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The first equation corresponds to an FIR filter with input x ( n ) and output w ( n ) , and the second equation corresponds to an all-pole filter with input w ( n ) and output y ( n ) . Therefore, this pair of equations represents a

cascade of two systems,

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This appendix summarizes mathematical topics used in the study of data structures THE FLOOR AND CEILING FUNCTIONS The floor and ceiling functions return the two nearest integers of a given real number The floor of x, denoted by x , is the greatest integer that is not greater than x The ceiling of x, denoted by x , is the smallest integer that is not smaller than x Here are the main properties of these two functions (The symbol stands for the set of all integers) Theorem A1 Properties of the Floor and Ceiling Functions 1 x = max{m | m x}, and x = min{n | n x} 2 x x < x + 1, and x 1 < x x 3 x 1 < x x x < x + 1 4 If n and n x < n + 1, then n = x .

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THE SECOND PRINCIPLE OF MATHEMATICAL INDUCTION The Second Principle of Mathematical Induction, also called strong induction, is nearly the same as the first principle. The only difference is in the inductive step. Theorem A.4 The Second 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 predecessors {P1 , P2, P3 , . . . , Pn 1 }. Then all of the statements P1, P2 , P3, . . . are true. So to verify the inductive step with strong induction, we may assume that all n 1 statements P1 , P2 , P3, . . . , Pn 1 are true. EXAMPLE A.5 Strong Induction

as illustrated in Fig. 8-6. The computational requirements for a direct form I structure are as follows: Number of multiplications: p Number of additions: p Number of delays: p

The directform IJ structure is obtained by reversing the order of the cascade of B(z) and 1/A(z) as illustrated in Fig. 8-7. With this implementation, x ( n ) is first filtered with the all-pole filter l / A ( z ) and then with B(z):

If n and n 1 < x < n, then n = x 5 If x , then x = x = x 6 If x , then x < x < x 7 x x and x = x 8 x + 1 = x and x + 1 x LOGARITHMS The logarithm with base b of a positive number x is the exponent y on b for which b y = x For example, the logarithm of 1000 base 10 is 3 because 103 = 1000 This is written log10 1000 = 3 The logarithm with base 2 is called the binary logarithm and is written lg x = log 2 x For example, lg 8 = 3.

Prove that the Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, . . . are asymptotically exponential. More precisely, we prove that Fn = O(2 n ), where the Fibonacci numbers Fn are defined as F0 = 0, F1 = 1, and Fn = Fn 1 + Fn 2. So our sequence of statements is P1 : P2 : F1 F2 21 22

If we denote the output of the all-pole filter 1 /A(z) by w(n),this structure is described by the following pair of coupled difference equations:

As a mathematical function, the logarithm is the inverse of the exponential function with the same base: y = log b x by = x For example, 3 = lg 8 because 23 = 8..

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