A time domain approach,” IEEE Trans. Inf. Theory, vol. 40, no. 2, pp. 340–349, Mar. 1994. [42] L. Tong, G. Xu, B. Hassibi, and T. Kailath, “Blind identification and equalization based on second-order statistics: A frequency-domain approach,” IEEE Trans. Inf. Theory, vol. 41, no. 1, pp. 329–334, Jan. 1995. [43] J. Wang, T. Chen, and B. Huang, “Closed-loop identification via output fast sampling,” J. Process Control, vol. 14, no. 5, pp. 555–570, 2004. , “Cyclo-period estimation for discrete-time cyclostationary sig[44] nals,” IEEE Trans. Signal Process., to be published. [45] P. P. Vaidyanathan, Multirate Systems and Filterbanks. Englewood Cliffs, NJ: Prentice-Hall, 1993.
Jiandong Wang received the B.E. degree in automatic control from Beijing University of Chemical Technology, Beijing, China, in 1997 and the M.Sc. degree in electrical and computer engineering from the University of Alberta, Edmonton, AB, Canada, in 2003. Currently, he is pursuing the Ph.D. degree at the University of Alberta. From 1997 to 2001, he was a control engineer with the Beijing Tsinghua Energy Simulation Company. His current research work involves system identification, cyclostationary signal processing, and multirate systems. Mr. Wang received the Izaak Walton Killam Memorial Scholarship, the Alberta Ingenuity Ph.D. Studentship, and the iCore Graduate Student Scholarship at the University of Alberta, all since May of 2004.
WANG et al.: SPECTRAL THEORY OF CYCLOSTATIONARY SIGNALS IN MULTIRATE SYSTEMS
Tongwen Chen received the B.Sc. degree from Tsinghua University, Beijing, China, in 1984 and the M.A.Sc. and Ph.D. degrees from the University of Toronto, Toronto, ON, Canada, in 1988 and 1991, respectively, all in electrical engineering. From 1991 to 1997, he was with the faculty of the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada. Since 1997, he has been with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, where he is presently a Professor of electrical engineering. He held visiting positions at the Hong Kong University of Science and Technology and Kumamoto University, Kumamoto, Japan. His current research interests include process control, multirate systems, robust control, network-based control, digital signal processing, and their applications to industrial problems. He co-authored, with B.A. Francis, the book Optimal Sampled-Data Control Systems (New York: Springer, 1995). Currently, he is an Associate Editor for Automatica, Systems and Control Letters, and DCDIS Series B. Dr. Chen received a University of Alberta McCalla Professorship for 2000/2001 and a Fellowship from the Japan Society for the Promotion of Science for 2004. He was an Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL from 1998 to 2000. He is a registered Professional Engineer in Alberta.
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Biao Huang received the B.Sc. and M.Sc. degrees from Beijing University of Aeronautics and Astronautics, Beijing, China, in 1983 and 1986, respectively, and the Ph.D. degree from University of Alberta, Edmonton, AB, Canada, in 1997. He is currently a professor with the Department of Chemical and Materials Engineering, University of Alberta. His research interest is in the areas of system identification, process control, and control monitoring.
