Digital Signal Processing Reference
In-Depth Information
This type IV filter with
N
=
7 has a linear phase
θ(ω)
=−
3
.
5
ω
+
π/
2anda
constant group delay
τ
=
3
.
5 samples.
The transfer function of the type IV linear phase filter in general is given by
e
−
j
[
(Nω
−
π)/
2]
⎨
⎩
ω
⎬
⎭
h
N
n
sin
n
(N
+
1
)/
2
+
1
1
2
H(e
−
jω
)
=
2
−
−
2
n
=
1
(5.21)
The frequency responses of the four types of FIR filters are summarized below:
e
−
j
[
(N/
2
)ω
]
⎨
⎩
⎬
⎭
h
N
2
h
N
n
cos
(nω)
N/
2
H(e
jω
)
=
+
2
2
−
n
=
1
for type I
e
−
j
[
(N/
2
)ω
]
⎨
⎩
ω
⎬
⎭
h
N
n
cos
n
(N
+
1
)/
2
+
1
2
1
2
H(e
−
jω
)
=
2
−
−
n
=
1
for type II
e
−
j
[
(Nω
−
π)/
2]
⎨
⎩
⎬
⎭
h
N
n
sin
(nω)
N/
2
H(e
−
jω
)
=
2
2
−
n
=
1
for type III
e
−
j
[
(Nω
−
π)/
2]
⎨
⎩
ω
⎬
⎭
h
N
n
sin
n
(N
+
1
)/
2
+
1
1
2
H(e
−
jω
)
=
2
−
−
2
n
=
1
for type IV
(5.22)
5.2.1 Properties of Linear Phase FIR Filters
The four types of FIR filters discussed above have shown us that FIR filters
with symmetric or antisymmetric coefficients provide linear phase (or equiva-
lently constant group delay); these coefficients are samples of the unit impulse
response. It has been shown above that an FIR filter with symmetric or anti-
symmetric coefficients has a linear phase and therefore a constant group delay.
The reverse statement, that an FIR filter with a constant group delay must have
symmetric or antisymmetric coefficients, has also been proved theoretically [4].
These properties are very useful in the design of FIR filters and their applica-
tions. To see some additional properties of these four types of filters, we have
evaluated the magnitude response of typical FIR filters with linear phase. They
are shown in Figure 5.2.
The following observations about these typical magnitude responses will be
useful in making proper choices in the early stage of their design, as will be
Search WWH ::
Custom Search