# 0.4 Lab 4 - sampling and reconstruction

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Questions or comments concerning this laboratory should be directedto Prof. Charles A. Bouman, School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907;(765) 494-0340; bouman@ecn.purdue.edu

## Introduction

It is often desired to analyze and process continuous-time signals using a computer.However, in order to process a continuous-time signal, it must first be digitized.This means that the continuous-time signal must be sampled and quantized, forming a digital signal that can be stored in a computer.Analog systems can be converted to their discrete-time counterparts, and these digital systems then process discrete-time signalsto produce discrete-time outputs. The digital output can then be converted back to an analog signal, or reconstructed , through a digital-to-analog converter. [link] illustrates an example, containing the three general components described above: a sampling system,a digital signal processor, and a reconstruction system.

When designing such a system, it is essential to understand the effects of the sampling and reconstruction processes.Sampling and reconstruction may lead to different types of distortion, including low-pass filtering, aliasing, and quantization.The system designer must insure that these distortions are below acceptable levels,or are compensated through additional processing.

## Sampling overview

Sampling is simply the process of measuring the value of a continuous-time signal at certain instants of time.Typically, these measurements are uniformly separated by the sampling period, ${T}_{s}$ . If $x\left(t\right)$ is the input signal, then the sampled signal, $y\left(n\right)$ , is as follows:

$y\left(n\right)={\left(x,\left(,t,\right)|}_{t=n{T}_{s}}\phantom{\rule{4pt}{0ex}}.$

A critical question is the following: What sampling period, ${T}_{s}$ , is required to accurately represent the signal $x\left(t\right)$ ? To answer this question, we need to look at thefrequency domain representations of $y\left(n\right)$ and $x\left(t\right)$ . Since $y\left(n\right)$ is a discrete-time signal, we represent its frequency content with the discrete-time Fourier transform (DTFT), $Y\left({e}^{j\omega }\right)$ . However, $x\left(t\right)$ is a continuous-time signal, requiring the use of the continuous-time Fourier transform (CTFT), denoted as $X\left(f\right)$ . Fortunately, $Y\left({e}^{j\omega }\right)$ can be written in terms of $X\left(f\right)$ :

$\begin{array}{ccc}\hfill Y\left({e}^{j\omega }\right)& =& \frac{1}{{T}_{s}}\sum _{k=-\infty }^{\infty }{\left(X,\left(,f,\right)|}_{f=\frac{\omega -2\pi k}{2\pi {T}_{s}}}\hfill \\ & =& \frac{1}{{T}_{s}}\sum _{k=-\infty }^{\infty }X\left(\frac{\omega -2\pi k}{2\pi {T}_{s}}\right)\phantom{\rule{4pt}{0ex}}.\hfill \end{array}$

Consistent with the properties of the DTFT, $Y\left({e}^{j\omega }\right)$ is periodic with a period $2\pi$ . It is formed by rescaling the amplitude and frequency of $X\left(f\right)$ , and then repeating it in frequency every $2\pi$ . The critical issue of the relationship in [link] is the frequency content of $X\left(f\right)$ . If $X\left(f\right)$ has frequency components that are above $1/\left(2{T}_{s}\right)$ , the repetition in frequency will cause these components to overlap with (i.e. add to) the components below $1/\left(2{T}_{s}\right)$ . This causes an unrecoverabledistortion, known as aliasing , that will prevent a perfect reconstruction of $X\left(f\right)$ . We will illustrate this later in the lab. The $1/\left(2{T}_{s}\right)$ “cutoff frequency” is known as the Nyquist frequency .

To prevent aliasing, most sampling systems first low pass filter the incoming signalto ensure that its frequency content is below the Nyquist frequency. In this case, $Y\left({e}^{j\omega }\right)$ can be related to $X\left(f\right)$ through the $k=0$ term in [link] :

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