Digital Signal Processing Reference
In-Depth Information
q
=
q
0
+ 2
q
0
+ 15
q
0
+ 150
q
1
0
2
1
−
2
q
0
1
+
2
q
0
k
p
=
q
1
−
k
p
k
=
In order to satisfy the passband edge specification, the digital passband edge
ω
p
=
θ
for Case I
filters. The digital stopband edge
ω
a
is then determined using the analog ratio
k
. (Here, frequency
warping from digital to analog domain, and vice versa, given by Eqn. (
18
) needs to be taken into
account.) Similarly,
ω
a
=
φ
for Case II filters, and
ω
p
can be determined by using ratio
k
. Also,
using given ripple specifications along with the boundary frequencies described in Table
1
, one can
determine the transfer function of the FIR masking filters
F
0
(
e
jω
)
and
F
1
(
e
jω
)
.
2.
Generation of seed FRM digital filter particle
: The seed FRM digital filter particle is formed as
follows:
• A particle with
B
1
coordinates is formed in which each coordinate serves as an index of the
corresponding CSD LUT for each multiplier coefficient constituent in the interpolation digital
subfilters. For FRM IIR digital filters, the multiplier coefficients correspond to the bilinear-LDI
allpass digital networks
G
0
(
z
)
and
G
1
(
z
)
.
• A particle with
B
2
coordinates is formed in which each coordinate serves as an index of the
corresponding CSD LUT for each multiplier coefficient in the FIR masking digital subfilters
F
0
(
z
)
and
F
1
(
z
)
.
3.
Generation of Initial Swarm
: An initial swarm of
K
particles is formed by generating a random
cloud around the seed particle as discussed in section
8.1
.
4.
Fitness Evaluation
: The fitness function for CSD FRM IIR digital filters is defined in accordance
with
f itness
magnitude
= −
20
log
[
max
(
ε
p
,
ε
a
)]
(62)
f itness
group
−
delay
=
ς
p
(63)
f itness
=
f intess
magnitude
−
f itness
group
−
del ay
(64)
where
[
W
p
|
H
(
e
jω
) −
1
|]
ε
p
=
max
(65)
|{z}
ω
∈∆
ω
p
[
W
a
|
H
(
e
jω
)|]
ε
a
=
max
(66)
|{z}
ω
∈∆
ω
a
ς
p
=
max
[
W
gd
|
τ
(
ω
) −
µ
τ
|]
(67)
|{z}
ω
∈∆
ω
p
