A property of FIR (finite-impulse response) lossless systems is introduced, leading to substantial improvement in the sign procedure for perfect-reconstruction QMF (quadrature mirror filter) banks. The property enables the designer to initialize the coefficients of a lattice structure (which characterizes the analysis bank), in such a way as to speed up to the convergence. A design example is provided. Compared to other methods, the proposed method is shown to converge faster, and always leads to much improved attenuation characteristics for a given filter length

Nguyen, T. Q.

Saramaki, T.

Vaidyanathan, P. P.

Caltech Authors - Main

IMPROVED APPROACH FOR DESIGN OF PERFECT RECONSTRUCTION 
FIR QMF BANKS, WITH LOSSLESS LATTICE STRUCTURES 
P. P. Vaidyanathant, T .  Q. Nguyent and T. Saramaki’ 
$Dept. EE, Caltech, Pasadena, CA 91125 D3.9 
*Dept. EE, Tampere Univ. Tech., Finland 
Abstract: A new property of FIR lossless systems is 
introduced, leading to  substantial improvement in the de- 
sign procedure for perfect reconstruction QMF banks. The 
property enables the designer to initialize the coefficients of 
a lattice structure (which characterizes the analysis bank), 
in such a way as to  speed up the convergence. 
I. Introduction 
Quadrature mirror filter banks (QMF) and their ap- 
plications are well-known in the multirate digital signal 
processing community [l-31. For a more complete refer- 
ence list see [3]. A maximally decimated QMF bank (see 
Fig. 1, [4]) is said to have the perfect-reconstruction (PR) 
property if the reconstructed signal ?(n) is equal to  a de- 
layed (and possibly scaled) version of .(.). A method was 
outlined in [4] for enforcing the P R  property, based on the 
losslessness of the Alias-component (AC [SI) matrix.2 Let 
H k ( z ) ,  0 2 k 5 M -  1 denote the M analysis filters, and 
define the M polyphase components of Hk(z )  to be E ~ L ( z ) ,  
0 5 5 M - 1 as in [3,4]. It is shown in [4] that a per- 
fect reconstruction FIR QMF bank can be designed if the 
M x M matrix E(z) f i [Ek~(z) ]  is lossless (this condition be- 
ing equivalent to the losslessness of the AC matrix). With 
E(z) constrained to be lossless, the QMF bank has the P R  
property if and only if the synthesis filter coefficients f k ( n )  
are time-reversed versions of hk(n) .  
In a subsequent work 161, a lattice structure has been 
presented (based on a state-space approach)in order to 
parametrize the complete family of M x M FIR lossless 
matrices E(z)  with a given degree N - 1 (the term degree 
stands for the McMillan degree [8], i.e., the minimum num- 
ber of scalar delays z-’ required for its realization). This 
parameterization has the following features: 1) the param- 
eters are angles 8 k ;  2) the number of parameters used is 
minimal, 3) regardless of the parameter values, the trans- 
fer matrix E(z) is guaranteed to  be lossless, and 4) every 
M x M FIR lossless E(z)  with a given degree is covered 
by the structure. 
For FIR systems, losslessness of E(z) is equivalent to 
Work supported in part by the National Science Foundation 
grants DCI 8552579 and MIP 8604456. 
2Unless noted otherwise, the notations, terminology, and def- 
initions used here are same as in our previous publications 
I3,41. 
the property E(z)E(z) = I. If E(z) has real coefficients, 
the lossless property becomes the LBR property [9]. For 
simplicity we shall restrict our discussions to  this special 
case. In this case the number of angles f?k required for 
parameterization is given by 
With the analysis filters so characterized in terms of E(z), 
the design problem is t o  optimize f?k such that  an objec- 
tive function 4 (sum of stopband energies of H k ( Z )  [4]) is 
minimized. Since 4 is highly nonlinear, the optimization 
time becomes large as M and N increase. One aim of this 
paper is to  show how this growth can be arrested t o  some 
extent, by using an important property of FIR lossless sys- 
tems. The other purpose is to obtain an initialization for 
the optimization, based on M t h  band filter theory, so as 
to further reduce the convergence-time. 
