UploadFile_6417

Frequency Sampling Design Techniques 345
system function. Then from (6.12) we have
and
with
H (z) =
M−1
∑
n=0
h (n) z −n = 1 − z−M
M
H(e jω )= 1 − e−jωM
M
M−1
∑
k=0
M−1
∑
k=0
H(k)
1 − z −1 e j2πk/M (7.33)
H(k)
1 − e −jω e j2πk/M (7.34)
{
(
H (k) =H e j2πk/M) H (0) , k =0
=
H ∗ (M − k) ,k=1,...,M − 1
Foralinear-phase FIR filter we have
h(n) =±h(M − 1 − n), n =0, 1,...,M − 1
where the positive sign is for the Type-1 and Type-2 linear-phase filters,
while the negative sign is for the Type-3 and Type-4 linear-phase filters.
Then H (k) isgiven by
where
( ) 2πk
H r =
M
and
⎧ ( )( M − 1 2πk
⎪⎨ −
2 M
̸ H (k) = ( M − 1
⎪⎩ +
2
H (k) =H r
( 2πk
M
)
e j̸ H(k)
⎧
⎨H r (0) , k =0
(
⎩H 2π(M−k)
r
M
)
, k =0,...,
) ⌊ 2π
M − 1
M (M − k) ,k= 2
or
⎧(
± π ) ( )( M − 1 2πk
−
2 2 M
⎪⎨ (
̸ H (k) = − ± π ) ( M − 1
+
2 2
⎪⎩
)
,k=0,...,
) 2π
(M − k) ,
M
)
,k=1,...,M − 1
⌊ ⌋ M − 1
⌊ M − 1
k =
2
2
⌋
+1,...,M − 1
⌊ ⌋ M − 1
2
⌋
+1,...,M − 1
, (Type-1 & 2)
(7.35)
(7.36)
(7.37)
, (Type-3 & 4)
(7.38)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.