sinc

Normalized sinc function

Syntax

sinc(x)

Description

sinc(x) returns sin(pi*x)/(pi*x). The symbolic sinc function does not implement floating-point results, only symbolic results. Floating-point results are returned by the sinc function in Signal Processing Toolbox™.

Examples

Sinc Function of Symbolic Inputs

syms x
sinc(x)

ans =
sin(pi*x)/(x*pi)

Show that sinc returns 1 at 0, 0 at other integer inputs, and exact symbolic values for other inputs.

V = sym([-1 0 1 3/2]);
S = sinc(V)

S =
[ 0, 1, 0, -2/(3*pi)]

Convert the exact symbolic output to high-precision floating point by using vpa.

vpa(S)

ans =
[ 0, 1.0, 0, -0.21220659078919378102517835116335]

Fourier Transforms Involving Sinc Function

Although sinc appears in tables of Fourier transforms, fourier does not return sinc in output.

Show that fourier transforms a pulse in terms of sin and cos.

fourier(rectangularPulse(x))

ans =
(cos(w/2)*1i + sin(w/2))/w - (cos(w/2)*1i - sin(w/2))/w

Show that fourier transforms sinc in terms of heaviside.

syms x
fourier(sinc(x))

ans =
(pi*heaviside(pi - w) - pi*heaviside(- w - pi))/pi

Plot Sinc Function

Plot the sinc function by using fplot.

syms x
fplot(sinc(x))

Rewrite Sinc Function to Other Functions

Rewrite the sinc function to the exponential function exp by using rewrite.

syms x
rewrite(sinc(x),'exp')

ans =
((exp(-pi*x*1i)*1i)/2 - (exp(pi*x*1i)*1i)/2)/(x*pi)

Differentiate, Integrate, and Expand the Sinc Function

Differentiate, integrate, and expand sinc by using the diff, int, and taylor functions, respectively.

Differentiate sinc.

syms x
diff(sinc(x))

ans =
cos(pi*x)/x - sin(pi*x)/(x^2*pi)

Integrate sinc from -Inf to Inf.

int(sinc(x),[-Inf Inf])

ans =
1

Integrate sinc from -Inf to x.

int(sinc(x),-Inf,x)

ans =
sinint(pi*x)/pi + 1/2

Find the Taylor expansion of sinc.

taylor(sinc(x))

ans =
(pi^4*x^4)/120 - (pi^2*x^2)/6 + 1

Prove Identity Involving Sinc Function

Prove an identity by defining the identity as a condition and using the isAlways function to check the condition.

Prove this identity.

$$\text{sinc}\left(x\right)=\frac{1}{\Gamma (1+x)\Gamma (1-x)}.$$

syms x
cond = sinc(x) == 1/(gamma(1+x)*gamma(1-x));
isAlways(cond)

ans =
  logical
   1

Input Arguments

x — Input

number | vector | matrix | array | symbolic number | symbolic variable | symbolic array | symbolic function | symbolic expression

Input, specified as a number, vector, matrix, or array, or a symbolic number, variable, array, function, or expression.