IEEE TRANSACTIONS ON
`
`
`SIGNAL PROCESSING
`
`A PUBLICATION OF THE IEEE SIGNAL PROCESSING SOCIETY
`
`OCTOBER 1997
`
`VOLUME 45
`
`NUMBER 10
`
`ITPRED
`
`(ISSN 1053—587X)
`
`17-0CT-199? BSDS 3833433"
`IEEE TRRNSF‘CTIDNS 0N SIGNHL PROCESSING
`
`VOL 45 Pain 19
`
`1/1
`
`PAPERS
`
`45563-219900
`PIC“
`
`PB ..
`
`'
`
`Underwater Acoustics Signal Prncessing
`Extension of the Pisarenko Method to Sparse Linear Arrays ................................................... J.-J. Fuchs 2413
`
`Digital Signal Processing
`
`Design of Computationally Efficient Elliptic IIR Filters with a Reduced Number of Shift-and-Add Operations in
`Multipliers ................................................................................ M. D. Lutovac and L. D. Milic’ 2422
`New Preceding for Intersymbol Interference Cancellation Using Nonmaximally Decimated Multirate Filterbanks with
`ideal FIR Equalizers ........................................................................................... X.-G. Xia 243]
`Magnitude Response Peak Detection and Control Using Balanced Model Reduction and Leakage to a Target
`........................................................................... K. D. Benson and W. A. Sethares 2442
`
`On Properties of Information Matrices of Delta-Operator Based Adaptive Signal Processing Algorithms ................
`......................................................................................... Q. Li and H. Fan 2454
`Study of the Transient Phase of the Forgetting Factor RLS ............................................ G. V. Mousrakides 2468
`
`Statistical Signal and Array Processing
`Joint Estimation of Time Delays and Directions of Arrival of Multiple Reflections of a Known Signal .................
`.................................................................................... M. Wax and A. Leshem 2477
`
`Efficient Method for Estimating Directions-of—Arrival of Partially Polarized Signals with Electromagnetic Vector
`Sensors ........................................................................... K.-C. Ho, K.-C. Tan, and B. T. G. Tan
`
`2485
`
`Multidimensional Signal Processing
`On Homomorphic Deconvolution of Bandpass Signals ................................ A. L. Marenco and V. K. Madisetti 2499
`
`Emerging Techniques
`. Neural Fuzzy Motion Estimation and Compensation ............................................ H. M. Kim and B. Kosko 2515
`Learning and Generalization of Noisy Mappings Using a Modified PROBART Neural Network ............ N. Srinivasa
`2533
`
`
`
`(Contents Continued on Back Cover)
`
`
`
`Supplied by the British Library 17 Nov 2020, 09:20 (GMT)
`
`Petitioner's Exhibit 1011
`
`Page 001
`
`Petitioner's Exhibit 1011
`Page 001
`
`

`

`(Contents Continued from Front Cover)
`___—__—_—___——-.—————u————————-—-——
`
`CORRESPONDENCE
`
`Signal Processing for Advanced Communications
`Digital Filters with Implicit lnterpolated Output .............................................................. G. Braileanu 2551
`Cross-Coupled DOA Trackers ........................................... A. Pérez-Neira. M. A. Lagunas. and R. L. Kirlin 2560
`
`Underwater Acoustics Signal Processing
`An Integrated Hybrid Neural Network and Hidden Markov Model Classifier for Sonar Signals ..........................
`........7
`A. KundttandG. C. Chen 2566
`Maximum Likelihood Time Delay Estimation in Non-Gaussian Noise ...... P. M. Schultlieiss. H. Messer. and G. Shor 2571
`
`Digital Signal Processing
`Improved Clustered Look-Ahead Pipelining Algorithm with Minimum Order Augmentation .................. K. Chang 2575
`Fixed-Point Error Analysis of Discrete Wigner—Ville Distribution ................ K. M. M. Prablru and R. S. Sundaram 2579
`An Improved Stochastic Gradient Algorithm for Principal Component Analysis and Subspace Tracking ................
