first-order hold
A mathematical model of the practical reconstruction of sampled signals that could be done by a conventional digital-to-analog converter and an analog circuit called an integrator.
first-order hold: linear ramp between sampled points
A first-order hold is a signal reconstruction method that connects consecutive sample values with straight lines instead of holding each value flat until the next sample arrives. When a digital-to-analog converter outputs discrete voltage steps, a first-order hold circuit slopes between them, creating a piecewise linear approximation of the original continuous signal. This is distinct from a zero-order hold, which maintains constant voltage between samples.
The circuit itself is simple: an integrator or ramp generator that reads the difference between the current and previous sample, then linearly interpolates across the sample interval. The output voltage rises or falls at a constant rate determined by this difference and the sample period. For a sample period of 100 microseconds with voltage changing 5 volts between samples, the output would ramp at 50 volts per millisecond. This linear transition happens during each interval, creating the characteristic sawtooth-like envelope around the reconstructed waveform.
First-order hold finds use in audio and control systems where smooth variation matters more than precision. A zero-order hold produces audible steps or artifacts in audio playback; a first-order hold reduces these. In control loops, the linear ramp approximates continuous plant behavior more closely than flat steps, improving closed-loop stability. However, it introduces its own artifacts: the slope creates high-frequency components that lag behind the true signal, making first-order hold unsuitable for applications requiring accurate phase response.
The trade-off is computational and practical simplicity against reconstruction fidelity. Higher-order schemes like cubic splines produce better approximations but require buffering multiple samples and more complex arithmetic. First-order hold sits in the middle ground: easier to implement than higher orders, visibly better than zero-order hold for many applications, but still far from perfect reconstruction. Engineers choose based on bandwidth requirements, latency tolerance, and the cost of filtering afterward.