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barcode lib ssrs (b) Repeat part ( a ) for the bilinear transformation. in Software
(b) Repeat part ( a ) for the bilinear transformation. Code 128B Reader In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Code 128 Code Set C Maker In None Using Barcode generator for Software Control to generate, create Code128 image in Software applications. (a) With impulse invariance, an analog filter with a system function
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Decode Bar Code In VB.NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in .NET applications. Read DataMatrix In C#.NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in .NET applications. FILTER DESIGN
(b) The mapping between the splane and the zplane 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 lefthalf splane. 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 lefthalf splane will be mapped to a pole or a zero inside the unit circle in the zplane (i.e., H(z) is minimum phase). The system function of a continuoustime filter H a ( s ) of order N 2 2 may be expressed as a cascade of two lowerorder 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 continuoustime 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 firstorder 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 splaneto: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 lowpass analog filters into highpass digital filters, and highpass analog tilters into lowpass digital filters. LeastSquares Filter Design
Suppose that the desired unit sample response of a linear shiftinvariant 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 ) ,

