# 0.2 Lab 2 - discrete-time systems

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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

A discrete-time system is anything that takes a discrete-time signal as input and generates a discrete-time signal asoutput. A more general behavioral view of systems is anything that imposes constraints on a set of signals. The concept of a system is very general. It may be used to model the response of an audio equalizer or theperformance of the US economy.

In electrical engineering, continuous-time signals are usually processed by electrical circuits described by differential equations.For example, any circuit of resistors, capacitors and inductors can be analyzed using mesh analysis to yield a system of differentialequations. The voltages and currents in the circuit may then be computedby solving the equations.

The processing of discrete-time signals is performed by discrete-time systems.Similar to the continuous-time case, we may represent a discrete-time system either by a set ofdifference equations or by a block diagram of its implementation. For example, consider the following difference equation.

$y\left(n\right)=y\left(n-1\right)+x\left(n\right)+x\left(n-1\right)+x\left(n-2\right)$

This equation represents a discrete-time system . It operates on the input signal $x\left(n\right)$ to produce the output signal $y\left(n\right)$ . This system may also be defined by a system diagram as in [link] .

Mathematically, we use the notation $y=S\left[x\right]$ to denote a discrete-time system $S$ with input signal $x\left(n\right)$ and output signal $y\left(n\right)$ . Notice that the input and output to the system are the completesignals for all time n . This is important since the output at a particular timecan be a function of past, present and future values of $x\left(n\right)$ .

It is usually quite straightforward to write a computer program to implement a discrete-time system from its difference equation.In fact, programmable computers are one of the easiest and most cost effective ways of implementingdiscrete-time systems.

While equation [link] is an example of a linear time-invariant system, other discrete-time systems may be nonlinear and/or time varying.In order to understand discrete-time systems, it is important to first understand their classificationinto categories of linear/nonlinear, time-invariant/time-varying, causal/noncausal, memoryless/with-memory, and stable/unstable.Then it is possible to study the properties of restricted classes of systems, such as discrete-time systems which are linear, time-invariantand stable.

## Background exercises

Submit these background exercises with the lab report.

## Example discrete-time systems

Discrete-time digital systems are often used in place of analog processing systems.Common examples are the replacement of photographs with digital images, and conventional NTSC TV with directbroadcast digital TV. These digital systems can provide higher quality and/orlower cost through the use of standardized, high-volume digital processors.

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