Sampling theorem; aliasing and aperture effect remedies

Sampling theorem

At the transmitter: a band-limited signal of finite energy with no frequency components above $W$ Hz is completely described by its sample values taken at uniform intervals of $T_s \le 1/(2W)$ seconds, i.e. at a rate $f_s \ge 2W$.

At the receiver: such a signal can be completely recovered from its samples taken at the rate of $2W$ samples per second (or more) by passing them through an ideal low-pass filter of bandwidth $W$.

The rate $2W$ is the Nyquist rate. Proof sketch: ideal sampling gives $G_\delta(f) = f_s\sum_n G(f - nf_s)$; the copies separate when $f_s \ge 2W$, so a LPF extracts $G(f)$.

Aliasing

When $f_s < 2W$, or the signal is not strictly band-limited, the spectral copies overlap. High-frequency components fold back and appear as false low frequencies $|f - f_s|$. Example: a 7 kHz tone sampled at 10 kHz appears as 3 kHz after reconstruction.

   copy n=0     copy n=1
  /‾‾‾‾‾‾\   /‾‾‾‾‾‾\
 /        \ /        \
/          X          \   overlap = aliasing
-W   0    fs/2   fs

Remedies

1. Use an anti-aliasing (pre-alias) low-pass filter before the sampler to strictly band-limit the signal to $W$.
2. Sample slightly above the Nyquist rate ($f_s > 2W$) to create a guard band, so a practical reconstruction filter with finite roll-off can be used. Example: speech 3.4 kHz sampled at 8 kHz.

Aperture effect

In flat-top sampling each sample is held for width $\tau$. The sampled spectrum becomes $f_sH(f)\sum_n G(f - nf_s)$ with $H(f) = \tau\,\text{sinc}(f\tau)e^{-j\pi f\tau}$. Because $|H(f)|$ falls with frequency, high-frequency parts of the message are attenuated and slightly delayed: this amplitude distortion is the aperture effect.

Remedies

1. Keep the pulse width small, $\tau \ll T_s$ (duty cycle $\tau/T_s \le 0.1$ makes the loss under about 0.5%).
2. Use an equalizer after the reconstruction LPF with response

$$H_{eq}(f) = \frac{1}{H(f)} = \frac{1}{\tau\,\text{sinc}(f\tau)}\ \text{for}\ |f| \le W$$

which boosts high frequencies to cancel the sinc loss.