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because if we dont use Fourier Transform and have the variable t then we are working in time domain and sometimes u cant analyze the signal in time domain and its easier to analyze it in frequency (f) domain.thats what i know.
Transformations are used to make operation easier, an rough example woud be given by logarithms . We know the Log of a multiplication is the sum of logs individually, so to sum is easier than multiply, that's why we use in telecommunications the Db . We can imagine that DB is a transformation to make operations easy. Likewise is the fourier transform, it gives you sometime a more suitable analisys of signals working in frequency domain than it is in time domain. For instance it transform convolutions operation in time domain , in so much easier multiplication in frequency domain. It is a rough idea. Fourier transform is a wide concept and cannot be explain in few words...
it's used to convert any periodic signals or even part of signals from time domain to frequency domain, the main advantage of that is to know the band width of you signal and compare ot with charactrestics of your channels or media (band width) then you will know if signal will be treanmitted without any lose or destorion of its chape .. that the simplest and main usage of fourir
It is to analyse the signal in a different domain, basically we represent all signals in time domain and certain manipulation of the signal could be done in time domain, but if we need to process the signal and analyse it for any communication or implementation purpose we had to go for another domain that is the frequency domain where manipulation of signal is comparatively easy.
Well see if u want to convert any signals from time domain to frequency domain then u need transforms. We cam use fourier series but fourier series only helps in evaluating periodic signals. For aperiodic signals we need Fourier transform.
this is a very basic question where most of people get doubt consider a signal with a very small time period (i cant plot here )but if u want to analyze the properties of the signal at a particular time if it is in time domain u cant get it where as in frequency domain if u reprsent the same signal it will be expanded and u can view the complete properties of the signal at this juncture if u think why do we use fourier series c wat a we do in fourier seeries we represent a given signal in terms of sin and cosine and a constant the cofficients of sin and cosine gives the power of signal at that particular frequency and the constant term gives the dc level of it.
But, i will explain in different way.. but not the different matter...
see there are two ways of analyzing the signal:
1) Time domain -> which explains about quantity of signal. like amplitude of the signal. but you can't tell wat frequency that signal consists of.
2) Frequency Domain -> which explains about the quality of the signal. like the frequency quantity of the signal. but here you can't say about the amplitude of the signal.
So, Both domains are very important in their fields. but If you want to know whole thing about the given signal... then you need a tool which can transform from one domain to another, which this Fourier transform does.
So In DSP, Fourier transform is a basic tool with which you can do anything & everything.
for example: Filter and Speech, Image & picture related processings etc.
So, at last i can say that,
in Time Domain you play with Quantity &
in Frequency Domain you play with Quality of the signal.
with these you can do everything in the DSP world.
Originally Fourier itroduced his transform for solving linear differential equations.
You know that LDE and systems of LDE describe continuous time linear systems like LRC chains. Chains consisting of resistors, capacitors, inductivities.
By the way, convolution is integral representation of system of LDE.
Idea was to represent signal as a sum of "easy for use" signals, find solution for all these signals and combine results. These "easy for use" signals are eigenfunctions of system to be solved. In case of LDE they are complex exponential functions (A*exp(j*(lambda))). These functions remains the same wenn transferring through linear systems. Only parameters change.
So, if we represent input signal as a sum of complex exponential functions (its forward fourier transform), set of LDE became a set of linear equations. Solving it and combining(this is inverse fourier transform) output exponential functions (remember, their magnitudes and phases are changed) we obtain output signal.
This is connection beetween several terms
Convolution <--> System of LDEs <-- (Fourier transform) --> System of linear equation on fourier domain
I think its better to find some more detailed description in some books.
The same is for discrete time with next changes
convolution becames discrete convolution
System of LDEs becames system of finite difference equations
fourier transform --> discrete fourier transform
So, main reason of introducing fourier transform --> make easy of analyzing, simulating and handling linear systems.
Additionally, it coincides in some way with intuitively clear term of frequency (in audio processing, for example)
Its simple, you cant see the various frequencies present in the a length of signal by analyzing the waveform in Time domain (time vs amplitude) to analyze the waveform in Frequency domain we need to find the spectrum of the time domain signal this can be achieved by Transforms invented by many, some of the famous ones are DFT, DCT, FFT etc . Each has their specific property. for example DCT has a energy compaction property so it is used in most of the compression algorithms and FFT is the optimized form of DFT so it is mostly used. And in filtering applications.
claudiocamera is right. FT is just a mathematical tool developed to make the lives of lazy engineers easier. In fact, FT was treated as unorthodox in the past by mathematicians until a more solid proof for it was derived.
In this case of FT, the operation that is of greatest concern is convolution, which is used in all filters as well as linear systems. Convolution in the time domain requires integration, which can get messy. But convolution in the frequency domain is simple multiplication.