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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.

Example of a typical digital signal processing system.

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 ( t ) is the input signal, then the sampled signal, y ( n ) , is as follows:

y ( n ) = x ( t ) t = n T s .

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

Y ( e j ω ) = 1 T s k = - X ( f ) f = ω - 2 π k 2 π T s = 1 T s k = - X ω - 2 π k 2 π T s .

Consistent with the properties of the DTFT, Y ( e j ω ) is periodic with a period 2 π . It is formed by rescaling the amplitude and frequency of X ( f ) , and then repeating it in frequency every 2 π . The critical issue of the relationship in [link] is the frequency content of X ( f ) . If X ( f ) has frequency components that are above 1 / ( 2 T s ) , the repetition in frequency will cause these components to overlap with (i.e. add to) the components below 1 / ( 2 T s ) . This causes an unrecoverabledistortion, known as aliasing , that will prevent a perfect reconstruction of X ( f ) . We will illustrate this later in the lab. The 1 / ( 2 T s ) “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 ( e j ω ) can be related to X ( f ) through the k = 0 term in [link] :

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Source:  OpenStax, Purdue digital signal processing labs (ece 438). OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10593/1.4
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