sinc

Sinc function

Syntax

y = sinc(x)

Description

y = sinc(x) returns an array, y, whose elements are the sinc of the elements of the input, x. The output y is the same size as x.

Examples

Ideal Bandlimited Interpolation

Perform ideal bandlimited interpolation of a random signal sampled at integer spacings.

Assume that the signal to interpolate, x, is 0 outside of the given time interval and has been sampled at the Nyquist frequency. Reset the random number generator for reproducibility.

rng default

t = 1:10;
x = randn(size(t))';
ts = linspace(-5,15,600);
[Ts,T] = ndgrid(ts,t);
y = sinc(Ts - T)*x;

plot(t,x,'o',ts,y)
xlabel Time, ylabel Signal
legend('Sampled','Interpolated','Location','SouthWest')
legend boxoff

Input Arguments

x — Input array

scalar value | vector | matrix | N-D array

Input array, specified as a real-valued or complex-valued scalar, vector, matrix, or N-D array. When x is nonscalar, sinc is an element-wise operation.

Data Types: single | double
Complex Number Support: Yes

Output Arguments

y — Sinc of input

scalar value | vector | matrix | N-D array

Sinc of the input array, x, returned as a real-valued or complex-valued scalar, vector, matrix, or N-D array of the same size as x.

More About

sinc

The sinc function is defined by

$$\mathrm{sinc}t=\{\begin{array}{cc}\frac{\sin \pi t}{\pi t}& t\ne 0,\\ 1& t=0.\end{array}$$

This analytic expression corresponds to the continuous inverse Fourier transform of a rectangular pulse of width 2π and height 1:

$$\mathrm{sinc}t=\frac{1}{2\pi }{\displaystyle {\int }_{-\pi }^{\pi }{e}^{j\omega t}d\omega .}$$

The space of functions bandlimited in the frequency range $$\omega =(-\pi ,\pi ]$$ is spanned by the countably infinite set of sinc functions shifted by integers. Thus, you can reconstruct any such bandlimited function g(t) from its samples at integer spacings:

$$g(t)={\displaystyle \sum _{n=-\infty }^{\infty }g(n)\mathrm{sinc}(t-n)}.$$

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

The sinc function fully supports tall arrays. For more information, see Tall Arrays.

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

The sinc function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced before R2006a

R2026a: Enhanced accuracy for integer inputs

Since R2026a, the sinc function uses sinpi instead of sin to calculate the sinc output. The change results in higher accuracy for integer inputs.

For an input x as integers from 1 to 5, this code shows the difference between sinc outputs. The sinc function returns exact zeros, agreeing with the sinc definition.

x = 1:5;
y = struct("sin",sin(pi*t)./(pi*t), ...
    "sinpi",sinpi(t)./(pi*t), ...
    "sinc",sinc(t))

y = struct with fields:
      sin: [3.8982e-17 -3.8982e-17 3.8982e-17 -3.8982e-17 3.8982e-17]
    sinpi: [0 0 0 0 0]
     sinc: [0 0 0 0 0]