General Multirate Building Structures with ... - Semantic Scholar

General Multirate Building Structures with Application to Nonuniform Filter Banks  Tongwen Cheny Dept. of Electrical and Computer Engineering University of Alberta Edmonton, Alberta, Canada T2N 1N4 Email: [email protected] Li Qiu Dept. of Electrical and Electronic Engineering Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong Email: [email protected] Erwei Bai Dept. of Electrical and Computer Engineering University of Iowa Iowa City, IA, 52242 USA Email: [email protected] First version: November 26, 1996; Revised: August 12, 1997

Abstract

This paper proposes a general linear dual-rate structure for multirate signal processing, which encompasses the usual multirate building blocks { expanders, LTI lters, decimators, and their cascade combinations { as special cases. Structural properties of such dual-rate systems are studied in detail; in particular, several equivalent implementation structures are proposed: (1) cascade connections of a block expander, a linear periodically time-varying system, and a block decimator; (2) vector sample-rate changers; (3) cascade connections of an expander, certain linear switching time-varying system, and a decimator. Using such general building blocks in multirate signal processing allows more design freedom and therefore can achieve what are otherwise impossible. This is illustrated in nonuniform multirate lter banks: Using the general building blocks as synthesis systems, the incompatibility for alias cancelation and structural dependency constraint for design,  This

research was supported by the Natural Sciences and Engineering Research Council of Canada and

the Hong Kong Research Grants Council.

y Corresponding

author.

1

both due to fractional decimation ratios in dierent channels, are eliminated. Hence optimal design of synthesis systems is possible.

Keywords: digital signal processing, multirate ltering, nonuniform lter banks, fractional decimation, optimization.

1 Introduction Traditional multirate building blocks in digital signal processing [8, 33] are decimators (downsamplers), expanders (upsamplers), and LTI lters, with possibly some summing junctions. An example is the fractional sample-rate changer shown in Figure 1, where " m is the ex-

u-

"m

-

F

- #n

y-

Figure 1: A sample-rate changer. pander by a factor m, # n the decimator by n, and F a suitable LTI lter. The output sample rate is m=n times the input sample rate. Such rate changers are not only useful in their own right [30], e.g., sample-rate conversion for bandlimited signals, they are also fundamental building blocks for multirate lter banks with uniform [8, 33] or nonuniform [26, 17] bands. Multirate signal processing has been studied a great deal, see recent books [18, 1, 21, 33, 9, 35, 32] and their references. The rate changer in Figure 1 is in fact a dual-rate system with the input-output property that shifting the input (u) by n samples results in shifting the output (y ) by m samples. Such a property is de ned as (m; n)-shift invariance, which is a generalization of time invariance for single-rate systems { if m = n = 1, a linear, (m; n)-shift-invariant system reduces to an LTI system. Consider two classes of dual-rate systems: the class associated with the structure in Figure 1, namely, the systems which are characterized by " m, LTI lter F , and # n, and the class of linear, causal, dual-rate systems shown in Figure 2, in which the notation \G : (m; n)" means that G is (m; n)-shift-invariant. Is the latter more general than the former? Or, is it always possible to realize a causal, dual-rate linear system in Figure 2 by the structure in Figure 1 with the expander " m, some LTI F , and the decimator # n? The answer is positive if the two integers m and n are coprime and negative otherwise [29], in this case m and n have some non-trivial common factor. As an simple example, if m = n = 2, a linear, (m; n)-shift-invariant system becomes a linear periodically time-varying (LPTV) system with 2

u

- G : (m; n)

y

-

Figure 2: A general dual-rate system. period 2; but it can be shown that the structure in Figure 1 in this case always represents an LTI system. The class of dual-rate systems in Figure 2 are more general than the structure in Figure 1 and are especially interesting when m and n have common factors. In this paper, we study such (m; n)-shift-invariant systems in detail and use them as the fundamental building blocks for multirate systems. One main contribution of this paper is to show that such (m; n)-shiftinvariant systems are realizable by using either block expanders and decimators and LPTV systems or traditional expanders and decimators and certain linear switching time-varying systems. Advantages of the more general setup have been already observed: Khansari and LeonGarcia [15, 16] and Nayebi et. al. [25] used LPTV lters and block decimators and expanders in lter-bank systems; in particular, Khansari and Leon-Garcia [16] showed that using general synthesis systems in a FIR lter bank, perfect reconstruction is possible if and only if the analysis lters have no common zero (note that this condition is necessary but not sucient for perfect reconstruction [36] using traditional building blocks); Xia and Suter [37, 38] later systematically studied block sampling in a more general setting using vector lter banks; Shenoy [29] showed that using general structures one can achieve what is otherwise impossible in design of fractional rate changers and also pointed out that the structural dependency in designing nonuniform lter banks [17, 4, 5] disappears when general structures are used for analysis and synthesis. As an application of the general dual-rate systems, consider the three-channel nonuniform lter bank shown in Figure 3, where the analysis and synthesis lters Hi and Fi are all LTI and causal. Hoang and Vaidyanathan [13] showed that the decimation integers, f2; 3; 6g, form an incompatible set; i.e., it is impossible to achieve alias cancelation by designing the six lters, let alone perfect reconstruction. Also, a design diculty called structural dependency arises in this setup [17, 4, 5]; this can be seen later after blocking the system to get the 6  6 equivalent transfer matrix. However, if appropriate dual-rate systems are used for analysis and synthesis as shown in Figure 4, the diculties encountered using the structure in Figure 3 no longer exist. This is 3

u

-

H0

- #2

v0-

" 2 - F0

-

H1

- #3

v1-

? " 3 - F1 - j

-

H2

- #6

v2-

? y" 6 - F2 - j

Figure 3: A nonuniform lter bank.

u

- H0 : (3; 6) v0- F0 : (6; 3) ? - H1 : (2; 6) v1- F1 : (6; 2) - j ? y- H2 : (1; 6) v2- F2 : (6; 1) - j

Figure 4: A nonuniform lter bank with general structures. because four of the six dual-rate lters used in Figure 4, H0 ; H1; F0 and F1 , have common factors in their m and n and hence are more general; they allow extra freedom in system design. In this case, optimal design based on model-matching theory, which was advocated in [30] for multirate lter design and in [6] for uniform lter-bank design, can be accomplished with relative ease { detailed discussion to be given later. Brie y, the paper is organized as follows. Section 2 de nes precisely what we mean by dual-rate systems and studies their fundamental properties such as shift invariance and causality; the blocking technique is used to convert the dual-rate systems into equivalent LTI systems. Section 3 shows that any linear dual-rate system can be realized by cascading a block expander, an LPTV system, and a block decimator, establishing the connection to block sampling used in [16]. Section 4 gives a connection between dual-rate systems and vector decimators and expanders, and multivariable LTI systems, which were studied in vector lter banks in [37, 38]. In Section 5, we introduce a special class of LPTV systems called linear 4

switching time-varying (LSTV) systems and prove the fact that any linear dual-rate system can be realized by cascading a traditional expander, an LSTV system, and a traditional decimator. Section 6 applies the general structures to nonuniform lter banks and shows that several limitations encountered when using traditional building blocks can be eliminated; an optimal design example is also given. Finally, in Section 7 we oer some concluding remarks.

