Digital Signal Processing Reference
In-Depth Information
of the filter equal to the number of coefficients in (5.6) is
N
+
1. If we are
given
H(z
−
1
)
+
0
.
5
z
−
6
, its order is 6, although only three
terms are present and the correct number of coefficients equal to the length of
the filter is 7, because
h(
0
)
=
0
.
3
z
−
4
+
0
.
1
z
−
5
0. It becomes necessary to
point out the notation used in this chapter, because in some textbooks, we may
find
H(z
−
1
)
=
h(
1
)
=
h(
2
)
=
h(
3
)
=
=
N
−
1
0
h(n)z
−
n
representing the transfer function of an FIR filter,
in which case the length of the filter is denoted by
N
and the degree or order
of the polynomial is
(N
n
=
−
1
)
. (Therefore students have to be careful in using
the formulas found in a chapter on FIR filters, in different topics; but with some
caution, they can replace
N
that appears in this chapter by
(N
−
1
)
so that the
formulas match those found in these topics.)
The notation often used in MATLAB,
is
H(z
−
1
)
h(
2
)z
−
1
=
h(
1
)
+
+
h(
3
)z
−
2
+
1
)z
−
N
, which is a polynomial of degree
N
, and has
+···+
h(N
(N
+
1
)
coefficients. In more compact form, it is given by
N
H(z
−
1
)
+
1
)z
−
n
=
h(n
(5.7)
n
=
0
The notation and meaning of angular frequency used in the literature on discrete-
time systems and digital signal processing also have to be clearly understood by
the students. One is familiar with a sinusoidal signal
x(t)
A
sin
(
w
t)
in which
w
=
2
πf
is the angular frequency in radians per second,
f
is the frequency in
hertz, and its reciprocal is the period
T
p
in seconds. So we have
w
=
2
π/T
p
radians per second. Now if we sample this signal with a uniform sampling
period, we need to differentiate the period
T
p
from the sampling period denoted
by
T
s
. Therefore, the sampled sequence is given by
x(nT
s
)
=
=
A
sin
(
w
nT
s
)
=
A
sin
(
2
πnT
s
/T
p
)
A
sin
(
w
/f
s
)
. The frequency
w
(in radians
per second) normalized by
f
s
is almost always denoted by
ω
andiscalledthe
normalized frequency (measured in radians). The frequency
w
is the
analog fre-
quency
variable, and the frequency
ω
is the
normalized digital frequency
.On
this basis, the sampling frequency
ω
s
=
2
π
radians. Sometimes,
w
is normalized
by
πf
s
or 2
πf
s
so that the corresponding sampling frequency becomes 2 or 1
radian(s). Note that almost always, the sampling period is denoted simply by
T
in the literature on digital signal processing when there is no ambiguity and the
normalized frequency is denoted by
ω
=
A
sin
(
2
πf/f
s
)
=
=
w
T
. The difference between the angular
frequency in radians per second and the normalized frequency usually used in
DSP literature has been pointed out in several instances in this topic.
5.2 LINEARPHASEFIRFILTERS
Now we consider the special types of FIR filters in which the coefficients
h(n)
of the transfer function
H(z
−
1
)
=
n
=
0
h(n)z
−
n
are assumed to be symmetric
or antisymmetric. Since the order of the polynomial in each of these two types
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