Two issues concerning alias-free, parallel, quadrature mirror filter (QMF) banks are addressed in this correspondence. First, a property concerning alias-free analysis/synthesis systems is established; second, a scheme is proposed, by which a synthesis bank can be modified in order to take care of aliasing errors caused by linear channel-distortion in a simple manner. Applications of the stated results are outlined

Vaidyanathan, P. P.

English

Caltech Authors - Main

Finally, by examining the typical LC admittance SC + l/(sL) 
which may be rewritten as 
we see that the admittance of an LC network may be written in 
terms of CX - networks as 
Y,,(s) =&-(js’)/(js). 
This allows, we believe, a more straightforward derivation of the 
properties of LC and RC networks than by proceeding from LC 
to RC, or by using arguments based on energy. 
Iv. &AMPLE 
A single capacitor in parallel with a negative-valued reactance 
(and, hence, a positive susceptance) has the admittance SC + jB, 
where B is positive. The zero is on the ju axis at w = - B/C, 
and the slope of the susceptance is positive, being equal to C. 
V. CONCLUSION 
We have derived the properties of CX-networks. The argu- 
ment is “elementary.” Also, we have related these results to the 
properties of LC and RC networks. 
REFERENCES 
111 V. B&witch, Clussicol Network Theory. San Francisco: Holden-Day, 
1968. 
PI D. S. Humpheqs, The Analysis, Design and Synthesis of Electric Filters. 
En&wood Cliffs, NJ: Prentice-Hall, 1970. 
A Note on General Parallel QMF Banks 
P. P. VAIDYANATHAN 
Abstract -Two issues concerning alias-free, parallel, quadrature mirror 
filter (QMF) banks are addressed in this correspondence. First, a property 
concerning alias-free analysis/synthesis systems is established; second, a 
scheme is proposed, by which a synthesis bank can be modified in order to 
take care of aliasing errors caused by linear channel-distortion in a simple 
manner. Applications of the stated results are outlined. 
I. INTRODUCTION 
Quadrature mirror filter (QMF) banks have received the atten- 
tion of a number of researchers in the past [l]-[9], and their 
importance in communications and sub-band coding is well 
known [2]. Fig. 1 shows a parallel r-channel QMF bank, the 
adjective “parallel” being meant to distinguish it from tree- 
structured QMF banks [2], [3]. The transfer functions 
f&J(z), HI(Z),. . . , H,- i (z) represent the analysis-bank filters, and 
E)(z), F,(z),. . . , F,- i(z) represent the synthesis-bank filters. The 
filtered signals are decimated and transmitted; at the receiver 
end, the signals are interpolated, filtered, and recombined as 
shown. 
In general, the reconstructed signal n(n) suffers from aliasing, 
amplitude, and phase distortions. Smith and Barnwell have shown 
Manuscript received April 10, 1986. This work was supported in part by the 
National Science Foundation under Grant ECS 84-04245 and in part by 
Caltech’s program in Advanced Technology, sponsored by Aerojet General, 
General Motors, GTE, and TRW. 
P. P. Vaidyanathan is with the Department of Elect&al Engineering, Cali- 
fornia Institute of Technology, Pasadena, CA 91125, USA. 
IEEE Log Number 8611175. 
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-34, NO. 1, JANUARY 1987 101 
Fig. 1. A parallel QMF bank. 
[5] how these distortions can be formulated in a unified manner. 
For the two-channel case, the theory and filter-design procedures 
are well understood [2]-[4], and methods for perfect cancelation 
of aliasing have been well known. For the case of r channels, if r 
is a power of two, tree structures can be used with perfect 
alias-free properties; for arbitrary r, interesting and valuable 
filter-design methods have been reported [6], [7], with the synthe- 
sis bank chosen so as to approximately cancel aliasing. For the 
special case of uniform DFT r-channel parallel structures, 
schemes for exact cancelation of aliasing have been reported in 
[8] and [9], but the resulting synthesis bank has higher order than 
the structures in [6] and [7]. 
In this correspondence, we consider parallel QMF banks as in 
Fig. 1, with no specific assumptions about the nature of Hk(z) 
