Sampling & Zero Order Hold

Firstly let's look at why Sampling is carried out?

Sampling makes an analog signal discrete so that a controller can intercept data only at those sampled intervals of time cause a controller knows only digital data. It gives us the frequency at which a signal is sampled.

Let's have a closer look at Sampling of a signal:

First, the continuous signal is multiplied by the impulse signal at discrete time intervals. The frequency of the pulse signals determines the frequency of the output sampled signal. So the frequency of the impulses must be enough so that we don't miss out on information from the analog signal. 

The figure of the sampling process is shown below:

Consider f*(t) is the function of the output signal. 

f(t) as the input signal and 𝛿(t) as the pulse train 

So here the output of the sampler can be expressed as 

In time domain

In frequency domain

For the reconstruction of the signal, we have the Zero Order Hold (ZOH).

ZOH is a mathematical model of the practical model reconstruction done by a conventional DAC.

It describes the effect of converting a discrete-time signal to a continuous-time signal and holding each value for one sample interval 

The impulse response of a ZOH 

Gn(t) = u(t) -  Us(T-t)  Where T is the sampling period

Converting into Laplace transform equation becomes 

The equation can also be written in frequency domain 

Thus the real Sampler and Zero order Hold can be replaced by a model equivalent continuous-time system that consists of an impulse sampler and T.F. of 

Thank you & See you in the next article of Mapping from S-plane to Z-plane.