数字信号处理5.pdf

FIR Filter: Finite Impulse
Response Filter
( 9 hours including final review )
TOPICS
1. Property of FIR Filter
2. FIR Filter Design Using Window Method
3. FIR Filter Design Using Kaiser Window
4. Frequency Sampling Method
5. Equiripple FIR Filter Design (Parks-McClellan)
6. Coefficients Quantization Effects
7. FIR Filter Application
Dr. Yibiao YU Infinite Impulse Response Filter 2005
1
FIR Filter Structure
Impulse Response System Function
ì ¹ 0 0 £ n £ N­1 N ­1
h(n) =í H( Z) = å h (n )Z ­ n
î = 0 others n = 0
x (n) Z ­1 Z ­1 Z ­1
h(0) h(1) h(2) h(N ­ 2) h(N ­ 1)
y(n )
Dr. Yibiao YU Infinite Impulse Response Filter 2005
Linear Phase Property
•Linear Phase
(1) H (e j w ) = H ( e j w ) e ­ j aw
q H (w) = ­aw
(2) Y (e j w ) = H (e j w ) X (e jw )
j w jw ­ j aw It is linear
= H ( e ) X ( e )e
If it equals to
j w ­ jaw
1, what is y(n) = Y 0 (e ) e
p
1 j w j wn Independent of
(3) y 0 (n ) = ò Y0 (e )e dw
2p ­ p filter phase
p
1 jw j w( n ­ a) The filter phase
(4) y(n) = ò Y0 (e )e dw
2p ­ p only cause a
= y 0 (n ­ a) time delay
Dr. Yibiao YU Infinite Impulse Response Filter 2005
2
Linear Phase Property
•Example
x (n ) = sin( ) + cos( )
­ j5w jp cos( 5 w)
jw ìe 0 £ w £ jw ìe 0 £ w £
H 1(e ) = í H 2 (e ) = í
î0 other î0 other
Linear phase Non-linear
filter phase filter
y(n) = sin((n ­ 5)) + cos((n ­ 5))
y (n ) = sin(0 .2 pn) ­ cos(0 .4 pn)
Dr. Yibiao YU Infinite Impulse Response Filter 2005
Linear Phase Property
Input signal
output signal
of H1(z)
output signal
of H2(z)
Dr. Yibiao YU Infinite Impulse Response Filter 2005
3
Linear Phase Property
•Condition of a linear phase FIR
Mid-point
h(n) = ±h(N ­1­ n)
symmetry
n If h ( n ) = h ( N ­ 1 ­ n )
N / 2 ­1 N ­1
Even N: H(e jw ) = å h (n)e ­ jw n + å h (n )e ­ jw n
n = 0 n = N / 2
N / 2 ­1 N ­1
= [ å 2h (n) cos( w (n ­ ))]e ­ jw ( N ­1) / 2
n = 0 2
( N