Given an implementation of an r-input p-output sys- 
tem, we shall label the outputs of the delay elements as 
the state variables and obtain the description 
x ( n  + 1) = Ax(n) + Bu(n) (2) 
y ( n )  = Cx(n) + Du(n). (3) 
The quadrupule [A, B, C, D] is the ‘state-space descrip- 
tion’ of the implementation. 
11. New results on the realization of 
FIR lossless systems 
In Sec. 111, we shall design the FIR filters Hk(z )  as 
follows: first choose Ho(z) to  be an FIR M t h  band filter 
[lo] with ‘good’ stopband attenuation, and based on this 
H o ( z ) ,  initialize ‘as many parameters Bk as possible’ where 
8 k  are the parameters described earlier. The quantity of 
fundamental interest therefore is the number of freedoms 
that  are left over, after constraining Ho(z)  to  be a specific 
function. We shall study this issue here. 
First, let eo(z) be an M x 1 FIR LBR vector, i.e., 
eo(z) = [ e o o ( z )  eol(z) . . . eo,M--l(z)IT (4) 
&(z)eo(z) = 1. (5) 
with 
1471 
CH2561-9/88/0000-1471 $1.00 0 1988 IEEE 
Then eo(.) can be realized as in Fig. 1,  where F(z) is 
M x M FIR LBR and $0 is a real column vector with 
unit norm, i.e., = 1. T F  synthesis procedure, which 
identifies the coefficients of F(z) and the-elements of 40, 
can be found in [l l] .  As shown in [ I l l ,  F(z) and $0 are 
not unique, for a given eo(z). Fig. 1 says that  we can write 
eo(z) = F(z)$o. (6) 
As a second step, let ek(z), 0 5 k 5 M - 1 be a set of 
FIR LBR vectors such that 
&(z)e l (z )  = 6[k - 4 (7) 
where 6 [ k  - 1 )  is the Kronecker delta function. Then there 
exists a single M x M FIR LBR matrix F(z) such that  all 
of the ek(z)'s can be realized by using this F(z) as 
ek(z) = F(z)qk (8) 
qTql = 6 [ k  - 11. 
where qk are appropriate unit-norm real vectors such that 
(9) 
This is a consequence of the results in [6],[7]. Here is a 
brief outline of the proof: define the M x M matrix 
S ( z )  = [ eo(z) el(z) . . . e ~ - l ( z ) ] .  (10) 
Then S(z)  is FIR LBR in view of (7). So it can be realized 
(see [6]) as in Fig. 2 where F(z) is an appropriately defined 
M x M FIR LBR matrix and Q is an appropriate real 
orthogonal matrix. In other words, we have 
S(z )  = F(z)Q. (11) 
If we take qk to  be the kth column of Q, the above result 
(8) follows immediately. Once again, the matrices F(z) and 
Q are not unique for a given set of ek(z), 0 5 k 5 M - 1. 
We shall now prove the following main result. 
Theorem 2.1. Let ek(z), 0 5 k 5 M-1 be a set of M x  1 
FIR LBR vectors satisfying (7). Let eo(z) be realized as 
in (6) using the procedure in [ l l ] .  Then all the vectors 
ek(z) can be realized as ek(z) = F(z)qk where $ k  are 
appropriate unit-norm vectors. 
