0.4 Lab 4 - sampling and reconstruction

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:

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 $X\left(f\right)$ :

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 $1/\left(2T_s\right)$ , the repetition in frequency will cause these components to overlap with (i.e. add to) the components below $1/\left(2T_s\right)$ . 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 $1/\left(2T_s\right)$ “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] :