2 General Dual-Rate Systems Dual-rate systems are to be used as fundamental building blocks for multirate systems. In this section we study basic concepts of linear dual-rate systems such as shift invariance, causality, and their various representations. Let ` be the space of discrete-time signals de ned on the set of all integers. A signal x in ` is written f


; x(?2); x(?1) j x(0); x(1); x(2);


g; the vertical bar separating the time from k = 0. A linear system G is regarded as a linear transformation mapping ` to itself, written y = Gu. It is a dual-rate system if the output and input have dierent sample rates { in this paper we shall make the assumption that the ratio of the two rates is a rational number, say, m=n; i.e., the output sample rate is m=n times the input sample rate. A linear, dual-rate system G can be always represented by a kernel function g (k; l):

y = Gu , y(k) =

X

l

g(k; l)u(l); 8k:

(1)

Shift Invariance To de ne shift invariance precisely, let U be the unit time delay on ` with transfer function z?1 . For a linear, dual-rate system G with output and input sample-rate ratio m=n, we de ne G to be (m; n)-shift-invariant if GU n = U m G: (The two integers m and n need not be coprime.) This means that shifting the input by n samples results in shifting the output by m samples. In terms of the kernel functions, (m; n)-shift invariance is characterized by the following relation

g(k + m; l + n) = g(k; l); 8k; l: Figure 2 represents an (m; n)-shift-invariant G using block diagrams. 5

This shift invariance guarantees that appropriate blocking of the input and output gives rise to a multivariable (vector) LTI system [20, 23, 3, 24, 34]. (A similar blocking structure was also used earlier in convolutional codes [10].) For an integer p > 0, de ne the p-fold blocking operator, Lp , via x = Lp x (underlining denotes blocking): 9 8 2 3 3 2 > > x(p) x(0) > > > > 7 > > 7 6 6 = < x ( p + 1) x (1) 7 7 6 6 7 6 7 6 ;


f


j x(0); x(1);


g 7! >


6 .. 7 ; 6 : (2) . 7 .. > > > 4 5 5 4 . > > > > ; : x(2p ? 1) x(p ? 1) Lp maps ` to `p , the external direct sum of p copies of `. If the underlying period for x is h, the underlying period for the blocked signal x is ph. The inverse L?p 1 , mapping `p to `, amounts to reversing the operation in (2). Let G be linear, dual-rate, and SISO (single-input, single-output). Block the input and output appropriately to get the blocked system G := Lm GL?n 1 , which has n inputs and m outputs. It is a well-known fact that G is LTI i G is (m; n)-shift-invariant [20, 24, 34]. Hence if G is (m; n)-shift-invariant, G has an m  n transfer matrix: 3 2 G^ 00(z) G^ 01(z)


G^ 0;n?1 (z) 6 G^ 10(z) G^ 11(z)


G^ 1;n?1 (z) 777 6 6 ^ G(z) = 6 .. .. .. 7: 5 4 . . . ^ ^ ^ Gm?1;0 (z) Gm?1;1(z)


Gm?1;n?1 (z) The entries in this matrix relate to the kernel function g (k; l) of G as follows: X (3) G^ij (z) = g(i + km; j )z ?k : k

Causality Causality of a dual-rate system G re ects implementability of the system in real time. Let G have input u and output y. Because the ratio of the sample rates of y and u is m=n, we can take the sample periods of u and y to be mh and nh, respectively, where h is some real number. Assuming both u and y are synchronized at time t = 0, we have that u(k) occurs at time t = k(mh) and y (k) at t = k(nh). Thus G is causal if for any k, the output y (k) depends only on inputs occurred at t  k(nh), or on u(l) for all l satisfying lm  kn. Similarly, G is strictly causal if y (k) depends only on u(l) for all l such that lm < kn. In terms of the kernel function in (1), G is causal i

g(k; l) = 0 whenever kn < lm; 6

(4)

and is strictly causal i

g(k; l) = 0 whenever kn  lm: If the linear, dual-rate system G is both (m; n)-shift-invariant and causal, the blocked system G is LTI and causal. Moreover, the direct feedthrough matrix 2 3 D00 D01


D0;n?1 6 D10 D11


D1;n?1 777 6 ^ 6 G(1) = 6 .. (5) .. .. 7 . . . 4 5 Dm?1;0 Dm?1;1


Dm?1;n?1 must have a block lower-triangular structure due to causality. This is called causality constraint [19, 7] and can be derived from equations (4) and (3):

Dij = 0 whenever in < jm:

(6)

As an example, consider the rate changer in Figure 1 with m = 2, n = 3, and F being LTI and causal. It follows that the overall (dual-rate) system is (2; 3)-shift-invariant and causal. Introduce the 6-fold, type-1 polyphase decomposition [33] of F^ (z ),

F^(z) =

5 X

i=0

z?i F^i (z6);

the polyphase components F^i (z ) being causal. Then the blocked dual-rate system has the transfer matrix [14, 4, 5] " # F^0 (z) z ?1 F^4 (z) z ?1 F^2 (z) : (7) F^3 (z) F^1 (z) z ?1 F^5(z) Note that this matrix contains all six polyphase components due to the coprimeness of m and n (to be seen later); note also that the (0; 1), (0; 2), and (1; 2) entries are all strictly causal, which shows causality of the dual-rate system. From now on, we shall restrict our attention to linear, dual-rate systems which are shiftinvariant, causal, and moreover, whose blocked transfer matrices have real-rational functions of z as their entries. All such transfer matrices have state-space models

x(k + 1) = Ax(k) + Bu(k); y(k) = Cx(k) + Du(k);

(8) (9)

where x is the state vector and A; B; C; D are all real matrices of compatible dimensions with A square. The matrices B; C; D are partitioned according to the input and output dimensions 7

of G:

2 i

h

6 6

B = B0 B1


Bn?1 ; C = 66 4

C0 C1 .. .

