View source for Zero-order hold

{{Use American English|date = March 2019}}
{{Short description|Model of signal reconstruction in digital-to-analog (DAC) converters}}
{{One source|date=August 2021}}
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.

==Time-domain model==

[[Image:Zeroorderhold.impulseresponse.svg|thumb|Figure 1.  The time-shifted and time-scaled rect function used in the time-domain analysis of the ZOH.]]
[[Image:Zeroorderhold.signal.svg|thumb|Figure 2.  Piecewise-constant signal ''x''<sub>ZOH</sub>(''t'').]]
[[Image:Sampled.signal.svg|thumb|Figure 3.  A modulated Dirac comb ''x''<sub>s</sub>(''t'').]]
A zero-order hold reconstructs the following continuous-time waveform from a sample sequence ''x''[''n''], assuming one sample per time interval ''T'':
<math display="block">x_{\mathrm{ZOH}}(t)\,= \sum_{n=-\infty}^{\infty} x[n]\cdot \mathrm{rect} \left(\frac{t-T/2 -nT}{T} \right) </math>
where <math>\mathrm{rect}(\cdot) </math> is the [[rectangular function]].

The function <math>\mathrm{rect} \left(\frac{t-T/2}{T} \right)</math> is depicted in Figure 1, and <math>x_{\mathrm{ZOH}}(t)</math> is the [[piecewise-constant]] signal depicted in Figure 2.

==Frequency-domain model==

The equation above for the output of the ZOH can also be modeled as the output of a [[LTI system theory|linear time-invariant filter]] with impulse response equal to a rect function, and with input being a sequence of [[dirac delta function|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 delta function|Dirac impulses]], ''x''<sub>s</sub>(''t''), representing the discrete samples, ''x''[''n''], is [[low-pass filter]]ed 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, ''x''<sub>s</sub>(''t''), to a [[LTI system|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:
<math display="block">\begin{align}
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{align}</math>

The scaling by <math>T</math>, which arises naturally by time-scaling the delta function, has the result that the mean value of ''x<sub>s</sub>''(''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,<ref>{{cite book | title = Principles of Digital Audio | author = Ken C. Pohlmann | publisher = McGraw-Hill | year = 2000 | edition = fifth | ISBN = 0-07-144156-5}}</ref> while many others omit the time-scaling and the ''T'', resulting in a low-pass filter model with a DC gain of ''T'', and hence dependent on the units of measurement of time.

[[Image:Zeroorderhold.impulseresponse.svg|thumb|Figure 4. Impulse response of zero-order hold ''h''<sub>ZOH</sub>(''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 hypothetical [[filter (signal processing)|filter]] or [[LTI system]] that converts the sequence of modulated Dirac impulses ''x<sub>s</sub>''(''t'')to the piecewise-constant signal (shown in Figure 2):
<math display="block">x_{\mathrm{ZOH}}(t) = \sum_{n=-\infty}^{\infty} x[n] \cdot \mathrm{rect} \left(\frac{t - nT}{T} - \frac{1}{2} \right) </math>
resulting in an effective [[impulse response]] (shown in Figure 4) of:
<math display="block">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 \le t < T  \\
0           & \text{otherwise}
\end{cases} </math>

The effective frequency response is the [[continuous Fourier transform]] of the impulse response.

<math display="block">H_{\mathrm{ZOH}}(f) = \mathcal{F} \{ h_{\mathrm{ZOH}}(t) \} = \frac{1 - e^{-i 2 \pi fT}}{i 2 \pi fT} = e^{-i \pi fT} \mathrm{sinc}(fT) </math>
where <math>\mathrm{sinc}(x) </math> is the (normalized) [[sinc function]] <math>\frac{\sin(\pi x)}{\pi x}</math> commonly used in digital signal processing.

The [[Laplace transform]] [[transfer function]] of the ZOH is found by substituting ''s'' = ''i'' 2 ''π'' ''f'':
<math display="block">H_{\mathrm{ZOH}}(s) = \mathcal{L} \{ h_{\mathrm{ZOH}}(t) \} \,= \frac{1 - e^{-sT}}{sT} \ </math>

The fact that practical [[digital-to-analog converter]]s (DAC) do not output a sequence of [[dirac delta|dirac impulses]], ''x''<sub>s</sub>(''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''<sub>ZOH</sub>(''t'') (a [[piecewise constant]] function), means that there is an inherent effect of the ZOH on the effective frequency response of the DAC, resulting in a mild [[roll-off]] of gain at the higher frequencies (a 3.9224&nbsp;dB loss at the [[Nyquist frequency]], corresponding to a gain of sinc(1/2) = 2/π). This drop is a consequence of the ''hold'' property of a conventional DAC, and is ''not'' due to the [[sample and hold]] that might precede a conventional [[analog-to-digital converter]] (ADC).

==See also==
* [[Nyquist–Shannon sampling theorem]]
* [[First-order hold]]
* [[Discretization#discrete function|Discretization of linear state space models (assuming zero-order hold)]]

==References==
{{reflist}}

[[Category:Digital signal processing]]
[[Category:Electrical engineering]]
[[Category:Control theory]]
[[Category:Signal processing]]