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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.

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hd(n) =

a[5] 33

sin(n - a ) w , .

, the smallest) element also to be in its correct location, so the array must be sorted The complexity function O(n2) means that, for large values of n, the number of loop iterations tends to be proportional to n2 That means that, if one large array is twice the size of another, it should take about four times as long to sort The inner j loop iterates n 1 times on the first iteration of the outside i loop, n 2 times on the second iteration of the i loop, n 3 times on the third iteration of the i loop, and so on.

n(t7 - a )

a[6] 22

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Because this filter is unrealizable (noncausal and unstable), it is necessary to relax the ideal constraints on the frequency response and allow some deviation from the ideal response. The specifications for a low-pass filter will typically have the form

For example, when n = 7, there are six comparisons made on the first iteration of the i loop, five comparisons made on the second iteration of the i loop, four comparisons made on the third iteration of the i loop, and so forth, so the total number of comparisons is 6 + 5 + 4 + 3 + 2 + 1 = 21 In general, the total number of comparisons will be (n 1) + (n 2) + (n 3) + + 3 + 2 + 1 This sum is n(n 1)/2 (See Theorem A7 on page 323) For large values of n, that expression is nearly n2/2 which is proportional to n2..

as illustrated in Fig. 9-1. Thus, the specifications include the passband cutoff frequency, w,, the stopband cutoff frequency, w,, the passband deviation, 6,. and the stopband deviation, 6,. The passband and stopband deviations

a[7] 88

Fig. 9-1.

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public static void sort(int[] a) { boolean sorted=false; for (int i = a.length; i > 0; i -= 2) { for (int j = 1; j < i; j++) { if (a[j-1] > a[j]) { swap(a,j-1,j); } } for (int j = i-2; j > 0; j--) { if (a[j-1] > a[j]) { swap(a, j-1, j); } } } }

Filter specifications for a low-pass filter,

a[8] 77

are often given in decibels (dB) as follows:

Property 9 allows the existence of arrays of arrays. Technically, Java allows multidimensional arrays only with primitive types. But for objects, an array of arrays is essentially the same thing. Since arrays themselves are objects, an array of arrays is an array of objects, and some of those component objects could also be nonarrays. (See Example 2.1.) Note that a consequence of Property 13 is that changing a reference component value to null has no effect upon the length of the array; null is still a valid value of a reference component. EXAMPLE 2.1 Some Array Definitions

a,. = -2010g(6,~)

99 22

The interval [w,, w,] is called the trunsitiotr hand. Once the filter specifications have been defined, the next step is to design a filter that meets these specificatlons.

Here are some valid array definitions:

and the design of an FIR filter involves finding the coefficients h ( n ) that result in a frequency response that satisfies a given ser of filter specifications. FIR filters have two important advantages over 1IR filters. First, they are guaranteed to be stable, even after the filter coefficients have been quantized. Second, they may be easily constrained to have (generalized) linear phase. Because FIR filters are generally designed to have linear phase, in the following we consider the design of linear phase FIR filters.

99 88

Let hd(n) be the unit sample response of an ideal frequency selective filter with linear phase,

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