Aliasing and aperture effects; types of sampling

Aliasing

Aliasing is the overlap of the shifted spectral copies of a sampled signal that occurs when the sampling rate is less than the Nyquist rate ($f_s < 2W$), or when the signal is not strictly band-limited. A high-frequency component $f$ then appears falsely as a low frequency $|f - nf_s|$ and the original signal cannot be recovered. Remedies: anti-aliasing low-pass filter before the sampler, and sampling a little above $2W$ (guard band).

Aperture effect

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

Types of sampling

1. Ideal (impulse) sampling

$$g_\delta(t) = \sum_n g(nT_s)\delta(t - nT_s), \qquad G_\delta(f) = f_s\sum_n G(f - nf_s)$$

Samples are zero-width impulses. Theoretical only, since impulses cannot be generated.

2. Natural sampling

$$s(t) = g(t)c(t), \qquad S(f) = \frac{\tau}{T_s}\sum_n \text{sinc}(nf_s\tau)G(f - nf_s)$$

Pulses of width $\tau$ whose tops follow the signal; done by an analog switch (chopper). No aperture distortion.

3. Flat-top sampling

$$s(t) = \sum_n g(nT_s)h(t - nT_s), \qquad S(f) = f_sH(f)\sum_n G(f - nf_s)$$

Pulses of width $\tau$ with constant height equal to the sample value (sample-and-hold). Practical; has aperture effect.

Waveforms

g(t)       .-~~~-.
         .'       '.___
Ideal    ^  ^  ^  ^  ^     impulses, height=g(nTs)
         |  |  |  |  |
Natural  ▄  █  █  ▆  ▂     tops follow curve
Flat-top ■  ■  ■  ■  ■     flat tops, held
         0 Ts 2Ts 3Ts 4Ts