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which, when rounded up, gives N = 6. For the 3-dB cutoff frequency of the Butterworth filter, we will select 52, so that
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that is, so that the Butterworth filter satisfies the passband specifications exactly (this will provide for some allowance for aliasing in the stopband). With
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Therefore, the magnitude of the frequency response squared is
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and the 12 poles of Ha(s)Ha(-s) = lie on a circle of radius R, = 0.7090, at angles
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FILTER DESIGN as illustrated in the following figure:
[CHAP. 9
Thus, the poles of the Butterworth filter are the three complex conjugate pole pairs of H,(s)H,(-s) left-half s-plane:
that are in the
Therefore, with
forming second-order polynomials from each conjugate pole pair, we have
The next steps, which are algebraically very tedious, are lo perform a partial fraction expansion of H,,(s),
perform the transformation
and then recombine the terms. The result is
H (z) =
0.0105z~-' 0.0168z-~ 0 . 0 0 4 2 ~ -+ 0.0001z-~ ~ 0.0007~ 3.3431~-I 5 . 0 1 5 0 ~ - ~ 4.21532-2 + 2 . 0 7 0 3 ~ -- 0.5593zr5 O.0646zr6 ~
The magnitude of the frequency response in decibels is plotted in the following figure.
CHAP. 91
FILTER DESIGN
at and , As a final check on the design, evaluating H ( e J U ) w = 0 . 2 ~ w = 0 . 3 ~we find that
Therefore, the filter exceeds the passband specifications and comes close to meeting the stopband specifications. If this filter is unacceptable, the design could be modified by decreasing Q, to improve the stopband performance.
Repeat Prob. 9.39 using the bilinear transformation.
Using the bilinear transformation to design a Butterworth filter according to the specifications given in Prob. 9.39, we first use the transformation
to map the passband and stopband frequencies of the digital filter to the cutoff frequencies of the analog filter. With Ts = 2, we have
= tan(0. I n ) == 0.3249
R, = tan 2 = t a n ( 0 . 1 5 ~L- 0.5095 )
As we found in Prob. 9.39, the required filter order IS N = 6. For the 3-dB cutoff frequency of the analog Butterworth filter, we may choose any frequency in the range
0.3667 5 R,. 5 0.39 10
If we select Q,. =0.3667, the passband specifications will be met exactly, and the stopband specifications will be exceeded. Conversely. if we set R, = 0.3910. the stopband specifications will be met exactly, and the passband specifications exceeded. Picking a frequency between the two extremes will produce an improvement in both bands. Because the stopband deviation is twice that of the deviation in the passband. we will set 0,.= 0.3667 in order to improve the stopband performance. From Table 9-4, we find the coel'ticients in the system function of a sixth-order normalized (52, = I) Butterworth filter to be
To obtain a Butterworth filter with a cutofl' frequency R, = 0.3666, we perform the low-pass-to-low-pass transformation r
FILTER DESIGN This gives Hob) =
[CHAP. 9
+ 1.4 165s5+ 1 .0033sJ + 0.4505s3 + 0. 1349s2+ 0.0256s + 0.0024
(0.3666)'
Finally, we apply the bilinear transformation I - z-' I 2-'
which yields the digital filter H (z) = 0 . 0 0 9 0 ~ - ~ 0.0120~--' 0 . 0 0 9 0 ~ - ~ 0 . 0 0 3 6 ~ - ~ 0.0006~-' 0.0006 0.0036z-' I - 3 . 2 9 4 2 ~ ' 4.8985zr2 - 4.0857zr3 1 . 9 9 3 2 ~ - ~0 . 5 3 5 3 ~ -+ 0 . 0 6 1 5 ~ - ~ ~
At the passband cutoff frequency, o, = 0.217, the magnitude of the frequency response is
and at the stopband cutoff frequency, o,= 0.317, the magnitude of the frequency response is
Therefore, this filter meets the given specifications.
Use the bilinear transformation to design a first-order low-pass Butterworth filter that has a 3-dB cutoff frequency w,. = 0 . 2 ~ .
If a digital low-pass filter is to have a 3-dB cutoff frequency at w,. = 0.217, the analog Butterworth filter should have a 3-dB cutoff frequency
Q. = tan($)
= tan(0. I n ) = 0.3249
For a first-order Butterworth filter.
Therefore, the system function is H0(s) = s Q,.
With Q, = 0.3249, applying the bilinear transformation
we have
H (z) =
0.3249 0.3249(1 + Z-I) 0.2452(1 + z-I) 1 - z-' 1 - 0.5095~-I (1 - 2-I) O.3249(1 z-I) 0.3249 1 z-'
A second-order continuous-time filter has a system function
H,(s) = -+ s-a s-b
where a < 0 and b
0 are real.
( a ) Determine the locations of the poles and zeros of H ( z ) if the filter is designed using the bilinear transformation with T, = 2.
(b) Repeat part (a)for the impulse invariance technique, again with T, = 2.
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