Lecture22.pdf

Least Integral-Squared Error Design of FIR Filters

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• Let H d (e j ) denote the desired frequency response • Since H d (e j ) is a periodic function of  with a period 2, it can be expressed as a Fourier series  j H d (e )   hd [n]e jn n   where 1  j jn hd [n]  H ( e )e d,    n    d 2  

Copyright © 2005, S. K. Mitra

Least Integral-Squared Error Design of FIR Filters • In general, H d (e j ) is piecewise constant with sharp transitions between bands • In which case, hd [n] is of infinite length and noncausal • Objective - Find a finite-duration ht [n] of length 2M+1 whose DTFT H t (e j ) j H ( e ) in approximates the desired DTFT d some sense 2 Copyright © 2005, S. K. Mitra

Least Integral-Squared Error Design of FIR Filters • Commonly used approximation criterion Minimize the integral-squared error 1  j j 2   H t (e )  H d (e ) d 2   where j

M

H t (e )   ht [n]e

 jn

n M

3 Copyright © 2005, S. K. Mitra

Least Integral-Squared Error Design of FIR Filters • Using Parseval’s relation we can write 

2

n   M

2

   ht [n]  hd [n]

 M 1

  ht [n]  hd [n]   n M

4

n 

hd2 [n] 



2 h  d [ n]

n  M 1

• It follows from the above that  is minimum when ht [n]  hd [n] for  M  n  M •  Best finite-length approximation to ideal infinite-length impulse response in the mean-square sense is obtained by truncation Copyright © 2005, S. K. Mitra

Least Integral-Squared Error Design of FIR Filters • A causal FIR filter with an impulse response h[n] can be derived from ht [n] by delaying: h[n]  ht [n  M ] • The causal FIR filter h[n] has the same magnitude response as ht [n] and its phase response has a linear phase shift of M radians with respect to that of ht [n] 5 Copyright © 2005, S. K. Mitra

Impulse Responses of Ideal Filters • Ideal lowpass filter -

sin c n hLP [n]   n ,    n  

• Ideal highpass filter -

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 1  c , n  0   hHP [n]   sin( c n)  n , n  0  Copyright © 2005, S. K. Mitra

Impulse Responses of Ideal Filters • Ideal bandpass filter -

sin( c 2n)  sin( c1n) , n  0 n  n hBP [n]   c 2 c1   , n  0    7 Copyright © 2005, S. K. Mitra

Impulse Responses of Ideal Filters • Ideal bandstop filter -

 1  (c 2  c1 ) , n  0   hBS [n]   sin( c1n) sin( c 2n)  n  , n0  n  8 Copyright © 2005, S. K. Mitra

Gibbs Phenomenon • Gibbs phenomenon - Oscillatory behavior in the magnitude responses of causal FIR filters obtained by truncating the impulse response coefficients of ideal filters Magnitude

1.5 1 0.5 0

9

N = 20 N = 60

0

0.2

0.4

0.6 /

0.8

1 Copyright © 2005, S. K. Mitra

Gibbs Phenomenon

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• As can be seen, as the length of the lowpass filter is increased, the number of ripples in both passband and stopband increases, with a corresponding decrease in the ripple widths • Height of the largest ripples remain the same independent of length • Similar oscillatory behavior observed in the magnitude responses of the truncated versions of other types of ideal filters Copyright © 2005, S. K. Mitra

Gibbs Phenomenon • Gibbs phenomenon can be explained by treating the truncation operation as an windowing operation: ht [n]  hd [n]  w[n] • In the frequency domain

H t ( e j ) 

11

1 2



j j ( ) H ( e )  ( e ) d  d



• where H t (e j ) and  (e j ) are the DTFTs of ht [n] and w[n] , respectively Copyright © 2005, S. K. Mitra

Gibbs Phenomenon j H ( e ) is obtained by a periodic • Thus t continuous convolution of H d (e j ) with  ( e j )

12 Copyright © 2005, S. K. Mitra

Gibbs Phenomenon • If  (e j ) is a very narrow pulse centered at   0 (ideally a delta function) compared to variations in H d (e j ), then H t (e j ) will approximate H d (e j ) very closely • Length 2M+1 of w[n] should be very large • On the other hand, length 2M+1 of ht [n] should be as small as possible to reduce computational complexity 13 Copyright © 2005, S. K. Mitra

Gibbs Phenomenon • A rectangular window is used to achieve simple truncation: 1, 0  n  M wR [n]   0, otherwise j • Presence of oscillatory behavior in H t (e ) is basically due to: – 1) hd [n] is infinitely long and not absolutely

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summable, and hence filter is unstable – 2) Rectangular window has an abrupt transition to zero Copyright © 2005, S. K. Mitra

Gibbs Phenomenon • Oscillatory behavior can be explained by j examining the DTFT R (e ) of wR [n] : 30

Rectangular window

Amplitude

20 10

15

side lobe

M=4

0 -10 -1

j

main lobe

M = 10

-0.5

0 /

0.5

1

• R (e ) has a main lobe centered at   0 • Other ripples are called sidelobes Copyright © 2005, S. K. Mitra

