barcode lib ssrs Frequency Sampling Filter Design in Software

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9.3.2 Frequency Sampling Filter Design
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Another method for FIR filter design is the frequency sampling approach. In this approach, the desired frequency response, Hd(eJ"), first uniformly sampled at N equally spaced points between 0 and 27r: is
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CHAP. 91
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FILTER DESIGN
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These frequency samples constitute an N -point DFT, whose inverse is an FIR filter of order N - 1:
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The relationship between h(n) and hd(n) (see Chap. 3) is
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Although the frequency samples match the ideal frequency response exactly, there is no control on how the samples are interpolated between the samples. Because filters designed with the frequency sampling method are not generally vely good, this method is often modified by introducing one or more transition samples as illustrated in Fig. 9-3. These transition samples are optimized in an iterative manner to maximize the stopband attenuation or minimize the passband ripple.
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Transition Band
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Fig. 9-3. Introducing a transition sample with an amplitude of A , in the frequency sampling method.
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9.3.3 Equiripple Linear Phase Filters
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The design of an FIR low-pass filter using the window design technique is simple and generally results in a filter with relatively good performance. However, in two respects, these filters are not optimal:
First, the passband and stopband deviations, 6, and 6,, are approximately equal. Although it is common to require S, to be much smaller than S, these parameters cannot be independently controlled in the , window design method. Therefore, with the window design method, it is necessary to overdesign the filter in the passband in order to satisfy the stricter requirements in the stopband. Second, for most windows, the ripple is not uniform in either the passband or the stopband and generally decreases when moving away from the transition band. Allowing the ripple to be uniformly distributed over the entire band would produce a smaller peak ripple.
An equiripple linear phase filter, on the other hand, is optimal in the sense that the magnitude of the ripple is minimized in all bands of interest for a given filter order, N. In the following discussion, we consider the design of a type I linear phase filter. The results may be easily modified to design other types of linear phase filters. The frequency response of an FIR linear phase filter may be written as
FILTER DESIGN
[CHAP. 9
where the amplitude. A(eJw), a real-valued function of w. For a type I linear phase filter, is h(n) = Iz(N - n) where N is an even integer. The symmetry of h(n) allows the frequency response to be expressed as
where L = N / 2 and
The terms cos(kw) may be expressed as a sum of powers of cos w in the form
where T&) is a kth-order Chebyshev polynomial [see Eq. (9.9)]. Therefore, Eq. (9.4) may be written as
Thus, A(ejw)is an Lth-order polynomial in coso. With Ad(ei") a desired amplitude, and W(eiw)a positive weighting function, let
be a weighted approximation error. The equiripple filter design problem thus involves finding the coefficients a ( k ) that minimize the maximum absolute value of E(ejU)over a set of frequencies, F,
For example, to design a low-pass filter, the set F will be the frequencies in the passband. [O, w,,], and the n], illustrated in Fig. 9-4. The transition band, (w,, o,), is a don't care region, and it is not as stopband,
Don't Care Fig. 9-4. The set R in the equiripple filter design problem, consisting of the passband [0, w,] and the stopband [op. , ] . The transition band ( o , , w,) is a don't care region. o
CHAP. 9 1
FILTER DESIGN
considered in the minimization of the weighted error. The solution to this optimization problem is given in the alternation theorem. which is as follows:
Alternation Theorem: Let 3 be a union of closed subsets over the interval [O. n]. For a
positive weighting function w ( d W ) ,a necessary and sufficient condition for
to be the unique function that minimizes the maximum value of the weighted error I E(eJW)I over the set 3is that the E(eJW) have at least L 2 alternations. That is to say, there must be at least L 2 extremalfiequencies.
over the set 3such that
= E(eJWk) - ~ ( e J ~ ~ + l ) = 0, I , . . . L k
~ l ~ ~ ( e j= m~a )x ! ~ ( e j ~ ) k = 0, I , ~
W F
... . L + 1
Thus, the alternation theorem states that the optimum filter is equiripple. Although the alternation theorem specifies the minimum number of extremal frequencies (or ripples) that the optimum filter must have, it may 3 extremal frequencies. A low-pass have more. For example, a low-pass filter may have either L 2 or filter with L 3 extrema is called an extraripplefilter. From the alternation theorem, it follows that
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