Nyquist sampling theorem; flat top sampling

Nyquist sampling theorem

A signal band-limited to $W$ Hz (no components above $W$) is completely described by, and can be exactly recovered from, its samples taken uniformly at a rate

$$f_s \ge 2W$$

The minimum rate $f_s = 2W$ is the Nyquist rate and $T_s = 1/(2W)$ is the Nyquist interval.

Explanation: ideal sampling multiplies $g(t)$ by an impulse train $\delta_{T_s}(t)$. In the frequency domain this repeats the spectrum every $f_s$:

$$G_\delta(f) = f_s\sum_{n=-\infty}^{\infty} G(f - nf_s)$$

If $f_s \ge 2W$ the copies do not overlap, so a low-pass filter of cut-off $W \le f_c \le f_s - W$ recovers $G(f)$ exactly. If $f_s < 2W$ the copies overlap (aliasing) and recovery is impossible.

Flat-top sampling

In flat-top sampling each sample is held constant for a pulse width $\tau$ (sample-and-hold), giving a PAM wave with flat tops. It is modelled as ideal sampling followed by convolution with a rectangular pulse $h(t)$ of width $\tau$:

$$s(t) = \sum_n g(nT_s)\,h(t - nT_s) = \big[g(t)\,\delta_{T_s}(t)\big] * h(t)$$

Taking the Fourier transform (convolution becomes multiplication):

$$\begin{aligned} S(f) &= \Big[f_s\sum_n G(f - nf_s)\Big] H(f) \\ H(f) &= \tau\,\text{sinc}(f\tau)\,e^{-j\pi f\tau} \end{aligned}$$

 g(t)--(x)-->ideal samples-->[h(t): hold tau]-->s(t)
        |
   impulse train

Meaning of the result

• The spectrum is still repeated every $f_s$, but each copy is multiplied by $H(f)$, a sinc shape.
• So the high frequencies of the message are attenuated: this amplitude distortion is the aperture effect.
• Distortion is small when $\tau \ll T_s$.
• At the receiver it is corrected with an equalizer of response $1/H(f)$ (in the band $|f| \le W$) after the reconstruction low-pass filter.

Flat-top sampling is the one used in practice (sample-and-hold before the ADC) because a constant amplitude during $\tau$ is easy to quantize and generate.