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private static int indexOfMedian(int[] a, int i, int j, int k) { // Returns the index of the median of {a[i], a[j], a[k]} if (a[i] <= a[j] && a[j] <= a[k]) return j; if (a[i] <= a[k] && a[k] <= a[j]) return k; if (a[j] <= a[i] && a[i] <= a[k]) return i; if (a[j] <= a[k] && a[k] <= a[i]) return k; if (a[k] <= a[i] && a[i] <= a[j]) return i; return j; }

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This verifies that the first loop invariant is true on every iteration of the outer loop Since swap(a,i,j) simply interchanges a[i] with a[j], the second loop invariant follows from the first The third loop invariant follows from the second and by mathematical induction During the first iteration of the main loop, the inner loop finds a[j] to be the largest element in the array The swap(a,i,j) puts that largest element in the last location a[i], so a[i] must be >= all the a[j] Prior to the ith iteration of the main loop, we have by the inductive hypothesis that the subarray a[i+1.n) is sorted and all the values in the subarray a[0.i] are smaller than a[i+1] Then after the ith iteration, a[i] is one of those smaller elements, so a[i] a[i+1] .. a[n-1].

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For a stopband attenuation of 20 log(0.O I ) = -40 dB. we may use a Hanning window. Although we could also use a Hamming or a Blackman window, these windows would overdesign the filter and produce a larger stopband attenuation at the expense of an increase in the transition width. Because the specification calls for a transition width of Aw = w, - w, = 0.02n, or Af = 0.01, with NAf = 3.1 for a Hanning window (see Table 9.2), an estimate of the required filter order is

For the quick sort with reversion to the insertion sort on arrays of size < 8, the only changes needed are in the sort() method:

The last step is to find the unit sample response of the ideal low-pass filter that is to be windowed. With a cutoff frequency of w,. = (w, w,)/2 := 0.2n, and a delay of cr = N/2 = 155, the unit sample response is

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Again, the proof is essentially the same as that for the corresponding theorem for the bubble sort On the first iteration of the outer i loop, the inner j loop iterates n 1 times On the second, it iterates n 2 times This progression continues, for a total of (n 1) + (n 2) + + 2 + 1 = n(n 1)/2..

In addition to the windows listed in Table 9-1, Kaiser developed a family of windows that are defined by

private static void sort(int[] a, int p, int q) { if (q - p < 2) { return; } if (q - p < 8) { insertionSort(a, p, q); return; } int m = partition(a, p, q); sort(a, p, m); sort(a, m+1, q); } // steps 2 & 3 // step 4

is where a = N / 2 , and l o ( . ) a zeroth-order modified Bessel function of the first kind, which may be easily generated using the power series expansion

public static void sort(int[] a, int[] ndx) { for (int i = 1; i < a.length; i++) { int ndxi = ndx[i], j; for (j = i; j > 0 && a[ndx[j-1]] > a[ndxi]; j--) { ndx[j] = ndx[j-1]; } ndx[j] = ndxi; } }

The parameter determines the shape of the window and thus controls the trade-off between main-lobe width and side-lobe amplitude. A Kaiser window is nearly optimum in the sense of having the most energy in its main lobe for a given side-lobe amplitude. Table 9-3 illustrates the effect of changing the parameter /3. There are two empirically derived relationships for the Kaiser window that facilitate the use of these windows to design FIR filters. The first relates the stopband ripple of a low-pass filter, a, = -20 log(6,), to the parameter B,

This requires a generalization of the insertion sort:

Parameter Side Lobe (dB) Transition Width (NAf Stopband Attenuation (dB)

CHAP. 14]

The second relates N to the transition width Af and the stopband attenuation a,,

public static void insertionSort(int[] a, int p, int q) { for (int i = p+1; i < q; i++) { int ai = a[i], j; for (j = i; j > 0 && a[j-1] > ai; j--) { a[j] = a[j-1]; } a[j] = ai; } }

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