It is shown that a large class of real-coefficient doubly-complementary IIR transfer function pairs can be implemented by means of a single complex allpass filter. For a real input sequence, the real part of the output sequence corresponds to the output of one of the transfer functions G(z) (for example, lowpass), whereas the imaginary part of the output sequence corresponds to its "complementary" filter H(z)(for example, highpass). The resulting implementation is structurally lossless, and hence the implementations of G(z) and H(z) have very low passband sensitivity. Numerical design examples are included, and a typical numerical example shows that the new implementation with 4 bits per multiplier is considerably better than a direct form implementation with 9 bits per multiplier. Multirate filter bank applications (quadrature mirror filtering) are outlined

Mitra, Sanjit K.

Regalia, Phillip A.

Vaidyanathan, P. P.

English

It is shown that a wide class of real-coefficient, doubly-complementary IIR transfer-function pairs can be implemented by means of a single complex allpass filter. For a real input sequence, the real part of the output sequence of the complex allpass filter corresponds to one of the transfer functions G(z) (for example, low-pass), whereas the imaginary part of the output sequence corresponds to its "complementary" filter H(z) (for example, highpass). Since the resulting implementation is structurally lossless, G(z) and H(z) have very low passband-sensitivity. Numerical design examples are included to demonstrate the ideas

Regalia, P.

Mitra, S. K.

