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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.

Next, we assume that the function returns the correct value for all integers less than some n > 0. Then the second part

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Use the initial value theorem to show that if hmi,(n) is a minimum phase sequence, and if h ( n ) is a causal sequence with the same Fourier transform magnitude, then

The initial value theorem states that for a causal sequence, the initial value may be found from the z-transform as follows: h(0) =: I % H(z) Let hmin(n) a minimum phase sequence, and let h ( n ) be the nonminimum phase sequence that is formed by be reflecting a zero from inside the unit circle at z = a to its conjugate reciprocal location at z = I/a*:

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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

return n*f(n-1);

have the same Fourier transform magnitude. Because ( z - ' - a * ) / ( l - az-I) is an allpass filter. h ( n ) and hmin(n) Using the initial value theorem, we may compare the value of h(0) to h,l,i,,(0):

and because la1 < I, lh(0)I < lhm,,(0)l. Because the magnitude of Ih(O)l is reduced each time that a zero of Hmin(z) is flipped outside the unit circle,

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:

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will return the correct value n! because (by the inductive hypothesis) the call f(n-1) will return (n 1)! and n! = n (n 1). Note that we are using the strong principle of mathematical induction here (also called the second principle of mathematical induction). In this version, the inductive hypothesis allows us to assume that all the preceding statements are true. In the weak (or first ) principle, we are allowed to assume that only the single preceding statement is true. But since these two principles are equivalent (i.e., they are both valid methods of proof), it is usually better to apply strong induction. The Euclidean algorithm is correct. We can use (strong) induction to prove this fact. (See page 322.) If m and n are equal, then that number is their greatest common divisor. So the function returns the correct value in that case because of the part

for any sequence h ( n ) that has a Fourier transform with the same magnitude as that of hmi.(n).

Prove the minimum energy delay property for minimum phase sequences. Let hm,,(n) be a minimum phase sequence, and let a k be a zero of Hmin(z). Then Hmin(z) be written as may

if (n < 2) { return 1; }

if (m == n) { return n; } If m and n are not equal, then the function returns either gcd(m,n-m) or gcd(m-n,n). To see that this too is the correct value, we need only realize that all three pairs (m,n), (m,n-m), and (m-n,n) will always have the same greatest common divisor. This fact is a theorem from

HmIn(z) = ( 1

Next, we assume that the function returns the correct value for all integers less than some n > 0. Then the second part

-akz-'Fmin(z)

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