dirac

Dirac delta function

Syntax

d = dirac(x)

d = dirac(n,x)

Description

d = dirac(x) represents the Dirac delta function of x.

d = dirac(n,x) represents the nth derivative of the Dirac delta function at x.

Examples

Handle Expressions Involving Dirac and Heaviside Functions

Compute derivatives and integrals of expressions involving the Dirac delta and Heaviside functions.

Find the first and second derivatives of the Heaviside function. The result is the Dirac delta function and its first derivative.

syms x
diff(heaviside(x), x)
diff(heaviside(x), x, x)

ans =
dirac(x)
 
ans =
dirac(1, x)

Find the indefinite integral of the Dirac delta function. The results returned by int do not include integration constants.

int(dirac(x), x)

ans =
sign(x)/2

Find the integral of the sine function involving the Dirac delta function.

syms a
int(dirac(x - a)*sin(x), x, -Inf, Inf)

ans =
sin(a)

Use Assumptions on Variables

dirac takes into account assumptions on variables.

syms x real
assumeAlso(x ~= 0)
dirac(x)

ans =
0

For further computations, clear the assumptions on x by recreating it using syms.

syms x

Evaluate Dirac Delta Function for Symbolic Matrix

Compute the Dirac delta function of x and its first three derivatives.

Use a vector n = [0,1,2,3] to specify the order of derivatives. The dirac function expands the scalar into a vector of the same size as n and computes the result.

syms x
n = [0,1,2,3];
d = dirac(n,x)

d =
[ dirac(x), dirac(1, x), dirac(2, x), dirac(3, x)]

Substitute x with 0.

subs(d,x,0)

ans =
[ Inf, -Inf, Inf, -Inf]

Plot Dirac Delta Function

You can use fplot to plot the Dirac delta function over the default interval [-5 5]. However, dirac(x) returns Inf at x equal to 0, and fplot does not plot the infinity.

Declare a symbolic variable x and plot the symbolic expression dirac(x) by using fplot.

syms x
fplot(dirac(x))

To handle the infinity at x equal to 0, use numeric values instead of symbolic values. Set the Inf value to 1 and plot the Dirac delta function by using stem.

x = -1:0.1:1;
y = dirac(x);
idx = y == Inf; % find Inf
y(idx) = 1;     % set Inf to finite value
stem(x,y)

Input Arguments

x — Input

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

Input, specified as a number, symbolic number, variable, expression, or function, representing a real number. This input can also be a vector, matrix, or multidimensional array of numbers, symbolic numbers, variables, expressions, or functions.

n — Order of derivative

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

Order of derivative, specified as a nonnegative number, or symbolic variable, expression, or function representing a nonnegative number. This input can also be a vector, matrix, or multidimensional array of nonnegative numbers, symbolic numbers, variables, expressions, or functions.

More About

Dirac Delta Function

The Dirac delta function, δ(x), has the value 0 for all x ≠ 0, and ∞ for x = 0. The Dirac delta function satisfies the identity

$${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\delta (x)dx}=1\text{.}$$

This is a heuristic definition of the Dirac delta function. A rigorous definition of the Dirac delta function requires the theory of distributions or measure theory.

For any smooth function f and a real number a, the Dirac delta function has the property

$${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\delta (x-a)f(x)}=f(a)\text{.}$$

Tips

• For complex values x with nonzero imaginary parts, dirac returns NaN.

• dirac returns floating-point results for numeric arguments that are not symbolic objects.

• dirac acts element-wise on nonscalar inputs.

• The input arguments x and n must be vectors or matrices of the same size, or else one of them must be a scalar. If one input argument is a scalar and the other one is a vector or a matrix, then dirac expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.