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THE EUCLIDEAN ALGORITHM The Euclidean Algorithm computes the greatest common divisor of two positive integers. Appearing as Proposition 2 in Book VII of Euclid s Elements (c. 300 B.C.), it is probably the oldest recursive algorithm. As originally formulated by Euclid, it says to subtract repeatedly the smaller number n from the larger number m until the resulting difference d is smaller than n. Then repeat the same steps with d in place of n and with n in place of m. Continue until the two numbers are equal. Then that number will be the greatest common divisor of the original two numbers. Figure 9.4 applies this algorithm to find the greatest common divisor of 494 and 130 to be 26. This is correct because 494 = 26 19 and 130 = 26 5.

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Therefore, H(z) may be written as the product of a minimum phase system with an allpass system as follows:

+ z-I)(2 - 3 z - I )

CHAP. 9]

Each step in the algorithm simply subtracts the smaller number from the larger. This is done recursively by calling either gcd(m,n-m) or gcd(m-n,n):

This system is not minimum phase because the factor ( 2 - 3 z - ' ) corresponds to a zero outside the unit circle at z = However, we may express this factor as the product of a minimum phase term with a linear phase term as follows. First, we reflect the zero about the unit circle and replace it with a pole:

6 7 8 9 10

$ and a zero at z = ::

1 2 3 4 5 6 7 8 9

H ( z )=

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

3 + z-' (2z-' - 3)(2 - 32-') ( I - 4z-')(2z-l - 3 )

public static int gcd(int if (m==n) { return n; } else if (m<n) { return gcd(m,n-m); } else { return gcd(m-n,n); } }

53 .1

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Find a real-valued causal sequence with ~ ( 0> 0 and )

For example, the call gcd(494,130) makes the recursive call gcd(364,130), which makes the recursive call gcd(234,130), which makes the recursive call gcd(104,130), which makes the recursive call gcd(104,26), which makes the recursive call gcd(78,26), which makes the recursive call gcd(52,26), which makes the recursive call gcd(26,26), which returns 26. The value 26 is then successively returned all the way back up the chain to the original call gcd(494,130), which returns it to its caller.

I X ( @ ' ) [ ~ = (1

Each step in the algorithm simply subtracts the smaller number from the larger. This is done recursively by calling either gcd(m,n-m) or gcd(m-n,n):

We begin by expressing IX (ej'")12in terms of complex exponentials:

INDUCTIVE PROOF OF CORRECTNESS Recursive functions are usually proved correct by the principle of mathematical induction. This principle states that an infinite sequence of propositions can be proved to be true by verifying that (i) the first statement is true, and (ii) the truth of every other statement in the sequence can be derived from the assumption that its preceding statements are true. Part (i) is called the

I X ( P ' " ) I ~ = (I+ a 2 )

1 2 3 4 5 6 7 8 9

basis step and part (ii) is called the inductive step. The assumption that the preceding statements are true is called the inductive hypothesis. The recursive factorial function is correct. To prove this fact, we first verify the basis. The call f(0) returns the correct value 1 because of the first part:

- u z f l ) ( l- a z )

.r(n) = 6 ( n )- u6(n - I)

public static int gcd(int if (m==n) { return n; } else if (m<n) { return gcd(m,n-m); } else { return gcd(m-n,n); } }

if (n < 2) { return 1; }

To solve this problem, we begin by expressing ( ~ ( e j ' " )in ~ l terms of complex exponentials as follows:

The minimum phase system is then formed by extracting the poles and zeros that are inside the unit circle:

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