A picture to understand the relationship between DFT and DTFT, DFS

Posted May 25, 20203 min read

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After learning digital signal processing, many students are confused by several nouns, such as DFT, DTFT, DFS, FFT, FT, FS, etc. FT and FS belong to the content of the signal and system course, which is for continuous time signals Handling, there is not much discussion here, only to explain the relationship between the first four.

First of all, I am not an expert in digital signal processing, so here I only explain the problem with the most obvious and easy-to-understand nature from the perspective of students, without involving any formula operations.

After learning convolution, we all know that the time domain convolution theorem and frequency domain convolution theorem, here only need to remember two points:1. The multiplication in one domain is equal to the convolution in the other domain; 2. The impulse function The convolution will produce a mirror image of the waveform at each pulse position.(In any signal and system textbook, these two properties are proved by detailed formulas)

The following two properties are used to illustrate the relationship between DFT, DTFT, DFS, and FFT:

First look at the picture:

First of all, figure(1) and figure(2). For an analog signal, as shown in figure(1), to analyze its frequency components, it must be transformed into the frequency domain. This is through Fourier transform, or FT(Fourier Transform) The result is the spectrum of the analog signal, as shown in Figure(2); Note 1:Both the time and frequency domains are continuous!

However, the computer can only process digital signals. First, the original analog signal needs to be discretized in the time domain, that is, it is sampled in the time domain. The sampling pulse sequence is shown in(3), and the frequency spectrum of the sampling sequence is shown in(4).), It can be seen that its spectrum is also a series of pulses. The so-called time domain sampling is to multiply the signal in the time domain, and the discrete time signal x \ [n ]can be obtained after(1) ×(3), as shown in(5); from the previous property 1, time The multiplication of the domain is equivalent to the convolution in the frequency domain. Then, Figures(2) and(4) are convoluted. According to the previous property 2, it is known that a mirror image will appear at each pulse point, so Figure(6) It is the DTFT(Discrete time Fourier Transform) of the discrete time signal x \ [n ]shown in Figure(5), that is, the discrete time Fourier transform. Here, the words "discrete time" are emphasized. Note 2:At this time, the time domain is discrete, while the frequency domain is still continuous.

After the above two steps, the signal we obtained still cannot be processed by the computer, because the frequency domain is both continuous and periodic. We naturally think that since the time domain can be sampled, why can't it be sampled in the frequency domain? Isn't it discretized in both time and frequency domains? That's right, the next step is to sample the frequency domain. The frequency spectrum of the sampled signal in the frequency domain is shown in Figure(8), and its time-domain waveform is shown in Figure(7). Now we are sampling in the frequency domain, that is, multiplying in the frequency domain. Figure(6) × Figure(8) to get the figure(10), then according to the property 1, this time is the frequency domain multiplication, the time domain convolution, figure(5) Convolve with figure(7) to get figure(9). As expected, the mirror image will appear periodically at each pulse point. We take the principal value interval of the periodic sequence in Figure(10) and record it as X(k), which is the DFT(Discrete Fourier Transform) of the sequence x \ [n ], that is, the discrete Fourier transform. It can be seen that 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, IDFT the graph(10), and return to the time domain, namely the graph(9), which is different from the x \ [n ]shown in the original discrete signal graph(5), it is x \ [n ]The cyclical extension! That's right, so if you look for a definition of IDFT, does it limit the value range of n? The implication of this limitation is that by taking the main value interval of the period extension sequence, x \ [n ]can be restored!

What about FFT? The proposal of FFT is only for quick calculation of DFT, its essence is DFT! The commonly used signal processing software MATLAB or DSP software package contains algorithms that are FFT instead of DFT.

DFS is proposed for time-domain periodic signals. If you perform DFS on the periodic extension signal shown in Figure(9), you will get Figure(10). As long as the main value interval is intercepted, it is completely one-to-one correspondence with DFT. The exact relationship. This can also be easily seen by comparing the definitions of DFS and DFT. Therefore, the essence of DFS and DFT is the same, but the method described is different.

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