Fourier transforms: periodicity maps to discreteness

Motivation for the definition of Fourier transform is often given as to capture the individual frequency components of periodic function(or, in the discrete-time case, a periodic series).

The frequency components (basis in Linear Algebra terms?) that express the periodic function are of the form $e^{j\omega t}$ (or, $e^{j\omega n}$ in case of discrete-time case).

Note that, in this article, we refer to independent variable of the original (call it original, for lack of a better word) function to be time, while the independent variable of the transform function to be frequency. And, we refer to a function of a real variable to be a continuous function, while we refer to a function defined on the set of integers to be a discrete function. Combining the two, we refer to:

an original function of a real variable as a continuous-time function
an original function defined on the set of integers as a discrete-time function
a transform function of a real variable as a continuous-frequency function
a transform function defined on the set of integers as a discrete-frequency function

A continuous-time periodic function $x(t)$ of period $T$ has a Fourier series expansion (subject to certain conditions) as follows:

$x(t) = \sum\limits_{k=-\infty}^{\infty}a_ke^{j\frac{2\pi}{T}k}$ (Synthesis equation)

where $a_k$ are referred to as the Fourier series coefficients. Now, the values of the Fourier series coefficients are given by:

$a_k = \frac{1}{T}\int\limits_Tx(t)e^{-j\frac{2\pi}{T}t}\,dt$ (Analysis equation)

An observation that can be made is that the Fourier expression of a periodic continuous-time function is a discrete-frequency aperiodic function.

Next, consider a discrete-time periodic function $x[n]$ with period $N$ . The Fourier series analysis and synthesis equations in this case are given by:

$x[n] = \frac{1}{N}\sum\limits_{k=\langle N\rangle} a_k e^{j\frac{2\pi}{N}nk}$
$a_k = \frac{1}{N}\sum\limits_{n=\langle N\rangle} x[n] e^{-j\frac{2\pi}{N}nk}$

In this case, we have a discrete-time periodic function and we find that its Fourier expression is a periodic discrete-frequency function. Note that, we only add finite number ( $N$ ) of terms in the synthesis equation. We regard the Fourier series coefficients to be infinite in number that just repeat with a period of $N$ .

Now, we consider the most general case which is an aperiodic continuous-time function $x(t)$ . The Fourier transform (analysis) and inverse Fourier transform (synthesis) equations are given by:

$x(t) = \frac{1}{2\pi}\int\limits_{-\infty}^{\infty} X(\omega)e^{j\omega t}\,dt$
$X(\omega) = \int\limits_{-\infty}^{\infty} x(t)e^{-j\omega t}\,d\omega$

The Fourier transform of an aperiodic continuous-time function is a continuous-frequency aperiodic function.

We could consider the final case of an aperiodic discrete-time function $x[n]$ whose synthesis and analysis equations are as follows:

$x[n] = \frac{1}{2\pi}\int\limits_{2\pi}X(\omega)e^{j\omega n}\,d\omega$
$X(\omega) = \sum\limits_{n=-\infty}^{\infty}x[n]e^{-j\omega n}$

The Fourier transform (called discrete-time Fourier transform) of an aperiodic discrete-time function is a continuous-frequency periodic function.

What is interesting to note is how periodicity maps to discreteness.