Caltech Authors - Main

DESIGN OF DOUBLY-COMPLEMENTARY IIR DIGITAL FILTERS, 
USING A SINGLE COMPLEX ALLPASS FILTER 
P. P. VAIDYANATHAN,' P. REGALIA: AND S. K. MITRA2 
IDept. of EE, Caltech, Pasadena, CA 91125, USA 
2Dept. of ECE, Univ. of Calif., Santa  Barbara, CA 93106, USA 
ABSTRACT 
It is shown that a wide class of real-coefficient, 
doubly-complementary IIR trader-function pairs  can 
be implemented by means of a single complex allpass 
filter. For a  real input sequence, the real part of the out- 
put sequence of the complex allpass filter corresponds 
to one of the transfer  functions G(z )  (for example, low- 
pass), whereas the imaginary part of the output se- 
quence corresponds to  its "complementary" filter H ( z )  
(for example, highpass). Since the resulting implemen- 
tation is structurally lossleas, G(z)  and H(z)  have very 
low passband-sensitivity. Numerical design examples 
are included to  demonstrate  the ideas. 
I. INTRODUCTION 
In  this  pzper we consider digital infinite impulse re- 
sponse (IIR)  stable transfer  functions of the form 
N N 
G(z) = P ( z ) / D ( z )  = PnZ-" / dnz-" (1) 
n=O n=O 
where pn and dn are real. Without loss of generality, it 
is assumed that IG(.'")l 5 1 for all w ,  i.e., we assume 
G(z )  to  be a  BR  function [l]. A  transfer function which 
is "complementary" to G(z)  is any function H ( z )  = 
Q(z ) /D(z )  such that 
lH(dw)12 + IG(.'")l2 = 1 (2) 
for all w .  Equation (2) is called the power complemen- 
tary property. In a  number of filtering applications, the 
numerators P(z )  and Q(z)  have linear-phase (a com- 
mon reason being that  the transmission zeros of G(z)  
and H ( z )  are typically on the unit circle). Accord- 
ingly, these polynomials are  symmetric or antisymmet- 
ric. When P ( z )  is symmetric and Q(z )  is antisymmet- 
ric, it is well known [2],[3] that G ( z )  and H ( z )  can be 
implemented in the form 
1 
G(z )  = 5 [ A d z )  + A 2 ( 4  (3) 
fw = [Adz) - A d z ) ]  1 (4) 
where A l ( z )  and A ~ ( z )  are  stable allpass functions with 
real coefficients. If Nl and N2 denote the orders of 
A l ( z )  and Az(2) then Nl+ N2 = N. Implementation of 
G(z) and H ( z )  in  terms of allpass functions in the above 
manner leads to extremely low passband-sensitivities 
because of structural boundedness [1],[2]. Moreover, a 
total of N multipliers is sufficient  in order to generate 
both  the transfer  functions G(z)  and H ( z ) ,  and accord- 
ingly we have a very efficient implementation. 
A wide class of transfer  functions fall under the class 
which can  be implemented as above. Typical examples 
are odd order  Butterworth, Chebyshev, and elliptic digi- 
tal lowpass filters. Now, equations (3),(4) are equivalent 
to  the inverse equations 
Al(2 )  = G(z) + H ( z )  (5) 
A ~ ( z )  G(z) - H ( z )  . (6) 
Any pair of transfer  functions satisfying (2) and ( 5 )  is 
said to be a Udoubly-complementary" pair. 
In  the above discussion, P ( z )  was assumed symmet- 
ric and Q(z)  antisymmetric. Next, if both P(z )  and 
Q ( z )  are  symmetric,t  the above allpass decomposition 
is not valid. This situation  arises when neither G(z) nor 
H ( z )  has a zero at w = 0. Typical examples are even or- 
der  digital Chebyshev and elliptic lowpass and highpass 
filters. In  the next section we  show  how a different type 
of decomposition can be performed in terms of complex 
allpass functions. 
II. THE DECOMPOSITION OF A BR FUNCTION 
INTO COMPLEX ALLPASS SECTIDNS 
Let G(z) be  a BR function as in (1). Assume there 
are no uncancelled common factors between P(z )  and 
D(z) .  Let H ( z )  be the complementary BR function 
satisfying (2) and given by 
N N 
H ( z )  = Q(z)  / D(z)  = qnz-" / dnz-" . (7) 
n=O n=O 
4 It is not possible for both P(z )  and Q(z)  to  be an- 
tisymmetric, for this would imply that  the left side of 
(2) is zero at w = 0. 
ICASSP 86, TOKYO 
48. 4. 1 
CH2243-4/86/0000-2547 $1.00 0 1986 IEEE 2547 
Fig. 1. Implementing the complementary pair. 
G(z)= p y ( z )  
(real ) X ( Z )  
Fig. 2. Realizing G(z) by one allpass function. 
The  coe5cients pn, qn, and dn are all real-valued. As- 
suming the  numerators P(z )  and Q(z) to  be  symmetric 
polynomials, we have Pn = p ~ + ,  and qn = qN-,,, and 
hence we havet 
P(z) = ZNP(z) , O(Z) = zNQ(z) . (8) 
If the power-complementary property of (2) holds, then 
by analytic  continuation we have 
P ( z ) P ( z )  + Q(z)Q(z) = E ( z ) D ( z )  (9) 
for all z. In view of (8), this becomes 
P2(z)  + Q2(z) = z - ~ D ( z - ' ) D ( z )  (IO) 
which can  be  rewritten as 
[P(z)  + jQ(z) ]  [P(z)  - jQ(z) ]  = z - ~ D ( z - ' ) D ( z )  . (11) 
With  three polynomials P(z ) ,  Q(z) ,  and D(z)  with  real- 
