Signal reconstruction and reconstructors

Signal reconstruction is the process of recovering a continuous-time signal from its samples $x(kT)$, so that it can drive a continuous plant. In a digital control system it is done after the D/A converter by a hold circuit (signal reconstructor).

Ideal reconstruction

If the signal is band-limited and $\omega_s > 2\omega_m$, an ideal low-pass filter with cut-off $\omega_s/2$ recovers $x(t)$ exactly:

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

This filter is non-causal (it needs future samples), so it cannot be built.

Practical reconstructors (hold circuits)

A hold circuit extrapolates the signal between $kT$ and $(k+1)T$ from the present and past samples using a polynomial:

$$h(kT+\tau) = a_n\tau^n + a_{n-1}\tau^{n-1} + \cdots + a_1\tau + a_0, \quad 0 \le \tau < T$$

with $h(kT) = x(kT)$, so $a_0 = x(kT)$. An $n$th-order hold uses the last $n+1$ samples.

• Zero order hold ($n=0$): $h(kT+\tau) = x(kT)$; staircase output. $G_{h0}(s) = \dfrac{1-e^{-Ts}}{s}$
• First order hold ($n=1$): straight-line extrapolation from two samples: $h(kT+\tau) = x(kT) + \dfrac{x(kT)-x((k-1)T)}{T}\,\tau$ $G_{h1}(s) = \dfrac{Ts+1}{T}\left(\dfrac{1-e^{-Ts}}{s}\right)^2$

 ZOH:  _|‾‾|__|‾‾‾|_   (steps)
 FOH:  /  /  /         (sloped segments)

The ZOH is the most widely used because it is simple and cheap, and its phase lag ($\omega T/2$) is small when $T$ is small. Higher-order holds track smooth signals better, but they need more memory and hardware and give more phase lag at high frequencies.