Digital Signal Processing Reference
In-Depth Information
transition band unconstrained (i.e. it does not affect the minimization of
the error).
The next step is to use (7.18) to reformulate the above expression as a
polynomial optimization problem. To do so, we replace the frequency re-
sponse
H
d
l
g
r
,
y
i
d
.
,
©
,
L
s
(
e
j
ω
)
with its polynomial equivalent and set
x
=
cos
ω
; the pass-
band interval
[
0,
ω
]
and the stopband interval
[
ω
s
,
π
]
are mapped into the
p
intervals for
x
:
I
p
=[
cos
ω
p
,1
]
I
s
=[
−
1, cos
ω
]
s
respectively; similarly, the desired response becomes:
1
ω
∈
I
p
D
(
x
)=
(7.21)
0
ω
∈
I
s
and the weighting function becomes:
1
ω
∈
I
p
W
(
x
)=
(7.22)
δ
/δ
ω
∈
I
s
p
s
The new set of specifications are shown in Figure 7.12. Within this polyno-
mial formulation, the optimization problem becomes:
I
s
W
(
x
)
P
(
x
)
−D
(
x
)
=
max
E
(
x
)
≤
δ
p
max
(7.23)
x
∈
I
p
∪
where
P
(
x
)
is the polynomial representation of the FIR frequency response
as in (7.18).
1
I
p
0
I
s
−
1
cos
(
0.6
π
)
cos
(
0.4
π
)
1
Figure 7.12
Filter specifications as in Figure 7.1 formulated here in terms of poly-
nomial approximation, i.e. for
x
=
cos
ω
,
ω
∈
[
0,
π
]
.
