It is shown in this correspondence that the system function H(z) of a linear time invariant (LTI) causal digital filter with real impulse response coefficients can be obtained from the real part of its frequency response HR(ejω) given in the form of a rational trigonomentric function, using algebraic methods rather than complex contour integration techniques

Mitra, S. K.

Vaidyanathan, P. P.

English

P.P. Vaidyanathan

S.K. Mitra

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On the construction of a digital transfer function from its real part on unit circle

Caltech Authors - Main

198 PROCEEDINGS OF THE IEEE, VOL. 70, NO. 2 ,  FEBRUARY 1982 
Step 2: To establish the validity of (4), note  that if n > 1 and k 2 1 
a(n, k) > a(n - 1, k). (8) 
In  other words, the number  of  coefficients in a  polynomial  of degree n 
is greater than the number of coefficients in a Polynomial of degree 
n - 1. The condition k > 1 simply requires that the polynomials have 
at least one variable whereas the condition n 2 1 is required so that the 
right side of (8) makes sense. 
Any polynomial Pn (x, y )  of degree n in  the k + 1 variables (x, y )  may 
always be written  uniquely as 
then 
Pn (x, V )  = qn(*) + qn-1 (x) Y + 4n-2 (x) Y' + * * * + qO(x) Y" (9)  
where q i ( x )  is a  polynomial of degree at most i. Therefore, 
a ( n , k + l ) = a ( n , k ) + a ( n - l , k ) + . . . + a ( l , k ) + l  (10) 
or alternatively 
a(n, k + 1) = [a@, k) + - . - a(Z + 1, k)] + .(I, k + 1).  (11) 
Using (8), it follows  from (11) that 
a(n, k + 1) > [a(n - I ,  k) + a  - - + a(1, k)] + a(Z, k +  1).  (12) 
However, (10) implies that the term in brackets is equal to a(n - I ,  
k + 1) - 1. Therefore, 
a(n, k + 1) > a(n - I ,  k + 1) + a(1, k + 1) - 1. (13) 
Thus, (4) is true under the hypothesis of the theorem. 
Step 3: Any reducible polynomial in 9 (n,  k) is always contained 
within the finite union of the sets  YO,^,, andfB(I) for Z = l , 2 , . * . ,  
n - 1. Smce each of  these sets has measure zero inRQ("9 k) ,  then  does 
the set S and  Theorem 2 follows. 
111. L)ISCUSSrON 
The result  presented in Theorem 2 may be easily extended to  other 
classes of polynomials. For example, it is straightforward to modify 
the proof of the theorem to show that the set of all reducible poly- 
nomials of degree n in k variables with complex coefficients correa- 
pon&toasetofmeasuIezeromRZQ(n~k)providedR>1andn>1. 
Another class of polynomials which is often encountered consists of 
those which have a given degree in each variable. Specifically, let 
9 (n, k) be  the set of all polynomials which have degree ni in xi  for 
i = 1, . - , k. A polynomialp,(x) in 9 (n, k) is therefore of the form 
With p(n, k) = (nl  + l)(nz + 1) - * (nk + 1) the number of coefficients 
required to specify the polynomial p,, (x), it is easily shown that 
8(1, k) + B(n - I ,  k) < B(n, k) (15) 
provided k > 1. Therefore, it follows in a  style similar to  that  in  the 
proof of Theorem 2 that  the set of all reducible polynomials m 9 (n, k) 
corresponds to a set of measure zero in R B(n*k). 
ACKNOWLEDGMENT 
The authors would like to acknowledge the invaluable contributions 
made by Prof. J. R Munlrers in the proof of Theorem 2. 
REFERENCES 
[ 11 A Mostowski and M. Stark, Introduction to Higher Algebm, New 
[ 2 ]  N. K. Bose, "Problems and progress in multidimensional systems 
York: Pergamon Reds, (1964). 
[ 31 M. H. Hayes, '"I%e Reconstruction of i multidimensional  sequence 
theory," Roc. IEEE, pp. 824-840. June 1977. 
from  the phase or magnitude of ita Fourier transform," to  appear 
in IEEE Tmnr Acoust., Speech, Signal Proteas. 
141 J. M. Ortega and W. C. Rheinboldt, Ircrotfvc Solution of Nonlinear 
Equutiom in S c r m J  Varbbles, New York: Academic Press (1970). 
On the  Construction of a Digital Transfer Function 
from Its Real Pat on Unit circle 
P. P. VAIDYANATHAN AND S. K. MITRA 
Abstmct-It is shown m this cmmpondence that the system function 
H(z) of a linear time invariaut (Ln) causal digital filter  with real im- 
pulse mponse coefficients cpa be obtained from the d part of its 
€requency response HR(elw) given m the form of a ratio& trigona- 