Proof: Let us,assume that  the M x 1 system eo(z) has 
been realized as'in (6) using the procedure i? [ll]. Assume 
that  the  M-input M;output core system F(z) has state 
space description [A, B,, e, Dc], i.e., 
F(2) + [A,B, ,C,D,] .  (12) 
The state-space descript*ion ofeo(z) is then [A, B o ,  C, Do] 
where Bo = B,$o and Do = D&. Thus 
eo(z) = F(z)$o + [A,Bc$~,C,Dc40] .  (13) 
Next, with S(z)  defined as in (IO),  consider its realization 
as in Fig. 2. Let [A,B, ,C,  D,] be the state-space descrip- 
tion of the core system F(z). Then the state space descrip- 
tion of S(z )  is given by [A,B,  C, D] where B = B,Q and 
D = DcQ. Now denoting the columns of Q by qk, we see 
that  e k ( z )  has the following relalization 
ek(z) = F(z)qk + [A,B,qk,C,D,qk]. (14) 
In particular, eo(z) has the realization 
eo(z) = F(z)qo -+ [A,B,qo,C,D,qo]. (15) 
From the results of [6] and [7] we know that  the two real- 
izations (13) and (15 )  for eo(z) are minimal (in the sense 
that  the A matrix has the smallest size, say, N-1 x N -  1) 
and are related by a similarity transformation [8] so that 
A B o  T-'AT T-'B,q, ] . (16) [C DO]  = [ CT D,qo 
In summary, eo(z) = F(z)qo, with F(z) realized as 
F(z) + [A, T-'B,, e, D,]. (17) 
By the same argument, ek(z) has a realization of the form 
F(z)qk with F(z) realized as in (17). Now consider the 
following ( N  - 1 + M )  x ( N  - 1 + M )  matrices 
It is known that  0 is orthogonal [GI. As a result we have 
ATA + C T e  = I. (19) 
Combining this with the fact that  C represents a realiza- 
tion of the LBR matrix F(z), we conclude [12, page 9191 
that  C is also orthogonal. As a result the columns of 
L J  
and [ T z B c  1 span the same subspace of rank M (viz., 
L " . I  
the  space orthogonal to  the column-space of [ i] . This 
implies the existence of an M x M matrix R such that 
T - ~ B ,  = B c ~ ,  D, = D<R. (20) 
F(z) -+ [A,B,R,e ,D,R] .  (21) 
F(z) = F(z)R. (22) 
Thus F(z) has the realization 
Combining (12) and (21) we get the relation 
Since ek(z) has the realization (14), we conclude that it 
can also be realized as 
e k ( Z )  = @(z)&,  q k  = Rqk (23) 
In view of the fact that ek(z) and F(z)  are both lossless, it 
can also be verified that $ k  has unit norm. This concludes 
the proof of the theorem. 
We now have a simplified procedure for synthesiszing 
an M x M FIR LBR system S(z) .  The procedure is to 
synthesize only one column, say eo(z), using the simple 
1472 
method in [ll]. This gives a realization which can be char- 
acterized as in (6), where F(z) is the cascaded lattice struc- 
ture which results as a by-product [11] while synthesizing 
eo(z). The remaining columns ek(z) are synthesized sim- 
ply by identifying unit-norm vectors 
The inverse matrix on the RHS of (24) is equal to Fk(z), 
see [9]. Even though the RHS of (24) appears to  be a 
function of z it is guaranteed to  turn out to be a constant 
upon simplification. If we now define the M x M matrix 
$1 ... Q M - ~ ]  then S ( z )  = F(z)Q, which can be 
readily implemented, since the structure realizing F(z) is 
already available as the by-product mentioned above. 
- A  
111. New analysis-bank design procedure 
with lossless E(z) 
We are interested in designing the M FIR filters Hk(z ) ,  
0 5 k 5 M - 1 such that the matrix E(z) of polyphase 
components defined in Sec. I is LBR. For simplicity assume 
the coefficients of Hk(z )  to be real. One consequence [4] of 
requiring E(z) to be lossless is that the AC matrix H(z) 
is lossless. Each H k ( z )  in particular satisfies 
M-1 
Hk(zW')Hk(z-'W-') = c (25) 
1=0 
A where c is a constant. In other words, Gk(z)=Hk(z-')Hk(z) 
is an M t h  band (or Nyquist ) linear-phase FIR filter [lo] 
with nonnegative real frequency response Gk(eiw). 
Suppose we have designed a low pass FIR M t h  band 
filter Go(z) with G"(eiw) 2 0. Let Ho(z) be an arbitrary 
spectral factor of Go(z). Let the polyphase components of 
Ho(z) be denoted by EOl(z),  0 5 I 5 A4 - 1. The property 
(25) ensures[4] that the vector 
eo(z)+oo(z) ~ o l ( z )  .. EO,M--I(Z)]T 
is FIR LBR. Assume that  this vector has been realized in 
the form (6) using the procedure in [ll]. The vector e:(.) 
is the first row of the polyphase component matrix E(z). 