and Fk (z). They could be FIR or IIR; the analysis bank may or 
may not be of the uniform DFT type [2], and r may not be a 
power of 2. By employing the properties of decimators and 
interpolators, one can arrive at the most general expression for 
the reconstructed signal X(z) 
g(z) =~$;X(ZW-‘)~~‘H,(ZW-‘)F,(Z) (1) 
k=O 
where W= e-j2”jr. The terms corresponding to / # 0 in (1) 
represent the aliasing terms. The first question we wish to address 
here is this: For a given (but arbitrary) set of transfer functions 
Hk(z),O < k < r -1, let the transfer functions Fk(z),O < k < r - 
1, be chosen such that a(n) is free from aliasing, for arbitrary 
x(n). If we now interchange the Fk(z)‘s with the Hk(z)‘s, i.e., if 
we let the analysis filters be Fe(z), Fl(z); . ., Frel(z), and the 
synthesis filters be H,(z), HI(z), . . . , H,- 1( z), is the recon- 
structed signal still free from aliasing? We show that the answer 
is in the affirmative. It is possible to address this question, based 
on the concept of network-transpositionJor multirate systems [2], 
but we take a direct approach here. If X(z) is free from aliasing 
in Fig. 1, then the following set of r - 1 equations hold, in view 
of (1): 
r-1 
kFoFk(z)Hk(zW-‘) =0, l<Z<r-1. (4 
Since (2) holds for all z, in particular it holds if z is replaced with 
z W’. Thus, the following r - 1 equations also hold: 
r-1 
kgoFk(zW’)Hk(z) =O, l<l<r-1. (3) 
As W’ = 1, the above set can be rephrased as 
r-1 
kGoHk(z)Fk(zWm’)=O, lgl<r-1 (4) 
which is precisely (2) with Fk( z)‘s and Hk(z)‘s interchanged. 
Thus, after the interchange is made, conditions necessary to 
cancel the aliasing still hold. Once aliasing is canceled, the 
0098-4094/87/0100-0101$01.00 01987 IEEE 
102 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CA!;-34, NO. 1, JANUARY 1987 
l Analysis bank l 4 + ’ Analysis bank 
l < * 
Synthesis bank Synthesis bank, 
Fig. 2. The tree structure for a 9-channel QMF bank. Fig. 3. A modified tree structure with more balanced complexity for analysis 
and synthesis banks. 
transfer function for Fig. 1 is 
The reconstructed signal continues to be given by (5) even after 
interchanging Hk (z)‘s with the Fk (z)‘s. We state these properties 
compactly as follows. 
The Swapping Property: In the parallel QMF bank of Fig. 1, 
let aliasing be completely absent. Then, interchanging each analy- 
sis filter Hk (z) with the corresponding Fk(z) leaves the recon- 
structed signal 2(n) unchanged. 
Note that if Fk(z) in Fig. 1 were such that aliasing is not 
perfectly canceled, then the above interchange of Fk(z)‘s and 
Hk (z)‘s does not leave the reconstructed signal unchanged. 
A possible application of this property is in QMF structures 
with composite r that is not a power of 2. For a parallel structure 
with large r which is not a power of 2, aliasing can be perfectly 
canceled only by a choice of synthesis filters Fk (z) having much 
higher order than the &(z)‘s [8], [9]. (If Hk( z)‘s are FIR with 
order N - 1, the orders of Fk(z)‘s are roughly proportional to 
(r - l)( N - l).) It is thus beneficial to build up the QMF bank in 
the form of tree structures [2], with alias-free building blocks 
having small (prime) number of channels. Fig. 2 shows a nine- 
channel QMF bank in terms of three-channel building blocks. 
Now, such a tree structure is disbalanced in the sense that the 
synthesis filters still have higher order than the analysis filters 
(even though not as much as in a parallel structure). This 
inbalance in complexity, if undesirable, can be reduced by re- 
structuring the filter bank as shown in Fig. 3. Such restructuring 
leaves the final output n(n) unchanged, as evidenced by repeated 
application of the above swapping property. 
A second issue we wish to address in this correspondence is 
related to channel distortion. In general, the transmission medium 
between the analysis and synthesis banks introduces both linear 
and nonlinear distortions, due to various coding schemes [2] 
involved. If the distortion is dominantly linear, it can be repre- 
sented by a channel transfer function, denoted by C,(z) for the 
kth channel. Even if k(z) is free from aliasing when channels 
