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Data hold circuit and ZOH derivation

Short answer

A data hold circuit reconstructs a continuous signal from samples. A zero-order hold keeps each sample constant for one period T, giving impulse response g(t)=u(t)-u(t-T). Its Laplace transform yields the transfer function G_h0(s) = (1-e^(-Ts))/s.

What do you mean by data hold circuit? Derive the transfer function of Zero Order Hold (ZOH).

A data hold circuit converts a sequence of samples into a continuous signal by holding or extrapolating the sample value(s) between sampling instants. The simplest is the zero order hold (ZOH), which keeps each sample constant for one period: h(kT+τ)=x(kT)h(kT+\tau) = x(kT), 0≤τ<T0 \le \tau < T, giving a staircase output.

Derivation of ZOH transfer function

Apply a unit impulse at t=0t=0. The ZOH holds the value 1 for one sampling period and then drops to zero, so its impulse response is a rectangular pulse:

gh0(t)=u(t)−u(t−T)g_{h0}(t) = u(t) - u(t-T) 
 g(t)
  1 |------+
    |      |
  0 +------+-------> t
    0      T

Taking the Laplace transform and using the shift theorem L[u(t−T)]=e−Ts/s\mathcal{L}[u(t-T)] = e^{-Ts}/s:

Gh0(s)=1s−e−Tss=1−e−Tss\begin{aligned} G_{h0}(s) &= \frac{1}{s} - \frac{e^{-Ts}}{s} \\ &= \frac{1-e^{-Ts}}{s} \end{aligned}

This is the transfer function of the zero order hold.

✓ Answer checked by Claude Opus 5.5 (high effort)

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