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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
any
can be approximated arbitrarily closely by
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 notresolved 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 thatsinusoids are Eigenfunctions of linear, time-invariant (LTI) systems. This is to say that if we pass any particular sinusoid through aLTI system, we get a scaled version of that same sinusoid on the output. Then, since Fourier analysis allows us to redefine thesignals in terms of sinusoids, all we need to do is determinehow any given system effects all possible sinusoids (its transfer function ) and we have a complete understanding of the system. Furthermore, sincewe are able to define the passage of sinusoids through a system as multiplication of that sinusoid by the transfer function atthe 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 itspower.
Now, after hopefully having sold you on the value of this method
of analysis, we must examine exactly what we mean by Fourieranalysis. 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 thesetransforms 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
Transform | Time Domain | Frequency Domain | Convolution |
---|---|---|---|
Continuous-Time Fourier Series | ${L}^{2}(\left[0 , T\right))$ | ${l}^{2}(\mathbb{Z})$ | Continuous-Time Circular |
Continuous-Time Fourier Transform | ${L}^{2}(\mathbb{R})$ | ${L}^{2}(\mathbb{R})$ | Continuous-Time Linear |
Discrete-Time Fourier Transform | ${l}^{2}(\mathbb{Z})$ | ${L}^{2}(\left[0 , 2\pi \right))$ | Discrete-Time Linear |
Discrete Fourier Transform | ${l}^{2}(\left[0 , N-1\right])$ | ${l}^{2}(\left[0 , N-1\right])$ | Discrete-Time Circular |
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