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# Signal sampling mechanism

2017-12-23 04:47
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Sampling is a process when continuous time signal is represented by series of discrete samples while reconstruction is reverse process when these samples re recreating adequate continuous time signal. Bellow the overall process is illustrated.

Sampling is a process when continuous time signal is recorded every T seconds by multiplying by an impulse train.

If signal is sampled in frequency domain. For this we need signal transformed in to frequency domain. The frequency spectrum has to be band-limited. The impulse train after transformation becomes impulse train with scale in heigh 1/T. Multiplication in time domain becomes a convolution in frequency domain. After convolving the signal is scaled and shifted. The signal in frequency domain becomes periodical.

In order to reconstruct the signal x(t) from sampled spectrum first we need to extract the original spectrum. For this ideal filter is needed to take single spectrum from spectrum signal train. Low pass filter fill cut other copies of spectrum. The filter should have cut off at f=1/(2T) and Gain of T:

After this reconstruction can be done. Ideal filter in frequency domain is a sinc function in time domain. This means that multiplying spectra by box in frequency domains means convolving with a sinc function in time domain. Because y(t) is a series of impulses, then reconstructed x(t) is a superposition of scaled and shifted sincs:

The fundamental result is that any band-limited signal can be completely discretized by equally spaced samples. And remember that sampling frequency has to be at least two times bigger than bandlimit of signal to avoid alias (overlap) of sampled signal spectra. Otherwise reconstruction of original signal becomes impossible.