We are now left with the task of finding the remaining 
M - 1 rows ez(z),k > 0 of E(z) such that  E(z) is loss- 
less and such that Hk(z) ,  k > 0 are 'good' bandpass filters 
with desired specifications. Recall that losslessness of E(z) 
is equivalent to the condition (7). But we know from The- 
orem 2.1 that if (7) holds, and if eo(z) has been realized 
as in (6), then all o f e k ( z )  can be realized as in (23). In 
other words, we have 
ET(z) = F(z)Q (26) 
where F(z) and the 0th column 4 0  of Q are already fixed 
by the choice of Ho(z).  The the number of freedoms that 
remain for the design of H k ( Z ) ,  1 5 k 5 M - 1 (under the 
constraint that  E(z) be LBR) is then equal to the number 
of freedoms we can exercise in choosing the columns $k, 
1 5 k 5 M - 1. Since E(z) and- F(z) are LBR, Q is of 
course M x M orthogonal. So Q has ( y )  freedoms [6]. 
The first column $0 which is a unit norm vector has M -  1 
freedoms, which are already used up  by the choice of Ho(z).  
We are therefore left with 
N f =  (7) -(M-1) 
freedoms. Summarizing, we have 
Lemma 3.1. Let H o ( z )  be a given FIR transfer function 
whose M x 1 polyphase-component vector eO(z) is LBR. 
Then the number of design freedoms available in choosing 
a set of M - 1 transfer functions H ~ ( z ) ,  1 5 k 5 M - 1 
under the constraint that  the complete M x M polyphase 
matrix E(z) of Hk(z) ,  0 5 k 5 M - 1 is LBR, is given by 
(27) regardless of the filter-lengths. 
Consider, for example, the special case of M = 2. As 
shown in [4], the matrix E(z) for the designs in [13] turns 
out to be lossless. Since N f  = 0 in this case, we see that  
once H o ( z )  is chosen to  be a spectral factor of a zero-phase 
half-band filter, there is no freedom in the  choice of Hl(z ) !  
This is completely compatible with the fact that  Hl(z) in 
[13] is constrained to  be equal to  Z - " ~ H O ( - Z - ' ) .  
Perhaps the most important part of the Lemma is 
the phrase 'regardless of the  filter length'. As a conse- 
quence of this, we can optimize the frequency responses of 
Hk(z),l 5 k 5 M - 1 (for a given Ho(z ) )  by varying only 
N f  parameters which is much smaller than Np in (1). 
A design procedure based on the above results would 
typically run as follows: first design a linear-phase FIR 
M t h  band filter Go(z) with nonnegative response Go(eiw) 
using, for example, the technique in [IO]. Then identify a 
spectral factor of Go(z), and label i t  as one of the analysis 
filters, say H0(z) .3  The choice of the length of Go(z) of 
course depends on the attenuation requirement of Ho(z). 
Next, figure out  the polyphase-component vector eo(z). 
This is guaranteed to  be LBR. Synthesize i t  using the 
technique in [ll]. The result- is in the form (6) where 
40 is a unit-norm vector and F(z) is characterized[ll] by 
( N  - 1)(M - 1) planar rotation angles e k ,  where N - 1 
is the  order of the highest-degree polyphase component. 
At this point, F(z) and $0 in (6) are completely specified. 
Optimize the N f  remaining degrees of freedom t o  minimize 
the stopband energies of Hk(z ) ,  1 5 k 5 A4 - 1. 
A second alternative for the design procedure would be 
to  optimize all the Np freedoms (see (l)), so that  all the 
filters (including Ho(z) )  tend to come out to  be equally 
good. This of course takes longer computer-time, but is 
still much faster than the 'random initialization' used in 
A design example based on the method described 
above is shown in Fig. 3. Here M = 3, and the analysis 
filters have order 55. The filter Hl(z) (rather than Ho(z ) )  
31t is possible to avoid general spectral factorization by use of a 
modified design of Go(z)  based on the eigenfilters approach [15]. 
See [16] for details. 
141. 
1473 
was obtained as a spectral factor of a 3rd band linear- 
phase filter of order 110. Explicit spectral factorization was 
avoided by using the multistage eigenfilter method reported 
in [16], for the design of the 3rd band filter. The second 
alternative described in the preceding paragraph was then 
used to optimize all the three analysis filter responses. Fig. 
3 shows the  final optimized responses. Compared to ear- 
lier methods [4],[6], the new method converges much faster, 
and always leads to  much improved attenuation character- 
istics for a given filter length. 