mentric function, using algebraic methods  rather  than complex contour 
integration techniqua 
I. INTRODUCTION 
For a causal linear time  invariant  (LTI)  continuous  time  system  with 
real impulse response, the  transfer  function H,(s) or  any other network 
function (like the driving point impedance) can be calculated from a 
component of the function such as real  part of the function or from  the 
magnitude or phase response of the function [ 11. Even though rela- 
tions exist in the form of integrals (like the Hilbert Transform [2]), 
there  are also available more convenient algebraic methods for this pur- 
pose [3]. For discrete time systems such as digital filters,  one can still 
find  relationships involving complex contour  integrals, such as discussed 
in [ 41. But there  does not seem to be available an algebraic method, as 
for continuous time systems. In this correspondence, we develop an 
algebraic procedure for retrieving the system function H(z)  of a causal, 
stable, LTI digital filter from the real part of its frequency response 
HR(elw), specified  as a  ratio of two  trigonomentric  functions. 
Consider a causal, stable LTI system described by 
H(z) = A(z) /B(z)  (1) 
with real impulse response sequence. Let the real part of the system 
function H(z) ,  evaluated on  the  unit circle be given  by 
HR(eiw) = N(eiw)/D(eiw) (2) 
where N(eiw) and D(eiw) are trigonomentric functions of the radian 
frequency w. It is shown in [4] that 
Thus H(z) can be evaluated everywhere in its region of convergence 
Iz I > 1, using (3), where the  contour of integration is the unit circle. 
In what follows, we develop an algebraic method for calculation of 
H(z) for IZI > 1 from HR(ejw). The process involves two steps: the 
fust one is to identify the poles of H(z);  second is to identify the 
numerator  polynomial  and  determine  the scale factor. 
11. CALCULATION F Porn OF H ( Z )  
We note  that 
and hence 
as the coefficients of B(z) are real. Thus D(z) has  zeroes  occurring  in 
mirror image pairs, and the zeros that fall inside the unit circle are 
precisely .the poles of H(z) from stability and causality requirements. 
make the following substitution  in D(elw) in  order to get B(z) B(z-1): 
As HR(eIw) is usually given in terms of sine and cosine functions, we 
2" + z- 
cos ( n u )  + - sin ( n u )  + -. ( 6 )  
Once De) = B(z)  B(2-l)  is thus  obtained, we  solve for its roots and 
take those roots inside the unit circle as the poles of H(z). Thus B(z) in 
zn - z-n 
2  2i 
National Science Foundation under Grant ECS79-18028. 
Engineering, University o f  California, Santa Barbara, CA  93106. 
Mrnuseript received July 13,  1981. Thia work was supported by the 
The authors are with the Department of Electrical and Computer 
0018-9219/82/0200-0198$00.75 Q 1982 IEEE 
PROCEEDINGS OF THE IEEE, VOL. 70, NO. 2, FEBRUARY 1982 199 
(1) is determined to within a scale factor. Note that ( 6 )  is analogous to 
the analytic continuation  methods  for  continuous time systems, where 
w is  replaced  by s f i  to get Hu(s) from H,(jw). 
111. CALCULATION F THE NUMERATOR POLYNOMIAL ( z )  OF H ( z )  
An examination of well-known methods  adopted  for a similar prob- 
lem  in continuous time systems leads us to look for a decomposition of 
A ( z )  and B(z )  
A ( z )  = m l ( z )  + n l ( z ) ;  B ( z ) = m 2 ( z )  + n 2 ( z )  (7a) 
so that 
in  such a way that 
mi(z) = rni(z-') ni(z) = -ni(z-i), i = 1, 2. ( 8 )  
The. motivation behind such a decomposition is that, in  view  of (4), 
HR(elw) becomes 
Now,  comparing the  form  (7) with (2), we find that 
k(mlm2 -n lnz )=N(z )  (10) 
(where k is a suitable scale factor whose determination is trivial, as 
shown in the exampje later). 
Thus given HR(eJw), if we make the substitution ( 6 )  in N(eiw) we 
get a function of z that can be  equated toml(z)m2(z) -nl(z)nz(z) to 
within a scale factor. Now, since the poles of H(z)  are already found, 
m2 and n2 are known and  therefore, by equating like powers of z in ( 8 )  
we can determine A(z)  = m 1 ( z )  + n 1 (2). 
A suitable way of defining m i @ )  and n&z) so that (10) is  satisfied  is: 
