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(b) Repeat part ( a ) for the bilinear transformation.
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(a) With impulse invariance, an analog filter with a system function
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will be mapped lo a digital filter with a system function
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Rewriting this system function as a ratio of polynomials, it follows that the locations of the zeros of H ( z ) will depend on the locations of poles as well as the zeros of H,,(s), and there is no way 10 guarantee that the zeros lie inside the unit circle. A simple example showing that a minimum phase continuous-time filter will not necessarily be mapped to a minimum phase discrete-time tilter is the following:
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Using the impulse invariance method with T, = I, we have
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which has a zero at
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z = -(be-'
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7e-2) % - 1.256
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Therefore, although H,(s) is minimum phase, H ( z ) is not.
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CHAP. 91
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FILTER DESIGN
(b) The mapping between the s-plane and the z-plane with the bilinear transformation is defined by
Therefore, a pole or a zero at s = s k becomes a pole or a zero at
If Ha(s) is minimum phase, the poles and zeros of H,,(s) are in the left-half s-plane. In other words, if H,(s) has a pole or a zero at s = sk, where sk = UL j Q n ,
Therefore,
I:kI
= 1 1 + ( T s / 2 ) k 1 2 1 ( 2 / ~ ) + ~ k -2[ ( 2 / K ) + ~ k ] ~ f f i< I 1
11 - ( T A / ~ -~1(2/TT)- sn12 - [(2/Ts) - un12f fi: ) I~ and it follows that a pole or a zero in the left-half s-plane will be mapped to a pole or a zero inside the unit circle in the z-plane (i.e., H(z) is minimum phase).
The system function of a continuous-time filter H a ( s ) of order N 2 2 may be expressed as a cascade of two lower-order systems:
Ha(s)= H,I(s)H,~(s) Therefore, a digital filter may either be designed by applying a transformation directly to H,(s) or by individually transforming H a l ( s )and H a z ( s )into H I ( z ) and H:!(z),respectively, and then realizing H ( z ) as the cascade: H ( z ) = Hi(z)Hz(z) ( a ) If H l ( z ) and H2(z) are designed from H O l ( s )and Ha2(s) using the impulse invariance technique, with the filter that is designed by using the impulse invaricompare the cascade H ( z ) = H I ( z ) H 2 ( z ) ance technique directly on H,(s).
(b) Repeat part ( a ) for the bilinear transformation.
(a) Due to sampling, aliasing occurs when designing a digital filter using the impulse invariance method. Because the operations of sampling and convolution do not commute, the filter designed by using impulse invariance on H,(s) will not be the same as the filter designed by cascading the two filters that are designed using impulse invariance on H,,(s) and Ha2(s). In other words, if
where h(n) = h,(nT,), hl(n) = hal(nTx), h2(n) = ha2(nT,). As an example, consider the continuous-time and filter that has a system function tf,(s) = I I ( s + l)(s+2) s+ 1 I s+2
Using the impulse invariance technique on Ha(s) with T, = I, we have
On the other hand, writing H,(s) as a cascade of two first-order systems,
and using the impulse invariance method on each of these systems with T, = 1, we have
which is not the same as the previous filter.
FILTER DESIGN
[CHAP 9
( b ) With the bilinear transformation. ( T , = 2 )
and the two designs are the same.
What are the properties of the s-plane-to-:-plane mapping defined by
and what might this mapping be used for
This mapping is very similar to the bilinear trandormation which. with T, = 2, is
In fact, this mapping may be considered to be a cascade of two mappings. The first is the bilinear transformation, and the second is one that replaces 1 with -:. " - -2 - This mapping reflects points in the :-plane about the origin and. tbr points on the unit circle. corresponds to a shift of 1 8 0 : H(z1)Ic,=,,,,., H ( - : ) I :=,, ,,,, = /I(-c"") = H (e.l""+n' = ) Therefore, this mapping has the same properties as the bilinear transformation. except that the ;St axis is mapped
Because the unit circle is rotated hy IXO' , this mapping may be used to map low-pass analog filters into high-pass digital filters, and high-pass analog tilters into low-pass digital filters.
Least-Squares Filter Design
Suppose that the desired unit sample response of a linear shift-invariant system is
Use the Pad6 approximation method to find the parameters of'a filter with a system function
that approximates this unit sample response.
Using the Pad6 approximation method, with p = q = I . we want to solve the following set of linear equations for b(0). h( I), and a( I):
CHAP. 91
FILTER DESIGN
Using the last equation, we may easily solve for a ( l ) ,
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