Practical considerations in sampling; natural sampling

Practical considerations in sampling

1. Real signals are not strictly band-limited. An anti-aliasing low-pass filter is used before sampling to cut off components above $f_s/2$.
2. Sample above the Nyquist rate to provide a guard band, because practical reconstruction filters cannot have a sharp cut-off. Example: speech band-limited to 3.4 kHz is sampled at 8 kHz, not 6.8 kHz.
3. Finite pulse width: practical samples are pulses of width $\tau$ (natural or flat-top), not impulses. Flat-top samples cause the aperture effect, which needs equalization.
4. Non-ideal reconstruction filter: a practical LPF has a transition band, which needs the guard band above.
5. Timing jitter and sample-and-hold accuracy in the ADC must be small.

Natural sampling

In natural sampling the message is multiplied by a periodic train of rectangular pulses $c(t)$ of width $\tau$ and period $T_s$; the top of each pulse follows the message.

m(t) ---->(x)----> s(t) = m(t) c(t)
           ^
           |  c(t): pulses, width tau, period Ts
   _   _   _   _
  | | | | | | | |      (tops follow m(t))
 _| |_| |_| |_| |_

Derivation. $c(t)$ is periodic, so it has the Fourier series

$$c(t) = \sum_{n=-\infty}^{\infty} C_n\, e^{j2\pi n f_s t}, \qquad C_n = \frac{\tau}{T_s}\,\text{sinc}(n f_s\tau)$$

The sampled signal is

$$s(t) = m(t)\,c(t) = \sum_{n=-\infty}^{\infty} C_n\, m(t)\,e^{j2\pi n f_s t}$$

Using the frequency-shift property, $m(t)e^{j2\pi nf_st} \leftrightarrow M(f - nf_s)$:

$$S(f) = \sum_{n=-\infty}^{\infty} C_n\, M(f - n f_s) = \frac{\tau}{T_s}\sum_{n} \text{sinc}(n f_s \tau)\, M(f - n f_s)$$

Interpretation:

• The spectrum consists of copies of $M(f)$ centred at $nf_s$, each weighted by a constant $C_n$ that depends only on $n$.
• Each copy keeps the original shape of $M(f)$ (no distortion), unlike flat-top sampling, where the shape is multiplied by $\text{sinc}(f\tau)$.
• The baseband term ($n = 0$) is $\frac{\tau}{T_s}M(f)$. If $f_s \ge 2W$, the copies do not overlap, and an LPF of cut-off $W$ recovers $m(t)$ exactly (scaled by $\tau/T_s$).

S(f)
 |   /\         /\          /\
 |  /  \  ...  /  \  ...   /  \   (heights follow C_n)
 +-/----\-----/----\------/----\---> f
   -W 0 W   fs-W fs fs+W  2fs