# Formulas of the Fourier transform family

It has annoyed me that there doesn't seem to be a source online where the complete complex Fourier transform family is presented with every variable defined. The lack of definitions can be a nuisance when the transforms often use same symbols to denote different variables. So, what are the relevant transforms and the equations?

(Note: I will post the answers myself, but feel free to edit the answer to be as clear and concise as possible).

As a complement to the above answers, we can also classify the transformations based on the orthogonality condition(s) used to derive each pair of transformations.

Fourier Transformation: $$\frac{1}{2\pi}\int^{+\infty}_{-\infty}e^{ikx}dx = \delta(k)$$

Fourier Series and DTFT: $$\frac{1}{2\pi}\int^{\pi}_{-\pi}e^{ikx}dx = \delta_{k,0}$$ $$\frac{1}{2\pi}\sum^{+\infty}_{k=-\infty}e^{ikx} = \delta(x),\;\;\;\;k\in Z, x\in (-\pi,\pi)$$ I group Fourier Series and DTFT together, because of the duality. The continuous, periodic function in time domain of Fourier Series, can be seen as the frequency domain function in DTFT. Therefore, there is not much difference between the two, at least on mathematical ground.

DFT: $$\frac{1}{N}\sum^{N-1}_{n=0}e^{i\frac{2\pi}{N}kn} = \delta_{k,0},\;\;\;\;k=0,1,...N-1$$

Note: Take caution when dealing with the Delta functions.

• Kronecker Delta: It can be treated as a discrete elementary function, and commonly used under a summation.
• Dirac Delta: It cannot be derived by elementary function, although many introductory texts use $$\delta(x-a) = \lim_{n\to\infty}\sqrt{\frac{n}{\pi}}e^{-n(x-a)^2}$$ to explain it. And treat it as an ordinary function with a value of $$\infty$$ at $$0$$, and a value of $$0$$ otherwise. Delta function is a distribution and has it meaning definite when used under an integral.

Periodic function:

$$\hat{x}(t) = \hat{x}(t+T) \quad \text{for all } t$$

Fourier series:

$$\hat{x}(t) = \sum\limits_{k=-\infty}^{+\infty} \hat{X}_k \ e^{+j k (2\pi/T) t}$$

$$\hat{X}_k = \frac{1}{T} \int\limits_{t_0}^{t_0+T} \hat{x}(t) \ e^{-j k (2\pi/T) t} \ dt \quad \text{for all } t_0$$

Fourier Transform:

$$X(f) = \int\limits_{-\infty}^{+\infty} x(t) e^{-j 2 \pi f t} \ dt$$

$$x(t) = \int\limits_{-\infty}^{+\infty} X(f) e^{+j 2 \pi f t} \ df$$

Let us assume we are in one dimension, and we are not concerned about existence. There are two ordinal variables: time $t$ in the primal space (could be space), frequency $f$ in the dual Fourier space. Both can be either discrete or continuous. For the sake of simplicity, let us assume that the discrete is periodic. One transform, four ($2\times 2$) possiblities, summarized in the following table:

One can draw the different options (in time domain) as:

The duality is much more profound that than, if you are interested, have a look at Pontryagin duality, with Fourier transform notes from Terry Tao blog.

$$\large \text{FOURIER TRANSFORMS}$$

## Fourier series $X[k]$:

$$\large X[k] = \int_{0}^{T}x(t)e^{\huge \frac{-j2\pi kt}{T}}dt$$

$T$ = Length of the window or fundamental oscillation, $\large \frac{1}{T} =$ fundamental frequency , $T$ can also be denoted as $L$.

$t$ = time

$k$ = Index of the harmonic frequency, 1= fundamental etc...

$\large \frac{k}{T}$ = oscillation frequency. The multiples of k are the frequencies and corresponding negative frequencies the transform consists of.

## Fourier transform $X(\omega)$:

$$X(\omega) =\large \int_{-\infty}^{\infty}x(t)e^{\large -j\omega t} d\omega$$

$\omega = \large \frac{2\pi}T$

$T$ = Length of the oscillation, $1/T$ frequency of the oscillation. As can be seen, $T$ is a different variable than in the Fourier series. Oscillation frequency was $\large \frac{k}{T_{series}}$ in the series representation.

Note: Fourier transform can be derived from Fourier series by considering a case where T $\rightarrow \infty$, which makes the frequency domain continuous. Fourier transform is often given as a function of the oscillation frequency $f=\large \frac{1}{T}$, there is nothing preventing doing the same to Fourier series.

## Discrete Fourier transform $X[k]$:

$$\large X[k]=\sum_{n=0}^{N-1}x[n]e^{\huge \frac{-j2\pi kn}{N}}$$

$N$ = Length of the window in samples

$n$ = Number of the sample

$k$ = Index

$\large \frac{N}{f_{s}}$ = Length of the window in time(sec), $\large \frac{kN}{f_{s}}$ = oscillation frequency for each of k. $f_s$ = sampling frequency.

DFT can be derived from Fourier series, by simply considering a case where the signal is discrete and evaluated in terms of samples instead of time. Thus the signal is discrete in time and frequency, which gives rise to the name.

DTFT representation and inverse transforms to be added.

• DFT is not the DTFT. DFT can be described as evaluating the DTFT at $N$ equally spaced (normalized angular) frequencies between $\omega=0$ and $\omega=2\pi$. it can also be described directly as mapping one periodic sequence $\hat{x}[n]$ of period $N$ to another periodic sequence $\hat{X}[k]$ of the same period $N$ without reference to the DTFT. – robert bristow-johnson Dec 31 '15 at 17:24
• @robertbristow-johnson Is there an error somewhere in the equation? Or just a general remark? – Dole Dec 31 '15 at 18:34
• there was a typo error in the equation and the DFT is not a.k.a. the DTFT. – robert bristow-johnson Dec 31 '15 at 23:15
• @robertbristow-johnson Discrete time Fourier SERIES, not DTFT... – Dole Jan 1 '16 at 11:31