DTIC ADA289238: Structures for Time-Reversed Inversion in Filter Banks,

4 Pages·0.39 MB·English

by  Defense Technical Information Center

Is3cC Ab. 5
068
19941223
STRUCTURES FOR TIME-REVERSED INVERSION IN FILTER BANKS
! i (cid:127)P. P. Vaidyanathan' and Tsuhan Chen2
A(cid:127)
C 2 8 1994 !'Dept. Electrical Engr., Caltech, Pasadena, CA 91125
E, 2L8 1994 2AT&T Bell Laboratories, Holmdel, NJ 07733-3030
. where x(n) = [xi(n) ... XN(n) IT is the state vec-
Abstract tor [xi(n) are the outputs of delays], u(n) is the filter
Anticausal inversion of IIR transfer functions has input and y(n) the filter output. The transfer function
gained importance in recent years, in the efficient im- is given by G(z) = D + C(z - nA )n'B and has an
plementation of IIR digital filter banks. In this paper anticausal inverse [i.e., there is an anticausal inverse
we first introduce the idea of a causal dual, as an inter- z-transfor Oif /n(r z)]r if and only if the realization
mediate step in the implementation of anticausal IIR matrix1?isnonsngular [5]. Weassumethis. Consider
inverses. With time reversal operators at the input the following "dual" casual state space equations
and output of the casusal dual, we get the anticausal
inverse of the original structure. The causal dual elim- R(n + 1)] [[[X(1 .2] )[x(n)
inates the need for similarity transformations, during a
key step called blockwise state transfer, in implement- '(n) L ](n)
ing anticausal inverses. In the paper we identify effi-
cient structures for causal duals of standard structures
like the direct-form, cascade-form, coupled form, and
IIR lattice structures, including the tapped lattice. where 7. = 7Z-'. Imagine we run the original sys-
tem (1.1) for a duration of L samples, with input
1. INTRODUCTION u(O) ... u(L - 1) and get the output y(O) ... y(L - 1)
and final state x(L). If we run the state space equations
Anticausal inversion of IIR transfer functions has gained (1.2) by setting the initial state to be R(L) = x(L),
importance in recent years, in the implementation of and the input sequence to be the time reversed block
digital filter banks [1]-[6]. This is due to a certain y(L -l1)... y(0), we can show that the output of this
class of IIR digital filter banks, in which the synthesis system or the duration of this block is u(L- 1) ... u(O).
filters are required to be anticausal if a stable recon- In this way we can invert (1.1), i.e., invert G(z). Since
struction system (synthesis bank) is desired. Thus, the time reversal is blockwise, it can be realized in
Fig. 1.1 shows the polyphase form [7] of a commonly practice with finite latency. By repeating the above
used two channel filter bank. This has perfect recon- process block by block, we can implement the anti-
struction [2(n) = x(n)] if and only if Rj(z) = 1/Ed(z). causal inverse 1/G(z) of the original filter G(z), even
If Ej(z) does not have all zeros inside the unit circle, for infinitely long inputs.
then the causal impulse response of Rj(z) is unstable. It can be shown that the transfer function of the
In some cases (e.g., when Ej(z) is allpass with poles causal system (1.2) is 1/G(z'). We can schematically
inside the unit circle), 1/Ej(z) has all poles outside, express the desired anticausal inverse 1/G(z) as in Fig.
so that an anticausal impulse response of 1/Es(z) is 1.2, in terms of ideal time reversal (TR) operators. In
stable. This explains the interest in implementing an- a practical implementation of the "anticausal" inverse
ticausal inverses. This has been shown to be possible 1/G(z), the time reversals are done blockwise, and the
[3]-[6], even in real time with infinite duration inputs, causal system 1/G(z-1) implemented using (1.2). The
as long as we perform the processing in blocks of length initial condition for the state recursion (1.2) at the be-
L, and properly transmit the state variables from El(z) ginning of each block is taken as the final state from
to 1/Ej(z) at the end of each block. the previous block of the state recursion (1.1) for G(z).
Consider a causal stable Nth order filter G(z), im- This state transfer is crucial.
plemented using a minimal structure (direct-form, lat-
tice, etc.), with state space description Aim of the Paper
Note that 1/G(z-1) can be realized using any structure
x(n+ 1)] _ [A B] [xnhi (1.1) (direct-form, lattice, etc.). The state space description
y(n) J [ D n of these structures are related to (I, B, C, .) by sim-
ilarity transforms. If an arbitrary structure is used
for 1/G(z-1), then the state transfer described above
_ __
Work supported in parts by the Office of Naval Research should be done only after a similarity transformation
grant N00014-93-1-0231, Rockwell Intl., and Tektronix, Inc. at the beginning of each block.
