A

**square wave function**(also called a

*pulse wave*or

*Rademacher function*) is a periodic function that constantly pulses between two values. Common values include the digital signal (0, 1), (-1, 1) and (-½, ½). It is also an odd function, which means it is symmetric around the origin.

In function notation, the square wave function is represented as follows (for a pulse between 1 and -1, pulsing around π):

## Relationship to the Boxcar Function

The square wave function can also be thought of as a **periodic expansion of the boxcar function. ** The graph of a boxcar function has the shape of a boxcar sitting on a “railway track” (the x axis); A periodic expansion would have boxcars placed at constant intervals on the x-axis— creating the square wave function. This may be why this periodic expansion is sometimes called a *pulse train*.

## Approximations

The sine function and cosine function can be used to approximate a square wave function. As an example, the following graph is the sine function f(x) = sin 2πx + (1/3) sin 3 (2πx) + (1/5) sin 5 (2πx) + (1/7) sin 7 (2πx) + (1/9) sin 9 (2πx) + (1/11) sin 11 (2πx) + (1/13) sin 13 (2πx) + (1/15) sin 15 (2πx) + (1/17) sin 17 (2πx) + (1/19) sin 19 (2πx):

## Real Life Uses of the Square Wave Function

Square wave functions are often used in computer science and electrical engineering. For example, it’s used in engineering to **model forces** acting on a mechanical system or in an electric circuit.

## References

Braithwaite, C. (2014). General Inner Product & Fourier Series. Advanced Topics in Linear Algebra, Spring.

Desmos.com (graph)Dobrushkin, V. MATHEMATICA TUTORIAL for the Second Course. Part V: Square wave functions

. Retrieved December 12, 2019 from: http://www.cfm.brown.edu/people/dobrush/am34/Mathematica/ch5/step.html

Singh, M. Fourier Series.