First-order Hold - Basic First-order Hold

Basic First-order Hold

The first-order hold is the hypothetical filter or LTI system that converts the ideally sampled signal

\begin{align} x_s(t) & {} = x(t) \ T \sum_{n=-\infty}^{\infty} \delta(t - nT) \\ & {} = T \sum_{n=-\infty}^{\infty} x(nT) \delta(t - nT) \end{align}

to the piecewise linear signal

resulting in an effective impulse response of

h_{\mathrm{FOH}}(t)\,= \frac{1}{T} \mathrm{tri} \left(\frac{t}{T} \right) = \begin{cases} \frac{1}{T} \left( 1 - \frac{|t|}{T} \right) & \mbox{if } |t| < T \\ 0 & \mbox{otherwise} \end{cases} \
where is the triangular function.

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

where is the sinc function.

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

This is an acausal system in that the linear interpolation function moves toward the value of the next sample before such sample is applied to the hypothetical FOH filter. This acausality is also reflected in the impulse response of the FOH filter beginning to respond before impulse is applied.

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