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Suppose that H ( z ) and C(z) are rational and have minimum phase. Which of the following filters have minimum phase

n! n c n k = ---------------------- = -k! n k ! 1

(a) H ( z ) G ( z )

public static int c(int n, int k) { if (k==0 || k==n) { return 1; // basis } return c(n-1,k-1) + c(n-1,k); // recursion }

n 1 ----------2

(b) H ( z )

+G(z)

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The basis for the recursion covers the left and right sides of the triangle, where k = 0 and where k = n.

public static int c(int n, int k) { if (n < 2 || k == 0 || k == n) { return 1; } int c = 1;

If t l ( z ) and G ( z ) have minimum phase, neither H ( z ) nor G ( z )have any poles or zeros outside the unit circle. Because the poles and zeros of H ( z ) G ( z ) the union of the poles and zeros of H ( z ) and G(z),H ( z ) G ( z )will are not have any poles or zeros outside the unit circle and. therefore, has minimum phase.

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(6) If H ( z )and G ( z ) have minimum phase. i t is no1 necessarily true that H ( z ) G ( z )will have minimum phase. We may show this by a simple counter example. If

The binomial coefficients are the same as the combination numbers used in combinatorial mathematics and computed explicitly by the formula n k+1 n 2 -----------------------------k 3 n In this context, the combination is often written c n k = and is pronounced n choose k. k For example, 8 choose 3 is 8 = (8/1)(7/2)(6/3) = 56. 3 n n! c n k = ---------------------- = -1 k! n k ! n 1 ----------2 EXAMPLE 9.12 Iterative Implementation of the Binomial Coefficient Function

CHAP. 9]

G ( z )=

TRANSFORM ANALYSIS OF SYSTEMS may have a zero anywhere in the z-plane by choosing the appropriate values for A and B . For example, because H(z) G(z) has a zero at 0.5B 0.75A -

This version implements the explicit formula given above. The expression on the right consists of k factors, so it is computed by a loop iterating k times:

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to place a zero at z = 2, we may set A = I and solve the following equation for B:

Multiplying a sequence by d'moves the poles and zeros radially by a factor of a:

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for (int j = 1; j <= k; j++) { c = c*(n-j+1)/j; } return c; }

In order for Y ( z ) to be minimum phase, all of the poles and zeros must be inside the unit circle. Because the singularity (pole or zero) of X ( 2 ) that is the furthest from the unit circle is the zero at z = y ( n ) will be minimum phase if Iff1 <

The system function H(z) has a nonminimum phase factor, (1 - 2 ~ - ~which may be written as the product of a ), minimum phase term and an allpass factor as follows:

public static int c(int n, int k) { if (n < 2 || k == 0 || k == n) { return 1; } int c = 1;

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