Part A | 9.2
208 Part A Fundamentals
0 1 2 3 4 5 6 7 8 9 10
n
a) h (n)
0 1 2 3 4 5 6 7 8 9 1011
n
b) h (n)
0 1 2 3 4 5
6 7 8 9 10
n
c) h (n)
0 1 2 3 4 5
6 7 8 9 1011
n
d) h (n)
Fig. 9.13a–d Example of impulse
responses of linear-phase FIR digital
filters: (a) type I; (b) type II; (c) type
III; (d) type IV
Linear-Phase Form
An important subclass of FIR digital filters is the one
that includes linear-phase filters. Such filters are characterized by a constant group delay ; therefore, they must
present a frequency response of the following form
H.e
i!
/ D B.!/ exp.i!! C i'/ :
(9.58)
In the above B.!/ is real and and ' are constant.
We now proceed to show that linear-phase FIR filters present impulse responses of very particular forms.
Specifically, if h.n/ is to be causal and of finite duration,
for 0 Ä n Ä M, we must necessarily have that
D
M
2
:
(9.59)
Therefore, one obtains that
h.n/ D e
2i' h
.M n/ :
(9.60)
This is the general equation that the coefficients of
a linear-phase FIR filter must satisfy. In the common
case, where all the filter coefficients are real, one finally
obtains that the filter impulse response must be either
symmetric or antisymmetric. In effect, the frequency response of linear-phase FIR filters with real coefficients
becomes as follows
H.e
i!
/ D B.!/ exp
Â
i!
Â
M
2
Ã
C i
Â
k
2
ÃÃ
: (9.61)
As a result, solely four distinct cases need be considered
that are described by the equations above. Their types
have been standardized and are referred to in literature
as follows [9.2]:
Type I: k D 0 and M even
Type II: k D 0 and M odd
Type III: k D 1 and M even
Type IV: k D 1 and M odd.
Typical impulse responses of the four cases of
linear-phase FIR digital filters are depicted in Fig. 9.13.
9.2.2 Important IIR Filter Structures
Direct Form
Recursive filters have transfer functions of the following form
H.z/ D
N.z/
D.z/
D
P M
iD0 b i z
i
1 C
P N
iD1 a i z i
:
(9.62)
Since, in most cases, such transfer functions give rise to
filters with impulse responses having infinite durations,
recursive filters are also referred to as infinite-duration
impulse response (IIR) filters.
We can consider that H.z/ as above results from the
cascading of two separate filters of transfer functions
N.z/ and 1=D.z/. The N.z/ polynomial can be realized
with the FIR direct form, as shown in the previous section. The realization of 1=D.z/ can be performed as
depicted in Fig. 9.14, where the FIR filter shown will
208 Part A Fundamentals
0 1 2 3 4 5 6 7 8 9 10
n
a) h (n)
0 1 2 3 4 5 6 7 8 9 1011
n
b) h (n)
0 1 2 3 4 5
6 7 8 9 10
n
c) h (n)
0 1 2 3 4 5
6 7 8 9 1011
n
d) h (n)
Fig. 9.13a–d Example of impulse
responses of linear-phase FIR digital
filters: (a) type I; (b) type II; (c) type
III; (d) type IV
Linear-Phase Form
An important subclass of FIR digital filters is the one
that includes linear-phase filters. Such filters are characterized by a constant group delay ; therefore, they must
present a frequency response of the following form
H.e
i!
/ D B.!/ exp.i!! C i'/ :
(9.58)
In the above B.!/ is real and and ' are constant.
We now proceed to show that linear-phase FIR filters present impulse responses of very particular forms.
Specifically, if h.n/ is to be causal and of finite duration,
for 0 Ä n Ä M, we must necessarily have that
D
M
2
:
(9.59)
Therefore, one obtains that
h.n/ D e
2i' h
.M n/ :
(9.60)
This is the general equation that the coefficients of
a linear-phase FIR filter must satisfy. In the common
case, where all the filter coefficients are real, one finally
obtains that the filter impulse response must be either
symmetric or antisymmetric. In effect, the frequency response of linear-phase FIR filters with real coefficients
becomes as follows
H.e
i!
/ D B.!/ exp
Â
i!
Â
M
2
Ã
C i
Â
k
2
ÃÃ
: (9.61)
As a result, solely four distinct cases need be considered
that are described by the equations above. Their types
have been standardized and are referred to in literature
as follows [9.2]:
Type I: k D 0 and M even
Type II: k D 0 and M odd
Type III: k D 1 and M even
Type IV: k D 1 and M odd.
Typical impulse responses of the four cases of
linear-phase FIR digital filters are depicted in Fig. 9.13.
9.2.2 Important IIR Filter Structures
Direct Form
Recursive filters have transfer functions of the following form
H.z/ D
N.z/
D.z/
D
P M
iD0 b i z
i
1 C
P N
iD1 a i z i
:
(9.62)
Since, in most cases, such transfer functions give rise to
filters with impulse responses having infinite durations,
recursive filters are also referred to as infinite-duration
impulse response (IIR) filters.
We can consider that H.z/ as above results from the
cascading of two separate filters of transfer functions
N.z/ and 1=D.z/. The N.z/ polynomial can be realized
with the FIR direct form, as shown in the previous section. The realization of 1=D.z/ can be performed as
depicted in Fig. 9.14, where the FIR filter shown will