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:
If all of these are true then the signal can be represented as:
and the coefficents are:
An example of a fourier series approximation of a saw-tooth wave.
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