Theory:

Zero-order hold (ZOH) devices convert sampled signals to continuoustime signals for analyzing sampled continuous-time systems. The zero-order hold discretization of a continuous-time LTI model is depicted in the Fig. 1.


The ZOH device generates a continuous input signal u(t) by holding each sample value u(k) constant over one sample period.
$$ u(t) = u(k) ; kT \le t \le (k+1)T \tag{1} $$ First-order hold (FOH) differs from ZOH by the underlying hold mechanism.
To turn the input samples into a continuous input, FOH uses linear interpolation between samples.
$$ u(t) = u(k) + \frac {t-kT}{T} [u(k+1)-u(k)] ; kT \le t \le (k+1)T \tag{2} $$ The transfer functions for these circuits are given as
For ZOH
$$ G(s) = \frac {1-e^{-Ts}}{s} \tag{3} $$ For FOH
$$ G(s) = \frac {1+Ts}{T} \frac {(1-e^{-Ts})^2}{s^2} \tag{4} $$ Practical implementable transfer functions are obtained using approximations of exponential series. (Pade Approximations)