heaviside

Heaviside step function

Syntax

H = heaviside(x)

Description

H = heaviside(x) evaluates the Heaviside step function (also known as the unit step function) at x. The Heaviside function is a discontinuous function that returns 0 for x < 0, 1/2 for x = 0, and 1 for x > 0.

Examples

Evaluate Heaviside Function for Symbolic and Numeric Arguments

The heaviside function returns 0, 1/2, or 1 depending on the argument value. If the argument is a floating-point number (not a symbolic object), then heaviside returns floating-point results.

Evaluate the Heaviside step function for a symbolic input sym(-3). The function heaviside(x) returns 0 for x < 0.

H = heaviside(sym(-3))

H = $$0$$

Evaluate the Heaviside step function for a symbolic input sym(3). The function heaviside(x) returns 1 for x > 0.

H = heaviside(sym(3))

H = $$1$$

Evaluate the Heaviside step function for a symbolic input sym(0). The function heaviside(x) returns 1/2 for x = 0.

H = heaviside(sym(0))

H = 
$$\frac{1}{2}$$

For a numeric input x = 0, the function heaviside(x) returns floating-point results.

H = heaviside(0)

H = 0.5000

Use Assumptions on Symbolic Variables

heaviside takes into account assumptions on variables.

Create a symbolic variable x and assume that it is less than 0.

syms x
assume(x < 0)

Evaluate the Heaviside step function for the symbolic input x.

H = heaviside(x)

H = $$0$$

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

syms x

Plot Heaviside Function

Plot the Heaviside step function for x and x - 1.

syms x
fplot(heaviside(x), [-2, 2])

fplot(heaviside(x - 1), [-2, 2])

Evaluate Heaviside Function for Symbolic Matrix

Evaluate the Heaviside function for a symbolic matrix. When the input argument is a matrix, heaviside computes the Heaviside function for each element.

syms x
H = heaviside(sym([-1 0; 1/2 x]))

H = 
$$\left(\begin{array}{cc}0& \frac{1}{2}\\ 1& \mathrm{heaviside}\left(x\right)\end{array}\right)$$

Differentiate and Integrate Expressions Involving Heaviside Function

Compute derivatives and integrals of expressions involving the Heaviside function.

Find the first derivative of the Heaviside function. The first derivative of the Heaviside function is the Dirac delta function.

syms x
diff_H = diff(heaviside(x),x)

diff_H = $$\delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}\left(x\right)$$

Evaluate the integral ${\int }_{-\infty }^{\infty }{\mathit{e}}^{-\mathit{x}}\mathit{H}\left(\mathit{x}\right)\mathit{dx}$.

syms x
int_H = int(exp(-x)*heaviside(x),x,-Inf,Inf)

int_H = $$1$$

Find Fourier and Laplace Transforms of Heaviside Function

Find the fourier transform of the Heaviside function.

syms x
F = fourier(heaviside(x))

F = 
$$\pi \hspace{0.17em}\delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}\left(w\right)-\frac{\mathrm{i}}{w}$$

Find the laplace transform of the Heaviside function.

syms x
L = laplace(heaviside(x))

L = 
$$\frac{1}{s}$$

Change Value of Heaviside Function at Origin

The default value of the Heaviside function at the origin is 1/2.

H = heaviside(sym(0))

H = 
$$\frac{1}{2}$$

Other common values for the Heaviside function at the origin are 0 and 1. To change the value of heaviside at the origin, use sympref to set the value of the 'HeavisideAtOrigin' preference. Store the previous parameter value returned by sympref, so that you can restore it later.

oldparam = sympref('HeavisideAtOrigin',1);

Check the new value of heaviside at 0.

H = heaviside(sym(0))

H = $$1$$

The preferences set by sympref persist throughout your current and future MATLAB® sessions. To restore the previous value of heaviside at the origin, use the value stored in oldparam.

sympref('HeavisideAtOrigin',oldparam);

Alternatively, you can restore the default value of 'HeavisideAtOrigin' by using the 'default' setting.

sympref('HeavisideAtOrigin','default');

Input Arguments

x — Input

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

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