`................................................................. J. Dehaene. M. Moonen. and J. Vandewalle 2582
`Characterization of Signals by the Ridges of Their Wavelet Transforms ...................................................
`.............................................................. R. A. Cannona. W. L. Hwang. and B. Torrésani 2586
`A General Formulation for Iterative Restoration Methods ................................ J. P. Noonan and P. Natarajan 2590
`Reconstruction of a Signal Using the Spectrum-Reversal Technique .............................. T. Arai and Y. Yoshida 2593
`A New Method for Optimal Deconvolution ........................................ H. -S. thang. X. —J. Lin. and T.-Y. Chat 2596
`Experimental Evaluation of Echo Path Modeling with Chaotic Coded Speech ............................................
`.......................................................... J. M. H. Elmirghani. S. H. Milner. and R. A. Cryan 2600
`
`Statistical Signal and Array Processing
`Minimum-Redundancy Linear Arrays for Cyclostationarity-Based Source Location ................ G. Gelli and L. [:20 2605
`Blind Separation of Convolutive Mixtures and an Application in Automatic Speech Recognition in a Noisy Environ—
`ment ....................................................................................... F. Ehlers and H. G. Schuster 2608
`An Equivalence of the EM and ICE Algorithm for Exponential Family ..................................... J. P. Delmas
`2613
`Fast Adaptive Identification of Stable Innovation Filters ............................... A. P. Mullhaupr and K. S. Riedel 2616
`
`Multidimensional Signal Processing
`2620
`Stability Analysis of 2—D State—Space Digital Filters Using Lyapunov Function: A Caution ........ H. Kar and V. Singh
`2621
`A Shift-Invariant Discrete Wavelet Transform ........................................... H. Sari—Sarraf and D. Brzakovic
`_________—__—__—_—._—_———
`
`Abstracts of Manuscripts in Review .........................................................................................
`2627
`
`
`263|
`EDICS—Editor‘s Information Classification Scheme .......................................................................
`2632
`Information for Authors ......................................................................................................
`___________——_____—__—..._————————-
`
`ANNOUNCEMENTS
`
`2633
`Call for Papers—IEEE—SP International Symposium on Time-Frequency and Time-Scale Analysis ......................
`2634
`Call for Papers—Eighth IEEE Digital Signal Processing Workshop .......................................................
`2635
`Call for Papers—IEEE Signal Processing Society 1998 International Conference on Image Processing ..................
`—_____——___—_—.—————————-
`
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`Petitioner's Exhibit 1011
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`

`IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 10, OCTOBER 1997
`
`2.431
`
`New Precoding for Intersymbol Interference
`Cancellation Using Nonmaximally Decimated
`Multirate Filterbanks with Ideal FIR Equalizers
`
`Xiang-Gen Xia, Member, IEEE
`
`Abstract—In this paper, we propose a new precoding method
`for intersymbol
`interference (ISI) cancellation by using non-
`maximally decimated multirate filterbanks. Unlike the existing
`precoding methods, such as the TH and trellis precodings,
`the
`new precoding
`i) may be independent of the ISI channel;
`ii)
`is linear and does not have to implement any modulo
`operation;
`iii) gives the ideal FIR equalization at the receiver for any FIR
`ISI channel including spectral-null channels;
`iv) expands
`the transmission bandwidth in a minimum
`amount.
`
`The precoding is built on nonmaximally decimated multirate
`filterbanks. Based on multirate filterbank theory, we present
`a necessary and sufficient condition on an FIR ISI
`transfer
`function in terms of its zero set such that there is a linear FIR
`
`N X K precoder so that an ideal FIR equalizer exists, where
`the integers If and IV are arbitrarily fixed. The condition is
`easy to check. As a consequence of the condition, for any given
`FIR ISI transfer function (not identically 0), there always exist
`such linear FIR precoders. Moreover, for almost all given FIR
`ISI transfer functions, there exist linear FIR precoders with size
`N x {N —1}, i.e., the bandwidth is expanded by 1/_-’\7. In addition
`to the conditions on the ISI transfer functions, a method for the
`design of the linear FIR precoders and the ideal FIR equalizers
`is also given. Numerical examples are presented to illustrate the
`theory.
