Reconstruction of signal from samples

A sampled signal carries values only at $t = kT$. Reconstruction produces a continuous signal from these values, so that the output of a digital controller can drive an analog plant.

Spectrum of the sampled signal

The impulse-sampled signal has spectrum

$$X^*(j\omega) = \frac{1}{T}\sum_{n=-\infty}^{\infty} X(j(\omega + n\omega_s)), \quad \omega_s = \frac{2\pi}{T}$$

i.e. the original spectrum (primary component, $n=0$) plus copies (complementary components) centred at $\pm\omega_s, \pm 2\omega_s, \dots$

 |X*(jw)|
   _/\_      _/\_      _/\_
 --------+---------+---------> w
       -ws    0    ws
      copies  |  primary
         <-- ideal LPF -->
           -ws/2    ws/2

Condition for exact recovery

If $x(t)$ is band-limited to $\omega_m$ and $\omega_s > 2\omega_m$ (sampling theorem), the copies do not overlap. Then an ideal low-pass filter with gain $T$ and cut-off $\omega_s/2$ keeps only the primary component and gives back $x(t)$ exactly:

$$x(t) = \sum_{k} x(kT)\,\frac{\sin[\omega_s(t-kT)/2]}{\omega_s(t-kT)/2}$$

If $\omega_s < 2\omega_m$ the copies overlap (aliasing) and recovery is impossible.

Practical reconstruction

The ideal filter is non-causal, so practical systems use hold circuits, which build the signal from past samples only:

1. Zero order hold (ZOH): keeps $x(kT)$ constant until $(k+1)T$. Transfer function $G_{h0}(s) = \frac{1-e^{-Ts}}{s}$. Its magnitude $T\left|\frac{\sin(\omega T/2)}{\omega T/2}\right|$ is a rough low-pass filter; it passes some high-frequency components (ripple) and adds a lag of $\omega T/2$.
2. First order hold (FOH): extrapolates with the slope of the last two samples. $G_{h1}(s) = \frac{Ts+1}{T}\left(\frac{1-e^{-Ts}}{s}\right)^2$.

 samples:   .   .   .
 ZOH:      |‾‾‾|‾‾‾|‾‾‾   staircase
 FOH:      /  /  /        ramps

Choosing a small $T$ (high sampling rate) and following the hold with a smoothing filter makes the reconstructed signal close to the original.