FFT and Fourier Transform
Recently, digital signal processing technology has progressed dramatically. However, the leading player of digital signal processing would still be Fast Fourier Transform (FFT.) As you may know, FFT transformers a signal from the time domain to the frequency domain and is used for spectrum analysis or others. Very simply speaking, FFT carries out Fourier transform on digital signals but the name is FFT. Fourier transform for digital (more strictly "discrete") exists as Discrete Fourier Transform (DFT), which is not so famous as FFT.
Now, why is only FFT valued? The reason is the fastness as the name shows. There is no difference between FFT and DFT in the fact that they carry out Fourier transform on discrete signals. The following is the basic formula for both FFT and DFT.
The difference between FFT and DFT is the implementation method to realize the following formula. FFT can reduce the amount of calculation by limiting the number of points (N in the above formula). The reduction ratio is approximately N2 versus 2N x log2(N). When N is 1024, the ration is about 1/100. This was really a landmark event in 1969 when the FFT was invented, considering the performance of computers at the time. Although the current computers are much, much and much faster than computers in 1969, importance of FFT still remains and will remain because more spectrum analysis in a limit time frame are required or real-time control system using FFT is becoming common.
However, FFT is not almighty. Even in DFT is essentially same except for it does not have the limitation to be power-of-two. The inherent problem in FFT is that it treats a limited number of points or a limited time frame while Fourier transform is applied to unlimited frame of time or minus unlimited to plus unlimited. For example, we extract one period of a human voice, convert the sampling frequency making the period fit to 256 points and analyze the spectrum by FFT. Since all the harmonic components are represented, we expect we can synthesize a perfect human voice based on the spectrum but the sound synthesized simply based on the spectrum is like a tone of a poor organ.
This is not caused by DFT or FFT itself. The reason is that only a period cannot represent the human voice structured by complex elements. Human voice has a vibration, whose frequency is lower than the fundamental frequency, and frequency components whose frequencies are not harmonics of the fundamental frequency. And, these components add complicated expression to the human voice. The situation is same on music instruments. If we synthesize a tone by repeating one period of a sampled tone, it would be difficult to distinguish which the tone is sampled from.
In fact, FFT/DFT is very good tool to analyze a period of tone but the problem is that analyzing only a period is usually not enough to grasp the characteristics of the tone. In other word, FFT is a good tool, just same other tools or methods in the world, to extract and analyze a reality. Therefore, it may cause a terrible result if you misjudged the prerequisite. However, of course, it does not degrade the value of FFT/DFT because the fundamentals of science are to reduce matters to simpler ones and analyze them one by one.
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