First-order hold

First-order hold (FOH) is a mathematical model of the practical reconstruction of sampled signals that could be done by a conventional digital-to-analog converter (DAC) and an analog circuit called an integrator.

The Laplace transform transfer function of 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.

Delayed first-order hold, sometimes called causal first-order hold, is identical to 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 The effective frequency response is the continuous Fourier transform of the impulse response.

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[n]−x[n−1]) 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.

The Laplace transform transfer function of the predictive FOH is found by substituting s = i 2 π f: This a causal system.

Ideally sampled signal x s ( t ).
Piecewise linear signal x FOH ( t ).
Impulse response (non-causal) of first-order hold h FOH ( t ).
Delayed piecewise linear signal x FOH ( t ).
Impulse response of causal first-order hold h FOH ( t ).
Predictive FOH output signal x FOH ( t ).
Impulse response of predictive first-order hold h FOH ( t ).