Causal dual of a structure. A structure for output node y(n) as indicated, the transfer function
l/G(z-') which has precisely the state space descrip- is G(z) = psin z-2/(l - 2pcos z-1 + p2z-2). Upon
tion (A, B, C, D) will be called a causal dual of the analysis we find that the realization matrix is
structure (A, B, C, D) for G(z). If the causal dual is
used for implementing 1/G(z-1), no similarity trans- [A B 1 [pcos9 -psin0 11
formations are necessary during state transfer. In this TRcoup = = p sin 0 p cos 0
paper we will identify efficient structures for the causal L D 0 1 0
duals of standard structures, such as the direct-form,
cascade-form, coupled form, and the family of lattice The inverse of this matrix is
structures, including the tapped lattice.
2. CAUSAL DUAL FOR THE DIRECT-FORM Tcouple = L 0 1
- cot 0 p sin 0 + p cot 0 cos 0
Fig. 2.1(a) shows the direct form for an Nth order filter
G(z). From [5] we know that the inverse filter 1/G(z) Fig. 3.1(b) shows the structure with this realization
has an anticausal impulse response (more simply, G z) matrix (where m = p sin 0 + p cot 9 cos 9). This is the
has an anticausal inverse) if and only if PN 5 0; we causal dual of Fig. 3.1(a). Note that A (the top-left
therefore assume this. With the state variables xi(n) 2 x 2 matrix above) has all eigenvalues = 0. This is
as indicated in the figure, the matrix A is in compan- because 1/G(z-') is FIR.
ion form (e.g., see [7]). Writing down the realization The Tapped Coupled Form. Fig. 3.2(a) shows
matrix RZw, e can explicitly invert it, and verify that the tapped coupled form. This has the extra multipli-
the structure in7 ?F.1i g.a n2t h.1i (sbtch)a euhrafaosl rt huea lr eaTlhizua tiothn mcaatsraixl eres (ttaapp coceofeffi ciefn ts) oDrm, . c1T hai nda s c2 wthh ich ccaan mbeu usseedd
R(cid:127)-1, and is therefore the causal dual. Thus the causal to obtain arbitrary numerators. The realization matrix
dual can be obtained by making a simple set of changes now becomes
to the multipliers, and renumbering the state variables
in reverse order. For the special case of allpass filters p cos 0 -p sin 0 1
the causal dual is even simpler, since p, = q* -,. For 'Rtap = psino p cos0 (3.1)
real coefficient allpass filters, the direct form structure cl c DI
2
is its own causal causal dual - only the states need to
be renumbered. This is consistent with the anticausal which can be rewritten as
inversion of direct-form reported in [8].
Cascade-form structures. The cascade-form, [1 0 1 [pcosO-p sin 11]
which is a cascade connection of second order direct "Ztap - 1 oi psin pcos0 (3.2)
form structures, is well-known for its simplicity and [D a j6 0 1 0
relatively better robustness to quantization [10]. More Roopl.
generally, let Gi(z) and G (z) be causal transfer func-
2
tions in cascade (Fig. 2.2), implemented with struc- with a = (c, - Dpcos9)/psinO, 83= c2 + Dpsin0-
tures having realization matrices I1Z and R(cid:127)2. Then ap cos 0. Thus R-1 can be written as
from the block of L outputs y2(n) ... y2(n + L - 1) t an
and the state x (n + L) of G (z), we can recover the
2 2
block of L inputs yl (n)... yi (n +L -1) of the system - 1 1 0 0 1 (3.3)
G2(z). We can then use yi(n) ... yj (n + L - 1) (out- tap = cope 0 1 0-J/(3.3
put block of Gi(z)) and the state vector xi(n + L) of
the system Gi(z) to recover the primary input block
u(n) ... u(n + L - 1). This is equivalent to connecting The structure with this realization matrix can be ob-
the anticausal inverses in reverse order. tained from Fig. 3.1(b) by replacing the input i?(n)
In fact, it can be shown that the causal dual of a with -(D/18) 1(n) - (a/f6)52(n) + (1/)3)iý(n), as shown
cascade is simply the cascade of the individual causal in Fig. 3.2(b). This is therefore the causal dual of the
duals, in reverse order. If we implement this cascaded tapped coupled-form.
causal dual structure with blockwise time reversal of
its primary input and output, we obtain the anticausal 4. CAUSAL DUALS FOR LATTICE STRUCTURES
implementation of the complete system 1/G(z). Thus,
regardless of the number of filters connected in cascade, In the case of the direct form structure, we could obtain
time reversal is necessary only at the primary input the causal dual simply by inspection (Fig. 2.1). For
and output nodes, and not at the intermediate inputs arbitrary structures, the rules could be more compli-
and outputs. cated, as shown by the coupled form example. We will
see next that for the lattice structures, which are well-
3. CAUSAL DUAL FOR THE COUPLED-FORM known for many good robustness properties [11]-[12],
the causal dual can be obtained again by inspection!
