Fourier Transforms

Definition of Fourier Transform

The frequencies associated with coefficients of the Fourier Series, cn, is w0.  Since w0=2p/T, the spacing between consecutive coefficients decreases as T increases.  As T goes to infinity (i.e. the function becomes aperiodic), the spacing between coefficients goes to zero and we get a continuous "spectrum" rather than distinct values.  I used this information to derive the Fourier Transform from the Fourier Series.  Start with the Fourier Series for cn:

X(w) is the Fourier Transform of the aperiodic function x(t).  We can also derive the inverse Fourier Transform by starting with the Fourier Series definition of x(t)

Fourier Transform
Synthesis Equation
Analysis Equation

Relationship between Fourier Series and Fourier Transform

Consider a function xp(t) that is the periodic extension, with period=T, of the aperiodic function x(t).

The Fourier Transform (X(w)) of the function x(t) is related to the Fourier Series coefficients (cn) of the function xp(t) by:

 


Convergence of Fourier Transform

Without proof I stated that the Fourier Transform exists if three criteria are met

  •  

  • There are a finite number of discontinuities in a finite interval.

  • There are a finite number of maxima and minima in a finite interval.

Only the first one of these criteria will matter to us

Fourier Transform Properties

You might want to print out  Fourier Transform Pairs and Fourier Transform Properties.  With just a few pairs and properties I showed that you could find the Fourier Xform of many different functions that were not in the table.   With a little cleverness and consideration you can often find the Fourier Xform of a function without having to resort to integration by consulting the table.

Fourier Transform of a Periodic Function

It seems that a periodic function would not have a Fourier Transform because it violates the first of the convergence criteria.  However, if we allow for impulse functions, we can get around this restriction (this will allow us to use Fourier Transforms for both periodic and aperiodic functions).

Consider the a frequency domain function that is a simple impulse scaled by 2p (the scaling factor will be convenient a bit later).

We can find the corresponding time domain function by calculating the inverse Fourier Transform,

(The last step was performed using the sifting property of the impulse function.)  Note that the time domain function, x(t), is periodic.  So if we allow impulses in the Fourier domain we can have periodic functions in the time domain.  This was a special case, but we can represent any (subject to convergence criteria like those for the Fourier Series) periodic function with a Fourier Transform.  First consider a Fourier Transform that is an infinite sum of impulses (this is contrived, but it simplifies to something useful).

(This derivation also uses the sifting property.)  So, to find the Fourier Transform of a periodic signal, x(t), first find the Fourier Series coefficients, cn, then


In class I gave you a handout that relates Fourier Xforms of aperiodic functions (x(t)), the Fourier Series of the periodic extension of aperiodic functions (xp(t)) and the Fourier Xform of the periodic extension.  The handout is here.