diric
Dirichlet or periodic sinc function
Syntax
Description
returns the Dirichlet Function of degree
y
= diric(x
,n
)n
evaluated at the elements of the input array
x
.
Examples
Dirichlet Function
Compute and plot the Dirichlet function between and for N = 7 and N = 8. The function has a period of for odd N and for even N.
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')
Periodic and Aperiodic Sinc Functions
The Dirichlet and sinc functions are related by . Show this relationship for . Avoid indeterminate expressions by specifying that the ratio of sinc functions is for , where is an integer.
xmax = 4; x = linspace(-xmax,xmax,1001)'; N = 6; yd = diric(x*pi,N); ys = sinc(N*x/2)./sinc(x/2); ys(~mod(x,2)) = (-1).^(x(~mod(x,2))/2*(N-1)); 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 = 13; yd = diric(x*pi,N); ys = sinc(N*x/2)./sinc(x/2); ys(~mod(x,2)) = (-1).^(x(~mod(x,2))/2*(N-1)); subplot(2,1,1) plot(x,yd) title('D_{13}(x*pi)') subplot(2,1,2) plot(x,ys) title('sinc(13*x/2) / sinc(x/2)')
Input Arguments
x
— Input array
real scalar | real vector | real matrix | real N-D array
Input array, specified as a real scalar, vector, matrix, or N-D
array. When x
is nonscalar, diric
is an
element-wise operation.
Data Types: double
| single
n
— Function degree
positive integer scalar
Function degree, specified as a positive integer scalar.
Data Types: double
| single
Output Arguments
y
— Output array
real scalar | real vector | real matrix | real N-D array
Output array, returned as a real-valued scalar, vector, matrix, or
N-D array of the same size as x
.
More About
Dirichlet Function
The Dirichlet function, or periodic sinc function, is
for any nonzero integer N.
This function has period 2π for odd N and period 4π for even N. Its maximum value is 1 for all N, 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.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Version History
Introduced before R2006a
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