Digital Signal Processing Reference
In-Depth Information
inductors (I), and capacitors (C). That is, for the design considered here, the filters
are built up as passive RLC networks. Design using active components (op-amp's)
is also possible, but will not be considered in this subsection.
Based on the required filter specifications (e.g., cutoff sharpness and stop-band
attenuation) the goal is to can find H(s) and a suitable RLC combination. The first
step in the design process is to determine the filter order N from standard tables.
Since Tables are usually only available for filters with a cutoff frequency of
x
c
= 1 rad/s, the filter specifications must first be transformed using frequency
denormalization. This is achieved by dividing all the critical frequency points (cut-
off frequency, stop edge frequency, etc.) by x
c
. This process is illustrated in the
example which follows.
1.5.4.1 Example of a Low-pass Filter Design
Example 1 Design a LPF with G
m
= 1, af
c
= 50 Hz, and stop-band gain B-
40 dB w.r.t G
m
for frequencies C140 Hz. Find the minimum order n required, if
the ripple B 1 dB and R
L
¼
100X.
Solution: x
c
= 2p(50) = 100 p, x
1
= (2p)140 = 280p, hence x
1N
= x
1
/
x
c
= 2.8. Initially, a butterworth design will be trailed. Use the Gain vs. Nor-
malized frequency plots to determine the minimum order to achieve -40 dB for
x
N
C 2.8 (see Tables and the left hand side of Fig.
1.34
). It is seen that n = 4s
not adequate to meet the 40 dB attenuation at x
N
C 2.8 but n = 5 is. If a but-
terworth filter is used, therefore, one would need a fifth order filter to meet
specifications.
Stopband gain vs. norm. freq. for B−LPF
Stopband gain vs. norm. freq. for C−LPF (
r
= 0.5 dB)
0
0
n = 1
n = 1
−10
−10
n = 2
−20
−20
n = 2
−30
−30
n = 3
n = 3
−40
−40
o
o
−50
−50
n = 4
−60
−60
1
2
3
4
5
6 7 8 9
10
1
2
3
4
5
6 7 8 9
10
ω
/
ω
c
ω
/
ω
c
Normalized frequency,
Normalized frequency,
Fig. 1.34
Gain plots versus normalized LPF's for different filter orders
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