Can anyone tell me exactly how the Fourier transforms were discovered? What was the need at that time to research it?
What is the physical significance of them especially Discrete Fourier transform?
refer to this link for ur easy understanding
http://books.google.co.in/books?id=coq49_LRURUC&pg=PA86&lpg=PA86&dq=physical+significance+of+dft&source=bl&ots=pLInWekamR&sig=XTXKyv8qL-XZopsL8F5QBVxBVAo&hl=en&sa=X&ei=1RF8UZGaFILsrAenqYC4Bg&ved=0CC4Q6AEwAA#v=onepage&q=physical%20significance%20of%20dft&f=false
Fourier transforms were introduced to convert aperiodic signals from time domain to frequency domain, in this way one can observe the spectra of the aperiodic signals. Discrete time fourier transform give the continuous spectrum of discrete time signal.
Fourier transforms, were discovered, for move time-frecuency, since, in space of time are not observed the frequencies.
Also, we consider that FT is only for stationary signals, when talking about time-frequency representation. Otherwise, it is appropriate.
Refer to this link
http://fourier.eng.hmc.edu/e101/lectures/handout2_tex/node2.html
The real meaning of fourier transform is that to convert a signal which is in time domain into frequency domain and vise versa. The advantage to use fourier transform is there when a signal is in time domain of infinite length and you want to analyze its response for any given channel. Since the signal is of infinite time length so you convert it into frequency domain where it will be periodic for a frequency range (so you take its first part) and you also convert channel in freq domain and there you just convolute/ multiply (as required) with each other and easily gets the answer.
The book "Signals & Systems" by Oppenheim and Willsky has a four-page account on the historical development of the Fourier transforms, including why it was developed and what has been available already at these times. (IIRC the Fourier series was developed first.)
Fourier Transform may be seen as the further step after such mathematical tricks like Taylor series expansion or orthogonal polynomials. The task is to calculate, up to given precision, the values of a possibly very complicated function, with least effort. Taylor series is good only in a close vicinity of a given point, orthogonal polynomials are usable in wider domains but they fail to reproduce finite periodic signals (more precisely: even those with values contained in any finite section, periodic or not).
If you need infinite precision, then the correct Fourier series (discrete and finite) may become a Fourier Transform, also very helpful. Say you have to solve a difficult differential equation. The solution may be next to trivial for arbitrary Fourier component of a full solution (but inverse transform may be nevertheless hard again).
Physical significance: an X-ray picture of a single crystal is nothing else but the FT of its structure. The meaning of FT in quantum mechanics is more subtle, but priceless.
For over two centuries, Baron Jean Baptiste Joseph Fourier (03/21/1768 - 05/16/1830) found transform that bears his name.
But it has not been shown to be applicable to random signals. Hence the lack of interest in this discovery.
In 1929 Wiener shows that FT autocorrelation of a random signal is equal to the power spectral density. This means that the FT is applicable to random or reals signals. This is one of the most important theorem of signal processing.
In 1929 Wiener shows that FT autocorrelation of a random signal is equal to the power spectral density. This means that the FT is applicable to random or reals signals. This is one of the most important theorem of signal processing.
Search in google "history of fourier transform" and you will find a nice paper about the history of it and wavelet transform:
transcription of a part:
"
In 1807,
the French mathematician Joseph Fourier (Fig. 2.2) found that any periodic signal
can be presented by a weighted sum of a series of sine and cosine functions.
However, because of the uncompromising objections from some of his contemporaries
such as J. L. Lagrange (Herivel 1975), his paper on this finding never
got published, until some 15 years later, when Fourier wrote his own book, The
Analytical Theory of Heat (Fourier 1822). In that book, Fourier extended his
finding to aperiodic signals, stating that an aperiodic signal can be represented by
a weighted integral of a series of sine and cosine functions. Such an integral is
termed the Fourier transform."
Check this:
A Short Biography of Joseph Fourier and Historical Development of Fourier Series and Fourier Transforms
http://dx.doi.org/10.1080/0020739X.2011.633712
For the history of FFT, check this other:
Historical Notes on the Fast Fourier Transform
http://www.signallake.com/innovation/FFTHistoryJun67.pdf
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