Impulse train sampling, aliasing, reconstruction

Impulse-train sampling

Impulse-train sampling is the mathematical model of ideal sampling: $x(t)$ is multiplied by a periodic train of unit impulses of period $T$:

$$x_p(t) = x(t)\sum_{n=-\infty}^{\infty}\delta(t-nT) = \sum_{n=-\infty}^{\infty} x(nT)\,\delta(t-nT)$$

Each impulse carries the sample value $x(nT)$ as its area.

Aliasing in impulse-train sampling

The spectrum of the sampled signal is

$$X_p(j\omega) = \frac{1}{T}\sum_{k=-\infty}^{\infty} X\big(j(\omega - k\omega_s)\big), \qquad \omega_s = \frac{2\pi}{T}$$

i.e. copies of $X(j\omega)$ at every multiple of $\omega_s$.

• If $\omega_s > 2\omega_M$, the copies are separate.
• If $\omega_s < 2\omega_M$, the copy at $\omega_s$ begins at $\omega_s - \omega_M$, which is below $\omega_M$, so the copies overlap. Frequencies above $\omega_s/2$ fold back into the band and add to the true components. This is aliasing; the original spectrum can no longer be separated.

ws < 2wM:
     ____     ____     ____
    /    \   /    \   /    \
---/------\-/------\-/------\---> w
         -ws  \/ 0  \/  ws
              overlap = aliasing

Reconstruction from samples

If $\omega_s > 2\omega_M$, pass $x_p(t)$ through an ideal low-pass filter with gain $T$ and cutoff $\omega_c$, where $\omega_M < \omega_c < \omega_s - \omega_M$ (usually $\omega_c = \omega_s/2$). The filter keeps only the $k=0$ copy, so its output is $x(t)$. In time,

$$h(t) = \frac{T\sin\omega_c t}{\pi t}, \qquad x_r(t) = \sum_{n} x(nT)\,\frac{T\sin\omega_c(t-nT)}{\pi(t-nT)}$$

This is band-limited (sinc) interpolation. Practical systems use simpler approximations: zero-order hold (staircase) or first-order hold (straight-line interpolation) followed by smoothing.