Square Wave

Definition

A square wave is a type of waveform characterized by its sharp, 90° corners and a pattern that alternates between two values. It is a periodic function that produces a clear musical note when played as a sound wave. The waveform is defined by its ability to switch between two distinct levels, typically +1 and -1, in a regular, repeating pattern.

Example

An example of a square wave is a sequence of numbers that alternates between 10,000 and -10,000, repeated 50 times each. This pattern, when repeated 441 times, forms a square wave that produces the musical note A. Another common representation of a square wave is a periodic function that alternates between +1 and -1.

Fourier Series Representation

A square wave can be represented as a Fourier series with specific sinusoidal components. The Fourier coefficients for a square wave are all zero except for the ( b_n ) coefficients for odd ( n ) values, which are given by:

[ b_n = \frac{4}{n\pi} ]

This means that the square wave can be decomposed into a series of sinusoidal basis functions, where only the sine components with odd harmonics contribute to the waveform.

Visualization

The square wave can be visualized as a vector in the space of functions. In this space, it has a component length of ( \frac{4}{\pi} ) in the ( \sin(2\pi t) ) direction and a component length of ( \frac{4}{3\pi} ) in the ( \sin(6\pi t) ) direction. These are the first two components in what would be an infinite list of coordinates for the square wave in this basis.

Figure 13.25 You can think of the square wave as a vector in the space of functions with a component length of 4/π in the sin(2πt) direction and component length of 4/3π in the sin(6πt) direction. The square wave has infinitely many more components beyond these two.

Fourier Coefficients Calculation

To calculate the Fourier coefficients of a square wave, one can use a function like fourier_coefficients(f, N). This function takes a periodic function ( f ) with period one and a number ( N ) of desired coefficients. It treats the constant function, as well as the functions ( \cos(2n\pi t) ) and ( \sin(2n\pi t) ) for ( 1 \leq n < N ), as directions in the vector space of functions and finds the components of ( f ) in those directions. The function returns the Fourier coefficient ( a_0 ), representing the constant function, a list of Fourier coefficients ( a_1, a_2, \ldots, a_N ), and a list of Fourier coefficients ( b_1, b_2, \ldots, b_N ) as a result.