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EXAMPLE 9 3 2 Suppose that we would like to design a low-pass filter with a cutoff frequency w, = n / 4 , a transition .. width Aw = 0.02n, and a stopband ripple 6, = 0.01. Because a, = -20 log(O.01)= -40. the Kaiser window parameter is

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public static void sort(int[] a) { sort(a, a.length); } public static void sort(int[] a, int n) { if (n < 2) { return; } sort(a, n-1); int temp = a[n-1], j; for (j = n-1; j > 0 && a[j-1] > temp; j--) { a[j] = a[j-1]; } a[j] = temp; }

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Although it is simple to design a filter using the window design method, there are some limitations with this method. First, it is necessary to find a closed-form expression for hd(n)(or it must be approximated using a very long DFT). Second, for a frequency selective filter, the transition widths between frequency bands, and the ripples within these bands, will be approximately the same. As a result, the window design method requires that the filter be designed to the tightest tolerances in all of the bands by selecting the smallest transition width and the smallest ripple. Finally, window design filters are not, in general, optimum in the sense that they do not have the smallest possible ripple for a given filter order and a given set of cutoff frequencies.

14.29 14.30 14.31

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

a[0]

These frequency samples constitute an N -point DFT, whose inverse is an FIR filter of order N - 1:

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The heap sort is not always preferred over the quick sort because it is slower in the average case. The heap sort is not always preferred over the merge sort because it is not stable. The postcondition of heapify (Algorithm 14.9 on page 266) establishes the loop invariant in step 3. That guarantees that the root s0 is the maximum element of the subsequence. Step 5 inserts that maximum at the end of the subsequence. So when the loop at step 4 is finished, the sequence will be sorted. The heapify algorithm restores the heap property to the complete segment ss by applying the heapifyDown() method from its root. The Las Vegas sort has complexity O(n n ). There are n! different permutations of a deck of n cards. Shuffling them is equivalent to selecting one permutation at random. Only one of the n! permutations leaves the cards in order. So the expected number of random shuffles required before the correct one occurs is n!/2. Then each permutation takes n 1 comparisons to see if it is the correct one. So the total complexity is O(n n!/2). By Stirling s Formula (page 325), O(n n!/2) = O(n!) = O(2 n ).

On the first iteration of the main loop, a[1] is compared with a[0] and interchanged if necessary. So a[1] after the first iteration. If we assume that the loop invariant is true prior to some kth iteration, then it must also be true after that iteration because during it a[k] is inserted between the elements that are less than or equal to it and those that are greater. It follows by mathematical induction that the loop invariant is true for all k.

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.

Fig. 9-3. Introducing a transition sample with an amplitude of A , in the frequency sampling method.

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