Aliasing via impulse-train sampling

Aliasing

Aliasing is the overlapping of spectral copies that happens when a signal is sampled below its Nyquist rate ($\omega_s < 2\omega_M$); high-frequency components then appear as lower frequencies and the signal cannot be recovered.

Frequency-domain analysis of impulse-train sampling

Let $x(t)$ be band-limited to $\omega_M$. Impulse-train sampling multiplies it by a periodic impulse train of period $T$:

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

The Fourier transform of the impulse train is also an impulse train, with spacing $\omega_s = 2\pi/T$:

$$P(j\omega) = \frac{2\pi}{T}\sum_{k=-\infty}^{\infty}\delta(\omega - k\omega_s)$$

Multiplication in time is convolution in frequency:

$$\begin{aligned} X_p(j\omega) &= \frac{1}{2\pi}\, X(j\omega) * P(j\omega) \\ &= \frac{1}{T}\sum_{k=-\infty}^{\infty} X\big(j(\omega - k\omega_s)\big) \end{aligned}$$

So the spectrum of the sampled signal is the original spectrum, scaled by $1/T$, repeated at every multiple of $\omega_s$.

Case 1: $\omega_s > 2\omega_M$ (no aliasing)

    /\        /\        /\
   /  \      /  \      /  \
--/----\----/----\----/----\---> w
    -ws        0        ws

The copies are separate. An ideal LPF (gain $T$, cutoff $\omega_M < \omega_c < \omega_s - \omega_M$) recovers $X(j\omega)$ exactly.

Case 2: $\omega_s < 2\omega_M$ (aliasing)

     ____    ____    ____
    /    \  /    \  /    \
---/------\/------\/------\---> w
        -ws   0     ws
         overlapped regions add

The $k=1$ copy starts at $\omega_s - \omega_M < \omega_M$, so it overlaps the baseband copy. In the band $\omega_s - \omega_M < |\omega| < \omega_M$, the spectrum is a sum of two copies. A component at $\omega_0 > \omega_s/2$ shows up at $\omega_s - \omega_0$.

Example: $x(t) = \cos(2\pi\cdot 700t)$ sampled at $f_s = 1000$ Hz. The copy at $1000 - 700 = 300$ Hz lies inside the LPF band, so the output of reconstruction is a 300 Hz tone instead of 700 Hz.

Aliasing is prevented by sampling at $\omega_s > 2\omega_M$ and using an anti-aliasing filter before sampling.