are ideal, this is not so with Ck(z)‘s present. Accordingly, for a 
given set of Hk( z)‘s, a new set of synthesis filters G,(z) should 
be designed to cancel aliasing. Fig. 4 depicts this sitaation. 
Fig. 4. The equivalent QMF structure with channel distortion. 
Clearly, the output signal is given by 
/i;(z) =f;~~~(zw-‘)~~~H~(z~-‘)C~(z’)G~(z). (6) 
k=O 
If Fk( z) is such that (2) holds, it would appear that the choice 
Gk(z) = Fk(z)/C,(zr) would result in ii(z)= k(z). Such a 
choice would result in an unstable synthesis bank if C,(z) had 
zeros on or outside the unit circle. In order to overcome this 
difficulty, note that if (2) holds, then so does the following 
equation (for arbitrary choice of S(z) independent of k): 
r-1 
c S(z’)F,(z)H,(zw-‘) =O, l<l<r-1. (7) 
k=O 
Thus, if Gk(z) are chosen according to 
r-1 
G/c(Z) = fi(Z) ,~,c,‘Z” 
i#k 
(8) 
then (2), which holds, implies that (7) holds with 
r-1 
S(z) = lrIoGtz). (9) 
Thus, if aliasing is absent in the ideal QMF structure of Fig. 1, 
then the structure with linear channel distortion is free of aliasing 
if the new synthesis filters are chosen according to (8). Such an 
alias-free system has the overall transfer function 
&,<z> r-1 
x(z> =Stz’) c F,(z)H,(z). 
k=O 
The order of G,(z) defined in (8) is typically much higher than 
that of Fk (z). This difficulty can be avoidled by carefully defining 
G,(z) to minimize cost as follows. Let [31 C. R. Galand and H. J. Nussbaumer, “New quadrature mirror filter 
structures,” IEEE Trans. Acoustics, Speech, Signal Process., vol. ASSP-32, 
pp. 522-531, June 1984. 
[41 V. K. Jain and R. E. Crochiere, “Quadrature mirror filter design in the 
time domain,” IEEE Tram. Acoustics, Speech, Signal Process., vol. 
ASSP-32, pp. 353-361, April 1984. 
where the polynomial Pk(z) has all zeros inside the unit circle, [51 M. Smith and T. Barnwell, III, “A unifying framework for 
and Qk (z) has zeros on and outside the unit circle. Assume, of analysis/synthesis systems based on maximally decimated filter banks,” 
course that the channels are stable, so that the polynomials 
in Proc. IEEE Int. Conf. on ASSP (Tampa, FL), Mar. 1985, pp. 521-524. 
WI P. L. Chu, “Quadrature mirror filter design for an arbitrary number of 
Dk(z) have all zeros inside the unit circle. Define Gk(z) accord- equal bandwidth channels,” IEEE Trans. Acoustics, Speech, Signal Pro- 
ing to cess., vol. ASSP-33, pp. 203-218, Feb. 1985. [71 J. H. Rothweiler, “Polyphase quadrature filters: a new subband coding 
F,(Z)Dk(Z’) technique,” in Proc. IEEE Int. Conf. on ASSP (Boston), Mar. 1983, pp. G,(z) = vk(z’) 1280-1283. Pk(Z’> 181 K. Swaminathan and P. P. Vaidyanathan, “Theory of uniform DFT, parallel quadrature mirror filter banks,” presented at IEEE Int. Conf. on 
where ASSP, Tokyo, Apr. 1986. 
[91 K. Swaminathan and P. P. Vaidyanathan, “Design of uniform DFT, 
r-1 parallel FIR quadrature mirror filters,” presented at IEEE Int. Symp. on 
V,(z) = ,Fo Q,(z). (13) Circuits and Systems, San Jose, CA, May 1986. 
l#k 
This is equivalent to taking S(z) to be the polynomial 
rather than a typically higher order rational function as in (9). In 
fact, if there are common factors among Q,(z)‘s, this can be 
HEBERTT SIRA-RAMIREZ 
exploited to further reduce the order of Gk (z) to some extent. 
If the ideal system of Fig. 1 is alias-free and such that the Absfruct -A variable-structure-controlled (VSC) Van der Pol oscillator 
overall transfer function (5) is allpass (which is a requirement if 
is shown to produce an ideal average response of perfect harmonic nature 
amplitude distortion is to be avoided), then (10) does not neces- 
when its motions are made to slide on a circle defined in the interior of its 
sarily represent an allpass distortion because S(z) is not guaran- 
limit cycle. The sinusoidal responses are robust with respect to parameter 
teed to be allpass. This situation can be corrected by defining 
perturbations and external disturbances. 
G,(z) in a different manner, provided that C, (z)‘s do not have I. INTRODUCTION 
zeros on the unit circle (under this condition; Qk(z)‘s have all 
zeros strictly outside the unit circle). If we now define G,(z) to 
