 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
barcode lib ssrs Frequency Sampling Filter Design in Software
9.3.2 Frequency Sampling Filter Design Recognize Code128 In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Make Code 128 Code Set A In None Using Barcode generation for Software Control to generate, create Code128 image in Software applications. 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 Decoding Code 128B In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. Code 128 Code Set B Drawer In C#.NET Using Barcode creator for .NET framework Control to generate, create ANSI/AIM Code 128 image in .NET framework applications. CHAP. 91
Code128 Generator In .NET Framework Using Barcode printer for ASP.NET Control to generate, create USS Code 128 image in ASP.NET applications. Code 128 Code Set A Generation In .NET Using Barcode creator for .NET Control to generate, create USS Code 128 image in Visual Studio .NET applications. FILTER DESIGN
Code 128B Generation In VB.NET Using Barcode maker for VS .NET Control to generate, create Code 128 Code Set A image in VS .NET applications. EAN13 Encoder In None Using Barcode encoder for Software Control to generate, create GTIN  13 image in Software applications. These frequency samples constitute an N point DFT, whose inverse is an FIR filter of order N  1: Printing Code 39 Full ASCII In None Using Barcode creator for Software Control to generate, create Code 39 Extended image in Software applications. Create EAN / UCC  13 In None Using Barcode creation for Software Control to generate, create GS1128 image in Software applications. The relationship between h(n) and hd(n) (see Chap. 3) is
Print Barcode In None Using Barcode encoder for Software Control to generate, create barcode image in Software applications. Data Matrix Printer In None Using Barcode maker for Software Control to generate, create Data Matrix 2d barcode image in Software applications. 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. 93. These transition samples are optimized in an iterative manner to maximize the stopband attenuation or minimize the passband ripple. ISBN Generation In None Using Barcode drawer for Software Control to generate, create International Standard Book Number image in Software applications. Creating USS Code 39 In None Using Barcode drawer for Word Control to generate, create Code39 image in Office Word applications. Transition Band
Barcode Recognizer In C# Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET applications. GTIN  13 Creation In Java Using Barcode generation for Java Control to generate, create EAN13 Supplement 5 image in Java applications. Fig. 93. Introducing a transition sample with an amplitude of A , in the frequency sampling method.
Code 128A Printer In ObjectiveC Using Barcode drawer for iPhone Control to generate, create Code128 image in iPhone applications. Bar Code Drawer In ObjectiveC Using Barcode printer for iPhone Control to generate, create barcode image in iPhone applications. 9.3.3 Equiripple Linear Phase Filters
Bar Code Maker In None Using Barcode printer for Font Control to generate, create barcode image in Font applications. Code39 Reader In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. The design of an FIR lowpass 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 realvalued 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 kthorder Chebyshev polynomial [see Eq. (9.9)]. Therefore, Eq. (9.4) may be written as Thus, A(ejw)is an Lthorder 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 lowpass filter, the set F will be the frequencies in the passband. [O, w,,], and the n], illustrated in Fig. 94. The transition band, (w,, o,), is a don't care region, and it is not as stopband, Don't Care Fig. 94. 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 lowpass have more. For example, a lowpass filter may have either L 2 or filter with L 3 extrema is called an extraripplefilter. From the alternation theorem, it follows that

