basics of fourier transform

To solve many engineering problems, we need to know the response
of a Linear Time Invariation 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:

DIRICHLET CONDITIONS

1. 
   1)f(t) has finite number of maxima and minima within T



2. 
   2)f(t) has finite number of discontinuities within T, and



3. 3)It is necessary that
   

These three conditions mean that you are able to calculate the area under the graph.

f (1) is not true then the signal power goes to infinity and it is not absolutely integrable. If (2) is not true then once again it is not absolutely integrable, and (3) is simply reaffirming conditions (1) and (2).

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


and the coefficients are:

In practice 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.

The Fourier series can also be represented in a complex form which is more compact and is more convenient 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