Derivation of the fourier transform

Let's begin by writing down the formula for the complex form of the Fourier Series:

as well as the corresponding Fourier Series coefficients:

As was mentioned in Chapter 2, as the period $T$ gets large, the Fourier Series coefficients represent more closely spaced frequencies. Lets take the limit as the period $T$ goes to infinity. We first note that the fundamental frequency approaches a differential

consequently

The $n$ th harmonic, $n{\Omega }_0$ , in the limit approaches the frequency variable $\Omega$

From equation [link] , we have

The right hand side of [link] is called the Fourier Transform of $x\left(t\right)$ :

Now, using [link] , [link] , and [link] in equation [link] gives

which corresponds to the inverse Fourier Transform . Equations [link] and [link] represent what is known as a transform pair . The following notation is used to denote a Fourier Transform pair

We say that $x\left(t\right)$ is a time domain signal while $X\left(j\Omega \right)$ is a frequency domain signal. Some additional notation which is sometimes used is

and

References