DSP Lecture 25

Lecture 25 EE-802 ADSP SEECS-NUST

EE 802-Advanced Digital Signal Processing

Dr. Amir A. KhanOffice : A-218, SEECS

9085-2162; [email protected]


Lecture Outline

• FIR Filter Design– Kaiser Window– Optimal Equiripple Design Technique


Kaiser Window FIR Design

• Main drawback of the conventional windows• single parameter M to control in the design procedure • trial and error method to exactly fit in the approximations

• Kaiser window design presents a more systematic approach

• Kaiser window

• Two controlling parameters give more flexibility :

M and

2 1/ 20

0

[ (1 [( ) / ] ) ] 0

[ ] ( )

0 otherwise

I nn M

w n I

/ 2M

0( )I where

is the zeroth-order modified Bessel function of the first kind


Kaiser Window PropertiesM fixed, as increases we have a narrower window for same M

M fixed, as increases, (1) main lobe becomes wider; (2) peak side lobe attenuation reduced

fixed, as increases, (1) main lobe becomes narrower; (2) Peak side lobe unaffected


Kaiser Window Design Procedure• Kaiser empirically derived some very useful design results

• Given the passband edge of a LPF with requirement

• Given the stopband cut-off frequency with requirement

• Compute :

• Find :

p1 ( ) 1jH e

s

( )jH e

1020logA ps

0.4

0.1102( 8.7) >50

0.5842 0.07886( 21) 21 50( 21)

0 21

A A

A AA

A

285.2

8AM


Kaiser Window Design Summary• For an ideal LP filter, the discontinuity in its frequency response is smeared as the

main lobe of the FT of the window slides across the discontinuity during convolution

• Width of resulting transition band depends on width of the main lobe

• Passband and stopband ripples depend on side lobes of the FT of the window

• The maximum passband and stopband deviations do not depend on M

• Can be changed only by changing the shape of the window used


Low-Pass Filter Design Ex.

2 1/20

0

sin ( ) [ (1 [( ) / ] ) ], 0

[ ] ( ) ( )

0, otherwise

c n I nn M

h n n I

－＝－

Approximation error (Desired – Designed)

Frequency responseof designed filter

1 20.4 , 0.6 , 0.01, 0.001p s

1 2min( , ) 0.001

1020log 60A

From equations ＝ 5.653 and M ＝ 37


High-Pass Filter Design Ex.

hp /2

0,

,

cj

j Mc

H ee

/ 2hp lp

j j M jH e e H e

hp

sin / 2 sin / 2,

/ 2 / 2cn M n M

h n nn M n M

2( ) jsH e

1 11 ( ) 1 jpH e

1 20.35 , 0.5 , 0.02s p

33.98A 2.6 and 24M

(0.35 0.5 ) / 2c ＋


High-Pass Filter Design Ex.

Approximation error

Frequency responseof designed filter with Type II-not correct


Multiband Filter

Ideal frequency response for multiband filter

mb

mb 11

sin / 2

/ 2

Nk

k kk

n Mh n G G

n M


Window Design Issues

• Windowing technique is simple and straight forward but

• Distortions in the passband and stopband are equal and cannot be specified simultaneously– there may be more stringent requirements in the stopband

• Approximation error (b/w desired and designed freq.

response) is not uniformly distributed– distortion is large near the discontinuities in the frequency

response

• Overall result is a relatively higher order filter

• Distributing the error uniformly results in a lower order

filter meeting the desired specifications

DSP Lecture 25

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