Digital Signal Processing Reference
In-Depth Information
H
j
()
H(
j
ω
)
ω
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.5
1
1.5
0
0.5
1
1.5
ωω
ωω
ωω
0
0
0
H(
j
ω
)
H(
j
ω
)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.5
1
1.5
0
0.5
1
1.5
ωω
ωω
ωω
0
0
0
Fig. 5.1
Typical characteristics of lowpass, highpass, bandpass and band-rejection filters
• filters unknown in analog technique having finite impulse response called
shortly FIR filters (sometimes also called non-recursive filters)—described in
Chap. 6
.
Typical algorithm of a digital IIR filter is given in the form [
1
,
3
,
5
]:
y
ð
n
Þ¼
X
N
1
a
ð
k
Þ
x
ð
n
k
Þ
X
M
b
ð
k
Þ
y
ð
n
k
Þ
ð
5
:
1
Þ
k
¼
0
k
¼
1
where y(n) is an output sample at discrete time instant n; x(n) is an input sample at
instant n, a(k); b(k) are filter coefficients.
Such a filter produces output as weighted sum of N recent input samples and
M preceding outputs.
Digital filters can be described with use of a transfer function, which is
determined using Z transform to both sides of Eq.
5.1
. This is easy if one
remembers that transfer functions are determined for zero initial conditions and
signal delay by one step (sample) is equivalent to multiplication by z
-1
. Conse-
quently, the transfer function of IIR filter is given by following equation:
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