Once we find ml + n 1 in the above manner, H(z)  is determined to 
within a constant factor. Since h(n)  is real, H(z) evaluated at z = 1 is 
purely real and must equal H R ( d W )  as w = 0 and this fact is  made use 
of to determine the constant  factor. 
IV. AN EXAMPLE 
Let 
1 + cos w + cos 2w N(eiW) 
HR(eiw) = -- 
17 - 8 cos 2w D(eiw) ' 
1) D(z )  = 17 - 8(z2 + ~ - ~ ) / 2  = 17 - 4z2 - 4z-2 and D(z)  has zeros at 
z = 1/2, 2, -1/2, -2. Thus we take B(z) = 1 - to be the de- 
nominator of H(z).  From here, we find 
2) Next, we find N(z )  from N(eiw) = 1 + cos w + cos 2~ by  substi- 
tuting ( 6 )  
22 z 2-l 2-2 
N(z )  = 1 +-+-+-+--. 
2 2 2  2 
3) We assume ~ ( z )  =a0 +ulz-l + U ~ Z - ~  since 2w is the largest 
multiple of w appearing. Then we write m (2) and n ( z )  as 
and find m1m2 - n l n 2  to be 
Then we equate  this  to N(z )  to get uo,  u 1 ,  and u2 yielding H(z)  = 
(1 +z- l  + ~ - ~ ) / l 2 ( 1  - f ~ - ~ )  where the scale factor is determined 
as described  earlier. 
MRENCES 
[ 11 D. M. Lipkin, "Novel expression for the relationship between the 
R o c .  IRE,vol. 47, p. 1383, Aug. 1959. 
real and imaginary parts of the transfer function of a linear filter," 
[ 2 ]  E. A. GuilIemin, m e  Mathematics of Circuit  Analysis. New York: 
Wiley, 1949. 
[ 31 S. Karni, Network meory: Analysis and Synthesis. Boston, MA: 
[ 4 ]  A. V. Oppenheim and R. W. Schafer, Digital Signal Processing. 
Allyn and Bacon, 1966. 
Englewood  Cliffs,  NJ:  Prentice  Hall. 
Circuit Diagnosis and Thevenin  Equivalents 
P. E. GRAY 
Absmct-Roytmnn and Swmy have used the properties of ortho- 
normal excitations to establish one  method of circuit diagnosis. In this 
letter the method is extended to determine the driviug-point or Thevenin 
equivalent impedance for each  node pair. 
I. INTRODUCTION 
The  Roytman  and Swamy [ 1) method of circuit diagnosis is concerned 
with the general, hear, time-invariant network of known topology 
exhibiting (n + 1) nodes. Using orthonormal excitations this method 
results in the voltage measurement which become the coefficients of 
the inverse of  the  node  admittance matrix [ YI-'. The method proceeds 
to compute matrix [ Y] from which the values of individual circuit 
elements are calculated. Adaptations of the method to test-fmture 
development shows the research to be eminently practical. 
The following discussion briefly summarizes the method and shows 
the resulting coefficients of [ VLk)] to be related to  the driving-point or 
Thevenin equivalent impedance at each node pair. 
11. CALCULATION F EQUIVALENT IMPEDANCE 
The Roytman-Swamy method can be  summarized 
where 
[ Y]-' is the inverse of the  node  admittance matrix. 
K ik) ]  is the matrix of node voltages of the kth node when only 
is the  unity matrix of orthonormal  current excitations. 
the nth node  is  excited  by  unity  current. 
[ Y ] is then calculated and  the values of the circuit elements determined. 
Extending  the method to calculate equivalent impedance at any node 
pair (pm) requires consideration of the coefficients of [ Ur)]. That 
is when m is the reference node  the corresponding equivalent unpedance 
at node pair ( p ,  n + 1) is a. When m is not the reference node the 
corresponding equivalent impedance becomes 
This relationship can be derived using the Scott [2] concept that a 
Manuscript  received July 6,1981. 
The  author is with the Electrical  Engineering Department, University 
of  Wisconsin,  Platteville,WI 53818. 
0018-9219/82/0200-0199$00.75 0 1982 IEEE 


and D(z) has zeros at z = 1/2, 2, -1/2, -2. Thus we take B(z) = 1 - to be the denominator of H(z). From here, we find 2) Next, we find N(z

I%e Reconstruction of i multidimensional sequence theory,&quot; Roc.

Introduction to Higher Algebm,

Ircrotfvc Solution of Nonlinear Equutiom in S c r m J Varbbles,

The author is with the Electrical Engineering Department,

Thia work was supported by the The authors are with the Department of Electrical and Computer 0018-9219/82/0200-0198$00.75 Q

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On the construction of a digital transfer function from its real part on unit circle

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