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Because the up-sampler inserts L - I zeros between each sample of x ( n ) , assume that H ( z ) is the system function of an FIR filter, and use the fact that many of the values of w ( n ) are equal to zero to derive a more efficient implementation of this system.

A direct implementation of the cascade of an up-sampler with an FIR filter using the transposed direct form is illustrated in the figure below.

CHAP. 10]

d The path length of a full tree of order d and height h is ------------------ h d h + 1 h + 1 d + 1 . For example, d 1 2 the path length of the full tree on Figure 10.4 on page 188 is 102.

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Note that the evaluation of each value of y ( n ) requires N multiplications and N - I additions. However, only one out of every L values that are being multiplied by the coefficients h ( n ) is nonzero. Therefore, it is more efficient to modify the structure so that the filtering is performed prior to the insertion of zeros. With the transposed direct form structure, we may commute the up-sampler with the branch multiplies as illustrated in the following figure:

(c) A casczlde of first- and second-order systems realized in direct form I1 (4 A cascade of first- and second-order systems realized in transposed direct form I1

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DEFINITIONS Here is the recursive definition of a binary tree: A binary tree is either the empty set or a triple T = (x, L, R), where x is a node and L and R are disjoint binary trees, neither of which contains x. The node x is called the root of the tree T, and the subtrees L and R are called the left subtree and the right subtree of T rooted at x. Comparing this definition with the one on page 186, it is easy to see that a binary tree is just an ordered tree of order 2. But be aware that an empty left subtree is different from an empty right subtree. (See Example 10.5 on Figure 11.1 Unequal binary trees page 191.) Consequently, the two binary trees shown Figure 11.1 are not the same. Here is an equivalent, nonrecursive definition for binary trees: A binary tree is an ordered tree in which every internal node has degree 2. In this simpler definition, the leaf nodes are regarded as dummy nodes whose only purpose is to define the structure of the tree. In applications, the internal nodes would hold data, while the leaf nodes would be either identical empty nodes, a single empty node, or just the null reference. This may seem inefficient and Figure 11.2 Equal binary trees more complex, but it is usually easier to implement. In Figure 11.2, the dummy leaf nodes in the tree on the right are shown as asterisks.

The tree diagram analysis of the St. Petersburg Paradox is shown in Figure 10.15. The flaw in this strategy is that there is a distinct possibility (i.e., a positive probability) that enough tails could come up in a row to make the required bet exceed the bettor s stake. After n successive tails, the bettor must bet $2n. For example, if 20 tails come up in a row, the next bet will have to be more than a million dollars!

Suppose that we are given 10 s of speech that has been sampled at a rate of 8 kHz and that we would like to filter it with an FIR filter h ( n ) of length L = 64. Using the overlap-save method with 1024-point DFTs, how many DFTs and inverse DFTs are necessary to perform the convolution

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