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sinc

Normalized sinc function

Syntax

Description

example

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

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Create a sinc function of a symbolic variable x.

syms x
sinc(x)
ans = 

sin(πx)xπ

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 = 

(010-23π)

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

vpa(S)
ans = (01.00-0.21220659078919378102517835116335)

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

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

syms x
fourier(rectangularPulse(x))
ans = 

sin(w2)+cos(w2)iw--sin(w2)+cos(w2)iw

Show that fourier transforms sinc in terms of heaviside.

fourier(sinc(x))
ans = 

πheaviside(π-w)-πheaviside(-w-π)π

Plot the sinc function by using fplot.

syms x
fplot(sinc(x))

Figure contains an axes object. The axes object contains an object of type functionline.

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

syms x
rewrite(sinc(x),'exp')
ans = 

e-πxii2-eπxii2xπ

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

Differentiate sinc.

syms x
diff(sinc(x))
ans = 

cos(πx)x-sin(πx)x2π

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(πx)π+12

Find the Taylor expansion of sinc.

taylor(sinc(x))
ans = 

π4x4120-π2x26+1

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

Prove the identity sinc(x)=1Γ(1+x)Γ(1-x).

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

Input Arguments

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

Version History

Introduced in R2018b

See Also