`
`I.
`
`INTRODUCTION
`
`NTERSYMBOL interference (ISI) is a common problem
`Iin telecommunication systems, such as terrestrial television
`broadcasting, digital data communication systems, and cellular
`mobile communication systems. The main reasons for the ISI
`are because of high-speed transmission and multipath fading.
`There have been considerable studies for these problems, such
`as [l]i[29] and [33}[40]. These studies can be primarily split
`into three categories:
`
`iii) precoding techniques, such as TomlinsoniHarashima
`(TH) precoding [7], [8], trellis precoding by Eyuboglu
`and Forney [9], [10], matched spectral null precoding
`in partial response channels [12], and other precoding
`schemes, for example, [l3]—[l7] and [40].
`
`The basic idea for DFE is that once an information symbol
`has been detected, the ISI that it causes on future symbols may
`be estimated and subtracted out prior to symbol detection. DFE
`usually consists of a feedforward filter and feedback filter.
`The feedback filter is driven by decisions of the output of the
`detector, and its coefficients are adjusted to cancel the ISI on
`the current symbol that results from past detected symbols. The
`coefficient adjustment may be done via a linear equalizer with
`LMS algorithms. The convergence of these iterative algorithms
`are dependent of the channel characteristics. When a channel is
`spectral null or frequency selective fading, these algorithms are
`very slow and, therefore, become computationally expensive.
`The performance of the existing linear equalizers significantly
`degrades over frequency selective fading channels. Although
`DFE has better performance than the existing linear equalizers
`when the frequency fading is in the middle of a passband,
`it does not offer much improvement in other fading cases.
`For more details, see,
`for example,
`[3] and [35]. In post
`equalization techniques, there are many research results (see,
`for example,
`[18]—[29] and [36]—[39] on blind equalizations
`where channel characteristics are assumed unknown. In blind
`
`equalization techniques, there are approximately three groups
`of results:
`
`i)
`ii)
`
`high-order statistics techniques;
`second-order cyclostationary statistics techniques with
`oversampling;
`iii) antenna array (smart antenna) multireceiver techniques;
`
`i)
`
`post equalization, such as least-mean—squared (LMS)
`equalizer and decision feedback equalization (DEE), for
`example, [1],[3], [18]7[29], and [36}[39];
`ii) multicarrier modulation to increase transmission sym-
`bol length, for example, [4]—[6]',
`
`Manuscript received September 2, 1995; revised April 18, 1997. The
`associate editor coordinating the review of this paper and approving it for
`publication was Prof. Roberto H. Barnberger.
`The author was with Hughes Research Laboratories, Malibu, CA 90265
`USA. He is currently with the Department of Electrical Engineering, Univer-
`sity of Delaware, Newark, DE 19716 USA (e—mail: xxia@ee.udel.edu).
`Publisher Item Identifier S 1053-587X(97)07357-l.
`
`where there is a considerable amount of overlaps between ii)
`and iii).
`A block diagram for TH precoding is shown in Fig. l, where
`the basic idea is to equalize the signal before transmission.
`With TH precoding there are two drawbacks: i) The transmitter
`needs to know the channel characteristics, and ii) the precoding
`is not reliable when the ISI channel H (.5) has spectral null or
`frequency selective fading characteristics, which is because
`the pre-equalizer mod[1/H[z), M] oscillates in a dramatic
`way when 117(2) is close to zero. The trellis precoding scheme
`proposed by Eyuboglu and Forney [9] whitens the noise at the
`equalizer output. This scheme combines precoding and trellis
`1053—587X1'97$10.00 © 1997 IEEE
`
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`
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`
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`
`

`

`2432
`
`IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 10, OCTOBER 1997
`
`SID]
`
`9 m
`
`XIII]
`
`Fig. 1. TH precoder.
`
`1 —I-
`
`'
`
`ui- '
`'r——~Fe
`
`'
`
`'
`
`Hi“.