Cm?1

3 7 7 7 7 5

;

and D is the same as the right-hand matrix in (5) satisfying the causality constraint in (6). The state-space model in (8-9) is for the blocked system. The corresponding dierence equations for the original dual-rate system are as follows:

x(k + 1) = Ax(k) +

nX ?1

y(km + i) = Cix(k) +

Bj u(kn + j );

j =0 nX ?1 j =0

Dij u(kn + j ); i = 0; 1;


; m ? 1:

These equations can be used to implement the dual-rate system: The input u is updated every mh seconds, the vector x every mnh seconds, and the output y every nh seconds. Such systems are implementable on, e.g., general-purpose DSP cards, with only nite memory. The state-space model in (8-9) is general and holds for both IIR and FIR systems. If G is FIR with order, say, p, a simpler model is given by

y(k) = M0u(k) + M1u(k ? 1) +


+ Mp u(k ? p); where M0 ; M1;


; Mp are all m  n constant matrices and M0 satis es the condition in (6) for causality. From here one can also write down the corresponding equations for the original system relating y to u. In this paper, we use the blocking technique consistently for treating dual-rate systems. An alternative technique is based on alias-component (AC) matrices in the frequency domain; we refer to Shenoy [28, 29], Shenoy et al. [30], and Reng [27] for recent studies along this direction. Finally, we make some remarks on using dual-rate systems as building blocks for multirate signal processing systems. The class of dual-rate systems we are interested are SISO, causal, and (m; n)-shift-invariant, where m and n range over all positive integers. Traditional building blocks are decimators, expanders, and LTI lters, all SISO. They are all special cases of dualrate systems: The decimator # n is (1; n)-shift-invariant and causal; the expander " m is (m; 1)-shift-invariant and causal; any LTI lter is (1; 1)-shift-invariant. However, general LPTV systems with period m are dual-rate systems [they are (m; m)-shift-invariant] but cannot be constructed using decimators, expanders, and one SISO, LTI lter. In this sense, we conclude that dual-rate systems are more general building blocks for multirate systems. 8

3 Realization via Block Decimation and Expansion In this section, we establish the connection between dual-rate systems discussed in the preceding section and the generalized elements: block decimators, block expanders, and LPTV systems; these generalized elements have already been studied in multirate lter banks [15, 16, 25, 37]. The result in this section also provides a way for realizing general dual-rate systems. Let us rst consider the special case when m and n are coprime. Then the structures in Figures 1 and 2 are equivalent [29]: Lemma 1 If m and n are coprime, any linear, (m; n)-shift-invariant system in Figure 2 is realizable by the cascade structure in Figure 1 of the expander " m, an LTI lter F , and the decimator # n. This is already a known result, which was proven, e.g., as a lemma in [5] from a dierent perspective. A linear, dual-rate system which is (m; n)-shift-invariant cannot be represented by the structure in Figure 1 for some LTI F , if m and n have non-trivial common factor [29]. In this case, as we will show, block decimators and expanders, and LPTV systems should be used instead. Two positive integers p and q characterize the block decimator shown in Figure 5, where q represents the block size and p the decimation ratio. For any integer k, the input-output

u

- # (p; q)

y

-

Figure 5: The block decimator. equation is

y(kq + i) = u(kpq + i); i = 0; 1;


; q ? 1: If one groups the input into blocks of size q (starting from time k = 0), this block decimator retains only every p-th block. It can be veri ed that the block decimator is (q; pq )-shiftinvariant and causal; moreover, if q = 1, it reduces to the traditional decimator: # (p; 1) =# p. The block expander is the dual system, shown in Figure 6, where q again is the block size and p the expansion ratio. For any integer k, the block expander is de ned via ( 0; 1;


; q ? 1; y(kpq + i) = u0;(kq + i); ii = = q; q + 1;


; pq ? 1: 9

u

- " (p; q)

y

-

Figure 6: The block expander. This corresponds to inserting p ? 1 blocks of zeros if the input and output are blocked with size q . The block expander is (pq; q )-shift-invariant but noncausal in general. If q = 1, it reduces to the traditional expander " p. (For the frequency-domain input-output relations, see [16, 37].) Let G be any linear, dual-rate system which is (m; n)-shift-invariant. If m and n are not coprime, we can nd the largest common factor q and write m = mq  , n = n q , so that m and n are coprime. Thus we can state the main result of this section:

Theorem 1 The dual-rate system G is realizable by the cascade structure, shown in Figure 7, of the block expander " (m;  q ), a single-rate, LPTV system F with period q , and the block decimator # (n; q ). u-

" (m;  q ) - F : (q; q )

- # (n; q)

y-

Figure 7: An equivalent structure for dual-rate systems. Note that if m and n are coprime, then q = 1. In this case, the structure in Figure 7 reduces to that in Figure 1 with F LTI; and the theorem reduces to Lemma 1.

Proof of Theorem 1 In the following we shall outline the main steps in the proof. First it can be veri ed that the overall system in Figure 7, S : u 7! y , is dual-rate, (m; n)-shift-

invariant. To show that this structure can be used to represent any dual-rate, (m; n)-shiftinvariant system G, it suces to show that by proper choice of the LPTV F , the transfer matrix for the blocked system S := Lm SLn?1 can match any given m  n transfer matrix G^ (z ) (the blocked system for G); or equivalently, the mn entries in S^(z ) can be arbitrarily selected. For ease of reference in equations, let us write the block expander in Figure 7 as Em;q  and the block decimator Dn;q . Then S = Dn;q FEm;q  . In order to nd S , note rst that 10

F := Lq FL?q 1 is LTI with q inputs and outputs: F = 64




F0;q?1

F00

2

.. .

3

.. .

Fq?1;0


Fq?1;q?1

7 5

:

(10)

All subsystems Fij are LTI and freely assignable. The following identities follow readily 2

Lq Dn;q L?q 1 = 64

Dn

...

Dn

3

2

7 5

?1 6 ; Lq Em;q  Lq = 4

Em

3

...

qq

Em

;

7 5

(11)

qq

where Dn is simply # n and Em is " m . Let Pm;q  be the m  m permutation matrix which takes the (column) vector h

0 q


(m ? 1)q 1 q + 1


(m  ? 1)q + 1


q ? 1 2q ? 1


mq  ?1

i

0

(prime denotes matrix transpose) to the vector h

i

0 1 2


mq  ?1 ;

similarly for Pn ;q . Then it can be shown that 2 6 Lm = Pm;q  4

Lm

...

Lm

L?n 1 6

3

2

7 5

Lq ; L?n 1 = L?q 1 64

3

...

