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For a narrowband filter that has most of its DFT coefficients equal to zero, the frequency sampling structure will be an efficient implementation. The frequency sampling structure is shown in Fig. 8-5. If h ( n ) is real, H ( k ) = H * ( N - k ) , and the structure may be simplified. For example, if N is even,

250 lgn = o(n), because 250 lgn grows more slowly than n. 0.086 n lgn = (n), because 0.086 n lgn grows faster than n.

'Note that here we are assuming that h ( n ) is of length N, instead of N I as in the previous sections. This is consistent with the convention that the frequency sampling filter is based on an N-point DFT of h(n).

15.19 15.20

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Keep in mind that these functions f(n), g(n), and so on, are usually used to describe how long it takes to run an algorithm. So if f(n) grows more slowly than g(n), then the algorithm with complexity f(n) is generally faster than the algorithm with complexity g(n). Less time is better. THE FIRST PRINCIPLE OF MATHEMATICAL INDUCTION The First Principle of Mathematical Induction, also called weak induction, is often used to prove formulas about positive integers. Theorem A.3 The First Principle of Mathematical Induction If {P1 , P2, P3 , . . . } is a sequence of statements such that: P1 is true. Each statement Pn can be deduced from its predecessor Pn 1 . Then all of the statements P1, P2 , P3, . . . are true. EXAMPLE A.4 Weak Induction

In the following sections, several different implementations of this system are presented, including the direct form structures, the cascade and parallel forms, and the transposed filter structures.

There are two direct form filter structures, referred to as direct form I and direct form 11. The direct form I structure is an implementation that results when Eq. (8.3)is written as a pair of difference equations as follows:

The adjacency matrix and the adjacency list are shown in Figure 15.63 on page 318. a. The breadth-first search visits ABDECHFIGKLJMONPQ; its spanning tree is shown on the left in Figure 15.64 on page 318. b. The depth-first search visits ABCFEIHDKLMJGNPQO; its spanning tree is shown on the right in Figure 15.64 on page 318.

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