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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.

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Thus, it follows that this system is linear and, therefore, additive and homogeneous. Because the real part of the sum of two numbers is the sum of the real parts, if y,(n) is the response of the system toxl(n), and yz(n) is the response to x2(n), the response to y(n) = yl(n) yz(n) is

public static void sort(int[] a) { for (int i = a.length-1; i > 0; i--) { for (int j = 0; j < i; j++) { if (a[j] > a[j+1]) { swap(a, j, j+1); } } } } public static void sort(int[] a) { boolean sorted=false; int i = a.length-1; while (i > 0 && !sorted) { for (int j = 0; j < i; j++) { sorted = true; if (a[j] > a[j+1]) { swap(a, j, j+1); sorted = false; } } --i; } }

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Mainlobe 'IT Width Fig. 9-2. The DTFT of a typical window, which is characterized by the width of its main lobe. A. and the peak amplitude of its side lobes, A, relative to the amplitude of W ( d ' " )at o = 0. Ideally, the main-lobe width should be narrow, and the side-lobe amplitude should be small. However, for a fixed-length window, these cannot be minimized independently. Some general properties of windows are as follows:

a[0]

1. As the length N of the window increases, the width of the main lobe decreases, which results in a decrease

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The loop invariant can be used to prove that the bubble sort does indeed sort the array After the first iteration of the main i loop, the largest element must have moved to the last position Wherever it began, it had to be moved step by step all the way to the right, because on each comparison the larger element is moved right For the same reason, the second largest element must have been moved to the second-from-last position in the second iteration of the main i loop So the two largest elements are in the correct locations This reasoning verifies that the loop invariant is true at the end of every iteration of the main i loop But then, after the last iteration, the n-1 largest elements must be in their correct locations That forces the nth largest (ie.

in the transition width between passbands and stopbands. This relationship is given approximately by

The proof is similar to that for the corresponding theorems for the bubble sort and the selection sort. On the first iteration of the outer i loop, the inner j loop iterates once. On the second, it iterates once or twice, depending upon whether a[1] > a[2]. On the third iteration, the inner j loop iterates at most three times, again depending upon how many of the elements on the left of a[3] are greater than a[3]. This pattern continues, so that on the kth iteration of the outer loop, the inner loop iterates at most k times. Thus the maximum total number of iterations is 1 + 2 + + (n 1) = n(n 1)/2. In this case, the inner loop will iterate only once for each iteration of the outer loop. So the total number of iterations of the inner loop is: 1 + 1 + 1 + +1 + 1 = n 1. For the quick sort pivoting on the last element, the only changes needed are in the partition() method:

2. The peak side-lobe amplitude of the window is determined by the shape of the window, and it is essentially independent of the window length. 3. If the window shape is changed to decrease the side-lobe amplitude, the width of the main lobe will generally increase.

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