The Dirichlet Function

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The functiondiriccomputes the Dirichlet function, sometimes called theperiodic sincoraliased sincfunction, for an input vector or matrixx. The Dirichlet function is defined by

$D (x) = {\begin{array}{c} \frac{\sin (N x / 2)}{N \sin (x / 2)}, & x \neq 2 π k, \\ (- 1)^{k (N - 1)}, & x = 2 π k, \end{array} k = 0, \pm 1, \pm 2, \pm 3, \dots$

where $N$ is a user-specified positive integer. For $N$ odd, the Dirichlet function has a period of $2 π$ ; for $N$ even, its period is $4 π$ . The magnitude of this function is $1 / N$ times the magnitude of the discrete-time Fourier transform of the $N$ -point rectangular window.

To plot the Dirichlet function between 0 and $4 π$ for $N = 7$ and $N = 8$ , use

x = linspace(0,4*pi,300); subplot(2,1,1) plot(x/pi,diric(x,7)) title(“N = 7”) subplot(2,1,2) plot(x/pi,diric(x,8)) title('N = 8') xlabel('x / \pi')

Figure contains 2 axes objects. Axes object 1 with title N = 7 contains an object of type line. Axes object 2 with title N = 8 contains an object of type line.

The Dirichlet Function

See Also