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where no is the amount of the shift and RN n ) is a rectangular window: (

A recursive function that returns the power xn:

A circular shift may be visualized as follows. Suppose that the values of a sequence x ( n ) , from n = 0 to n = N - I , are marked around a circle as illustrated in Fig. 6-1 or in an eight-point sequence. A circular shift to the

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

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right by no corresponds to a rotation of the circle no positions in a clockwise direction. An example illustrating the circular shift of a four-point sequence is shown in Fig. 6-2. Another way to circularly shift a sequence is to form the periodic sequence i ( n ) , perform a linear shift, T ( n - no), and then extract one period o f f ( n - no) by multiplying by a rectangular window. If a sequence is circularly shifted, the DFT is multiplied by a complex exponential,

double pow(double x, int n) { if (n == 0) { return 1.0; // basis } return x*pow(x,n-1); // recursion }

.r((n - no)) N R N ( ~ ) w;lk x ( k )

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; } }

(6.6)

A recursive function that returns the power xn and makes no more than lgn recursive calls:

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

Similarly, with a circular shift of the DFT, X ( ( k - k O ) ) ~ . sequence is multiplied by a complex exponential, the

double pow(double x, int if (n == 0) { return 1.0; } double p = pow(x,n/2); if (n%2 == 0) { return p*p; } else { return x*p*p; } } n) { // basis

Let h ( n ) and x ( n ) be finite-length sequences of length N with N-point DFTs H ( k ) and X ( k ) , respectively. The sequence that has a DFT equal to the product Y ( k ) = H (k)X ( k ) is

whereP(n) and h(n) are the periodic extensions of the sequences x(n) and h(n ), respective1y. ~ e c a u s h(n) = h(n) e for 0 5 n < N, the sum in Eq. (6.8) may also be written as

// recursion (n even) // recursion (n odd)

A recursive function that returns the power xn:

The sequence y(n) in Eq. (6.9) is the N-point circular convolution of h(n) with x(n), and it is written as

The circular convolution of two finite-length sequences h(n) and x(n) is equivalent to one period of the periodic convolution of the periodic sequences i ( n ) andf(n),

A recursive function that returns the integer binary logarithm of n:

double pow(double x, int n) { if (n == 0) { return 1.0; // basis } return x*pow(x,n-1); // recursion }

In general, circular convolution is not the same as linear convolution. and N -point circular convolution is different, in general, from M-point circular convolution when M # N .

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