Gibbs Phenomenon

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• Main lobe of R (e j ) characterized by its width 4 /(2M  1) defined by first zero crossings on both sides of   0 • As M increases, width of main lobe decreases as desired • Area under each lobe remains constant while width of each lobe decreases with an increase in M j • Ripples in H t (e ) around the point of discontinuity occur more closely but with no decrease in amplitude as M increases Copyright © 2005, S. K. Mitra

Gibbs Phenomenon • Rectangular window has an abrupt transition to zero outside the range  M  n  M , which results in Gibbs phenomenon in H t (e j ) • Gibbs phenomenon can be reduced either: (1) Using a window that tapers smoothly to zero at each end, or (2) Providing a smooth transition from passband to stopband in the magnitude specifications 17 Copyright © 2005, S. K. Mitra

Fixed Window Functions

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• Using a tapered window causes the height of the sidelobes to diminish, with a corresponding increase in the main lobe width resulting in a wider transition at the discontinuity • Hann: w[n]  0.5  0.5 cos( 2 n ),  M  n  M 2M  1 • Hamming: 2  n  M  n  M w[n]  0.54  0.46 cos( ), 2M  1 • Blackman: 2  n  n 4 w[n]  0.42  0.5 cos( )  0.08 cos( ) 2M  1 2M  1 Copyright © 2005, S. K. Mitra

Fixed Window Functions • Plots of magnitudes of the DTFTs of these windows for M = 25 are shown below: Hanning window Hann window

0

0

-20

-20

Gain, dBdB Gain,

Gain,dBdB Gain,

Rectangular window Rectangular window

-40 -60

-60 -80

-80 -100

-40

0

0.2

0.4

0.6

0.8

-100

1

0

0.2

0

-20

-20

Gain,dBdB Gain,

Gain, dBdB Gain,

0

-60 -80

19

-100

0.6

0.8

1

0.8

1

/ Blackman Blackman window window

/ Hamming Hamming window window

-40

0.4

-40 -60 -80

0

0.2

0.4

0.6 /

0.8

1

-100

0

0.2

0.4

0.6

/ Copyright © 2005, S. K. Mitra

Fixed Window Functions • Magnitude spectrum of each window characterized by a main lobe centered at  = 0 followed by a series of sidelobes with decreasing amplitudes • Parameters predicting the performance of a window in filter design are: • Main lobe width • Relative sidelobe level 20 Copyright © 2005, S. K. Mitra

Fixed Window Functions • Main lobe width  ML - given by the distance between zero crossings on both sides of main lobe • Relative sidelobe level As - given by the difference in dB between amplitudes of largest sidelobe and main lobe

21 Copyright © 2005, S. K. Mitra

Fixed Window Functions

• Observe H t (e j (c  ) )  H t (e j (c ) )  1 j c H t (e )  0.5 • Thus, • Passband and stopband ripples are the same 22 Copyright © 2005, S. K. Mitra

Fixed Window Functions • Distance between the locations of the maximum passband deviation and minimum stopband value   ML

• Width of transition band    s   p   ML 23 Copyright © 2005, S. K. Mitra

Fixed Window Functions • To ensure a fast transition from passband to stopband, window should have a very small main lobe width • To reduce the passband and stopband ripple d, the area under the sidelobes should be very small • Unfortunately, these two requirements are contradictory 24 Copyright © 2005, S. K. Mitra

Fixed Window Functions • In the case of rectangular, Hann, Hamming, and Blackman windows, the value of ripple does not depend on filter length or cutoff frequency c , and is essentially constant • In addition, c   M where c is a constant for most practical purposes 25 Copyright © 2005, S. K. Mitra

Fixed Window Functions • Rectangular window -  ML  4 /(2M  1) As  13.3 dB,  s  20.9 dB,   0.92 / M • Hann window -  ML  8 /(2M  1) As  31.5 dB,  s  43.9 dB,   3.11 / M • Hamming window -  ML  8 /(2M  1) As  42.7 dB,  s  54.5 dB,   3.32 / M • Blackman window -  ML  12 /(2M  1) As  58.1 dB,  s  75.3 dB,   5.56 / M 26 Copyright © 2005, S. K. Mitra

FIR Filter Design Example • Lowpass filter of length 51 and c   / 2 Lowpass Filter Designed Using Hamming window 0

Gain, dB

Gain, dB

Lowpass Filter Designed Using Hann window 0

-50

-50

-100

-100 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

1

/

/

Gain, dB

Lowpass Filter Designed Using Blackman window 0

-50

-100

27

0

0.2

0.4

0.6 /

0.8

1

Copyright © 2005, S. K. Mitra

FIR Filter Design Example • An increase in the main lobe width is associated with an increase in the width of the transition band • A decrease in the sidelobe amplitude results in an increase in the stopband attenuation

28 Copyright © 2005, S. K. Mitra