# Fourier Series

## Introduction

The Fourier Series is only briefly covered here as a backdrop to the Fourier Transform and the Discrete Fourier Transform. It is not covered in any depth as I assume that you have already encountered the Fourier Series before. If there are enough requests this section can be expanded to give a FULL breakdown of the Fourier Transform.

To solve many engineering problems, we need to know the response of a Linear Time Invariant system to some input signal. If the input signal can be broken up into simple signals and we know how the system responds to these simple signals, then we can predict how the system will behave to our input.

Therefore anything that can break a signal down into it's constituent parts would be very useful. One such tool is the Fourier Series

With the exception of some mathematical curiosities, any periodic signal of period T can be expanded into a trigonometric series of sine and cosine functions, as long as it obeys the following conditions:

1. f(t) has finite number of maxima and minima within T
2. f(t) has finite number of discontinuties within T, and
3. It is necessary that
These three conditions mean that you are able to calculate the area under the graph. If (1) is not true then the signal power goes to infinity and it is not absolutely integratable. If (2) is not true then once again it is not absolutely integratable, and (3) is simply reafirming conditions (1) and (2).

If all of these are true then the signal can be represented as:

and the coefficents are:

### In Practice

In practise things don't seem as complicated as the theory. For a start you only need a finite number of harmonics to construct f(t) with an acceptable error.

An example of a fourier series approximation of a saw-tooth wave.

• The first signal has only two harmonics,
• The second signal has five harmonics,
• The third signal has ten harmonics, and
• The original signal at the bottom.
With ten harmonics, the signal is virtually identical to the original.

The other problem is that the signal must be periodic; few real world signals are truly periodic. You can get around this by artifically changing a non-periodic signal (over time T) so that it becomes periodic. e.g. a pulse signal can be repeated to give the apperance of a periodic signal.

e.g. From the non-periodic input signal

we can produce this periodic signal, of period T0

The Fourier series can also be represented in a complex form which is more compact and is more convient when dealing with complex signals.

If we make use of a trig identity

and we define

we get the complex form of the fourier series to be

So now we know how to expand vitually any signal with the Fourier Series. We take this knowledge onto the Fourier Transform

Back to Contents or on to Fourier Transform

or on to Discrete Fourier Transform or on to Fast Fourier Transform