coeficients satisfying (11), it can be shown that D(z)  
cannot have real zeros  when there  are no common fac- 
tors between P(z )  and D(z)  (as assumed). Accordingly 
all zerw Qf D ( z )  are complex conjugate pairs, and hence 
N must be even and D(z)  has  the form 
M 
D ( z ) =  n( l -Z- lzk)( l -zz- lZi )I  M=--. (12) N 
2 k = l  
Thus, (11) becomes 
[ P ( 4  + iQ(~)l  [ P ( 4  - iQ(z ) l=  
M 
z-N n (1 - z-'zk)( 1 - (1 - zzk)( 1 - 22;) . (13) 
Since P(z )  and Q(z) are symmetric, the zeros of P(z )  + 
jQ(z)  occur in reciprocal pairs. The same is true of 
P(z )  - jQ(z) .  Moreover, if zk is a zero of P ( z )  + j Q ( z ) ,  
then zi is a zero of P(z)  - jQ (z )  because the coeffi- 
cients in P(z )  and Q(z)  are real. Consequently, we can 
uniquely factorize (13) as 
k = l  
3 The tilde  notation  means  that z should be replaced 
with z-', and the coefficients (if complex) should be 
conjugated. On the  unit circle, this  corresponds to con- 
jugating the function itself. 
M 
P(2) + jQ(z )  = pz-M (1 - Z - l z k )  ( I  - Z Z k )  (14) 
M 
P(z)  - jQ(z) = (1 - z-'~;) (1 - zz i )  (15) 
where /3 is a complex constant  (to be identified below). 
Dividing both sides of (14),(15) by D(z) ,  we thus arrive 
at  
k=l  
where Al(z) and &(z) are  stable allpass functions with 
complex coefiicients, defined as 
The inverses of the relations (16),(17) are 
G(z)  = [ A l ( 4  + A 2 ( z ) l /  2 ( 19) 
ff(Z) = [ A l b )  - A 2 ( z ) l /  2 j  (20) 
In summary, G(z )  and j H ( z )  satisfy (2) and (16) si- 
multaneously, and hence  form a doubly-complementary 
pair. Moreover, they  can  be implemented in terms 
of sums and differences of allpass functions. Figure 1 
shows a  schematic of the implementation. 
The allpass functions Al(z) and A ~ ( z )  have orders 
A4 each,  where M = N/2 .  The coefficients of these func- 
tions  are complex, and moreover the coefficients of &(z) 
are  the complex conjugates of those of Al(z).  Accord- 
ingly, with  real  inputs, in order to  realize the schematic 
of Figure 1, it is only necessary to implement Al(z). 
The real part of the  output sequence corresponds to the 
output of G(z )  and the imaginary  part  corresponds to 
the output of H ( z ) .  Figure 2 shows this implementa- 
tion. The simplest means of implementing Al(z )  is to 
cascade first-order sections of the form shown in Fig- 
ure 3. Four real multiplications  are involved  in the cir- 
cuit of Figure 3. The multiplier p in (IS), operating on 
v 
Fig. 3. Implementing a complex allpass section. 
48. 4. 2 
2548 ICASSP 86, TOKYO 
a real  input  sequence,  is equivalent to two real  multipli- 
cations. Thus  the  total number of real  multiplications 
per  transfer  function  in  Figure 2 is (4M+ 2)/2 = N + 1, 
as against the direct  form which requires 3N/2 + 1 real 
multiplications  for a linear-phase  numerator. 
Since the multipliers in Figure 2 can be quantized 
such that Al(z) remains  stable  allpass, it is  clear that 
G(z )  and H ( z )  remain BR  in spite of quantization. Ac- 
cordingly, the implementation exhibits low passband- 
sensitivity [1],[2], and  remains  stable  in  spite of multi- 
plier quantization. 
It is also possible to obtain cascaded lattice imple- 
mentations of (18) by employing a complex version of 
the Gray and Markel lattice structures [4]. Further- 
more,  a  large  number of equivalent structures for com- 
plex allpass  functions  can  be  generated by appropriate 
extension of the multiplier  extraction  approach [5]. The 
relative merits of these various schemes remain to be 
explored: 
III. A DESIGN EXAMPLE 
Given a BR function G(z)  = P(z) /D(z) ,  with 
symmetric P(z) ,  assume that there  xists H ( z )  = 
Q(z ) /D(z )  with  symmetric  numerator Q(z ) ,  satisfying 
(2). Then Q ( z )  can be computed easily without go- 
ing through  elaborate  spectral  factorization schemes, by 
noting that (10) implies 
Q2(z) = z - ~ D ( z - ' ) D ( z )  - P2(z)  . (21) 
Let the righthand  side of (21), which is known,  be de- 
noted as R(z)  = ci!!orkz-k. Then the coefficients of 
Q(z)  can  be recursively computed as 
Once Q(z)  is found, we evaluate the zeros of the poly- 
nomial p ( ~ )  + jQ(z),  in  order to identify zk in (14). In 
order to find /3, note  that from (16) 
Since all quantities in (25) except P are known, P can be 
evaluated.  Thus, Al(z) has been completely identified, 
and  the implementation of Figure 2 can  be  obtained. 
As an example,  consider the design of an  8th-order 
lowpass elliptic BR digital  transfer  function: 
48. 
-0.042K5 + 0.074K6 - 0 . 0 2 9 ~ - ~  + O.O26z-' 
- 6 . 9 0 6 ~ - ~  + 3.019& - 0 . 8 0 9 ~ - ~  + 0 . 1 0 4 ~ - ~  * (26) 
Once Q(z)  is  computed, we have 