`
`yw
`
`Blocking with block size N W”
`
`(a)
`
`w n
`[
`
`] Unblocking with unblock size N
`
`w n
`[
`
`]
`
`(b)
`
`Fig. 3. Blocking and unblocking.
`
`Next, we want
`
`to
`
`find
`
`brief
`
`solutions
`
`for
`
`these
`
`questions. When K : 1, (30(3) 2 1, (31(3) 2
`: GN_1(z) : 0,
`the precoding scheme in Fig. 2 is
`equivalent to the fractionally spaced equalizer studied, for
`example,
`[36]—[39], where the receiver needs to sample a
`signal N times faster than the baud sampling. When K = 1,
`the precoding concept has appeared in [39] by Tsatsanis and
`Giannakis, where the precoder G1(z) : clJ : 0,17 -
`-
`-
`, N— 1
`for N constants C; was used. As we can see,
`the case of
`K : 1 is a very special case in our precoding scheme, and
`moreover, our new precoding scheme in Fig. 2 provides other
`potential precoders Gl(z),l : 0,1,---7 N — 1 rather than
`only constants C], which allows one to search the optimal one
`with respect to an individual channel.
`When K 2 N and there are N interference channels instead
`of a single channel H (z) in Fig. 2, a detailed analysis was
`given by Nguyen [31]. When K > N, as mentioned in
`[31], PR is impossible, but partial alias cancellation filterbanks
`were proposed in [31]. The applications discussed in [31] are
`in wide-band radio communications, where only part of the
`signal frequencies is of interest to the user.
`In this paper,
`we are interested in applications in the ISI channels with PR
`systems in Fig. 2 and, therefore, the case of K < N. This
`also implies that unlike the existing precoding techniques, the
`new precoding expands the transmission bandwidth, which is
`what we lose for the new precoding method, and fortunately,
`we will show that the bandwidth expansion can be as small
`as possible in theory.
`An intuitive way to reduce the ISI generated from a lowpass
`H (z) is to smoothly interpolate $[n] with a large enough
`number of interpolations between samples of a7[n] so that
`the interpolated one has the lowpass property. However, two
`drawbacks about
`this approach may occur. One is that
`it
`usually requires a large amount of increasing of data rate
`(number of interpolations between samples). The other is
`that a good frequency band structure for a nonlowpass, such
`as bandpass, filter H (z) is required for PR.
`In this paper,
`we want to solve the above three problems systematically.
`Given two integers 0 < K < N, we present a necessary
`and sufficient condition (see Theorem 1) on an FIR filter
`H (3) such that there exists an FIR nonmaximally decimated
`multirate filterbank with N channels and decimation by K so
`that $[n] can be recovered from fln] in Fig. 2 with an FIR
`synthesis bank. The condition we found is basically very weak.
`In fact, it can be proved that for any given FIR filter H (2:)
`not identically 0, there always exists an FIR nonmaximally
`decimated multirate filterbank in Fig. 2 for recovering $[n]
`
`,2._|"
`
`l".-:»,
`
`'7;
`
`I'-‘.ii'r’r
`
`Lila.
`
`A“
`
`Fig. 2. Nonmaximally decimated multirate filterbank in a communication
`channel with ISI.
`
`shaping. There are also similar drawbacks about this approach.
`i)
`The transmitter also needs to know the ISI channel
`characteristics.
`
`ii) The trellis shaping depends on the ISI channel.
`iii) The trellis precoding technique may not be suitable for
`spectral-null channels either.
`
`In the matched spectral null precoding scheme [12] in partial
`response channels, certain error control codes are chosen to
`match the spectral nulls of partial response channels in order to
`lose less signal information through the channel. This approach
`is mainly for magnetic recording systems.
`We now propose a multirate filterbank as a precoder before
`transmission (shown in Fig. 2), where l K indicates downsam-
`pling by K, and T N indicates upsampling by N, i.e., inserting
`N — 1 zeros between two adjacent samples, and H (z) is the
`ISI transfer function. Later, we will see a multirate filterbank
`decoder for the receiver to eliminate the ISI. If input signal
`$[n] in Fig. 2 can be completely recovered from the received
`signal flu] through an FIR linear system, we call that the
`system in Fig. 2 has perfect reconstruction (PR) or an FIR
`ideal linear equalizer. In what follows, we use “precoder” and
`“multirate filterbank” interchangeably.
