Nyquist sampling theorem; natural sampling

Nyquist sampling theorem

A signal $g(t)$ band-limited to $W$ Hz can be exactly reconstructed from its uniformly spaced samples if the sampling rate satisfies $f_s \ge 2W$. The minimum rate $2W$ is the Nyquist rate and $1/(2W)$ the Nyquist interval. Below this rate the spectral copies overlap and aliasing occurs.

Natural sampling

In natural (chopper) sampling the signal is multiplied by a periodic train of rectangular pulses $c(t)$ of width $\tau$ and period $T_s$. During each pulse the sample follows the shape of the signal:

$$s(t) = g(t)\,c(t)$$

g(t) ----->(x)-----> s(t)  (tops follow g(t))
            ^
  c(t): |‾|  |‾|  |‾|  width tau, period Ts

Derivation

1. Expand the pulse train in a Fourier series ($f_s = 1/T_s$):

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

2. Multiply by $g(t)$:

$$s(t) = \sum_{n} C_n\, g(t)\, e^{j2\pi n f_s t}$$

3. By the frequency-shift property, $g(t)e^{j2\pi nf_st} \leftrightarrow G(f - nf_s)$, so

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

Interpretation

• The spectrum is the message spectrum $G(f)$ repeated at multiples of $f_s$.
• Each copy keeps the exact shape of $G(f)$; only its height is scaled by the constant $C_n$. Hence there is no aperture distortion (unlike flat-top sampling).
• The $n = 0$ copy is $(\tau/T_s)G(f)$, so a low-pass filter with cut-off between $W$ and $f_s - W$ recovers $g(t)$ (scaled by $\tau/T_s$), provided $f_s \ge 2W$.
• The envelope of the copy heights follows $\text{sinc}(f\tau)$, which falls as $n$ grows.

Drawback: the pulse tops vary, so quantizing them is difficult; therefore flat-top sampling is used in PCM.