L?n 1

7 7 5

Pn?;q1 :

(12)

Based on equations (10-12), we can write ?1 S = Lm Dn;qFEm;q  Ln    ?1 Lq L?1 = Lm L?q 1 Lq Dn;q L?q 1 Lq FL?q 1 Lq Em;q  Lq n 32 2 F00


F0;q?1 Lm Dn 76 6 .. .. . . = Pm;q 54  4 . . . Fq?1;0


Fq?1;q?1 Lm Dn

Em L?n 1 6

3

32 76 54

...

Em L?n 1

7 7 5

Pn?;q1 :

From here, the inner three matrices together can be viewed as a q  q block system, the (i; j ) block being

Lm Dn Fij Em L?n 1 ;

which is m  n and whose entries are arbitrarily assignable by proper choice of the LTI Fij because m  and n are coprime (Lemma 1). Hence the inner three matrices together can match 11

any given transfer matrix; so can S because Pm;q  and Pn  ;q are permutation matrices. Thus the proof is completed. 2 The proof supports a procedure to compute the LPTV F if the dual-rate system G is given. To illustrate, consider the case with m = 4 and n = 6: Given the 4  6 blocked transfer matrix [G^ ij (z )], we want to realize G via the structure in Figure 7 with m  = 2, n = 3, and q = 2, or more precisely, nd the LPTV F with period 2. Write the blocked F (F = L?2 1 FL2) as: " # ^00(z ) F^01 (z ) F ^ F (z) = F^ (z) F^ (z) :

Then from the proof of Theorem 1 we set

10

11

G = P2;2 RP3?;21 with

"

#

"

L2D3 0 00 R01 R= R R10 R11 := 0 L2 D3

#"

F00 F01 F10 F11

(13) #"

E2L?3 1 0

0

#

E2L?3 1 :

Since Rij = L2 D3Fij E2L?3 1 , its transfer matrix is of the form in (7) in terms of type-1 polyphase components [33] of F^ij (z ), denoted F^ijk (z ) for k = 0; 1;


; 5. Equation (13) is equivalent to P2?;21 GP3;2 = R, which reads 2 G^00(z) G^02(z) G^ 04(z) G^ 01(z) G^ 03(z) G^ 05(z) 3 6 ^ 7 ^ ^ ^ ^ ^ 6 G20 (z ) G22 (z ) G24 (z ) G21 (z ) G23 (z ) G25 (z ) 7 6 ^ 10(z ) G^ 12 (z ) G^ 14(z ) G^ 11(z ) G^ 13(z ) G^ 15(z ) 75 4 G G^30(z) G^32(z) G^ 34(z) G^ 31(z) G^ 33(z) G^ 35(z) 3 2 F^000 (z) z ?1 F^004 (z) z ?1 F^002 (z) F^010 (z) z ?1 F^014 (z) z ?1F^012 (z) 7 6 ^3 ?1 ^5 ?1 ^ 5 ^1 ^3 ^1 (14) = 664 F^000 (z ) ?F100^(4z ) z ?1 F^002 (z ) F^010 (z ) ?F101^(4z ) z ?1 F^012 (z ) 775 : F10(z) z F10(z) z F10(z) F11 (z) z F11(z) z F11(z) F^103 (z) F^101 (z) z ?1 F^105 (z) F^113 (z) F^111 (z) z ?1 F^115 (z) Hence F^00(z) = F^000 (z6) + z ?1 F^001 (z6) + z ?2F^002 (z 6) + z ?3 F^003 (z 6 ) + z ?4 F^004 (z6 ) + z ?5 F^005 (z6) = G^ 00(z 6 ) + z ?1 G^ 22(z 6 ) + z 4 G^ 04(z 6) + z ?3 G^ 20(z 6 ) + z 2 G^ 02(z 6) + z G^ 24(z 6): (15) This relates F00 to the given dual-rate system G in terms of its blocked transfer matrix. We remark that if G is causal, so is F00: Causality of G requires that G^ 04(z ), G^ 02(z ), and G^ 24 (z ) be strictly causal; hence the terms such as z 4 G^ 04(z 6) in (15) are still causal. Similarly, one can determine other F^ij (z ) from their polyphase components and (14). 12

4 Realization via Vector Sample-Rate Changers Theorem 1 also establishes a connection through blocking between dual-rate systems and vector sample-rate changers studied by Xia and Suter [37, 38]. For any dual-rate system G which is (m; n)-shift-invariant, introduce m , n , and q as in Theorem 1. De ne a vector  is the vector expander and # n the vector sample-rate changer as in Figure 8, where " m

u-

" m

- F

- # n

y-

Figure 8: The vector sample-rate changer. decimator, both with q channels 2 " m 6 ... " m = 4

" m

3

2

7 5

; # n = 64

# n

3

...

# n

7 5

;

and F is a multivariable (vector) LTI system with q inputs and q outputs. The structure in Figure 8 relates to the dual-rate system G through q -fold blocking:

Theorem 2 The blocked dual-rate system, Lq GL?q 1, is realizable by the structure in Figure 8 for some LTI F. Proof By Theorem 1, G can be represented by the structure in Figure 7 for some LPTV F with period q : G = Dn;q FEm;q  , following the notation in the proof of Theorem 1. Thus

?1 Lq GL?q 1 = Lq Dn;q FEm;q  Lq ?1 = (Lq Dn;q L?q 1 )(Lq FL?q 1 )(Lq Em;q  Lq ) 2 3 2 3 " m # n 7 7 ?1 6 ... ... = 64 5; 5 (Lq FLq ) 4 " m # n the last equality follows from (11). Identifying F = Lq FL?q 1 , which is LTI because F is LPTV with period q , completes the proof. 2

The converse of Theorem 2 is also true: One can represent the vector sample-rate changer in Figure 8 through blocking a dual-rate system. Thus the two structures are equivalent in this sense. 13

Connecting with Theorem 1, we also conclude that the structure in Figure 7 based on block sampling and the one in Figure 8 based on vector processing are equivalent through blocking. At this point, one may wonder why Xia and Suter [37] showed that their vector lter banks, which are built with structures in Figure 8, are more general than those in [16] using block sampling. This is because LTI lters were used in [16] for constructing the lter banks. Had LPTV lters been used appropriately according to Theorem 1, the two lter bank structures would have been equivalent.

