2 Informal derivation of the Fourier transform

Recall that if $f\left(t\right)$ is a period $T$ function, which we will temporarily re-write as $f_T\left(t\right)$ for emphasis, then we can expand it in a complex Fourier series,

$f_T\left(t\right)={\sum }_{n=-\infty }^{\infty }{c}_n{e}^{\text{i}n{\omega }_{0}t}$ (1)

where ${\omega }_0=\frac{2\pi }{T}$ . In words, harmonics of frequency $n{\omega }_0=n\frac{2\pi }{T}n=0,\pm 1,\pm 2,\dots$ are present in the series and these frequencies are separated by

$n{\omega }_0-\left(n-1\right){\omega }_0={\omega }_0=\frac{2\pi }{T}.$

Hence, as $T$ increases the frequency separation becomes smaller and can be conveniently written as $\Delta \omega$ . This suggests that as $T\to \infty$ , corresponding to a non-periodic function, then $\Delta \omega \to 0$ and the frequency representation contains all frequency harmonics.

To see this in a little more detail, we recall ( HELM booklet  23: Fourier series) that the complex Fourier coefficients $c_n$ are given by

$c_n=\frac{1}{T}{\int \; <!--nolimits\; -->}_{-\frac{T}{2}}^{\frac{T}{2}}{f}_T\left(t\right)e^{-\text{i}n{\omega }_{0}t}dt.$ (2)

Putting $\frac{1}{T}$ as $\frac{{\omega }_0}{2\pi }$ and then substituting (2) in (1) we get

$f_T\left(t\right)={\sum }_{n=-\infty }^{\infty }\left\{\frac{{\omega }_0}{2\pi }{\int \; <!--nolimits\; -->}_{-\frac{T}{2}}^{\frac{T}{2}}{f}_T\left(t\right)e^{-\text{i}n{\omega }_{0}t}dt\right\}{e}^{\text{i}n{\omega }_{0}t}.$

In view of the discussion above, as $T\to \infty$ we can put ${\omega }_0$ as $\Delta \omega$ and replace the sum over the discrete frequencies $n{\omega }_0$ by an integral over all frequencies. We replace $n{\omega }_0$ by a general frequency variable $\omega$ . We then obtain the double integral representation

$f\left(t\right)={\int \; <!--nolimits\; -->}_{-\infty }^{\infty }\left\{\frac{1}{2\pi }{\int \; <!--nolimits\; -->}_{-\infty }^{\infty }f\left(t\right)e^{-\text{i}\omega t}dt\right\}{e}^{\text{i}\omega t}d\omega .$ (3)

The inner integral (over all $t$ ) will give a function dependent only on $\omega$ which we write as $F\left(\omega \right)$ . Then (3) can be written

$f\left(t\right)=\frac{1}{2\pi }{\int \; <!--nolimits\; -->}_{-\infty }^{\infty }F\left(\omega \right)e^{\text{i}\omega t}d\omega$ (4)

where

$F\left(\omega \right)={\int \; <!--nolimits\; -->}_{-\infty }^{\infty }f\left(t\right)e^{-\text{i}\omega t}dt.$ (5)

The representation (4) of $f\left(t\right)$ which involves all frequencies $\omega$ can be considered as the equivalent for a non-periodic function of the complex Fourier series representation (1) of a periodic function.

The expression (5) for $F\left(\omega \right)$ is analogous to the relation (2) for the Fourier coefficients $c_n$ .

The function $F\left(\omega \right)$ is called the Fourier transform of the function $f\left(t\right)$ . Symbolically we can write

$F\left(\omega \right)=\mathcal{F}\left\{f\left(t\right)\right\}.$

Equation (4) enables us, in principle, to write $f\left(t\right)$ in terms of $F\left(\omega \right)$ . $f\left(t\right)$ is often called the inverse Fourier transform of $F\left(\omega \right)$ and we denote this by writing

$f\left(t\right)={\mathcal{F}}^{-1}\left\{F\left(\omega \right)\right\}.$

Looking at the basic relation (3) it is clear that the position of the factor $\frac{1}{2\pi }$ is somewhat arbitrary in (4) and (5). If instead of (5) we define

$F\left(\omega \right)=\frac{1}{2\pi }{\int \; <!--nolimits\; -->}_{-\infty }^{\infty }f\left(t\right)e^{-\text{i}\omega t}dt.$

then (4) must be written

$f\left(t\right)={\int \; <!--nolimits\; -->}_{-\infty }^{\infty }F\left(\omega \right)e^{\text{i}\omega t}d\omega .$

A third, more symmetric, alternative is to write

$F\left(\omega \right)=\frac{1}{\sqrt{2\pi }}{\int \; <!--nolimits\; -->}_{-\infty }^{\infty }f\left(t\right)e^{-\text{i}\omega t}dt$

and, consequently:

$f\left(t\right)=\frac{1}{\sqrt{2\pi }}{\int \; <!--nolimits\; -->}_{-\infty }^{\infty }F\left(\omega \right)e^{\text{i}\omega t}d\omega .$

We shall use (4) and (5) throughout this Section but you should be aware of these other possibilities which might be used in other texts.

Engineers often refer to $F\left(\omega \right)$ (whichever precise definition is used!) as the frequency domain representation of a function or signal and $f\left(t\right)$ as the time domain representation. In what follows we shall use this language where appropriate. However, (5) is really a mathematical transformation for obtaining one function from another and (4) is then the inverse transformation for recovering the initial function. In some applications of Fourier transforms (which we shall not study) the time/frequency interpretations are not relevant. However, in engineering applications, such as communications theory, the frequency representation is often used very literally.

As can be seen above, notationally we will use capital letters to denote Fourier transforms: thus a function $f\left(t\right)$ has a Fourier transform denoted by $F\left(\omega \right)$ , $g\left(t\right)$ has a Fourier transform written $G\left(\omega \right)$ and so on. The notation $F\left(\text{i}\omega \right)$ , $G\left(\text{i}\omega \right)$ is used in some texts because $\omega$ occurs in (5) only in the term $e^{-\text{i}\omega t}$ .