Consider a continuous signal and its sampled counterpart with a sampling period T.

Applying a zero-order hold to the sampled signal results in a piecewise constant signal.

Cascading the zero hold with a reconstruction filter gives the reconstructed signal.

An ideal low-pass filter removes all spectrum replicas in the frequency domain for optimal reconstruction.

The time-domain impulse response of this ideal filter is a sinc function.

Convolution of the sampled signal with the sinc function results in the time-domain signal.

This process, known as band-limited interpolation, ensures accurate signal reconstruction.

Convolution of the sampled signal with the triangular impulse response results in a smoother, peak-free time-domain signal.

This is referred to as a first-order hold, commonly called Linear Interpolation.

In the frequency domain, the curve is smoothed at the center and compressed at the sides, reducing side replicas more effectively than a zero-order hold.