barcode lib ssrs Fig. 9-8. Frequency response of Chebyshev type I filter for orders N = 5 and N = 6. in Software

Creation Code 128A in Software Fig. 9-8. Frequency response of Chebyshev type I filter for orders N = 5 and N = 6.

Fig. 9-8. Frequency response of Chebyshev type I filter for orders N = 5 and N = 6.
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The system function of a type I Chebyshev filter has the form
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where H,(O) = ( 1 - c2)-'I2 if N is even, and H,(O) = 1 if N is odd. Given the passband and stopband cutoff frequencies, Q , and Q,, and the passband and stopband ripples, 8 , and 6, (or the parameters E and A), the steps involved in designing a type I Chebyshev filter are as follows: Find the values for the selectivity factor, k, and the discrimination factor, d. Determine the filter order using the formula
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Form the rational function
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, where E = [(l - s , ) - ~ 1 1 l / ~ and construct the system function H,(s) by taking the N poles of G,(s) that lie in the left-half s-plane.
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EXAMPLE 9.4.2 If we were to design a low-pass type I Chebyshev filter to meet the specificationsgiven in Example 9.4.1 where we found d = 0.0487 and k = 0.6, the required filter order would be
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or N = 4. Therefore, with
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and then where 52, = 2n(6000).
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FILTER DESIGN
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[CHAP. 9
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A type I1 Chebyshev filter, unlike a type 1 filter, has a monotonic passband and an equiripple stopband, and the system function has both poles and zeros. The magnitude of the frequency response is
where N is the order of the filter, 52, is the passband cutoff frequency, R, is the stopband cutoff frequency, and E is the parameter that controls the stopband ripple amplitude. Again, as the order N is increased. the number of ripples increases and the transition width becomes narrower. Examples are given in Fig. 9-9 for N = 5 , 6 .
( b ) Even order (N = 6). (a)Odd order (N = 5). Fig. 9-9. Frequency response of a Chebyshev type I1 filter for orders N = 5 and N = 6.
The system function of a type I1 Chebyshev filter has the form
The poles are located at
Q ak = Sk
where sk for k = 0, I, . . . , N - 1 are the poles of a type 1 Chebyshev filter. The zeros bk lie on the j52-axis at the frequencies for which TN(Q,/ R ) = 0. The procedure for designing a type 11 Chebyshev filter is the same as for a type I filter, except that = ( 8 . - 1)- ' I 2 ~ ~
Elliptic Filter
An elliptic filter has a system function with both poles and zeros. The magnitude of its frequency response is
where UN(52/Qp) is a Jacobian elliptic function. The Jacobian elliptic function U N ( x ) a rational function of is order N with the following property:
Elliptic filters have an equiripple passband and an equiripple stopband. Because the ripples are distributed uniformly across both bands (unlike the Butterworth and Chebyshev filters, which have a monotonically decreasing
CHAP. 91
FILTER DESIGN
passband and/or stopband), these filters are optimum in the sense of having the smallest transition width for a given filter order, cutoff frequency Q,, and passband and stopband ripples. The frequency response for a 4th-order elliptic filter is shown in Fig. 9- 10.
Fig. 9-10. The magnitude of the frequency response of a sixth-order elliptic filter.
The design of elliptic filters is more difficult than the Butterworth and Chebyshev filters, because their design relies on the use of tables or series expansions. However, the filter order necessary to meet a given set of specifications may be estimated using the formula
where d is the discrimination factor, and
where with k being the selectivity factor.
9.4.2 Design of ZZR Filters from AnalogFilters
The design of a digital filter from an analog prototype requires that we transform ha(t) to h(n) or Ha(s) to H(z). A mapping from the s-plane to the z-plane may be written as
where s = m(z) is the mapping function. In order for this transformation to produce an acceptable digital filter, the mapping m(z) should have the following properties: The mapping from the jS2-axis to the unit circle, Izl = 1, should be one to one and onto the unit circle in order to preserve the frequency response characteristics of the analog filter. 2. Points in the left-half s-plane should map to points inside the unit circle to preserve the stability of the analog filter. 3. The mapping m(z) should be a rational function of z so that a rational Ha(s) is mapped to a rational H(z).
Described below are two approaches that are commonly used to map analog filters into digital filters.
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