`With the precoder proposed in Fig. 2, there are three ques-
`tions to be answered:
`
`i) What is the condition on H (7:) such that there exists
`a multirate filterbank with N channels and decimation
`
`ii)
`
`iii)
`
`by K in Fig. 2 so that $[n] can be recovered from flu]
`through an FIR linear system?
`If the condition on H (z) in the first question is satisfied,
`how does one design a multirate filterbank in Fig. 2 to
`eliminate the ISI?
`
`If both of these two problems are solved, how does the
`receiver recover the input signal $[n] from the received
`flit]?
`
`Authorized licensed use limited to: Boston Spa Staff [DSC]. Downloaded on November 17,2020 at 07:47:08 UTC from IEEE Xplore. Restrictions apply.
`
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`
`Petitioner's Exhibit 1011
`
`Page 005
`
`Petitioner's Exhibit 1011
`Page 005
`
`

`

`
`
`XIA: NEW PRECODING FOR INTERSY 30L INTERFERENCE CANCELLATION
`
`2433
`
`x[n]
`
`.
`.
`_
`Blocking wuth block SlZe N
`
`H(z)
`
`.
`.
`.
`Unblocklng With unblock snze N
`
`*[nl
`
`Fig. 4. Equivalence of an LTI system and its blocked version.
`
`Blocking with
`
`block size K
` 'Unblocking with
`
`Blocking with
`block size K
`
`”(Z)G(Z) lunblocksizeN
`
`from £[n]. A nonmaximally decimated filterbank precoder
`requires a higher transmission rate with the increasing amount
`proportional to the difference N — K. It
`is clear that the
`smallest N — K is 1. In other words, a multirate filterbank
`with N channels and decimation by N — 1 has the smallest
`increasing of a transmission rate, and therefore, it is usually
`desired. We show that a multirate filterbank with N channels
`
`and decimation by N — 1 exists in Fig. 2 for PR if and only if
`any two sets of zeros of the polynomials H(zW/}V) of 7:71
`for l : 0,1,---, N — 1 do not
`intersect, where WN :
`C’Zflfl/N. This condition is true almost surely. Various
`examples are presented. With the above conditions, we also
`derive some results on the submatrices of a pseudo-circulant
`polynomial matrices [32]. Constructions of FIR nonmaximally
`decimated multirate filterbanks and their FIR syntheses for
`the reconstruction for a given H (z) in Fig. 2 are provided.
`Numerical examples are presented to illustrate the theory,
`which also indicates that
`the technique we developed for
`eliminating the ISI is robust.
`In Section II, we
`This paper is organized as follows.
`present necessary and sufficient conditions on H (z) We also
`discuss the construction of nonmaximally decimated multirate
`filterbanks for eliminating ISI.
`In Section III, we present
`examples and the reconstruction method. In Section IV, we
`consider applications of the ISI cancellation.
`
`H. A NECESSARY AND SUFFICIENT CONDITION
`
`In this section, we study necessary and sufficient conditions
`on the ISI transfer functions H (z) in Fig. 2 such that there
`exists a nonmaximally decimated multirate filterbank with
`N channels and decimation by K and such that an ideal
`FIR linear equalizer exists. We also present a design method
`for an FIR nonmaximally decimated multirate filterbank for
`eliminating the ISI. Throughout this paper, boldface lower-
`case letters denote vector-valued sequences, capital
`letters
`denote transfer functions, and boldface capital letters denote
`function matrices (or polynomial matrices). We first consider
`the case when K and N (0 < K < N) are two arbitrarily
`fixed integers.
