Transcript Document

Discrete-Time Filter Design by Windowing
Quote of the Day
In mathematics you don't understand things.
You just get used to them.
Johann von Neumann
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall
Inc.
Filter Design by Windowing
• Simplest way of designing FIR filters
• Method is all discrete-time no continuous-time involved
• Start with ideal frequency response


1
j
 jn
j
jn
Hd e   hd ne
hd n 
H
e
e
d
d

2  
n  
 
 
• Choose ideal frequency response as desired response
• Most ideal impulse responses are of infinite length
• The easiest way to obtain a causal FIR filter from ideal is
hd n 0  n  M
hn  
else
 0
• More generally
hn  hd nwn
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where
1 0  n  M
wn  
else
0
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Windowing in Frequency Domain
• Windowed frequency response
 

  

1
j
j    
H e j 
H
e
W
e
d
d

2  
• The windowed version is smeared version of desired response
• If w[n]=1 for all n, then W(ej) is pulse train with 2 period
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Properties of Windows
• Prefer windows that concentrate around DC in frequency
– Less smearing, closer approximation
• Prefer window that has minimal span in time
– Less coefficient in designed filter, computationally efficient
• So we want concentration in time and in frequency
– Contradictory requirements
• Example: Rectangular window
   e
We
j
M
n0
 jn
1  e jM1
 jM / 2 sinM  1 / 2


e
sin / 2
1  e j
• Demo
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Rectangular Window
• Narrowest main lob
– 4/(M+1)
– Sharpest transitions at
discontinuities in
frequency
• Large side lobs
– -13 dB
– Large oscillation
around discontinuities
• Simplest window
possible
1 0  n  M
wn  
else
0
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Bartlett (Triangular) Window
• Medium main lob
– 8/M
• Side lobs
– -25 dB
• Hamming window
performs better
• Simple equation
0  n  M/2
 2n / M

wn  2  2n / M M / 2  n  M

0
else

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Hanning Window
• Medium main lob
– 8/M
• Side lobs
– -31 dB
• Hamming window
performs better
• Same complexity as
Hamming
1 
 2n 
 1  cos
 0  n  M
wn  2 
 M 

0
else

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Hamming Window
• Medium main lob
– 8/M
• Good side lobs
– -41 dB
• Simpler than Blackman

 2n 
0.54  0.46 cos
 0nM
wn  
 M 

0
else

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Blackman Window
• Large main lob
– 12/M
• Very good side lobs
– -57 dB
• Complex equation

 2n 
 4n 
0.42  0.5 cos
  0.08 cos
 0nM
wn  
 M 
 M 

0
else

• Windows Demo
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Incorporation of Generalized Linear Phase
• Windows are designed with linear phase in mind
– Symmetric around M/2
wM  n 0  n  M
wn  
else
 0
• So their Fourier transform are of the form
 
 
 
W ej  We ej e jM / 2
where We ej is a realand even
• Will keep symmetry properties of the desired impulse response
• Assume symmetric desired response
 
 
Hd ej  He ej e jM / 2
• With symmetric window
 
A e e j

  

1
j
j   

H
e
W
e
d
e

2  
– Periodic convolution of real functions
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Linear-Phase Lowpass filter
• Desired frequency response
 jM / 2

  c
e
j
Hlp e  
c    

 0
 
• Corresponding impulse
response
sinc n  M / 2
hlp n 
n  M / 2
• Desired response is even
symmetric, use symmetric
window
hn 
sinc n  M / 2
wn
n  M / 2
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Kaiser Window Filter Design Method
• Parameterized equation
forming a set of windows
– Parameter to change mainlob width and side-lob area
trade-off
2
 
n

M
/
2


 I0  1  
 

 M/2  
wn   
 0nM

I0 

0
else

– I0(.) represents zeroth-order
modified Bessel function of
1st kind
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Determining Kaiser Window Parameters
• Given filter specifications Kaiser developed empirical equations
– Given the peak approximation error  or in dB as A=-20log10 
– and transition band width   s  p
• The shape parameter  should be
0.1102A  8.7
A  50


0.4
  0.5842A  21  0.07886A  21 21  A  50

0
A  21

• The filter order M is determined approximately by
M
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A8
2.285
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Example: Kaiser Window Design of a Lowpass Filter
• Specifications p  0.4, p  0.6, 1  0.01, 2  0.001
• Window design methods assume 1  2  0.001
• Determine cut-off frequency
– Due to the symmetry we can choose it to be c  0.5
• Compute
A  20log10   60
  s  p  0.2
• And Kaiser window parameters
  5.653
M  37
• Then the impulse response is given as
2


 n  18.5  

I0 5.653 1  


18
.
5

 
hn   sin0.5n  18.5 
 0nM

n  18.5
I0 5.653

0
else

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Example Cont’d
Approximation Error
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General Frequency Selective Filters
• A general multiband impulse response can be written as
hmb n 
sin k n  M / 2


G

G

k
k 1
n  M / 2
k 1
Nmb
• Window methods can be applied to multiband filters
• Example multiband frequency response
– Special cases of
• Bandpass
• Highpass
• Bandstop
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