The variable-structure control of dynamic systems undergoing 
be 
sliding motions on nonlinear manifolds offers a richer variety of 
design alternatives than those possible with linear hyperplanes. 
(15) 
These possibilities are based on the fact that a larger class of 
static relationships can be synthesized, among the state variables, 
when nonlinear surface values are used in the switching logic 
where Q,(z) is Q,(z) with coefficients in reversed order, then controlling the system. The dynamic behavior of the controlled 
G,(z) are indeed stable, and this is equivalent to defining S(z) system is totally determined by the nature of the nonlinear 
in (7) as surface, making the controlled system robust with respect to 
external disturbances and parameter perturbations. 
Sliding surface reachability and invariance are essential in- 
gredients of the sliding motion design for variable-structure sys- 
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-34, NO. 1, JANUARY 1987 103 
Harmonic Response of Variable-Structure-Controlled 
Van der Pol Oscillators 
Since S(z) in (16) is (stable and) allpass, the overall transfer 
tems (VSS’s). These tasks are accomplished by opportune switch- 
function (10) of the alias-free system is now allpass, if (5) is 
ings among feedback laws, which guarantee state trajectories 
allpass. In this manner, aliasing caused due to linear channel 
invariably directed towards the sliding surface. In the sliding 
distortion can be avoided, at the same time avoiding additional 
regime, one of the outstanding characteristics of the controlled 
amplitude distortion. 
motion is that radically new properties are obtained compared 
with those of the individual structures responsible for its creation. 
II. CONCLUSIONS 
The reader is referred to several books (U&in [l], Utkin [2], 
Two important issues pertaining to alias-free parallel QMF 
Itkis [3]) and survey articles (Utkin [4], Utkin [5]) for a complete 
structures have been analyzed in this correspondence. Some 
account of the theory and its many practical applications. 
applications have been outlined; it is felt that the swapping 
Using the theory of variable-structure control [l], the possibil- 
property may have further applications in the efficient design of 
ity of creating harmonic limit cycles in controlled Van der Pol 
QMF banks, but details remain to be explored. 
oscillators is explored. An ideally sinusoidal response is obtained 
REFERENCES Manuscript received April 29, 1986; revised September 8, 1986. This work was supported in part by the Joint Electronics Program under Contract 
[l] A. Croisier, D. Esteban, and C. Galand, “Perfect channel splitting by use N00014-84-C-0149. 
of interpolation/decimation/tree decomposition techniques, presented at The author is with the Control Systems Department, Universidad de Los 
Intl. Conf. Information Sci. Syst., Patras, Greece, 1976. Andes, Metida, Venezuela, currently on leave at the Coordinated Science 
[2] R. E. Crochiere and L. R. Rabiner, Multirate Digital Signal Processing Laboratory, University of Illinois, Urbana, IL 61801. 
Englewood Cliffs, NJ: Prentice-Hall, 1983. IEEE Log Number 8611583. 
009%4094/87/0100-0103$01.00 01987 IEEE 


A note on general parallel QMF banks

A Note on General Parallel QMF

Clussicol Network Theory. San Francisco: Holden-Day,

Design of uniform DFT, r-1 parallel FIR quadrature mirror filters,” presented at IEEE Int. Symp. on V,(z) = ,Fo Q,(z). (13) Circuits and Systems,

of interpolation/decimation/tree decomposition techniques, presented at The author is with the Control Systems Department, Universidad de Los Intl. Conf. Information Sci.

oscillators is explored. An ideally sinusoidal response is obtained REFERENCES Manuscript received April 29, 1986; revised

Quadrature mirror filter design for an arbitrary number of Dk(z) have all zeros inside the unit circle. Define Gk(z) accord- equal bandwidth channels,”

where the polynomial Pk(z) has all zeros inside the unit circle, [51

https://authors.library.caltech.edu/5897/1/VAIieeetcs87g.pdf

A note on general parallel QMF banks

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main