In signal processing, a continuous-time signal can be sampled using an impulse-train sampling technique, followed by the zero-order hold method. Impulse-train sampling involves the use of a periodic impulse train, which consists of a series of delta functions spaced at regular intervals determined by the sampling period. When a continuous-time signal is multiplied by this impulse train, it generates impulses with amplitudes corresponding to the signal's values at the sampling points.

In the frequency domain, sampling is represented by the convolution of the original signal's spectrum with the spectrum of the impulse train. The impulse train's spectrum consists of shifted replicas of the original signal's spectrum, spaced at intervals equal to the sampling frequency. This convolution produces a periodic function, resulting in the sampled signal's spectrum being composed of these shifted replicas, scaled by the inverse of the sampling period.

The zero-order hold method is used to reconstruct the signal after sampling. It retains each sampled value constant until the next sampling period, creating a piecewise constant signal. This method approximates the original continuous-time signal by holding the amplitude of each sample until the arrival of the next sample, effectively creating a staircase-like waveform.

This piecewise constant signal is then processed through a system characterized by a rectangular impulse response. This system smooths out the transitions between the held values, resulting in a steady output that approximates the original signal. The zero-order hold method is particularly useful in digital-to-analog conversion, where it provides a simple and effective way to generate a continuous-time signal from discrete samples.

In essence, impulse-train sampling and the zero-order hold method together form a fundamental process in digital signal processing, enabling the conversion of continuous-time signals into discrete-time signals and their subsequent reconstruction. This process is crucial in various applications, including digital audio, telecommunications, and data acquisition systems, ensuring accurate representation and recovery of analog signals in the digital domain.