Zero-order hold

The zero-order hold (ZOH) is a mathematical model of the practical signal reconstruction done by a conventional digital-to-analog converter (DAC). That is, it describes the effect of converting a discrete-time signal to a continuous-time signal by holding each sample value for one sample interval. It has several applications in electrical communication. A zero-order hold reconstructs the following continuous-time waveform from a sample sequence x[n], assuming one sample per time interval T: where is the rectangular function. The function is depicted in Figure 1, and is the piecewise-constant signal depicted in Figure 2. The equation above for the output of the ZOH can also be modeled as the output of a linear time-invariant filter with impulse response equal to a rect function, and with input being a sequence of dirac impulses scaled to the sample values. The filter can then be analyzed in the frequency domain, for comparison with other reconstruction methods such as the Whittaker–Shannon interpolation formula suggested by the Nyquist–Shannon sampling theorem, or such as the first-order hold or linear interpolation between sample values. In this method, a sequence of Dirac impulses, xs(t), representing the discrete samples, x[n], is low-pass filtered to recover a continuous-time signal, x(t). Even though this is not what a DAC does in reality, the DAC output can be modeled by applying the hypothetical sequence of dirac impulses, xs(t), to a linear, time-invariant filter with such characteristics (which, for an LTI system, are fully described by the impulse response) so that each input impulse results in the correct constant pulse in the output. Begin by defining a continuous-time signal from the sample values, as above but using delta functions instead of rect functions: The scaling by , which arises naturally by time-scaling the delta function, has the result that the mean value of xs(t) is equal to the mean value of the samples, so that the lowpass filter needed will have a DC gain of 1.