Zero-order hold

Thezero-order hold(ZOH) is a mathematical model of the practicalsignal reconstructiondone by a conventionaldigital-to-analog converter(DAC).That is, it describes the effect of converting adiscrete-time signalto acontinuous-time signalby holding each sample value for one sample interval. It has several applications in electrical communication.

Time-domain model

Figure 1. The time-shifted and time-scaled rect function used in the time-domain analysis of the ZOH.

Figure 2. Piecewise-constant signalx_ZOH(t).

Figure 3. A modulated Dirac combx_s(t).

A zero-order hold reconstructs the following continuous-time waveform from a sample sequencex[n], assuming one sample per time intervalT: $$x_{\mathrm {ZOH} }(t)\,=\sum _{n=-\infty }^{\infty }x[n]\cdot \mathrm {rect} \left({\frac {t-T/2-nT}{T}}\right)$$ where$\mathrm {rect} (\cdot )$is therectangular function.

The function$\mathrm {rect} \left({\frac {t-T/2}{T}}\right)$is depicted in Figure 1, and$x_{\mathrm {ZOH} }(t)$is thepiecewise-constantsignal depicted in Figure 2.

Frequency-domain model

The equation above for the output of the ZOH can also be modeled as the output of alinear time-invariant filterwith impulse response equal to a rect function, and with input being a sequence ofdirac impulsesscaled to the sample values. The filter can then be analyzed in the frequency domain, for comparison with other reconstruction methods such as theWhittaker–Shannon interpolation formulasuggested by theNyquist–Shannon sampling theorem,or such as thefirst-order holdor linear interpolation between sample values.

In this method, a sequence ofDirac impulses,x_s(t), representing the discrete samples,x[n], islow-pass filteredto recover acontinuous-time signal,x(t).

Even though this isnotwhat a DAC does in reality, the DAC output can be modeled by applying the hypothetical sequence of dirac impulses,x_s(t), to alinear, time-invariant filterwith such characteristics (which, for an LTI system, are fully described by theimpulse 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: $${\begin{aligned}x_{s}(t)&=\sum _{n=-\infty }^{\infty }x[n]\cdot \delta \left({\frac {t-nT}{T}}\right)\\&{}=T\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT).\end{aligned}}$$

The scaling by$T$,which arises naturally by time-scaling the delta function, has the result that the mean value ofx_s(t) is equal to the mean value of the samples, so that the lowpass filter needed will have a DC gain of 1. Some authors use this scaling,while many others omit the time-scaling and theT,resulting in a low-pass filter model with a DC gain ofT,and hence dependent on the units of measurement of time.

Figure 4. Impulse response of zero-order holdh_ZOH(t). It is identical to the rect function of Figure 1, except now scaled to have an area of 1 so the filter will have a DC gain of 1.

The zero-order hold is the hypotheticalfilterorLTI systemthat converts the sequence of modulated Dirac impulsesx_s(t)to the piecewise-constant signal (shown in Figure 2): $$x_{\mathrm {ZOH} }(t)=\sum _{n=-\infty }^{\infty }x[n]\cdot \mathrm {rect} \left({\frac {t-nT}{T}}-{\frac {1}{2}}\right)$$ resulting in an effectiveimpulse response(shown in Figure 4) of: $$h_{\mathrm {ZOH} }(t)\,={\frac {1}{T}}\mathrm {rect} \left({\frac {t}{T}}-{\frac {1}{2}}\right)={\begin{cases}{\frac {1}{T}}&{\text{if }}0\leq t<T\\0&{\text{otherwise}}\end{cases}}$$

The effective frequency response is thecontinuous Fourier transformof the impulse response.

$$H_{\mathrm {ZOH} }(f)={\mathcal {F}}\{h_{\mathrm {ZOH} }(t)\}={\frac {1-e^{-i2\pi fT}}{i2\pi fT}}=e^{-i\pi fT}\mathrm {sinc} (fT)$$ where$\mathrm {sinc} (x)$is the (normalized)sinc function${\frac {\sin(\pi x)}{\pi x}}$commonly used in digital signal processing.

TheLaplace transformtransfer functionof the ZOH is found by substitutings=i2πf: $$H_{\mathrm {ZOH} }(s)={\mathcal {L}}\{h_{\mathrm {ZOH} }(t)\}\,={\frac {1-e^{-sT}}{sT}}\$$

The fact that practicaldigital-to-analog converters(DAC) do not output a sequence ofdirac impulses,x_s(t) (that, if ideally low-pass filtered, would result in the unique underlying bandlimited signal before sampling), but instead output a sequence of rectangular pulses,x_ZOH(t) (apiecewise constantfunction), means that there is an inherent effect of the ZOH on the effective frequency response of the DAC, resulting in a mildroll-offof gain at the higher frequencies (a 3.9224 dB loss at theNyquist frequency,corresponding to a gain of sinc(1/2) = 2/π). This drop is a consequence of theholdproperty of a conventional DAC, and isnotdue to thesample and holdthat might precede a conventionalanalog-to-digital converter(ADC).

See also

• Nyquist–Shannon sampling theorem
• First-order hold
• Discretization of linear state space models (assuming zero-order hold)