Transcript Transform Analysis of LTI Systems

Transform Analysis of LTI Systems
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Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall
Inc.
Quick Review of LTI Systems
• LTI Systems are uniquely determined by their impulse

response
yn 
 xk hn  k  xk  hk
k  
• We can write the input-output relation also in the z-domain
Yz  HzXz
• Or we can define an LTI system with its frequency response
 
  
Y ej  H ej X ej
• H(ej) defines magnitude and phase change at each frequency
• We can define a magnitude response
Y e j  H e j X e j
• And a phase response
 
 
   
 
 
Y ej  H ej  X ej
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Ideal Low Pass Filter
• Ideal low-pass filter
 
Hlp e
j

  c
1


0 c    
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hlp n 
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sin cn
n
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Ideal High-Pass Filter
 
Hhp e
j
0
  c

1 c    
• Can be written in terms of a low-pass filter as
 
 
Hhp ej  1  Hlp ej
hhp n  n  hlp n  n 
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sin cn
n
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Phase Distortion and Delay
• Remember the ideal delay system
 
DTFT
hidn  n  nd  

Hid ej  e jnd
• In terms of magnitude and phase response
 
e   n
Hid e j  1
Hid
j
d
 
• Delay distortion is generally acceptable form of distortion
– Translates into a simple delay in time
• Also called a linear phase response
– Generally used as target phase response in system design
• Ideal lowpass or highpass filters have zero phase response
– Not implementable in practice
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Ideal Low-Pass with Linear Phase
 jnd

e
  c

Hlp e j  
c    
 0
• Delayed version of ideal impulse response
 
hlp n 
sin c n  nd 
n  nd 
• Filters high-frequency components and delays signal by nd
• Linear-phase ideal lowpass filters is still not implementable
• Group Delay
–
–
–
–
Effect of phase on a narrowband signal: Delay
Derivative of the phase
Linear phase corresponds to constant delay
Deviation from constant indicated degree of nonlinearity
  
  grdH e j  
   
d
argH e j
d
– arg[] defines unwrapped or continuous phase
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System Functions for Difference Equations
• Ideal systems are conceptually useful but not implementable
• Constant-coefficient difference equations are
– general to represent most useful systems
– Implementable
– LTI and causal with zero initial conditions
N
M
 a yn  k   b xn  k
k 0
k
k 0
k
• The z-transform is useful in analyzing difference equations
• Let’s take the z-transform of both sides
N
M
 a z Y z   b z Xz
k 0
k
k
 N

  ak z k  Y z  
 k 0

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k 0
k
k
 M

  bk z k Xz 
 k 0

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System Function
• Systems described as difference equations have system
functions of the form
M
Y z 
Hz  

Xz 
 bk z
k 0
N
k
a
z
 k
k 0
• Example
k

1c z 


M
b
  0 
 a0 
k
k 1
N
1
 1  d z 
k
1
k 1
1  z 
1  2z 1  z 2
Y z 
Hz  


1 1 3 2 Xz 
1 1 
3 1 

1

z  z
1

z
1

z



4
8
2
4



1 2


1 1 3 2 

1

z  z  Y z   1  2z 1  z 2 Xz 

4
8


1
3
yn  yn  1  yn  2  xn  2xn  1  xn  2
4
8
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Stability and Causality
• A system function does not uniquely specify a system
– Need to know the ROC
• Properties of system gives clues about the ROC
• Causal systems must be right sided
– ROC is outside the outermost pole
• Stable system requires absolute summable impulse response

 hn  
k  
– Absolute summability implies existence of DTFT
– DTFT exists if unit circle is in the ROC
– Therefore, stability implies that the ROC includes the unit circle
• Causal AND stable systems have all poles inside unit circle
–
–
–
–
Causal hence the ROC is outside outermost pole
Stable hence unit circle included in ROC
This means outermost pole is inside unit circle
Hence all poles are inside unit circle
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Example
• Let’s consider the following LTI system
5
yn  yn  1  yn  2  xn
2
• System function can be written as
Hz  
1
1 1 

1
1  z  1  2z
2




• Three possibilities for ROC
– If causal ROC1 but not stable
– If stable ROC2 but not causal
– If not causal neither stable ROC3
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ROC1 : z  2
1
 z 2
2
1
ROC3 : z 
2
ROC2 :
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Inverse System
• Given an LTI system H(z) the inverse system Hi(z) is given as
Hi z 
1
Hz
• The cascade of a system and its inverse yields unity
gn  hn  hi n  n
Gz  HzHi z  1
• If it exists, the frequency response of the inverse system is
 
Hi e j 
1
H e j
 
• Not all systems have an inverse: zeros cannot be inverted
– Example: Ideal lowpass filter
• The inverse of rational system functions
1  c z 

M
 b0
Hz    
 a0 
k
k 1
N
1
 1  d z 
k
1
1  d z 

 a0

 Hi z    
 b0 
k 1
N
k
k 1
M
1
 1  c z 
k
1
k 1
• ROC of inverse has to overlap with ROC of original system
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Examples: Inverse System
• Example 1: Let’s find the inverse system of
1  0.5z 1
1  0.9z 1
ROC: z  0.9 

Hz  
Hi z 
1
1  0.9z
1  0.5z 1
• The ROC of the inverse system is either z  0.5 or z  0.5
• Only z  0.5 overlaps with original ROC
• Example 2: Let’s find the inverse system of
z 1  0.5
1  0.9z 1
ROC: z  0.9 

Hz  
Hi z 
1
1  0.9z
1  0.5z 1
z  2 or z  2
• Again two possible ROCs
• This time both overlap with original ROC so both are valid
– Two valid inverses for this system
hi,1 n  22 u n  1  1.82
n
n 1
hi,2 n  22 un  1.82
n
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n 1
u n
un  1
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Infinite Impulse Response (IIR) Systems
• Rational system function
1  c z 

 b0
Hz    
 a0 
M
k
k 1
N
1
 1  d z 
k
1
k 1
• If at least one pole does not cancel with a zero
• There will at least one term of the form
anun or - anu n  1
• Therefore the impulse response will be infinite length
• Example: Causal system of the form yn  ayn  1  xn
1
ROC: z  a fromcausality
1
1  az
• The impulse response from inverse transform
hn  anun
Hz  
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Finite Impulse Response (FIR) Systems
• If transfer function does not have any poles except at z=0
– In this case N=0
M
Y z 
Hz  

Xz 
k
b
z
 k
k 0
N
k
a
z
 k

M
k
b
z
 k
k 0
k 0
• No partial fraction expansion possible (or needed)
• The impulse response can be seen to be
hn 
M
 b n  k 
k 0
k
• Impulse response is of finite length
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Example: FIR System
an 0  n  M
• Consider the following impulse response hn  
else
0
• The system function is
M1 M1

M
1
a
z
Hz   hnzn   anzn 
1  az1
n  
n 0
• Assuming a real and positive the zeros can be written as
zk  aej2k /M1 for k  0,1,...,M
• For k=0 we have a zero at z0=a
• The zero cancels the pole at z=a
• We can write this system as
yn 
M
k
a
 xn  k 
k 0
• Or equivalently from H(z) as
yn  ayn  1  xn  aM1xn  M  1
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