The delayed first-order hold, sometimes called causal first-order hold, is identical to the FOH above except that its output is delayed by one sample period resulting in a delayed piecewise linear output signal
resulting in an effective impulse response of
- 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 delayed FOH is found by substituting s = i 2 π f:
The delayed output makes this a causal system. The impulse response of the delayed FOH does not respond before the input impulse.
This kind of delayed 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 integrating (in continuous-time, H(s) = 1/(sT)) the DAC output.
Read more about this topic: First-order Hold
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