A figure shows the relationship between DFT, dtft, and DFS.

Source: Internet
Author: User


After learning digital signal processing, many of them were confused by the terms in it, such as DFT, dtft, DFS, FFT, FT, FS, etc, FT and fs are the content of the signal and system course. They are the processing of continuous time signals. Here we will not discuss much about them. I will only explain the relationship between the first four.

First of all, I am not a digital signal processing expert. Therefore, I am here only to explain the problem from the perspective of students with the most understandable nature, without involving any formula operations.

After learning convolution, we all know that there are time domain convolution theorem and frequency domain convolution theorem. Here we only need to remember two points: 1. the multiplication of one domain equals the convolution of another domain; 2. convolution with the pulse function generates a waveform image at each pulse position. (In any book on Signals and Systems, the two properties are proved by a detailed formula)

The following describes the relationships between DFT, dtft, DFS, and FFT by using these two properties:

First look at the image:

First, for a analog signal, as shown in figure (1) and figure (2), to analyze its frequency composition, it must be transformed to the frequency domain, this is obtained through Fourier transform (FT), so we have the analog signal spectrum (2). Note 1: the time domain and frequency domain are continuous!

However, computers can only process digital signals. First, they need to discretization the original analog signal in the time domain, that is, sampling it in the time domain, as shown in the sample pulse sequence (3, the spectrum (4) of the sampling sequence shows that its spectrum is also a series of pulses. The so-called time-domain sampling means that signals are multiplied in the time-domain. After (1) × (3), the discrete time signal X [N] can be obtained, as shown in (5; from the first property 1, the multiplication of the time domain is equivalent to the convolution of the frequency domain. Then, the image (2) and the image (4) are convolution. According to the previous property 2, the image appears at each pulse point, so figure (6) is displayed, which is the dtft (discrete) of the discrete time signal X [N] shown in figure (5 ).
Time Fourier Transform), that is, Discrete Time Fourier transformation, which emphasizes the word "Discrete Time. 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 is still not processed by the computer, because the frequency domain is continuous and cyclical. Naturally, we thought that since the time domain can be sampled, why can't the frequency domain be sampled? In this case, will the time domain and frequency domain be discretization? That's right. Next we will sample the frequency domain. The spectrum of the sampling signal in the frequency domain (8) shows its Time Domain Waveform (7 ). Now let's perform frequency-domain sampling, that is, multiplication of the frequency-domain. Figure (6) × figure (8) gets the figure (10). So according to the Nature 1, this time is the multiplication of the frequency-domain, and the time-domain convolution is done, figure (5) and figure (7) convolution result in figure (9). As expected, the image appears periodically at each pulse point. We take the main value range of the graph (10) Periodic Sequence and mark it as X (k). It is the DFT (discrete) of the sequence X [N ].
Fourier Transform), that is, discrete Fourier transformation. It can be seen that DFT only samples dtft in the frequency field and intercepts the main value for the convenience of computer processing. Some may be confused. If you perform idft on the image (10) and return to the time domain, that is, figure (9), it is different from the X [N] shown in the original discrete signal diagram (5, it is a periodic extension of X [N! That's right, so if you look for an idft definition, is the range of N restricted? This restriction means that the X [N] can be restored by taking the main value range of the periodic continuation sequence.

What about FFT? The concept of FFT is simply to calculate DFT quickly. Its essence is DFT! In our commonly used signal processing software MATLAB or DSP software package, the algorithms included are FFT rather than DFT.

DFS is proposed for time-domain periodic signals. If DFS is performed on the cyclic continuation signal shown in figure (9), the figure (10) is obtained. As long as the main value range is intercepted, it has a exactly one-to-one relationship with DFT. This can be easily seen by comparing the definitions of DFS and DFT. The nature of DFS is the same as that of DFT, but the methods described are different.

I don't know whether you understand the relationship between various T-grams after the above explanation? If you are not an algorithm designer, you only need to know how to use FFT to analyze the spectrum. The blogger will update an article about how to use FFT to analyze the spectrum of simple signals.

In fact, I personally think that the Tangle is so much to break the boundaries between the real simulated world and the computer digital world!

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