Fig. 3.1(a) shows the coupled form structure, whose The lattice structure is shown in Fig. 4.1(a). Here
robustness to quantization is well-known [10]. The GN(z) = Y(z)/U(z) is a causal Nth order allpass filter.
poles of this system are at pej° and pe-j°. With the The rectangular building block labelled km can take
2
several possible forms (Fig. 4.1(b)). While the nor- Real coefficient case. If GN(z) has real coeffi-
malized structure offers automatic scaling and normal- cients, then km are real. So the causal dual is identical
ization of noise gains [11], the two-multiplier version to the original structure, except that delays have to be
offers economy. For a given set of lattice coefficietns moved up.
{kmn}, the allpass function GN (z) is the same regardless Denormalized lattice sections. Suppose the lat-
of which of these building blocks is used. The lattice tice structure of Fig. 4.1(a) uses denormalized sections
coefficients km.a re possibly complex, but Ik,1 < 1, so (e.g., the two multiplier section of Fig. 4.1(b), or one-
GN(Z) is stable. The multiplier ik m= 1 -IkmI 2 is multiplier sections which are available for real coeffi-
always real. cient case [7]). Then the structure is related to the
With the state variables xm(n), the input u(n) and normalized lattice via a diagonal similarity transfor-
output y(n) indicated as shown in Fig. 4.1(a), let the mation. Using this, it is possible to show that if Fig.
realization matrix of this Nth order structure be de- 4.1(a) represents a denormalized lattice, then Fig. 4.2
noted as TZN. We will show that the causal dual (i.e., still represents its causal dual. That is, we simply con-
the structure with realization matrix 7Z(cid:127) 1) is as shown jugate coefficients and move the delays to upper rails.
in Fig. 4.2, where the boxes k< are the builing blocks The Tapped Lattice Structure. The tapped lat-
in the original structure Fig. 4.1, but with coefficients tice, which can be used to realize an arbitrary IIR
conjugated. Thus the causal dual structure is obtained transfer function H(z) is shown in Fig. 4.3(a). Here
from the original structure by conjugating the multipli- H(z) has denominator equal to the that of the allpass
ers, and moving the delays from the bottom rails to the filter GNw(z). The tap coefficients an can be chosen to
top rails. realize the numerator of H(z). We will show that the
For Fig. 4.2, define the state vector R(n) and the causal dual of this is given by Fig. 4.3(b). (This as-
state space description (A,, B, C, B), and let the real- sumes aN+1 # 0. If aN+1 = 0, the numerator of H(z)
ization matrix be denoted fZN. First consider the nor- has a smaller order than the denominator, and an an-
malized lattice. In this case 7ZN is unitary [13], so we ticausal inverse does not exist [5]. The causal dual
would then be of no interest.)
only have to show fZN = 7Zr (transpose conjugate). For fixed input u(n), the state variables xi(n) in
This is based on: Figs. 4.3(a) and 4.1(a) are identical. In Fig. 4.3(a)
Lemma 3.1. The (N + 1) x (N + 1) realization yi(n) = aQxi(n+ 1) + aN+ly(n). Thus the real-
matrix 7ZN for the lattice of Fig. 4.1(a) with normal- ization matrix 7Zarb for the arbitrary transfer function
ized building blocks can be expressed as a product of H(z) is related to the realization matrix 7Zau of the
N unitary matrices: allpass filter GN(z) as
1
So][ 0 ] IN 0] 0
0 2 R.arb = 0N+1 (4.3)
0IN -1 0 0 IN-2 0 1N
k I where a= [Ia a2 ... aN]. Thus
where the unitary matrix Em= Lkkn k*,"n¢ 2, [0 44
In arb all-a/aN+l 1/aN+] (44
Proof. It can be shown that realization matrix of I
the m-stage lattice and the (m - 1)-stage lattice are Since 1Z-'1 corresponds to Fig. 4.2, the above realiza-
related as I
tion matrix corresponds to the structure Fig. 4.3(b),
r 01 [0I which is therefore the causal dual of Fig. 4.3(a). This
m - 1 0 em has taps in a feedback loop. Whether this is stable
depends on the numerator of H(z).
