Zero-order hold

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The Zero-order hold (ZOH) is a mathematical model of the practical reconstruction of sampled signals done by conventional digital-to-analog converters (DAC). A mathematical model such as the ZOH (or possibly the first-order hold) is necessary because, in the sampling and reconstruction theorem, a sequence of dirac impulses, xs(t), representing the discrete samples, x(nT), is low-pass filtered to recover the original signal that was sampled, x(t). However, outputting a sequence of dirac impulses is decidedly impractical. Most conventional DACs output a voltage proportional to the discrete sample value and hold that voltage to a constant value for the duration of the sampling interval and then change that voltage rapidly to the value corresponding to the next discrete sample value.

Even though this is not what is physically done, an identical output can be generated by applying the hypothetical sequence of dirac impulses, xs(t), to a linear, time-invariant system, otherwise known as a linear 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.

Ideally sampled signal xs(t).
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Ideally sampled signal xs(t).

Thus, the Zero-order hold is the hypothetical filter or LTI system that converts the ideally sampled signal

x_s(t)\, = x(t) \ T \sum_{n=-\infty}^{\infty} \delta(t - nT) \
= T \sum_{n=-\infty}^{\infty} x(nT) \delta(t - nT) \
Piecewise constant signal xZOH(t).
Enlarge
Piecewise constant signal xZOH(t).

to the piecewise constant signal

x_{\mathrm{ZOH}}(t)\,= \sum_{n=-\infty}^{\infty} x(nT) \mathrm{rect} \left(\frac{t - nT}{T}-\frac{1}{2} \right) \
Impulse response of zero-order hold hZOH(t).
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Impulse response of zero-order hold hZOH(t).

resulting in an effective impulse response of

h_{\mathrm{ZOH}}(t)\,= \frac{1}{T} \mathrm{rect} \left(\frac{t}{T}-\frac{1}{2} \right) = \begin{cases} \frac{1}{T} & \mbox{if } 0 \le t < T \\ 0 & \mbox{otherwise} \end{cases} \
where \mathrm{rect}(x) \ is the rectangular function.

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

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) \
where \mathrm{sinc}(x) \ is the sinc function.

The Laplace transform transfer function of the ZOH is found by substituting s = i 2 π f:

H_{\mathrm{ZOH}}(s)\, = \mathcal{L} \{ h_{\mathrm{ZOH}}(t) \} \,= \frac{1 - e^{-sT}}{sT} \

The fact that practical digital-to-analog converters (DAC) do not output a sequence of dirac impulses, xs(t), (that, if ideally low-pass filtered, result in the original signal before sampling) but instead output a sequence of rectangular pulses, xZOH(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 dB loss at the Nyquist frequency). This is really 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) as is often misunderstood.

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