0.321 - 1.901z-' + 5 . 4 5 5 ~ - ~   9 . 7 2 4 8  + 11.706K' 
1 - 3.9822-' + 8 . 4 9 8 ~ ~ ~  - 1 1 . 4 8 6 ~ - ~  + 1 0 . 7 0 7 ~ - ~  
- 9 . 7 2 4 ~ - ~  + 5.455K6 - 1 . 9 0 1 ~ - ~  + 0 . 3 2 1 ~ - ~  
- 6 . 9 0 6 ~ - ~  + 3 . 0 1 9 ~ - ~  0 . 8 0 9 ~ - ~  + 0.104& * (27) 
We then compute the zeros of P(z)  + jQ(z) and 
identify Al(z): 
where 
/3 = 0.4698 + j0.8828  (29)
zl = 0.4344 - j0.2253, 22 = 0.4831 + j0.5675, 
z3 = 0.5244 - j0.7367, 24 = 0.5492 + j0.8075 . (30) 
In order to  demonstrate the low passband-sensitivity 
properties, the  structure of Figure 2 was simulated dig- 
itally,  with quantirted multipliers. The allpass  function 
Al(z) was implemented by cascading the first-order sec- 
tions in Figure 3; each multiplier Zk,r and Zk,i was quan- 
tieed to 4 binary  bits  in  sign-digit code. 
Figure 4 shows the magnitude responses of H ( z )  
and G(z)  in the ideal (Le., unquantized) case. Fig- 
ure 5 shows the response [G(cj")l for the 4-bit  allpass- 
based implementation of Figure 2. The low passband- 
sensitivity  is  evident  from the plot of passband  details, 
which are included in Figure 5(b). Next, the same  trans- 
fer  function G(z)  was implemented  in  direct-form with 
9-bits  per multiplier. The corresponding  responses  are 
shown in  Figure 6 which reveals a very poor  passband- 
sensitivity. Thus, a 4-bit allpass based implementation 
(Figure 2) has much better passband response than a 
%bit direct-form implementation. In addition, the im- 
plementation of Figure 2 requires only 9 real  multipliers 
per  transfer function,  whereas the direct-form  requires 
13 multipliers. 
4. 3 
ICASSP 86, TOKYO 2549 
CONCLUDING REMARKS 
A new class of structurally passive low-sensitivity 
implementations has been reported in this paper. The 
implementation is a single complex allpass section, giv- 
ing rise to two BR transfer functions. Application of 
these  results in signal-splitting and reconstruction, and 
in multi-rate signal processing are  currently being stud- 
ied. 
References 
[l] P. P. Vaidyanathan and S. K. Mitra, “Low pass- 
band sensitivity  digital filters: A generalized viewpoint 
and synthesis procedures,” Proceedings of the IEEE, 
pp. 404-423, Apr. 1984. 
[2] P. P. Vaidyanathan, S. K. Mitra, and Y. Neuvo, “A 
new approach to  the realization of low sensitivity IIR 
digital filters,” IEEE Trans. on Acoustics, Speech, and 
Signal Processing, v. ASSP-34, 1986 (to appear). 
131 A. Fettweis, “Wave digital lattice filters,” Int. J. Cir- 
cuit  Theory and Applications, pp.  203-211, June 1974. 
[4] A. H. Gray and J. D. Markel, “Digital lattice and 
ladder filter synthesis,’’ IEEE ‘ h n s .  Audio Electroa- 
coustics, v. AU-21, pp. 491-500, Dec. 1973. 
[5] S. K. Mitra and K. Hirano, “Digital allpass net- 
works,” IEEE Trans. on Circuits  and Systems, v.  CAS- 
21, pp. 688-700, Sept. 1974. 
161  A. G. Constantinides and R.  A. Valenzuela, “An 
efficient and modular transmultiplexer design,” IEEE 
’lkans. on Comm., v. 30, pp. 1629-1641, July 1982. 
[7] R. Ansari and B. Liu, “A class of low-noise compu- 
tationaIIy efficient recursive digital filters with applica- 
tions to sampling rate alterations,” IEEE Trans. Acous- 
tics, Speech, and Signal Proc., pp. 90-97, Feb. 1985. 
0. 
-20.000 
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LA 
0 
S 
Q 
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CT 
-188.018 
Normalized frequency 
-20.000 
m 
-m.000 
0 
S 
1 4 b i t  al lpass based I 
-lOB.OOO 
0.  0.100 0.2fl0 0.300 0.400 0.580 
Normalized frequency 
F i g .  5 ( a )  The quantized  implementation. 
Passband Details 
W 
VI 
8 
0 
1.003 
,“ 1.001 4 b i t  al lpass based 
a, 
L 
3 
c, 
C 
.P ideal 
2 
m 0.997 - 
Fig. 5(b )  The quantized  implementation. 
1.885 
1.003 
S 
0 
VI 
n 
U L” a 1.001 r ideal 
+-’ 0.999 
TJ 
S 
.r 
m ‘ 0.997 
0.995 
Normalized frequency 
F i g .  6 .  The quantized direct  form. 
F i g .  4.  The conlplementary response:, 
Work supported in part by NSF g r a n t s  ECS 84-04245 and ECS 85-08017 and i n  Part C a l t e c h I s  PAT grant. 
48. 4. 4 
2550 ICASSP 86, TOKYO 


Design of doubly-complementary IIR digital filters, using a single complex allpass filter

Design of doubly-complementary IIR digital filters using a single complex allpass filter, with multirate applications

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He is the recipient of the

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yanathan, “The discrete-time bounded real lemma in digital filtering,”

https://authors.library.caltech.edu/5847/1/VAIieeetcs87e.pdf

Design of doubly-complementary IIR digital filters using a single complex allpass filter, with multirate applications

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