Ft,dtft,dft's relationship (reproduced)

Many students learn the digital signal processing, by the inside of a few nouns confused, such as DFT,DTFT,DFS,FFT,FT,FS, FT and FS belong to the signal and system course content, is the processing of continuous time signal, here is not too much discussion, only explain the relationship between the first four.

First of all, I'm not a digital signal processing expert, so here I'm just standing on the students ' point of view to explain the problem in the most understandable nature, without involving any formula calculations.

Learning convolution, we all know that sometimes the domain convolution theorem and the frequency domain convolution theorem, here only need to remember two points: 1. Multiplying in one domain equals the convolution of another domain; 2. With the convolution of the pulse function, the image of a waveform is generated at the position of each pulse. (in any of the signals and system textbooks, these two properties have a detailed formula proof)

Below, we use these two properties to illustrate the connection between Dft,dtft,dfs,fft:

Look at the picture first:

First of all, figure (1) and figure (2), for an analog signal, (1), to analyze its frequency components, must be transformed to the frequency domain, which is obtained by Fourier transform, ft (Fourier Transform), so that the analog signal spectrum, (2) Note 1: Both the time domain and the frequency domain are continuous!

However, the computer can only process digital signals, the first need to be the original analog signal in the time domain discretization, that is, in the time domain to sample, the sampling pulse sequence (3), the sampling sequence of the spectrum (4), it can be seen that its spectrum is a series of pulses. The so-called time domain sampling, is to multiply the signal in the time domain, (1) x (3) can be obtained after the discrete-time signal x[n], (5) is shown, by the preceding property 1, the time domain is multiplied by the frequency domain convolution, then, the figure (2) and the figure (4) convolution, according to the previous properties of 2 know, will appear at each pulse point image , the figure (6), which is the DTFT (discrete time Fourier Transform), which is the discrete temporal signal x[n, shown in figure (5), is the discrete-time Fourier transform, which emphasizes "discrete time" four words. Note 2: The time domain is discrete at this time, and the frequency domain is still continuous.

After the above two steps, the signal we get is still not processed by the computer, because the frequency domain is both continuous and periodic. We naturally thought, since the time domain can be sampled, why the frequency domain cannot be sampled? Does this not disperse the time domain and the frequency domain? Yes, next to the frequency domain in the sampling, frequency domain sampling signal spectrum (8), its time domain waveform (7). Now we do frequency domain sampling, that is, the frequency domain multiplication, the figure (6) x (8) to get the figure (10), then according to the nature of 1, this time is multiplied by the frequency domain, the time domain convolution, the figure (5) and figure (7) convolution get the figure (9), as expected, the mirror will appear periodically We take the main value interval of the sequence of the graph (10) and record it as X (k), which is the DfT of sequence X[n] (discrete Fourier Transform), which is the discrete Fourier transform. It can be seen that the DfT is just for the convenience of computer processing, sampling the DTFT in the frequency domain and intercepting the main value. Some people may be puzzled, the figure (10) is idft, back to the time domain is the figure (9), it is different from the original discrete signal diagram (5) shown in the X[n], it is x[n] periodic extension! Yes, so you're going to look for a IDFT definition, is it a limit to the value range of n? The implication of this restriction is that you can restore x[n]! by taking the main value interval of the continuation sequence of the cycle.

What about the FFT? The FFT is presented solely to calculate the DfT quickly, its essence is dft! Our commonly used signal processing software, MATLAB or DSP packages, contains algorithms that are FFT rather than DFT.

DFS, is proposed for the time domain periodic signal, if the diagram (9) shown in the periodic extension signal Dfs, you will get the figure (10), as long as the main value interval is intercepted, then the DFT is a complete one by one corresponding to the exact relationship. This can be easily seen in the definition of DFS and DFT. So Dfs is the same as the nature of DFT, except that the methods described are different.

Do not know after the above explanation, do you understand the various T relationship? If you are not an algorithmic designer, as long as you know how to use FFT to analyze the spectrum, bloggers will recently update an article that specifically describes how to use FFT to analyze the spectrum of simple signals.

In fact, personally think, tangled up so much, is to break the reality of the world and the computer Digital world of the boundary!

Ft,dtft,dft's relationship (reproduced)