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Unformatted text preview: t he g ain factor of a normalized analog
elliptic lowpass filter of order n w ith a minimum passband gain G p dB, a nd maximum s topband g ain Gs dB. T he n ormalized passband edge is 1 r ad/s. (b)
(a) Gs J -.---,.,Y o o F ig. 7.27 Frequency transformation for highpass filters. C omputer E xample C 7.10
Design the lowpass elliptic filter for the specifications in Example 7.7 using functions
from Signal Processing Toolbox in MATLAB. T (s) = W p=10jWs=16.5jGp=-2jGs=-20j
[ num,den] = ellip(n,-Gp,-Gs, W p, ' s') (7.55) wp
MATLAB r eturns n = 3 and num= 0 2.7881 0 481.1626, d en= 1 7.261 106.9991
481.1626; t hat is,
2.78818 2 + 481.1626
( ) - 83 + 7.2618 2 + 106.99918 + 481.1626
To plot amplitude response, we can use the last three functions from Example C7.5. 7.7 0 Frequency Transformations E arlier we s aw how a lowpass filter transfer function of a rbitrary specifications
can b e o btained from a normalized lowpass filter using frequency scaling. Using
c ertain frequency transformations, we c an o btain t ransfer functions of highpass,
bandpass, and b andstop filters from a basic lowpass filter ( the p rototype filter)
design. For example, a highpass filter transfer function can be o btained from t he
p rototype l owpass filter transfer function by replacing s w ith w p / s . Similar transformations allow us t o design bandpass a nd b andstop filters from appropriate lowpass
p rototype filters.
T he p rototype filter may b e of any kind, such as Butterworth, Chebyshev,
elliptic, a nd so o n. We first design a suitable p rototype lowpass filter 'Hp (s). I n
t he n ext s tep, we replace 8 with a proper transformation T (s) t o o btain t he desired
highpass, b andpass, o r b andstop filter. 7.7-1 E xample 7 .8
Design a Chebyshev high pass filter with the amplitude response specifications illustrated in Fig. 7.28awithws = 100, w p = 165, G s = 0.1 ( -20dB), and Gp = 0 .794(-2dB).
S tep 1: D etermine t he p rototype l owpass filter
The prototype lowpass filter has wp = 1 and W. = 165/100 = 1.65. This means the
prototype filter in Fig. 7.27b has a passband 0 :::; w :::; 1 and a stopband w 2: 1.65, as
shown in Fig. 7.28b. Also, G p = 0.794 ( -2dB) and G . = 0.1 ( -20dB). We already
designed a Chebyshev filter with these specifications in Example 7.7. The transfer
function of this filter is [Eq. (7.54)J
OdB - 2 dB 0.794
0.7 ( b) (a) 0.5 0.3 Highpass Filters F igure 7.27a shows an amplitude response of a typical high pass filter. T he
a ppropriate l owpass p rototype response required for the design of a highpass filter
in Fig. 7.27a is d epicted in Fig. 7.27b. We m ust first determine this p rototype filter
transfer function 1 ip(s) w ith t he p assband 0 :::; W :::; 1 a nd the s topband W ~ wp/ws.
T he d esired t ransfer function of t he high pass filter t o satisfy specifications in Fig.
7.27a is t hen o btained by replacing s w ith T (s) in 'Hp(s), where - 20 dB 0.1 o 100 165 400 600 800 1000 o F ig. 7.28 Highpass Filter Design for Example 7.8. 1.65 5 26 7 f t (s...
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