5 Realization via Linear Switching Time-Varying Systems In the preceding two sections, we presented two structures which can be used to implement a general dual-rate system: The rst one uses block decimation and expansion and the second uses vector decimation and expansion. The problem with the rst is that real-time implementation of the block decimator and expander is technically demanding: The block decimator requires certain storage function and the block expander is noncausal, if the input and output samples are uniform in time. (These problems seem to disappear if input and output samples are allowed to be nonuniform in time.) The disadvantage of the second structure is that blocking is required to convert a SISO system to a multivariable (vector) system. If m and n are already coprime, Lemma 1 states that any (m; n)-shift-invariant system can be implemented by the structure in Figure 1 using only the traditional decimator and expander. If m and n are not coprime, can we make use of a similar structure with traditional decimator and expander and some SISO system F ? The answer is positive; but we need to consider a special class of LPTV systems for F , which are called linear switching time-varying (LSTV) systems. With these LSTV structures, the above mentioned disadvantages associated with Figures 7 and 8 are eliminated. For two positive integers r and s, an [r; s]-LSTV system F , represented in Figure 9, consists of r LTI subsystems, F0 ; F1;


; Fr?1, and a switching device as depicted in Figure 10. The

u

-

F : [r; s]

y

-

Figure 9: An [r; s]-LSTV system. switching device switches the input u to each subsystem for exactly s samples starting from F0 to Fr?1 and then repeats. In other words, divide the time into intervals of rs samples starting 14

from k = 0; during the rst interval, 0  k  rs ? 1, the inputs u(k) over js  k  (j +1)s ? 1 (0  j  r ? 1) are connected to Fj ; similarly for the remaining intervals; the outputs of all subsystems are summed up to form the overall output y . Note that at times when one subsystem is switched to u, the other subsystems are assumed to have zero inputs, and their outputs evolve according to the past history. This LSTV structure can be implemented with

u - e??

? &s??

?

e

-

F0

e

-

F1

? -j ? r r r

e

? y- Fr?1 - j

Figure 10: The structure of an [r; s]-LSTV system. the periodic switching device and the LTI subsystems; for example, it is possible to implement them on microprocessors. Let f (k; l) be the kernel representation of the LSTV F :

y = Fu , y(k) =

X

l

f (k; l)u(l); 8k:

Let fj (k) be the impulse response of the LTI subsystem Fj . Now think of [f (k; l)] as an in nite matrix; the in nite matrix representation of Fj is [fj (k ? l)], a Toeplitz matrix. The switching mechanism implies that [f (k; l)] can be obtained from [fj (k ? l)] as follows: Starting from l = 0, the rst s columns of [f (k; l)] are identical to the rst s columns in [f0 (k ? l)], the next s columns are identical to the corresponding ones in [f1 (k ? l)], and so on; and repeat this process for every rs columns in [f (k; l)]. In equations, we write

f (k; l) = fj (k ? l); l = js; js + 1;


; js + s ? 1 mod rs:

(16)

It follows readily from (16) that the LSTV system F is LPTV with period rs. Hence F := Lrs FL?rs1 is LTI with an (rs)  (rs) transfer matrix. To see how to write down this transfer matrix, let us consider the case with r = s = 2; then F^ (z ) is 4  4. Introducing the 15

type-1 polyphase representations [33] for F0 and F1 ,

F^j (z) =

we get the blocked

3 X

z?i F^ji (z4 ); j = 0; 1;

i=0 systems F j = L4Fj L?4 1

F^j0 (z) z ?1 F^j3 (z) 6 ^0 6 ^1 F^ j (z) = 66 FF^j2((zz)) FF^j1 ((zz)) 4 j j F^j3 (z) F^j2 (z) 2

(j = 0; 1) as pseudo-circulant matrices:

z ?1 F^j2(z) z ?1 F^j3 (z) F^j0 (z) F^j1 (z)

z ?1 F^j1 (z) 7 z ?1 F^j2 (z) 77 ; j = 0; 1: z ?1 F^j3 (z) 75 F^j0 (z) 3

Then it follows from (3) and (16) that F^ (z ) can be formed by putting together the rst two columns of F^ 0 (z ) and the last two columns of F^ 1 (z ): 3 2 F^00 (z) z ?1 F^03(z) z ?1 F^12 (z) z ?1 F^11(z) 7 6 ^1 ?1 ^3 ?1 ^ 2 ^0 F^ (z) = 664 FF^02 ((zz)) FF^01((zz)) z F^F0 (1z()z) zz ?1FF^23((zz)) 775 : 0 0 1 1 F^03 (z) F^02 (z) F^11(z) F^10 (z) Note that if this matrix is partitioned into two 4  2 submatrices, each submatrix is pseudocirculant column-wise but not row-wise. This observation generalizes in the obvious way for the LSTV structure in Figure 10:

Lemma 2 Let F j = LrsFj L?rs1; partition each transfer matrix as follows: F^j (z) = F^ 0j (z) F^ 1j (z)


F^ jr?1 (z) ; i

h

every submatrix being (rs)  s. Then the transfer matrix for the blocked LSTV system, F = Lrs FL?rs1 , is given by

F^(z) = F^ 00 (z) F^ 11(z)


F^ rr??11 (z) : h

i

Each submatrix in F (z ) is pseudo-circulant column-wise only.

Now we are setup to state and prove the main result of this section: Any general dual-rate system can be implemented by an LSTV system along with some traditional decimator and expander. Let G be any linear, dual-rate system which is (m; n)-shift-invariant; write m = mq  and n = nq for some integer factor q so that m and n are coprime: 16

u-

" m

- F : [q; m n] - # n

y-

Figure 11: An equivalent structure for dual-rate systems.

Theorem 3 The dual-rate system is realizable by the cascade structure, shown in Figure 11, of the expander " m  , some [q; m  n ]-LSTV system F , and the decimator # n . Proof It is easy to check that H := Dn FEm is dual-rate, (m; n)-shift-invariant, where Dn =# n and Em =" m . To show that this structure can be used to realize any (m; n)-shiftinvariant system G, it suces to show that the blocked system, H = Lm HL?n 1, has an m  n transfer matrix whose elements are all freely assignable by proper choice of F . Now

H = LmDn FEm L?n 1 :

(17)

To relate this to the blocked system of F in Lemma 2, de ne two linear transformations Kn and Jm as block diagonal matrices: h

h

i

i



1 0


0 1n ;


; 1 0


0 1n ; 9 3 2 3 > 1 1 > > 7 > 6 = 0 777 6 0 7 7 6 Jm = block diag .. 7 ;


; 6 .. 7 > : > 4 . 5 .5 > > 0 m 1 ; 0 m 1 The blocks in Kn are repeated m times and the blocks in Jm n times. Then it can be veri ed that 1 Lm Dn = Kn Lmn ; Em L?n 1 = L?mn (18) :  Jm

Kn = block diag

82 > > > > > 4 > > :

