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Use the impulse invariance method to design a digital filter from an analog prototype that has a system function
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To design a filter using the impulse invariance technique, we first expand H,,(s) in a partial fraction expansion as follows,
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+ (a - jb)
where
and Therefore, with
Hu(s)=
s+a+,jh+s+a-
using the mapping given in Eq. (9.10), we have
cos(bT,). Thus, the location of the zero in the discreteNote that the zero at s = -a is mapped to a zero at z = PT7 time filter depends on the position of the poles as well as the zero in the analog filter.
With the impulse invariance method, the unit sample response of a digital filter is formed by sampling the impulse response of the continuous-time filter,
Another approach is to use the step invariance method in which the step response of the digital filter is formed by sampling the step response of the continuous-time filter.
( a ) Design a digital filter with the step invariance method using the continuous-time prototype
(6) Determine whether or not the filter is the same as that which would be designed using the impulse
invariance method.
( a ) If the impulse response of a continuous-time filter is h,(r), its step response is
Therefore, because the Laplace transform of the step response is related to the system function H,(s) as follows,
then
CHAP. 91
FILTER DESIGN
To design a digital filter using step invariance, we first perform a partial fraction expansion of S,(s),
where
Therefore. Sampling s,,(t),
a I S,, (s) = - a2
-a + j h -a jh + b2 s + 2(a2+ h2)(s+ u + j h ) + 2(a2 + b2)(s+ a
s ( n ) = s,(nT,)
and finding the z-transform of s(n) corresponds to the substitution
Thus, the z-transform of the step response is
The system function of the digital filter is then
( b ) Using impulse invariance, we see from Prob. 9.36 that the system function is
Note that although H ( z ) has the same poles as the filter designed using step invariance, the system functions are not the same. Therefore, the two designs are not equivalent.
Suppose that we would like to design a discrete-time low-pass filter by applying the impulse invariance method to a continuous-time Butterworth tilter that has a magnitude-squared function
FILTER DESIGN The specifications for the discrete-time ti lter are
[CHAP. 9
Show that the design is not affected by the value of the sampling period that is used in the impulse invariance technique. In the absence of aliasing, the impulse invariance method is a linear mapping from H,,(jQ) to H(eJL" for Iwl 5 n. This mapping is H W " ) = H,,(;w "! = ,, IwI 5 n Let us assume that there is no aliasing (this will be approximately true if the filter order is large enough). The required filter order is then log d N Z log k where d, the discrimination factor, is (I
d = [
- 6,J2 -
6;z-1 k' = R
and k , the selectivity factor, is Q, Clearly, the discrimination factor does not depend on the sampling period T,. In addition, with w, = Q,T, and w, = Q,T,, it follows that k = - w,,l T, - w,, -to,,
which does not depend on the sampling period. Therefore, the required filter order is independent of T,. Next, if we expand the system function of the Butterworth filter in a partial fraction expansion. we have
where the poles, sk, are
With impulse invariance, the system function of the discrete-time filter becomes
and it follows that the poles of H(z) are at
= exp(sl T,) = exp{Q,T,Or
where Because w, = R, T , is the 3-dB cutoff frequency of the low-pass filter in the discrete-time domain, it is fixed by the tilter specifications. Therefore, the poles of H ( : ) will not be affected by the sampling period T,. For example, if we try to decrease T, to reduce aliasing, this would require an increase in Q, to preserve the cutoff frequency. Thus, 11 follows that the design is not affected by T,.
Use the impulse invariance method to design a low-pass digital Butterworth filter to meet the following specifications:
CHAP. 91
FILTER DESIGN
In the absence of aliasing, the impulse invariance method is a linear mapping from H,(jR) to H(eJW) Iwl 5 n, for which is given by 5 H(eJ") = ~ . ( . i R ) l ~ , ~ ~ , 1 4 x Therefore, in order to simplify the design, we will assume that there is no aliasing and then, after the design is completed. check to see that the filter satisfies the given specification:;. Because the parameter T, does not enter into the design using the impulse invariance method (see Prob. 9.38), for convenience we will set T, = 1 . The first step, then, is to design an analog Butterworth filter according to the following specifications:
To determine the filter order. we compute the discrimination factor,
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