Triangular Flat Top Sampling
- Signals and Systems 2023 (Spring) Assignment Requirements - Eighth Assignment
- Signals and Systems 2023 (Spring) Homework Reference Answers - 8th Homework
01 The eighth assignment
1. Triangular flat-top sampling
In the eighth assignment, there is an exercise that discusses the spectrum of a flat-top sampled signal. Firstly, the frequency spectrum of the time-domain signal g(t), G omega, corresponds to an isosceles triangle. The time-domain waveform of g(t) can be deduced by using the dual characteristics of Fourier transform, which is in the form of the square of the sinc function. Here is the time domain waveform with g(t) plotted out.
For the ideal adoption of g(t), the periodic impulse signal is multiplied by g(t), and the spectrum of the corresponding sampled signal is the periodic extension of the spectrum of g(t). The exercise defines the flat-top sampling of the triangular pulse of the signal, that is, modifying the impulse signal into a narrow triangular pulse signal. The width of the signal is tao, and the period is defined as half of the period corresponding to omega m, where omega m is the highest frequency of the previous g(t) spectrum.
▲ 图1.1.1 g(t) 信号波形
As for the flat-top sampling , that sounds a bit weird. This is because the triangular pulse has no "flat top" at all. The concept of flat top here means that only the height of the triangular pulse sampling waveform is the same as the height of the sampling point signal, and it still maintains the form of an equally symmetrical triangular pulse as a whole. That is, it is not multiplied by the signal. Influenced by the signal, the shape of the original isosceles triangle is changed. For this reason, there is a close relationship between this signal and the previous ideal sampled signal. Through their expressions, it can be found that each delta signal corresponds to ideal sampling, and the flat-top triangular pulse sampling corresponds to a displaced triangular pulse. Therefore, this sampled signal can be regarded as the convolution of f0 and the ideal sampled signal. The f0 signal here is a pulse signal in the triangular pulse signal. According to this relationship, the relationship between them can be obtained, and then the frequency spectrum of the triangular flat-top sampling signal can be obtained.
▲ 图1.1.2 三角平顶采样的波形
2. Triangular Flat Top Sampling Spectrum
For the ideal sampling of g(t), its corresponding frequency spectrum is the period extension of G omega, the period of extension is the sampling period, and the coefficient in front is one part of Ts. For triangular flat-top sampling, the frequency spectrum corresponding to a single triangular pulse can be written out, which is equal to 1/2 of tao, sinc squared, and 1/4 of omega tao. According to the analysis just now, the spectrum of the triangular flat-top sampling signal should be equal to the product of the spectrum of the above two signals. After multiplying them and simplifying, the spectrum of the triangular sampling signal is obtained. This is the mathematical expression corresponding to the frequency spectrum. From this expression, a graph of the frequency spectrum is drawn. The triangular waveform is actually a periodic triangular pulse formed after the period extension of g(t) triangular frequency spectrum. Since the sampling period is exactly equal to twice the highest frequency of the g(t) spectrum, a continuous triangular pulse spectrum is formed here. The product of the sinc squared signal corresponding to the spectrum of the narrow triangular pulse then yields the spectrum of the sampled signal. You can see that because they are multiplied together, each triangular pulse is actually deformed. This is the spectrum waveform of triangular flat-top sampling. So far, give the answer to this exercise.
3. Signal recovery
How to restore the sampled signal g(t) without distortion from the triangular flat-top sampled signal is discussed below. Restoring g(t) actually needs to restore the spectrum G omega corresponding to g(t), which requires two processes. One is to obtain the low-frequency spectrum from the cycle-extended spectrum through a low-pass filter with a bandwidth of omega m spectrum. Then, for the function multiplied before, it is also necessary to multiply the spectrum after the low-pass filter by the corresponding reciprocal, so that G omega can be obtained. The form of the complete filter is given here. Compared with the traditional ideal low-pass filter, its gain in the passband is no longer a constant, but a function about omega. Only in this way can the distortion caused by multiplying the triangular pulse spectrum be compensated.
※ Summary ※
This paper discusses the problem of triangular flat-top sampling in the eighth assignment, and at the same time gives a low-pass filter for recovering the original signal from the sampled signal.
■ Links to related literature:
- Signals and Systems 2023 (Spring) Assignment Requirements - Eighth Assignment
- Signals and Systems 2023 (Spring) Homework Reference Answers - 8th Homework
● Links to related diagrams: