# Documentation

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# diric

Dirichlet or periodic sinc function

## Syntax

y = diric(x,n) 

## Description

y = diric(x,n) returns a vector or array y the same size as x. The elements of y are the Dirichlet function of the elements of x. n must be a positive integer.

## Examples

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Compute and plot the Dirichlet function between and for N = 7 and N = 8.

x = linspace(-2*pi,2*pi,301); d7 = diric(x,7); d8 = diric(x,8); subplot(2,1,1) plot(x/pi,d7) ylabel('N = 7') title('Dirichlet Function') subplot(2,1,2) plot(x/pi,d8) ylabel('N = 8') xlabel('x / \pi')

The function has a period of for odd N and for even N.

The Dirichlet and sinc functions are related by . Illustrate this fact for .

xmax = 2; x = linspace(-xmax,xmax,1001)'; N = 6; yd = diric(x*pi,N); ys = sinc(N*x/2)./sinc(x/2); subplot(2,1,1) plot(x,yd) title('D_6(x*pi)') subplot(2,1,2) plot(x,ys) title('sinc(6*x/2) / sinc(x/2)')

Repeat the calculation for .

N = 9; yd = diric(x*pi,N); ys = sinc(N*x/2)./sinc(x/2); subplot(2,1,1) plot(x,yd) title('D_9(x*pi)') subplot(2,1,2) plot(x,ys) title('sinc(9*x/2) / sinc(x/2)')

## Diagnostics

If n is not a positive integer, diric gives the following error message:

Requires n to be a positive integer.

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### Dirichlet Function

The Dirichlet function, or periodic sinc function, is

${D}_{N}\left(x\right)=\left\{\begin{array}{ll}\frac{\mathrm{sin}\left(Nx/2\right)}{N\mathrm{sin}\left(x/2\right)}\hfill & x\ne 2\pi k,\text{ }k=0,±1,±2,±3,...\hfill \\ {\left(-1\right)}^{k\left(N-1\right)}\hfill & x=2\pi k,\text{ }k=0,±1,±2,±3,...\hfill \end{array}$

for any nonzero integer N. This function has period 2π for odd N and period 4π for even N. Its peak value is 1, and its minimum value is –1 for even N. The magnitude of the function is 1/N times the magnitude of the discrete-time Fourier transform of the N-point rectangular window.