Practical factors in sampling; sampling interval for X1(t)X2(t)

Practical factors in sampling

1. Signals are not strictly band-limited. Real signals (and noise) have some energy above $W$, which would alias. An anti-aliasing LPF is used before the sampler.
2. Non-ideal filters. Ideal brick-wall filters do not exist, so the sampling rate is chosen above $2W$ to leave a guard band for the filter's transition region (speech 3.4 kHz sampled at 8 kHz instead of 6.8 kHz).
3. Finite pulse width (aperture effect). Practical samples are pulses of width $\tau$, not impulses. Flat-top sampling multiplies the spectrum by $\tau\,\text{sinc}(f\tau)$, attenuating high frequencies; fixed by small $\tau/T_s$ and an equalizer $1/H(f)$.
4. Sample-and-hold limits. Acquisition time, droop of the hold capacitor and aperture jitter (uncertainty in sampling instant) cause errors, especially for fast signals.
5. Clock stability. Jitter in $f_s$ produces timing errors, so stable crystal clocks are used.
6. Choice of rate. Higher $f_s$ eases filtering but raises bit rate, storage and bandwidth; a compromise is chosen.
7. Noise and quantization. The sampled value is followed by quantization; ADC resolution and noise must suit the required SNR.

m(t)->[Anti-alias LPF]->[S/H, fs>2W]->[ADC]->...
...->[DAC]->[Recon. LPF]->[Equalizer]->m(t)

Maximum sampling interval for $Y(t) = X_1(t)X_2(t)$

Multiplication in time is convolution in frequency:

$$Y(f) = X_1(f) * X_2(f)$$

If $X_1$ occupies $|f| \le W_1$ and $X_2$ occupies $|f| \le W_2$, the convolution occupies $|f| \le W_1 + W_2$. So the bandwidth of $Y$ is $W_1 + W_2$.

$$\begin{aligned} f_{s,min} &= 2(W_1 + W_2) \\ T_{s,max} &= \frac{1}{2(W_1 + W_2)} \end{aligned}$$

Answer: maximum sampling interval $= \dfrac{1}{2(W_1 + W_2)}$ seconds.