`Before we go to the results, let us see some fundamentals
`on blocking and linear time invariant (LTI) systems. We then
`convert the system in Fig. 2 into a single multirate system. The
`output y[n] shown in Fig. 3(a) of the blocked y[n] with block
`size N is the vector-valued signal y[n] : (y[Nn]7 y[Nn —
`1],
`-
`-
`-
`, y[Nn — N —l— 1])T, where T indicates transpose. Con-
`versely, the output w[n] shown in Fig. 3(b) of the unblocked
`vector-valued signal w[n] = [w0[n], w1[n], ---, wN_1[n])T
`
`N many
`
`(b)
`
`Fig. 5.
`
`Equivalent systems of the system in Fig. 2.
`
`with unblock size N is w[n] = wk[l] when n = Nl—k for k =
`0,1,---7 N — 1. In particular, when w[n] : (y[Nn]7 y[Nn —
`1]7 ---, y[Nn — N + 1])T, then w[n] : y[n].
`Let H(z) : Zn h[n]z_" and Hj(z) be its jth forward
`polyphase component with N channels,
`i.e., H3(2) :
`En h[Nn—l—j]z’",0 g j g N — 1. With Hj(z),0 g j g
`N — 1, we form the following N X N pseudo-circulant matrix
`H(z) (see [30], [32])
`
`H0(z)
`H1(z)
`
`z‘lHN_1(z)
`H0(z)
`
`H<z> =
`
`5
`
`s
`
`HN_2(Z)
`HAT—1(2)
`
`HN_3(Z)
`HN_2(Z)
`
`2—1H1(z)
`2—1H2(z)
`
`s
`
`2—1HN_1(Z)
`110(2)
`
`(2.1)
`
`Then, we have the equivalence for an LTI system and
`blocking process shown in Fig. 4, where H(z) is from (2.1)
`and is called the blocked version of H(z)', see [30] and [32].
`For 0 S l S N — 1,
`let G17j(z) be the jth forward
`polyphase component of the lth filter Gl(z) in Fig. 2 with K
`channels, i.e., G17j(z) : Zn gl[Kn +j]z_", when Gl(z) :
`2n gl[n]z’", for 0 S j S K — 1. Let G(z) be the polyphase
`matrix of
`the filterbank (30(2), (31(2), ---, GN_1(Z)
`in
`Fig. 2: G(z) = [G17j(z)]NxK. Then,
`the system in Fig. 2
`is equivalent to the one in Fig. 5(a), which is also equivalent
`to the one in Fig. 5(b).
`
`Authorized licensed use limited to: Boston Spa Staff [DSC]. Downloaded on November 17,2020 at 07:47:08 UTC from IEEE Xplore. Restrictions apply.
`
`Supplied by the British Library 17 Nov 2020, 08:18 (GMT)
`
`Petitioner's Exhibit 1011
`
`Page 006
`
`Petitioner's Exhibit 1011
`Page 006
`
`

`

`2434
`
`IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 10, OCTOBER 1997
`
`Therefore, the PR of .rrfrr] from :f:[n] in Fig. 2 is equivalent to
`the one of the linear multirate system H(z)G(z) in Fig. 4(b).
`Notice that H(z)G(z) is an N x K function matrix of 2:71. To
`analyze it, we need a property on the pseudo-circulant matrix
`H(z) in (2.1). In fact, H(z) can be diagonalized as follows.
`Let WN be
`the N X N DFT matrix,
`i.e.,
`WA! é (Hiii'fhjgja kEN—l: where I-‘I‘iry = (€_2W VITUN.
`Let A(z) be the diagonal matrix
`
`A(,z) é diag (1, 2’1,
`
`Z’NH).
`
`the transpose of the matrix H(z) is the for-
`Notice that
`ward polyphase matrix of the N filters H(z), 2—1H(z),
`--, IVY—“2111(3)
`in N channels
`
`lfltr‘éla 7711505)a
`
`[NH-3(2)] :
`
`(1! Z—l'. "'1 z—:\ii+l)H(zj\i)'
`
`Replacing z by 2H}.— for I. = 0,1,” ‘ , N — l in the above
`equality, we have the following N X N matrix multiplications
`
`H(z) : WR,A(Z)H(ZN)
`
`(2.2)
`
`where we have (2.3), shown at the bottom of the page. Let
`
`V(z) % diag [11(3), thWN),
`
`Haw—1)].
