Practical sampling; band-pass sampling

Practical sampling

Ideal impulses cannot be generated, so in practice sampling is done with pulses of finite width using an electronic switch (natural sampling) or a sample-and-hold (S/H) circuit (flat-top sampling).

Sample-and-hold circuit:

           S (FET switch)          buffer
 x(t) ---[ buffer ]---o/ o----+----[ >  ]----> xs(t)
                      ^       |
       sampling pulse |      === C (hold
       (period T)            |   capacitor)
                            GND

Working:

1. Sample mode: a short sampling pulse closes the switch S (a FET). The capacitor $C$ charges quickly to the input value $x(nT)$ through the low output resistance of the input buffer.
2. Hold mode: the switch opens. The capacitor holds the voltage for the rest of the period $T$ because the output buffer has very high input impedance. The output is a staircase (flat-top samples), which the ADC then converts.
3. Mathematically, flat-top sampling = ideal sampling followed by a hold filter $h(t)$ (a pulse of width $\tau$). Its spectrum is $X_s(j\omega) = \frac{1}{T}H(j\omega)\sum_k X(j(\omega - k\omega_s))$, with $H(j\omega) = \tau\,\text{sinc}$-shaped. This causes the aperture effect (slight high-frequency loss), corrected with an equalizer $1/H(j\omega)$ at reconstruction.

With natural sampling (switch only, no capacitor), the pulse tops follow the signal and the spectrum copies are only weighted by constants, so no aperture distortion occurs.

Band-pass sampling theorem

For a band-pass signal occupying $f_L$ to $f_H$ with bandwidth $B = f_H - f_L$, sampling at $2f_H$ is not necessary. The signal can be recovered from samples taken at a rate as low as $2B$, provided

$$\frac{2f_H}{k} \le f_s \le \frac{2f_L}{k-1}, \qquad k = 1, 2, \dots, \left\lfloor\frac{f_H}{B}\right\rfloor$$

In particular, if $f_H$ is an integer multiple of $B$, then $f_s = 2B$ works. Example: a signal from 20 to 25 kHz ($B = 5$ kHz, $f_H = 5B$) can be sampled at 10 kHz instead of 50 kHz.