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9.1 The basis of a recursive function is its starting point in its definition and its final step when it is being called recursively; it is what stops the recursion. The recursive part of a recursive function is the assignment that includes the function on the right side of the assignment operator, causing the function to call itself; it is what produces the repetition. For example, in the factorial function, the basis is n! = 1 if n = 0, and the recursive part is n! = n (n 1) if n > 0. The call factorial(10) will generate 10 recursive calls. The call f(6) to the Fibonacci function will generate 14 + 8 = 22 recursive calls because it calls f(5) and f(4), which generate 14 and 8 recursive calls, respectively. A recursive solution is often easier to understand than its equivalent iterative solution. But recursion usually runs more slowly than iteration. Direct recursion is where a function calls itself. Indirect recursion is where a group of functions call each other.

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This periodic extension may be expressed as follows:

+ b))).

9.2 9.3 9.4 9.5

where ( n mod N ) and ((n))N are taken to mean "n modulo N ." That is to say, if n is written in the form n = kN + I where 0 5 I < N ,

For example, (( 13))8 = 5 and ( ( - 6 ) ) x = 2. A periodic sequence may be expanded using the DFS as in Eq. ( 6 . 1 ) . Because x ( n ) = 2 ( n ) for n = 0, 1, . . . , N - 1 , x ( n ) may similarly be expanded as follows:

A recursive function that returns the sum of the first n elements of an array:

9.1 A recursive function that returns the sum of the first n squares:

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Because the DFS coefficients are periodic, if we let X ( k ) be one period of i ( k ) and replace%(k) in the sum with X ( k ) , then we have

The sequence X ( k ) is called the N-point DFT of x ( n ) . These coefficients are related to x ( n ) as follows:

double sum(double[] a, int n) { if (n == 0) { return 0.0; // basis } return sum(a,n-1) + a[n-1]; // recursion }

A recursive function that returns the sum of the first n powers of a base b:

This expansion is valid for complex-valued as well as real-valued sequences. Comparing the definition of the DFT of x ( n ) to the DTFT, it follows that the DFT coefficients are samples of the DTFT:

A recursive function that returns the maximum among the first n elements of an array:

Alternatively, the DFT coefficients correspond to N samples of X ( z ) that are taken at N equally spaced points around the unit circle:

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double sum(double b, int n) { if (n == 0) { return 1; // basis } return 1 + b*sum(b,n-1); // recursion }

In this section, we list some of the properties of the DFT. Because each sequence is assumed to be finite in length, some care must be exercised in manipulating DFTs.

double max(double[] a, int n) { if (n == 1) { return a[0]; // basis } double m = max(a,n-1); // recursion if (a[n-1] > m) { return a[n-1]; } else { return m; } }

If x l ( n )and x 2 ( n ) have N-point DFTs X l ( k ) and X 2 ( k ) ,respectively,

+ b))).

In using this property, it is important to ensure that the DFTs are the same length. If x l ( n ) and x z ( n ) have different lengths, thc shorter sequence must be padded with zeros in order to make it the same length as the longer sequence. For example, if x l ( n ) is of length N I and x 2 ( n ) is of length N2 with N2 > N 1. x l ( n ) may be considered to be a sequence of length N2 with the last N2 - N I values equal to zero, and DFTs of length N2 may be taken for both sequences.

A recursive function that returns the maximum among the first n elements of an array and makes no more than lgn recursive calls:

If x ( n ) is real-valued, X ( k ) is conjugate symmetric,

A recursive function that returns the sum of the first n elements of an array:

and if x ( n ) is imaginary, X ( k ) is conjugate antisymmetric,

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