Sub-sampling theory and aperture effect

Sub-sampling (band-pass sampling) theory

For a band-pass signal occupying $f_L$ to $f_H$ with bandwidth $B = f_H - f_L$, it is not necessary to sample at $2f_H$. The signal can be recovered from samples taken at a rate as low as about $2B$, i.e. below the Nyquist rate of its highest frequency. This is called sub-sampling (or under-sampling). The allowed rates are

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

Aliasing is used deliberately to shift the band down to baseband without overlap. Example: an 80–115 kHz signal can be sampled at 76.7–80 kHz instead of 230 kHz. Sub-sampling is used in digital receivers to sample IF signals directly.

Aperture effect

In flat-top sampling the sample is held for a finite width $\tau$, so the spectrum is multiplied by $H(f) = \tau\,\text{sinc}(f\tau)$. This attenuates the higher frequencies of the message, an amplitude distortion called the aperture effect. It is reduced by making $\tau \ll T_s$ and corrected by an equalizer with response $1/H(f)$ after the reconstruction filter.