Sampling and reconstruction of low-pass signals and SystemView simulation

Sampling theorem is the theoretical basis for the digital transmission of analog signals and the most classic and important theorem in communication principles. The so-called signal sampling is to use the sampling pulse sequence p (t) to "extract" a series of discrete values ​​from the continuous signal xa (t). This kind of signal is usually called "sampling signal", which is expressed by x (t) in this article. The mathematical model can be simplified as the product of the pulse sequence and the original continuous signal.

1 Time domain sampling theorem and discussion

Let the Fourier transform of the continuous signal xa (t) to xa (jω), the Fourier transform of the sampling pulse sequence p (t) to p (jω), and the Fourier transform of the sampled signal x (t) to x (jω). If uniform sampling is used, the sampling period is ts, and the sampling frequency is ωs = 2πf. From the previous analysis, it can be seen that the sampling process can be completed by multiplying the sampling pulse sequence p (t) and the continuous signal xa (t), that is:

Equation (6) shows that after the signal is sampled in the time domain, its spectrum x (jω) is the shape of the continuous signal spectrum xa (jω), which is obtained by repeating the sampling frequency ωs at intervals, and the amplitude is p (t) Fourier series pn weighting. Because pn is just a function of n, xa (jω) does not change its shape during the repetition process.

Assuming that the frequency spectrum of the signal xa (t) is limited to a range of ωm to + ωm, if xa (t) is sampled at intervals ts, from equation (6), the spectrum x (jω) of the sampling signal x (t) is Repeat with ωs as the cycle. Obviously, if ωs <2ωm in the sampling process, x (jω) will have spectrum aliasing. Only when the condition of ωs≥2ωm is met in the sampling process, x (jω) will not produce spectrum aliasing. The end can completely restore the original continuous signal xa (t) from x (t), which is the core content of the low-pass signal sampling theorem.

2 Signal reconstruction

From the frequency domain, suppose the highest frequency of the signal does not exceed the folding frequency:

Equation (10) shows that as long as the sampling frequency is higher than twice the highest frequency of the signal, the continuous time function xa (t) can be expressed by his sampling value xa (nt) without losing any information. At this time, xa (t) can be obtained by multiplying and summing each sampling instantaneous value with equation (9). At each sampling point, since only the interpolation function equation (9) corresponding to the sampling value does not Is zero, so the signal value at each sampling point does not change, and the signal between the sampling points is composed of equation (10).

3 Simulation of low-pass signal sampling theorem on systemview

The previous analysis shows that it is not necessary to transmit the analog signal itself in order to transmit the analog signal, but only need to transmit the sampling value obtained according to the sampling theorem. Figure 1 shows the principle diagram of low-pass signal sampling and reconstruction.

The corresponding systemview simulation circuit established from the schematic diagram is shown in Figure 1. The analog signal source sampled in the figure is a sine wave with an amplitude of 1 v and a frequency of 100 Hz. The sampling pulse is a rectangular pulse with a pulse width of 1 μs. The sampler is replaced with a multiplier. In order to verify the conditions of signal sampling and recovery without distortion, different sampling frequencies of 10ohz, 200hz, 1000hz, etc. were selected to observe and analyze the original input signal waveform and the waveform after sampling recovery, so as to intuitively verify the low-pass signal sampling theorem.

The icon 1o in Fig. 2 is a sinusoidal signal source. Symbols 1 and 4 are baseband polyphase low-pass filters. The signal to be transmitted is filtered to reduce the bandwidth occupied by the actual transmission. In the paper, the Butterworth low-pass with a third-order parameter and a cut-off frequency of 100 hz is set in the paper. iir filter. Symbols 8 and 9 are gains of a factor of 1, and amplify the input signal. Figure 2 is the multiplier. Figure 3 is a sampling pulse that generates a periodic pulse train with a specific amplitude and frequency. By changing the frequency parameter in icon 3, it is equivalent to changing the sampling frequency fs. The fs is set to 100, 200, 1000 hz respectively to obtain the simulation results shown in Figure 3 to Figure 6.

The comparison of simulation results shows that the sampling signal has the best recovery effect when the sampling frequency is 1000hz, and can basically obtain the same waveform as the original signal. This intuitively proves the correctness of the sampling theorem: the sampled signal can be reconstructed under the condition of fs≥2fm; the higher the sampled signal frequency, the better the reconstruction effect; because the spectrum of the sampled signal is a periodic extension of the original signal Extension. Therefore, as long as it passes a low-pass filter with a cut-off frequency fc (fm≤fc≤fs-fm), the original signal can be recovered.

4 Conclusion

In this paper, a detailed theoretical derivation of the sampling theorem is made, and the conditions for sampling and signal recovery of the low-pass signal are obtained. Theoretical derivation shows that the spectrum of the signal obtained in actual sampling is very similar to the spectrum structure of ideal sampling, but the amplitude change rule is different. As long as the sampling theorem is satisfied, the original continuous signal can be correctly recovered from the sampled signal. The dynamic simulation model adopts the method of gradually increasing the sampling frequency to obtain a series of simulation results, which intuitively verifies the correctness of the sampling theorem of low-pass signals.