Equations (17) and (18) lead to 



H = Km Lm nq FL?m1nq Jm = Kn F Jm ;  n q )  (m  n q ) transfer matrix F^ (z ) studied in Lemma 2. where F has an (m Note that pre- and post-multiplying F^ (z ) by Kn and Jm , respectively, amount to extracting an m  n matrix from F^ (z ) by taking rows numbered 0; n ; 2n;


; (m ? 1)n and columns 17

numbered 0; m;  2m; 


; (n ? 1)m  . Since F^ (z ) can be partitioned into column pseudo-circulant submatrices (Lemma 2)

F^(z) = F^ 00 (z) F^ 11(z)


F^ qq??11 (z) ; h

i

H^ (z) has a corresponding partition i

h

H^ (z) = H^ 0 (z) H^ 1(z)


H^ q?1 (z) ;

(19)

the blocks being mutually independent. Note that H^ j (z ) relates to F^ jj (z ), which in term relates to Fj . Now it remains to show that all elements in H^ j (z ) are assignable by proper choice of Fj . Take H^ 0 (z ) as an example; this is an m  n matrix whose elements are those in F^ 00 (z ) with the following row and column index pairs: (in; j m  ) : i = 0; 1;


; m ? 1; j = 0; 1;


; n ? 1: Because all elements in the same column of F^ 00 (z ) are independent (Lemma 2), if two elements  )-th and (i2 n ; j2m )-th elements in F^ 00(z ), were related, it would of H^ 0 (z ), say, the (i1n ; j1m be due to the column pseudo-circulant property; in this case,

i1n ? j1 m = i2 n ? j2 m mod (m n q): Thus we can write (i1 ? i2 )n + (j2 ? j1)m  = k(m n q ) 0  i1; i2  m ? 1; 0  j1; j2  n ? 1;

(20) (21) (22)

k in (20) being an integer. The fact that m and n are coprime and (20) imply that m divides i1 ? i2 ; then from (21) we have i1 = i2. In this case (20) reduces to (j2 ? j1 )m  = km n q; which implies that n divides j2 ? j1, which implies with (22) in turn that j1 = j2. The above shows that the elements in H^ 0 (z ) are all independent; similarly for the other blocks in (19). 2  n q )-fold To get the (m  n q )-fold blocked system F , according to Lemma 2, we need the (m polyphase components of the subsystems Fj , j = 0; 1;


; q ? 1. Every one of those polyphase 18

components appear exactly once in H^ (z ). Hence, if the structure in Figure 11 is used to realize a dual-rate, (m; n)-shift-invariant system G, there exists a one-to-one correspondence between the dual-rate G and the set of subsystems of F , fF0 ; F1;


; Fq?1g. A procedure outlining this correspondence can be given based on the proof of Theorem 3.

6 Nonuniform Filter Banks In this section we study using the general structures in nonuniform lter banks to achieve what are otherwise impossible. Consider the three-channel nonuniform lter bank in Figure 3, built via traditional blocks. It is shown that this system is incompatible [13] and hence alias cancelation is impossible using LTI and causal lters, let along perfect reconstruction. Let the lter bank system u 7! y in Figure 3 be G. To see the structural dependency constraint, we need to compute the blocked transfer matrix for G := L6 GL?6 1 . Bring in the 6-fold, type-1 polyphase decompositions [33] for the analysis and synthesis lters:

H^ j (z) =

5 X

i=0

5

X z ?i H^ ji (z6); F^j (z) = z?i F^ji (z6 ); j = 0; 1; 2:

i=0

It follows from the procedures in [4, 5] that G^ (z ) = F^ (z )H^ (z ), where H^ (z ) and F^ (z ) are, respectively, the analysis and synthesis matrices: 3 2 H^ 00(z) z ?1 H^ 05(z) z ?1 H^ 04(z) z ?1 H^ 03(z) z ?1 H^ 02(z) z ?1 H^ 01(z) 6 ^2 H^ 01 (z) H^ 00(z) z ?1 H^ 05(z) z ?1 H^ 04(z) z ?1 H^ 03(z) 777 6 H0 (z ) 6 ^ 04(z ) H^ 03 (z ) 6 H H^ 02(z) H^ 01 (z) H^ 00(z) z ?1 H^ 05(z) 77 (23) H^ (z) = 66 H 0 ? 1 5 ? 1 4 ? 1 3 ? ^ 1 (z ) z H^ 1 (z ) z H^ 1 (z ) z H^ 1 (z ) z 1 H^ 12(z ) z ?1 H^ 11(z ) 77 6 6 ^ 13(z ) H^ 12 (z ) 4 H H^ 11(z) H^ 10 (z) z ?1 H^ 15(z) z ?1 H^ 14(z) 75 H^ 20(z) z ?1 H^ 25(z) z ?1 H^ 24(z) z ?1 H^ 23(z) z ?1 H^ 22(z) z ?1 H^ 21(z) 2 3 F^00 (z) z ?1 F^04 (z) z ?1 F^02 (z) F^10 (z) z ?1F^13 (z) F^20 (z) 6 ^1 ?1 F^ 5 (z ) z ?1 F^ 3 (z ) F^1 (z ) z ?1 F^ 4 (z ) F^ 1 (z ) 7 6 F0 (z ) z 7 0 0 1 1 2 6 2 (z ) 0 (z ) ?1 F^ 4 (z ) F^2 (z ) z ?1 F^ 5 (z ) F^ 2 (z ) 7 ^ ^ 6 F 7 F z F^ (z) = 66 F^03 (z) F^01 (z) z ?1 F^05 (z) F^13 (z) F^ 0(1z) F^23 (z) 77 : (24) 6 7 0 0 0 1 1 2 6 ^04 (z ) F^02 (z ) 4 F F^00 (z) F^14 (z) F^11 (z) F^24 (z) 75 F^05 (z) F^03 (z) F^01 (z) F^15 (z) F^12 (z) F^25 (z) Note the structural dependency: The rst three columns in F^ (z ) are mutually dependent and so are the next two columns; similarly, the rst three rows in H^ (z ) are mutually dependent and so are the next two rows. The structural dependency poses a diculty in design [5]. 19

Shenoy [29] rst observed that structural dependency disappears if more general structures are used. Now we propose to use general dual-rate systems as in Figure 4 to replace the analysis and synthesis subsystems; note the way Figure 4 is constructed: Each channel has the same decimation ratio as the corresponding channel in Figure 3, but all three analysis subsystems have the same n, the least common multiple of the three decimation integers f2; 3; 6g in Figure 3; this way H0, H1, F0 and F1 in Figure 4 are more general because they have common factors in their m and n. Denote the system u 7! y in Figure 4 by T . It is easily veri ed that T is LPTV with period 6; the equivalent blocked system (T = L6TL?6 1 ) is 2 h

T = L6 F0 F1 F2 =

h

=:

h

i

6 4

3

H0 7 H1 5 L?6 1 H2

L6 F0L?3 1 L6 F1 L?2 1 L6F2 2

F0 F1 F2

=: F H;

i

H0 6 4 H1 H2

3

i

?1 L 3 H0 L6 6 ?1 4 L2 H1 L6 ? H 2 L6 1 2

3 7 5

7 5

where H and F are both LTI and 6  6: H^ (z ) is the analysis matrix and F^ (z ) the synthesis matrix. Note that all entries in H^ and F^ are freely designable (no structural dependency). Hence T can perfectly match any causal, LPTV system with period 6; in particular, it can perfectly match any time-delay system by proper choice of the dual-rate systems. Therefore, perfect reconstruction for the structure in Figure 4 is possible. It is worth noting that if dual-rate structures are used to build nonuniform lter banks as suggested in Figure 4, the blocked lter-bank systems have the form F^ (z )H^ (z ), where no structural constraints exist among the elements in the two matrices F^ (z ) and H^ (z ). This situation is very similar to the polyphase matrix representation of uniform lter banks. Hence, design techniques for uniform lter banks are immediately applicable. A more interesting scenario is perhaps given in Figure 12 in which the analysis structure in Figure 3 is combined with the synthesis structure in Figure 4. Is perfect reconstruction possible in this case? The answer is positive and is illustrated by the simple example below. (Note that it suces to construct an analysis lter bank so that the corresponding analysis matrix H^ has no nite unstable zeros; then we just take the synthesis matrix F^ to be H^ ?1 multiplying some 20

u

-

H0

- #2

v0-

F0 : (6; 3)

-

H1

- #3

v1-

F1 : (6; 2)

? -j

-

H2

- #6

v2-

F2 : (6; 1)

? y-j

Figure 12: A lter bank with mixed structures. time delay.) Let the LTI analysis lters in Figure 12 be H^ 0(z) = 1; H^ 1(z) = z ?4 + z ?5 ; H^ 2 (z) = z ?3 : The associated analysis matrix can be computed from (23): 2 1 0 0 0 0 0 3 6 0 1 0 0 0 77 6 0 7 6 H^ (z) = 666 00 z ?0 1 z0?1 00 10 00 777 : 7 6 4 0 0 0 0 z ?1 z ?1 5 0 0 0 z ?1 0 0 De ne the synthesis matrix to be 2 z ?1 0 0 0 0 03 6 0 ?z ?1 0 1 0 0 77 6 6 ?1 0 0 0 0 777 : F^ (z) = 666 00 z 0 (25) 0 0 0 1 77 6 4 0 0 z ?1 0 0 0 5 0 0 ?z ?1 0 1 0 It follows that F^ (z )H^ (z ) = z ?1 I and hence the system achieves perfect reconstruction with time delay z ?6 . The synthesis matrix in (25) is not subject to the structural dependency constraint as in (24) and hence does not give rise to the traditional synthesis structure in Figure 3; it corresponds to some general dual-rate structures in Figure 12. The above observation can be generalized to nonuniform multirate lter banks with multiple channels and arbitrary fractional decimation ratios [26, 17, 5]: If the synthesis subsystems are replaced by appropriate dual-rate structures, incompatibility [13] and structural dependency [17, 5] disappear; and perfect reconstruction is always possible. 21

One advantage of eliminating structural dependency is to allow optimal design of synthesis systems; we conclude this section by considering such a design example. We use the structure in Figure 12 and pre-select the linear-phase, FIR analysis lters: H0 (order 40) is lowpass with cuto frequency ! = =2; H1 (order 30) is bandpass with passband =2  !  5=6; H2 (order 14) is highpass with cuto frequency 5=6. All three lters are designed using MATLAB function fir1 with their magnitude Bode plots given in Figure 13. 20

0

−20

−40

−60

−80

−100

−120 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 13: jH^ 0j (solid), jH^ 1j (dot), and jH^ 2j (dash) in dB versus !=2 . The synthesis systems now can be designed by minimizing the `2-induced norm [6] of the error system between an ideal time-delay system Td and the lter bank T :

Jopt := min kTd ? T k: F i

By blocking, we can reduce this optimization problem to an equivalent H1 model-matching problem [6, 5] kT^d(z) ? F^ (z)H^ (z)k1; Jopt = min ^ F (z)

where T^d (z ) is the 6-fold blocked time-delay system, H^ (z ) the associated 6  6 analysis matrix given in (23), F^ (z ) the associated 6  6 synthesis matrix with no structural dependency, and 22

the norm is the H1 norm [11, 12] (the peak maximum singular value of the frequency-response matrix). The later optimization problem is solvable [11, 12] by, e.g., the MATLAB software { -Analysis and Synthesis Toolbox [2]. For good reconstruction error (Jopt ), we take the ideal time delay to be T^d (z ) = z ?80 . The H1 optimization then yields a reconstruction error Jopt = 0:31%. This means [22] that the maximum alias and magnitude distortions, de ned appropriately as in [22], of the designed system are both  0:31% and the phase distortion is  sin?1 (0:31%) = 0:18. Of course, the synthesis systems designed use the general dual-rate structures.

7 Conclusion In this paper we proposed dual-rate systems as general building blocks for multirate signal processing. Several structures implementing such dual-rate systems were studied in detail. The work in this paper provides a uni ed framework for using several generalized elements such as block decimation and expansion, LPTV ltering, and vector sample-rate changers, which have already found applications in multirate lter banks [16, 25, 37]. The advantages of such general dual-rate systems in nonuniform lter banks were illustrated: Channel incompatibility [13] and structural dependency [17, 5] are removable by proper use of the dual-rate structures. The structures studied in this paper fall into the general class of LPTV systems. This fact is essential for using the blocking technique to relate to equivalent single-rate, LTI systems which are necessarily multivariable. The techniques in this paper are not directly applicable to more general structures which are not periodically time-varying; examples of such systems include the linear time-varying lter banks studied in [31] and the references therein. For the time-varying multirate systems, time-domain techniques are proven to be more eective.

Acknowledgment The authors are grateful to Dr. Masoud Khansari and Dr. Ram Shenoy for helpful discussions and especially to Dr. Shenoy for an advance copy of [29]; they would also like to thank the anonymous reviewers for their helpful comments/suggestions.

