Frequency Response of Rational Systems
Quote of the Day
If knowledge can create problems, it is not
through ignorance that we can solve them.
Isaac Asimov
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall
Inc.
Frequency Response of Rational System Functions
• DTFT of a stable and LTI rational system function
M
 bk e
 
He
j
 jk
k 0
N
 jk
a
e
 k

1c e 


M
b
  0 
 a0 
k 0
k
k 1
N
 j
 1  d e 
k
 j
k 1
• Magnitude Response
M
 
H e j 
b0
a0
 j
1

c
e

k
k 1
N
 j
1

d
e

k
k 1
• Magnitude Squared
 1  c e 1  c e 
M
 
He
j
2
  He 
 He
j 
j
 b0 
  
 a0 
2
k
k 1
N
*
j
k
 1  d e 1  d e 
k
k 1
Copyright (C) 2005 Güner Arslan
 j
351M Digital Signal Processing
 j
*
j
k
2
Log Magnitude Response
• Log Magnitude in decibels (dB)
 
20 log10 H e
j
M
b0
 20 log10
  20 log10 1  ck e  j
a0 k 1
N
  20 log10 1  dk e  j
k 1
 
Gain in dB  20 log10 H e j
 
Attenuation in dB  - 20 log10 H e j  Gain in dB
• Example:
– |H(ej)|=0.001 translates into –60dB gain or 60dB attenuation
– |H(ej)|=1 translates into 0dB gain
– |H(ej)|=0.5 translates into -6dB gain
• Output of system
 
 
 
20 log10 Y e j  20 log10 H e j  20 log10 X e j
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
3
Phase Response
• Phase response of a rational system function
 
H e
j
N
 b0  M
 j
      1  ck e
   1  dk e j
k 1
 a0  k 1




• Corresponding group delay
 
grdH e
j



N
d
d
 j

arg1  ck e

arg1  dk e j
k 1 d
k 1 d
M

– Here arg[.] represents the continuous (unwrapped) phase
– Work it out to get
dk
 
1 d
grdH e
j
M
k 1
Copyright (C) 2005 Güner Arslan
2
k
 Red e 
c

 2 Red e 
1 c
k
 j
2
k
M
 j
2
k
k 1
351M Digital Signal Processing
k

 Re ck e j
2


 2 Re ck e j

4
Unwrapped (Continuous) Phase
• Phase is ambiguousWhen
calculating the arctan(.)
function on a computer
– Values between - and +
– Denoted in the book as ARG(.)
  
   ARGH ej  
– Any multiple of 2 would give
the same result
 
  
H ej  ARGH ej  2r
– Here r() is an integer for any
given value of 
• Group delay is the derivative of
the unwrapped phase
d
j
grdH e

argH e j
d
  
Copyright (C) 2005 Güner Arslan
   
351M Digital Signal Processing
5
Frequency Response of a Single Zero or Pole
• Let’s analyze the effect of a single term
1  ck e
 j
2
j
 1  re e
• If we represent it in dB
 j
2
 1  r2  2r cos  


20 log10 1  re je  j  10 log10 1  r 2  2r cos  
• The phase term is written as

j
ARG1  re e
 j

 r sin   
 arctan

1  r cos  
• And the group delay obtained by differentiating the phase

j
grd1  re e
 j

r 2  r cos  
r 2  r cos  


2
2
j  j
1  r  2r cos  
1  re e
• Maximum and minimum value of magnitude

1  r


 2r cos    10log 1  r

 2r  20log
10log10 1  r2  2r cos    10log10 1  r2  2r  20log10 1  r
10log10
2
10
2
10
1r
Demo at Mississippi State and at Arizona State University
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
6