`
`(2.4)
`
`Then, the matrix H(3) in (2.3) can be rewritten as
`
`111(3) : V(z)w;.A(z).
`
`This completes the following diagonalization of H(,Z""") by
`combining (2.2)
`
`He“) = r marvewwz)
`where I means the inverse.
`
`(2.5)
`
`From now on, we assume all filters in Fig. 2 are FTR, and the
`PR of the system in Fig. 2 means the overall system function
`H(z)G(z) has an FIR inverse.
`The PR of the multirate system H(z)G(z) is equivalent
`to the one of the multirate system H(:5N)G(ZN). In fact, if
`H(z‘\i)G(z‘h‘i) has PR, then any input signal 21(3) can be
`reconstructed from H(ZN)G(ZN)X(Z). Thus, X(ZN) can be
`reconstructed from H(ZN)G(2N)X(ZN) with an FIR synthe-
`sis filterbank. In other words, any X(z) can be reconstructed
`from H(z)G(z)X(z) with an FIR synthesis filterbank. This
`implies the PR of H(z)G(z). Conversely, we assume the PR
`of H(z)G(3), which is equivalent to that there exists an FIR
`inverse, i.e., there is an FIR K X N polynomial matrix Q(z)
`such that
`
`Qrzmtzmrz) : 1n-
`
`where IR is the K X K identity matrix. Thus, we also have
`
`Qfizl‘i)H(zl‘r)G(/3N) : IK-
`
`It implies that H(2N)CA%(Z‘N) has an FIR inverse (or PR).
`
`We, thus, consider the PR of H(zJV )G(z‘w). By (2.5)
`
`H(2N)G(ZN) : [W2A(z)]IV(z)WRA(z)G(zN).
`
`(2.6)
`
`It is clear that [WRA(Z)]I = A(z_1)W_N, which is parau-
`nitary. Let
`
`64(2) 3 meme”).
`
`to the one of
`the PR of H(2)G(z) is equivalent
`Then,
`V(z)é(z). Notice that the size of the matrix V(2)G(z) is
`N x K.
`
`On the other hand, V(3)G(z) has an FIR inverse equivalent
`to that of the greatest common divisor (gcd) of all determinants
`of all K X K submatrices of the N X K matrix V(2)G(z)
`that is 0:576! for a nonzero constant c and an integer d; see,
`for example, [32]. Since V(z‘) is diagonal and of the form
`(2.4), the above condition for the PR can be simplified further
`as follows.
`
`Without loss of the generality, we assume
`
`P
`
`H(z) = Z h.[k]z_k
`FrZU
`
`75 0, MP] % O, and P 2 1. Let S de-
`where MO]
`note the set of all zeros of the polynomial 111(3) of 2'71:
`5 é {21, 33,
`21>} with H(z;) = 0, where 33,1 3 I. S
`P may not be necessarily distinct. For a constant C,
`let
`CS % {(351, (323,
`..,, cap}, which is a rotated version of S.
`We have the following result for the PR.
`Theorem I: There exists an FIR nonmaximally decimated
`multirate filterbank in Fig. 2 such that the system in Fig. 2 has
`an FIR ideal linear equalizer if and only if
`
`fl
`051142 c;(;rKg.N>1
`
`(5;, U 512 U - -- U Six) : (f)
`
`(2.7)
`
`Ir.
`where 5,, = Iris-5, r.- = 1,2, -
`Theorem 1
`tells us that there exists a multirate filterbank
`
`in Fig. 2 for the ideal linear equalization if and only if the
`intersection of the unions of any X sets of all N rotated
`zero sets of S with angles Err/NJ : 0,1,---, N — 1 of
`the ISI transfer function 11(2) is empty. When K : N, the
`intersection in (2.7) contains at least 5, which is not empty.
`This implies that when K = N, the system in Fig. 2 does not
`have PR in the sense of nonexistence of FIR inverses. This
`
`is not surprising because any maximally decimated multirate
`filterbank does not add any redundancy to the signal and,
`therefore, does not have any e