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# Fourier's Daring Leap

Fourier postulated around 1807 that any periodic signal (equivalently finite length signal) can be built up as an infinite linear combination of harmonic sinusoidal waves.

i.e. Given the collection

$B = { e j 2 π T n t } n = - ∞ ∞ B = { e j 2 π T n t } n = - ∞ ∞$
1

any

$f ( t ) ∈ L 2 [ 0 , T ) f ( t ) ∈ L 2 [ 0 , T )$
2

can be approximated arbitrarily closely by

$f ( t ) = ∑ n = - ∞ ∞ C n e j 2 π T n t . f ( t ) = ∑ n = - ∞ ∞ C n e j 2 π T n t .$
3

Now, The issue of exact convergence did bring Fourier much criticism from the French Academy of Science (Laplace, Lagrange, Monge and LaCroix comprised the review committee) for several years after its presentation on 1807. It was not resolved for also a century, and its resolution is interesting and important to understand from a practical viewpoint. See more in the section on Gibbs Phenomena.

Fourier analysis is fundamental to understanding the behavior of signals and systems. This is a result of the fact that sinusoids are Eigenfunctions of linear, time-invariant (LTI) systems. This is to say that if we pass any particular sinusoid through a LTI system, we get a scaled version of that same sinusoid on the output. Then, since Fourier analysis allows us to redefine the signals in terms of sinusoids, all we need to do is determine how any given system effects all possible sinusoids (its transfer function) and we have a complete understanding of the system. Furthermore, since we are able to define the passage of sinusoids through a system as multiplication of that sinusoid by the transfer function at the same frequency, we can convert the passage of any signal through a system from convolution (in time) to multiplication (in frequency). These ideas are what give Fourier analysis its power.

Now, after hopefully having sold you on the value of this method of analysis, we must examine exactly what we mean by Fourier analysis. The four Fourier transforms that comprise this analysis are the Fourier Series, Continuous-Time Fourier Transform, Discrete-Time Fourier Transform and Discrete Fourier Transform. For this document, we will view the Laplace Transform and Z-Transform as simply extensions of the CTFT and DTFT respectively. All of these transforms act essentially the same way, by converting a signal in time to an equivalent signal in frequency (sinusoids). However, depending on the nature of a specific signal i.e. whether it is finite- or infinite-length and whether it is discrete- or continuous-time) there is an appropriate transform to convert the signal into the frequency domain. Below is a table of the four Fourier transforms and when each is appropriate. It also includes the relevant convolution for the specified space.

Transform Time Domain Frequency Domain Convolution
Continuous-Time Fourier Series $L 2 ⁢ 0 T L 2 0 T$ $l 2 ⁢Z l 2$ Continuous-Time Circular
Continuous-Time Fourier Transform $L 2 ⁢R L 2$ $L 2 ⁢R L 2$ Continuous-Time Linear
Discrete-Time Fourier Transform $l 2 ⁢Z l 2$ $L 2 ⁢ 0 2⁢π L 2 0 2$ Discrete-Time Linear
Discrete Fourier Transform $l 2 ⁢ 0 N−1 l 2 0 N 1$ $l 2 ⁢ 0 N−1 l 2 0 N 1$ Discrete-Time Circular
Table 1: Table of Fourier Representations