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CHAP. 91
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FILTER DESIGN
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(a) The bilinear transformation is defined by the mapping
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Therefore, for the given filter, we have
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which has poles at
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z2 = I-h To find the zeros, it is necessary to combine the two terms in the system function over a common denominator. Doing this. we have
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Finding the roots of the numerator may be facilitated by noting that H,(s) has a zero at s = co,which gets mapped to z = - 1 with the bilinear transformation. Therefore. one of the factors of the numerator is ( 1 z-I). Dividing the numerator by this factor, we obtain the second factor, which is [(2 - a - b) - 2z-'1. Thus, H ( ; ) has zeros at
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(b) With the impulse invariance technique, for first-order poles, the mapping is
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Therefore, with T, = 2 we have
which has two poles at
z1=eL1
and only one zero. which is located at
z:!=e2h
z,, = i(e2"
+ eZb
The system function of a digital fi lter is
(a) If this filter was designed using impulse invariance with T, = 2, find the system function, H&), of an analog filter that could have been the analog filter prototype. Is your answer unique
(b) Repeat part (a) assuming that the bilinear transformation was used with T, = 2.
(a) Because H ( z ) is expanded in a partial fraction expansion, the poles at z = a k in H ( z ) are mapped from poles in Ha(s)according to the mapping ( y k = esl
Therefore, if T, = 2,
sk =
FILTER DESIGN
and one possible analog filter prototype is
[CHAP. 9
Because the mapping from the s-plane to the z-plane is not one to one, this answer is not unique. Specifically, note that we may also write
= @rT,+iZn
Therefore, with T, = 2, we may also have
and another possible analog Rlter prototype is
(6) With the bilinear transformation, because the mapping from the s-plane to the z-plane is a one-to-one mapping, with T, = 2, I +s z=I -S and the analog filter prototype that is mapped to H ( z ) is unique and given by
A continuous-time system is called an inlegrator. if the response of the system y,(r) to an input x u ( [ )is
The system function for an integrator is
( a ) Design a discrete-time "integrator" using the bilinear transformation, and find the difference equation relating the input x ( n ) to the output y ( n ) of the discrete-time system.
(b) Find the frequency response of the discrete-time integrator found in part ( a ) ,and determine whether or not this system is a good approximation to the continuous-time system.
(a) With a system function H&) = I/s, the bilinear transformation produces a discrete-time filter with a system function
The unit sample response of this filter is
and the difference equation relating the output y ( n ) to the input s ( n ) is
CHAP. 91
FILTER DESIGN
(b) Because the frequency response of the continuous-time system, H,(jR) = I / j R , is related to the discrete-time
filter through the mapping
the frequency response of the discrete-time system is
Note that because H (ejC") to zero at w = n ,then H ( e J wwill not be a good approximation to H,(jR) = goes ) 1 / j R except for low frequencies. However, if w < I, using the expansion <
sinxxx
x<<1
we have and we have, for the frequency response.
Therefore, for small w
H (ciw)% T,H,(jR)
Let H,(jQ) be an Nth-order low-pass Butterworth filter with a 3-dB cutoff frequency Q,.
(a) Show that H,(s) may be transformed into an Nth-order high-pass Butterworth filter by adding N zeros at s = 0 and scaling the gain.
( 6 ) What is the relationship between the corresponding digital low-pass and high-pass Butterworth filters that are designed using the bilinear transformation
(a) For an N th-order low-pass Butterworth with a system function H,(s),
Adding N zeros to H,(s) at s = 0, we have
Multiplying numerator and denominator by ( J R ,
yields
which corresponds to a magnitude-squared frequency response
by Scaling t f u ( j R ) R-N results in a filter that has a frequency response G , ( j R )with a squared magnitude
which is a high-pass filter with a cutoff frequency R,.. Specifically, note that IC,(jR)12 is equal to zero at R = 0, and that I C , ( ~ R+ ~ ~ R + oo, ) 1 as
FILTER DESIGN
[CHAP. 9
( h ) Applying the bilinear transformation to a low-pass Butterworth filter, we have
For the high-pass filter, on the other hand, we have
Therefore, we see that the poles of the low-pass digital Butterworlh filter are the same as those of the high-pass digital Butterworth filter. The zeros, however, which are at z = - I in the case of the low-pass filter, are at z = I in the high-pass filter. Thus, excepl for a difference in the gain, the high-pass digital Butterworth filter may be derived from the low-pass filler by flipping the N zeros in H ( z ) at z = - 1 to z = 1.
The impulse invariance method and the bilinear transformation are two filter design techniques that preserve stability of the analog filter by mapping poles in the left-half s-plane to poles inside the unit circle in the z-plane. An analog filter is minimum phase if all of its poles and zeros are in the left-half s-plane.
( a ) Determine whether or not a minimum phase analog tilter is mapped to a minimum phase discrete-time system using the impulse invariance method.
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