References [1] A. N. Akansu and R. A. Haddad, Multiresolution Signal Decomposition, Academic, New York, 1992. 23

[2] G. J. Balas, J. C. Doyle, K. Glover, A. Packard, and R. Smith, -Analysis and Synthesis Toolbox User's Guide, MuSyn Inc. and the MathWorks, Inc., 1991. [3] C. W. Barnes and S. Shinnaka, \Block-shift invariance and block implementation of discrete-time lters," IEEE Trans. Circuits and Systems, vol. 27, pp. 667-672, 1980. [4] T. Chen, \A lifting approach to analysis and design of nonuniform multirate lter banks," Proc. Canadian Conf. on Electrical and Computer Engineering, pp. 874-877, Montreal, Sept. 1995. [5] T. Chen, \Nonuniform multirate lter banks: Analysis and design with an H1 performance measure," IEEE Trans. Signal Processing, vol. 35, no. 3, pp. 572-582, 1997. [6] T. Chen and B. A. Francis, \Design of multirate lter banks by H1 optimization," IEEE Trans. Signal Processing, vol. 43, no. 12, pp. 2822-2830, 1995. [7] T. Chen and L. Qiu, \H1 design of general multirate sampled-data control systems," Automatica, vol. 30, no. 7, pp. 1139-1152, 1994. [8] R. E. Crochiere and L. R. Rabiner, Multirate Digital Signal Processing, Prentice-Hall, Englewood Clis, NJ, 1983. [9] N. J. Fliege, Multirate Digital Signal Processing, Wiley, Chichester, 1994. [10] G. D. Forney, Jr., \Convolutional codes I: algebraic structure," IEEE Trans. on Info. Theory, vol. 16, pp. 720-738, 1970. [11] B. A. Francis, A Course in H1 Control Theory, Springer-Verlag, Lecture Notes in Control and Information Sciences, vol. 88, Berlin, 1987. [12] M. Green and D. J. N. Limebeer, Linear Robust Control, Prentice-Hall, Englewood Clis, NJ, 1995. [13] P.-Q. Hoang and P. P. Vaidyanathan, \Non-uniform multirate lter banks: theory and design," Proc. IEEE Int. Symp. Circuits and Systems, Portland, OR, 1989, pp. 371-374. [14] C.-C. Hsiao, \Polyphase lter matrix for rational sampling rate conversions," Proc. IEEE Int. Conf. Accoust., Speech, Signal Processing, Dallas, April 1987, pp. 2173-2176. [15] M. R. K. Khansari and A. Leon-Garcia, \Perfect reconstruction conditions for subband decomposition with generalized subsampling," SPIE - Visual Communications and Image Processing Conf., Boston, 1992. 24

[16] M. R. K. Khansari and A. Leon-Garcia, \Subband decomposition of signals with generalized sampling," IEEE Trans. Signal Processing, vol. 41, no. 12, pp. 3365-3376, 1993. [17] J. Kovacevic and M. Vetterli, \Perfect reconstruction lter banks with rational sampling factors," IEEE Trans. Signal Processing, vol. 41, no. 6, pp. 2047-2064, 1993. [18] H.S. Malvar, Signal Processing with Lapped Transforms, Artech House, Boston, 1992. [19] D. G. Meyer, \A new class of shift-varying operators, their shift-invariant equivalents, and multirate digital systems," IEEE Trans. Automatic Control, vol. 35, pp. 429-433, 1990. [20] R. A. Meyer and C. S. Burrus, \A uni ed analysis of multirate and periodically timevarying digital lters," IEEE Trans. Circuits and Systems, vol. 22, pp. 162-168, 1975. [21] Y. Meyer, Wavelets: Algorithms and Applications, SIAM, Philadelphia, 1993. [22] S. Mirabbasi, B. A. Francis, and T. Chen, \Controlling distortions in maximally decimated lter banks," IEEE Trans. on Circuits and Systems II: Analog and Digital Signal Processing, vol. 44, no. 7, pp. 597-600, 1997. [23] S. K. Mitra and R. Gnanasekaran, \Block implementation of recursive digital lters: new structures and properties," IEEE Trans. Circuits and Systems, vol. 25, pp. 200-207, 1978. [24] T. Miyawaki and C. W. Barnes, \Multirate recursive digital lters { a general approach and block structures," IEEE Trans. Acoustics, Speech, Signal Processing, vol. 31, no. 10, pp. 1148-1154, 1983. [25] K. Nayebi, T. P. Barnwell, III, and M. J. T. Smith, \Block decimated analysis-synthesis lter banks," Proc. ISCAS, pp. 947-950, 1992. [26] K. Nayebi, T. P. Barnwell, III, and M. J. T. Smith, \Nonuniform lter banks: a reconstruction and design theory," IEEE Trans. Signal Processing, vol. 41, no. 3, pp. 1114-1127, 1993. [27] R. L. Reng, \Polyphase and modulation descriptions of multirate systems { a systematic approach," Proc. Int. Conf. DSP, 1995, pp. 212-217. [28] R. G. Shenoy, \Analysis of multirate components and application to multirate lter design," Proc. IEEE Int. Conf. Accoust., Speech, Signal Processing, 1994, vol. 3, pp. 121-124. 25

[29] R. G. Shenoy, \Multirate speci cations via alias-component matrices," to appear IEEE Trans. Signal Processing. [30] R. G. Shenoy, D. Burnside, and T. W. Parks, \Linear periodic systems and multirate lter design," IEEE Trans. Signal Processing, vol. 42, pp. 2242-2256, 1994. [31] I. Sodagar, K. Nayebi, T. P. Barnwell, III, and M. J. T. Smith, \Time-varying analysissynthesis systems based on lter banks and post ltering," IEEE Trans. Signal Processing, vol. 43, pp. 2512-2524, 1995. [32] G. Strang and T. Q. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, MA, 1996. [33] P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice-Hall, Englewood Clis, NJ, 1993. [34] P. P. Vaidyanathan and S. K. Mitra, \Polyphase networks, block digital ltering, LPTV systems, and alias-free QMF banks: a uni ed approach based on pseudocirculants," IEEE Trans. Acoustics, Speech, Signal Processing, vol. 36, no. 3, pp. 381-391, 1988. [35] M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Prentice-Hall, Englewood Clis, NJ, 1995. [36] M. Vetterli and C. Herley, \Wavelets and lter banks: Theory and design," IEEE Trans. Acoustics, Speech, Signal Processing, vol. 40, no. 9, pp. 2207-2232, 1992. [37] X.-G. Xia and B. W. Suter, \Multirate lter banks with block sampling," IEEE Trans. Signal Processing, vol. 44, no. 3, pp. 484-496, 1996. [38] X.-G. Xia and B. W. Suter, \Vector-valued wavelets and vector lter banks," IEEE Trans. Signal Processing, vol. 44, no. 3, pp. 508-518, 1996.

26