ZOH and FOH defined; ZOH transfer function

Zero order hold and first order hold

A hold circuit reconstructs a continuous signal from samples by extrapolating between sampling instants with a polynomial of order $n$.

• Zero order hold (ZOH): holds the latest sample constant until the next one: $h(kT+\tau) = x(kT)$, $0 \le \tau < T$. Output is a staircase. It needs only the present sample.
• First order hold (FOH): extrapolates a straight line using the slope of the last two samples: $h(kT+\tau) = x(kT) + \dfrac{x(kT)-x((k-1)T)}{T}\,\tau$. Transfer function $G_{h1}(s) = \frac{Ts+1}{T}\left(\frac{1-e^{-Ts}}{s}\right)^2$.

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

Transfer function of ZOH

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

$$g_{h0}(t) = u(t) - u(t-T)$$

 g(t)
  1 |------+
    |      |
  0 +------+-------> t
    0      T

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

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

Frequency response

With $s = j\omega$:

$$G_{h0}(j\omega) = T\,\frac{\sin(\omega T/2)}{\omega T/2}\,e^{-j\omega T/2}$$

so the ZOH acts as a low-pass filter with gain $T$ at low frequency, zero gain at multiples of $\omega_s$, and a phase lag of $\omega T/2$ (half-period delay).