Low-pass and band-pass sampling theorem

low pass sampling theorem

A band is limited to (0, $f_H$) in the low-pass signal m(t), if $f_s$≥2$f_H$uniform sampling at a rate of , then the sampling sequence {m(n $T_s$)} can restore m(t) without distortion. where $f_H$is the upper limit frequency, $f_s$is the sampling rate, $T_s$is the sampling interval.
Uniform sampling means sampling at equal intervals.
$f_s$=2$f_H$is the minimum sampling rate, also known as the Nyquist sampling rate.
$T_s$=$\frac{1}{f_s}$, it can be seen that the smaller the sampling rate, the larger the sampling interval. $T_s$=$\frac{1}{2f_H}$called the Nyquist sampling interval.
When the sampling rate $f_s$＜2$f_H$, spectral aliasing occurs.

bandpass sampling theorem

Band limited to ( $f_L$，$f_H$) within the band-pass signal m(t), bandwidth B= $f_H$-$f_L$, if $f_H$=nB+kB (where n∈N, 0≤k<1), then the minimum sampling rate fs f_s of m(t) can be recovered without distortion$f_s$=2B(1+$\frac{k}{n}$).

If the bandpass signal is sampled at the minimum sampling rate above, the sampled signal spectrum will have neither gaps nor aliasing . It should be noted that if the sampling rate is greater than this value, the requirements may not always be met.
Example: $f_L$=100.5MHz，$f_H$=100.9MHz，$f_H$=252B+0.25B
Solution: Bandwidth: B = $f_H$-$f_L$=0.4MHz=400kHz
bandpass sampling rate: $f_s$=2B(1+$\frac{k}{n}$)=2×400kHz×(1+$\frac{0.25}{252}$)=800.8kHz
If the above example uses the low-pass sampling theorem, the sampling rate is greater than or equal to 2 times $f_H$, at least 201.8MHz, which is much larger than 800.8kHz, so it is more difficult to realize, so the band-pass signal is solved by the band-pass sampling theorem.
In practice, the lower limit frequency $f_L$much larger than the bandwidth.
Bandpass sampling rate $f_s$and the lower limit frequency $f_L$The relationship curve is shown in the figure below.

The above figure is based on the formula $f_s$=2B(1+$\frac{k}{n}$) drawn, when k=0, $f_s$Take the minimum value 2B, when k=1, $f_s$Take the maximum value under different n.

Reference Video:
Sampling Theorem