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A continuous-time signal is sampled using impulse-train sampling followed by the zero-order hold method.

Impulse-train sampling uses a periodic impulse train, which is a series of delta functions spaced at multiples of the sampling period.

When the continuous-time signal is multiplied by the impulse train, impulses are produced with amplitudes matching the signal's value at the sampling points.

In the frequency domain, sampling results in the convolution of the spectrum of the original signal with the spectrum of the impulse train.

The spectrum of the impulse train contains shifted replicas of the spectrum of the signal, spaced at multiples of the sampling frequency.

This convolution results in a periodic function with the sampled spectrum having shifted replicas, scaled by the inverse of the sampling period.

The zero-order hold method keeps each sampled value constant until the next sampling period, creating a piecewise constant signal.

This piecewise constant signal is processed through a system with a rectangular impulse response, resulting in a steady output.