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This is 1 less than the number of times n can be divided by 2, so the value returned, 1 + lg(n/2), is correct First, we prove the conjecture that the call print(left, right) will print n! distinct strings, all having the same prefix string left, where n = rightlength() If n = 1, the method prints left+right and returns; that is 1! (distinct) string Assume that when rightlength() = n 1, the call print(left,right) prints (n 1)! distinct strings all having the same left prefix string Then, when rightlength() = n, the for loop makes n calls of the form print(left+temp,ss), where temp is a distinct character and ss = ssubstring(1,n) Since the length of ssubstring(1,n) is n 1, each of those calls will print (n 1)! distinct strings all having the same left+temp prefix string..

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It would appear that there is considerably more effort involved in performing convolutions using DFTs. However, significant computational savings may be realized with this approach if the DFTs are computed efficiently. As we will see in Chap. 7, the fast Fourier transform (FFT) provides such an algorithm. In spite of its computational advantages, there are some difficulties with the DFT approach. For example, if x ( n ) is very long, we must commit a significant amount of time computing very long DFTs and in the process accept very long processing delays. In some cases, it may even be possible that x ( n ) is roo long to compute the DFT. The solution to these problems is to use block convolution, which involves segmenting the signal to be filtered, x ( n ) , into sections. Each section is then filtered with the FIR filter h(n), and the filtered sections are pieced together to form the sequence y(n). There are two block convolution techniques. The first is overlap-add, and the second is overlap-save.

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Let x ( n ) be a sequence that is to be convolved with a causal FIR filter h(n)of length L:

These are more accurate recursive implementation of the sine and cosine functions:

The implementation symbol is used to show that a class implements an interface. EXAMPLE 1.7 Implementing the Comparable Interface

Assume that x ( n ) = 0 for n i0 and that the length of x ( n ) is much greater than L . In the overlap-add method, x ( n ) is partitioned into nonoverlapping subsequences of length M as illustrated in Fig. 6-4. Thus, x ( n ) may be written as a sum of shifted finite-length sequences of length M,

Change line 1 of Example 1.5 on page 6 to:

public static double s(double x) { if (Math.abs(x) < 0.00005) { return x - x*x*x/6; // basis }

x ( n )= x r i ( n - M i )

1 =O

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public class Ratio implements Comparable { public int compareTo(Object object) { if (object==this) { return 0; } else if (!(object instanceof Ratio)) { throw new IllegalArgumentException("Ratio type expected"); } Ratio that = (Ratio)object; normalize(this); normalize(that); return (this.num*that.den - that.num*this.den); } private static void normalize(Ratio x) { if (x.num == 0) { // x == Ratio.ZERO x.den = 1; } else if (x.den < 0) { // change sign of num and den: x.num *= -1; x.den *= -1; } }

CHAP. 9]

xi(n) =

x(n+Mi)

and then add these two methods:

return 2*s(x/2)*c(x/2); }

where yi(n) is the linear convolution of x;(n)with lz(n),

Because each sequence yi(n) is of length N = L M - I . it may be found by multiplying the N-point DFTs of xi(n)and h(n).The reason for the name overlap-add is that, for each i. the sequences yi(n)and y ; + ~ ( n ) overlap at (N - M) points and, in performing the sum in Eq. (6.14). these overlapping points are udded.

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

public static double c(double x) { if (Math.abs(x) < 0.00005) { return 1.0 - x*x/2; // basis } return 1 - 2*s(x/2)*s(x/2); // recursion }

Fig. 6-4. Partitioning a sequence into subsequences of length M for the ovelap-add method of block convolulion.

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