## Sampling with Zero-Order Hold

### English Definition

A signal can be sampled using piecewise step functions; The sample is extended directly across until the new sample is reached. This sampling of the function is called $x_0(t)$. The reconstructed function $x_r(t)$ can be recovered using a low pass filter created from the multiplication of the original step function generator $\mathcal{H}_1(\omega)$ and another function $\mathcal{H}_2(\omega)$

## Calculation

The are 3 steps to recover the original function from its resampling.

### 1) Multiply the function by an impulse train

First, multiply this function x(t) by an impulse train p(t) This should yield:

$x_p(t)=x(t)*p(t)$

$x_p(t)=x(t)\sum_{n=-\infty}^\infty \delta(t-nT)$ where T is the period of the function

### 2) Convolve Xp(t) with h1(t), the step function

We have a series of impulses in the time domain, but we want them to extend across, so we must convolve them to get $x_0(t)$ , or the actual sampling.

Beforehand, our $h_1(t)$ is given as

$h_1(t) = \begin{cases} 1 & 0 \le t < T \\ 0 & \mbox{else} \end{cases}$

$x_0(t)=x_p(t)*h_1(t)$

$x_0(t)=h_1(t)*x(t)\sum_{n=-\infty}^\infty \delta(t-nT)$

Now, convert $x(t)$ to $x(nT)$ for period T

$x_0(t)=h_1(t)*x(nT)\sum_{n=-\infty}^\infty \delta(t-nT)$

$x_0(t)=x(nT)h_1(t-nT)$ , which is our result, the shifted piecewise step functions.

### 3) Recovering the reconstructed signal from its resampling

To recover the signal $x(t)$, we must find a signal $h_2(t)$ so that the convolution of $h_1(t)$ and $h_2(t)$ represent a low pass filter with a gain of T and a cutoff frequency $\omega_c$ between $[\omega_m,\omega_s-\omega_m]$ to avoid aliasing.

In other words,

$x_r(t)=x_0(t)*h_1(t)*h_2(t)$

This is most easiest performed in the frequency domain with Fourier transforms. To most easily find $h_2(t)$, make sure that :

$H(\omega)=H_1(\omega)H_2(\omega)=\mathcal{F}(h_1(t)*h_2(t))$

and make sure that $H(\omega)$ has the correct low pass filter properties listed above.