First-order Hold - Predictive First-order Hold

Lastly, the predictive first-order hold is quite different. This is a causal hypothetical LTI system or filter that converts the ideally sampled signal

into a piecewise linear output such that the current sample and immediately previous sample are used to linearly extrapolate up to the next sampling instance. The output of such a filter would be

resulting in an effective impulse response of

where is the rectangular function and 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 predictive FOH is found by substituting s = i 2 π f:

This a causal system. The impulse response of the predictive FOH does not respond before the input impulse.

This kind of piecewise linear reconstruction is physically realizable by implementing a digital filter of gain H(z) = 1 − z−1, applying the output of that digital filter (which is simply x−x) to an ideal conventional digital-to-analog converter (that has an inherent zero-order hold as its model) and applying that DAC output to an analog filter with transfer function H(s) = (1+sT)/(sT).