Impulse train sampling and reconstruction

Impulse train sampling

Impulse train (ideal) sampling multiplies the continuous signal $x(t)$ by a periodic train of unit impulses spaced $T$ seconds apart. The result is a train of impulses whose strengths are the sample values:

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

$T$ is the sampling period and $\omega_s = 2\pi/T$ the sampling frequency.

Spectrum of the sampled signal

$p(t) = \sum\delta(t - nT) \leftrightarrow P(j\omega) = \frac{2\pi}{T}\sum_k\delta(\omega - k\omega_s)$. Multiplication in time is convolution in frequency:

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

So $X_p(j\omega)$ is $X(j\omega)$ repeated at every multiple of $\omega_s$, scaled by $1/T$.

 X(jw)            Xp(jw), ws > 2wm
   /\               /\      /\      /\
  /  \             /  \    /  \    /  \
-+-+--+-->  w   --+--+--+-+--+--+-+--+--> w
 -wm 0 wm          -ws    0  wm   ws

Reconstruction from samples

1. Condition: $x(t)$ must be band-limited to $\omega_m$, and $\omega_s > 2\omega_m$ (Nyquist). Then the copies do not overlap.
2. Ideal low-pass filter: pass $x_p(t)$ through an LPF with gain $T$ and cutoff $\omega_c$, where $\omega_m < \omega_c < \omega_s - \omega_m$ (usually $\omega_c = \omega_s/2$):

$$H(j\omega) = \begin{cases} T, & |\omega| < \omega_c \\ 0, & \text{otherwise}\end{cases} \quad\Rightarrow\quad X_r(j\omega) = X_p(j\omega)H(j\omega) = X(j\omega)$$

3. Time-domain view (interpolation): the filter's impulse response is $h(t) = \frac{T\sin(\omega_c t)}{\pi t}$. Convolving it with the impulse train gives

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

With $\omega_c = \omega_s/2$ this becomes $x_r(t) = \sum x(nT)\,\text{sinc}\left(\frac{t - nT}{T}\right)$: each sample is replaced by a sinc pulse, and the sum of these pulses fills in the values between samples exactly.

 x(t) --(x)--> xp(t) --> [ LPF: gain T, cutoff wc ] --> xr(t) = x(t)
         ^
     p(t) = sum delta(t - nT)

Practical note: an ideal LPF is non-causal, so real systems use a zero-order hold (D/A converter) followed by a smoothing filter, and sample a little above the Nyquist rate to leave a guard band.