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Thus, the DFT is evaluated by lint computing the three-point DFT of each row of the input array, followed by the four-point DFT of each column. The following figure shows how the four-point DFTs are interconnected to the three-point DFTs.

CHAP. 11]

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Because a 4-point DFT does not require any multiplications (see Prob. 7.1 1 ), and because each 3-point DFT requires only 4 complex multiplications, the 12-point prime factor algorithm requires 16 complex multiplies. For a mixed-radix FFT, there are, in addition, six twiddle factors. The cost for eliminating these six multiplications is an increase in complexity in indexing and in programming.

Figure 11.20 shows the evaluation of the expression in Example 11.18 using 2 for x, 3 for y, and 1 for z:

Assume that a complex multiply takes 1 p s and that the amount of time to compute a DFT is determined by the amount of time it takes to perform all of the multiplications. (a) How much time does it take to compute a 1024-point DFT directly (b) How much time is required if an FFT is used

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Figure 11.20 shows the evaluation of the expression in Example 11.18 using 2 for x, 3 for y, and 1 for z:

(a) Including possible multiplications by fI, computing an N-point DFT directly requires N complex multipli' cations. If it takes I p s per complex multiply, the direct evaluation of a 1024-point DFT requires

A BinaryTree CLASS Here is a class for binary trees that directly implements the recursive definition. (See page 200.) By extending the AbstractCollection class, it remains consistent with the Java Collections Framework. (See 4.)

(b) With a radix-2 FFT, the number of complex multiplications is approximately (N/2) log, N which, for N = 1024, is equal to 5 120. Therefore, the amount of time to compute a 1024-point DFT using an FFT is

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If the length of the DFT is increased by a factor of 4 to N = 4096, the number of multiplications necessary to compute the DFT directly increases by a factor of 16. Therefore, the time required to evaluate the DFT directly is 1 ~ ~ =.16.78 S 7 If, on the other hand, an FFT is used, the number of multiplications is 2,048 . log, 4,096 = 24,576 and the amount of time to evaluate the DFT is

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

A complex-valued sequence x ( n ) of length N = 8 192 is to be convolved with a complex-valued sequence h ( n ) of length L = 5 12.

( 6 ) Repeat part (a) using the overlap-add method with 1024-point radix-2 decimation-in-time FFTs to

public class BinaryTree<E> extends AbstractCollection { protected E root; protected BinaryTree<E> left, right, parent; protected int size; public BinaryTree() { } public BinaryTree(E root) { this.root = root; size = 1; } public BinaryTree(E root, BinaryTree<E> left, BinaryTree<E> right) { this(root); if (left != null) { this.left = left; left.parent = this; size += left.size(); } if (right != null) { this.right = right; right.parent = this; size += right.size(); } } public boolean equals(Object object) { if (object == this) { return true; } else if (!(object instanceof BinaryTree)) { return false; } BinaryTree that = (BinaryTree)object; return that.root.equals(this.root) && that.left.equals(this.left) && that.right.equals(this.right) && that.parent.equals(this.parent) && that.size == this.size; } public int hashCode() { return root.hashCode() + left.hashCode() + right.hashCode() + size; } public int size() { return size; } public Iterator iterator() { return new java.util.Iterator() { // anonymous inner class private boolean rootDone; private Iterator lIt, rIt; // child iterators

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