References: 
[l] R. E. Crochiere and L. R. Rabiner, Multirate digital 
signal processing, Prentice Hall, Inc., 1983. 
[2] A. Croisier, D. Esteban, and C. Galand, “Perfect 
channel splitting by use of interpolation/decimation/ 
tree decomposition techniques,” Int. Symp. on Info., 
Circuits and Systems, Patras, Greece, 1976. 
131 P. P. Vaidyanathan, “Quadrature mirror filter banks, 
M-band extensions and perfect-reconstruction tech- 
niques,” IEEE ASSP magazine, pp. 4-20, July 1987. 
[4] P. P. Vaidyanathan, “Theory and design of M-channel 
maximally decimated quadrature mirror filters with 
arbitrary M,  having perfect reconstruction property,” 
IEEE Trans. ASSP, pp. 476-492, April 1987. 
151 M. J. T. Smith and T.  P. Barnwell III, “A unify- 
ing framework for analysis/synthesis systems based 
on maximally decimated filter banks,” Proc. IEEE 
ICASSP, pp. 521-524, Tampa, FL, Mar. 1985. 
[6] P. P. Vaidyanathan, Zinnur Doganata and Truong 
Q. Nguyen, “More results on the perfect reconstruc- 
tion problem in M-band parallel QMF banks,” Proc. 
IEEE ISCAS, pp. 847-850, Philadelphia, May 1987. 
[7] P. P. Vaidyanathan, Zinnur Doganata and Truong 
Q. Nguyen, “General synthesis procedures for FIR 
lossless transfer matrices for perfect-reconstruction 
multirate filter bank applications,” Tech. Rep. Cal- 
tech, Aug 1987. 
[8] T. Kailath, Linear Systems, Prentice Hall, 1980. 
[9] P. P. Vaidyanathan and S. K. Mitra, “Low pass- 
band sensitivity digital filters: A generalized view- 
point and synthesis procedures,” Proc. IEEE, pp. 
404-423, April 1984. 
[lo] F. Mintzer, “On half-band, third-band and N t h  band 
FIR filters and their design,” IEEE Trans. ASSP, 
pp.734-738, Oct. 1982. 
[ll] P. P. Vaidyanathan, “Passive cascaded lattice struc- 
tures for low sensitivity FIR filter design, with ap- 
plications to filter banks,” IEEE Trans. CAS, pp. 
1045-1064, NOV. 1986. 
[12] P. P. Vaidyanathan , “The discrete-time bounded 
real lemma in digital filtering,” IEEE Trans. CAS, 
pp. 918-924, Sept. 1985. 
[13] M. J. T .  Smith and T. P. Barnwell 111, “Exact re- 
construction techniques for tree-structured subband 
coders,” IEEE Trans. on ASSP, pp. 434-441, June 
1986. 
[14] Truong Q. Nguyen and P. P. Vaidyanathan, “New 
lattice structures for the design of perfect recon- 
struction QMF banks with image-symmetry prop- 
erty,” IASTED ISSPA Proc., Brisbane, Australia, 
Aug. 1987. 
[15] P. P. Vaidyanathan and T. Q. Nguyen, “Eigenfilters: 
a new approach to least squares FIR filter design and 
applications including Nyquist filters,” IEEE Trans. 
CAS, pp. 11-23, Jan  1987. 
[16] T. Q. Nguyen, T. Saramaki and P. P. Vaidyanathan, 
“Eigenfilters for the design of special transfer func- 
tions with applications in multirate signal process- 
ing,” Proc. ICASSP 88, New York, NY, Apr. 1988. 
A 
Fig. 1. Realization of e,(z) 
Fig. 2. Realization of S (z) 
-20.0 
-40.0 
0 .  0.1 0.2 0.3  0.4 c .5 
NORMAL1 ZED FREQUENCY 
F ig .  3 .  Magnitude r e sponse  i n  DB f o r  
optimized a n a l y s i s  f i l t e r s .  
1474 


Improved approach for design of perfect reconstruction FIR QMF banks, with lossless lattice structures

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Improved approach for design of perfect reconstruction FIR QMF banks, with lossless lattice structures

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