For m = 1 we can explicitly verify that TI? = 0i. The
lemma follows from this by induction. 17 7 V References
Theorem 3.1. Causal dual for normalized lat- [1] Ramstad, T. A. "IIR filterbank for subband coding
tice. The causal dual of the normalized lattice struc- of images," Proc. of the IEEE ISCAS, pp. 827-830,
ture (Fig. 4.1(a), with normalized building blocks) is Finland, June 1988.
given by the lattice of Fig. 4.2. < [21 Husoy, J., and Ramstad, T. A. "Application of an
Proof. We can show that the realization matrices einsgff,c"i eSnti gpnaaral lPlerlo cIIeRss ifnilgte, rA buagnukst to1 9i9m0,a gpep s. u2b7b9a-n2d9 2c.od-
for the m-stage lattice and (m - 1)-stage lattice in Fig.
4.2 are related as [3] Babic, H., Mitra, S. K., Creusere, C. D., and Das, ..........
A., "Perfect reconstruction recursive QMF banks for
S 0 [ 0 (4.2) image subband coding," Proc. 25th Annual Asil. Conf.
7lm [O 0 1 Sig., Sys. and Comp., Nov. 1991.
[4] Mitra, S. K., Creusere, C. D., and Babic, H. "A
and we can explicitly verify that R(cid:127)1 = Of. From these novel implementation of perfect reconstruction QMF
banks using IIR filters for infinite length signals," Proc.
we can obtain fZN -7 -Z R_N(cid:127) ' * v IEEE ISCAS, pp. 2312-2315, San Diego, May 1992.
3! .
[5] Vaidyanathan, P. P., and Chen, T. "Role of an- 'Yin)
ticausal inverses in multirate filter-banks - Part I:
System-Theoretic Fundamentals, and Part II: The FIR u(n)- 1 PZ0l z1 y(n) m -cot 0
Case, Factorizations, and biorthogonal Lapped Trans- Sz / g
forms," IEEE Trans. SP (submitted).
opa) b
[6] Chen, T., and Vaidyanathan, P. P. "General theory (b
of time-reversed inversion for perfect reconstruction fil- -pi~n) n -colt a (n)
ter banks," Proc. 26th Annual Asilomar Conf. Sig., Fig. 3.1. (a) The coupled-f2orm structure, and (b) its causal dual.
Sys. and Comp., pp. 821-825, Oct. 1992. INd
[7] Vaidyanathan, P. P. Multirate systems and filter -y)
banks, Prentice Hall, 1993. y(n) m -ot e
[8] Creusere, C. D. PR Modulated polyphase filter D c1 t 2
banks using time-reversed subfilters, Ph.D Disserta- I -cot () ITxs-n I
tion, Dept. ECE, UCSB, 1993. uQn _ sn -co
[9] Vaidyanathan, P. P., and Chen, T. "Structures for I ,i'L.i4 b
anticausal inverses in multirate filter banks," in prepa- (a) ýJ( x n)-a -Dl
2
r a t ion . _p , n x (n) x ( n)
[10] Oppenheim, A. V., and Schafer, R. W. Discrete- -psin X1(n ) x2(n)
time signal processing, Prentice Hall, Inc., 1989. Fig. 3.2. (a) The tapped coupled form, and (b) its causal dual.
[11] Gray, Jr., A. H., and Markel, J. D. "A normalized
digital filter structure," IEEE Trans. ASSP, pp. 268- vN(n) (n)
u(n) -oo
277, June 1975. j jk jf7x(n)
KN n
121 Gray, Jr., A. H. "Passive cascaded lattice digital G -z) xN1
fiters," IEEE Trans. CAS, pp. 337-344, May 1980. z-l
[13] Vaidyanathan, P. P. "The discrete-time bounded Y(n)
ifliI-k
real lemma in digital filtering," IEEE Trans. GAS, pp.
918-924, Sept. 1985. or (b)
x Jfnormalized block 2-multiplier block
Fig. 4.1. (a) The general form of IIR allpass lattice, and
(b) two possible building blocks.
Analysis bank Synthesis bank -'fz_,
Fig. 1.1. A two-channel filter bank in polyphase form. '(n)z) e
- N~ N-i(n
anticausal inverse causal system
(b) TR definition: s(n)-. . r(n)=s(-n) Fig. 4.2. The causal dual of the IIR lattice structure.
Fig. 1.2. The anticausal inverse represented in terms
of a causal system and time-reversal operators.
u(n) y(n) 1 PN 7(n) U II"I (n) x n) (n)
x (n) Tp 1 G(a) Y(nl/(Z kN XNf kN N-1x *-I k x
. 9N z (Na zN-1./ ... y0o
Z- - -1 H'z'-Y'z'/U'z'1Z
~
(a) (b) Z)~'' _N.1 N tO 2
q p 1O/PN Y(n)
Fig. 21 (a) The direct-form structure, and (b) its causal dual. (b) k ,I/ k I ml
u(n) y(n)
Fig. 2.2. A cascaded system. Fig. 4.3. (a) The tapped [IR lattice and (b) its causal dual.
4