# MATLAB - Transforms

MATLAB provides command for working with transforms, such as the Laplace and Fourier transforms. Transforms are used in science and engineering as a tool for simplifying analysis and look at data from another angle.

For example, the Fourier transform allows us to convert a signal represented as a function of time to a function of frequency. Laplace transform allows us to convert a differential equation to an algebraic equation.

MATLAB provides the **laplace**, **fourier** and **fft** commands to work with Laplace, Fourier and Fast Fourier transforms.

## The Laplace Transform

The Laplace transform of a function of time f(t) is given by the following integral −

Laplace transform is also denoted as transform of f(t) to F(s). You can see this transform or integration process converts f(t), a function of the symbolic variable t, into another function F(s), with another variable s.

Laplace transform turns differential equations into algebraic ones. To compute a Laplace transform of a function f(t), write −

laplace(f(t))

### Example

In this example, we will compute the Laplace transform of some commonly used functions.

Create a script file and type the following code −

syms s t a b w laplace(a) laplace(t^2) laplace(t^9) laplace(exp(-b*t)) laplace(sin(w*t)) laplace(cos(w*t))

When you run the file, it displays the following result −

ans = 1/s^2 ans = 2/s^3 ans = 362880/s^10 ans = 1/(b + s) ans = w/(s^2 + w^2) ans = s/(s^2 + w^2)

## The Inverse Laplace Transform

MATLAB allows us to compute the inverse Laplace transform using the command **ilaplace**.

For example,

ilaplace(1/s^3)

MATLAB will execute the above statement and display the result −

ans = t^2/2

### Example

Create a script file and type the following code −

syms s t a b w ilaplace(1/s^7) ilaplace(2/(w+s)) ilaplace(s/(s^2+4)) ilaplace(exp(-b*t)) ilaplace(w/(s^2 + w^2)) ilaplace(s/(s^2 + w^2))

When you run the file, it displays the following result −

ans = t^6/720 ans = 2*exp(-t*w) ans = cos(2*t) ans = ilaplace(exp(-b*t), t, x) ans = sin(t*w) ans = cos(t*w)

## The Fourier Transforms

Fourier transforms commonly transforms a mathematical function of time, f(t), into a new function, sometimes denoted by or F, whose argument is frequency with units of cycles/s (hertz) or radians per second. The new function is then known as the Fourier transform and/or the frequency spectrum of the function f.

### Example

Create a script file and type the following code in it −

syms x f = exp(-2*x^2); %our function ezplot(f,[-2,2]) % plot of our function FT = fourier(f) % Fourier transform

When you run the file, MATLAB plots the following graph −

The following result is displayed −

FT = (2^(1/2)*pi^(1/2)*exp(-w^2/8))/2

Plotting the Fourier transform as −

ezplot(FT)

Gives the following graph −

## Inverse Fourier Transforms

MATLAB provides the **ifourier** command for computing the inverse Fourier transform of a function. For example,

f = ifourier(-2*exp(-abs(w)))

MATLAB will execute the above statement and display the result −

f = -2/(pi*(x^2 + 1))