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**Unformatted text preview: **Purdue University: ECE438 - Digital Signal Processing with Applications 1 ECE438 - Laboratory 2: Discrete-Time Systems October 6, 2010 1 Introduction A discrete-time system is anything that takes a discrete-time signal as input and generates a discrete-time signal as output. 1 The concept of a system is very general. It may be used to model the response of an audio equalizer or the performance of the US economy. In electrical engineering, continuous-time signals are usually processed by electrical cir- cuits described by differential equations. For example, any circuit of resistors, capacitors and inductors can be analyzed using mesh analysis to yield a system of differential equations. The voltages and currents in the circuit may then be computed by 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 of difference equations or by a block diagram of its implementation. For example, consider the following difference equation. y ( n ) = y ( n- 1) + x ( n ) + x ( n- 1) + x ( n- 2) (1) This equation represents a discrete-time system . It operates on the input signal x ( n ) to produce the output signal y ( n ). This system may also be defined by a system diagram as in Figure 1. Mathematically, we use the notation y = S [ x ] to denote a discrete-time system S with input signal x ( n ) and output signal y ( n ). Notice that the input and output to the system are the complete signals for all time n . This is important since the output at a particular time can be a function of past, present and future values of x ( n ). 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 implementing discrete-time systems. While equation (1) 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, Questions or comments concerning this laboratory should be directed to Prof. Charles A. Bouman, School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907; (765) 494- 0340; [email protected] 1 A more general behavioral view of systems is anything that imposes constraints on a set of signals. Purdue University: ECE438 - Digital Signal Processing with Applications 2 y(n) x(n) D D D + + + Figure 1: Diagram of a discrete-time system. (D = unit delay) it is important to first understand their classification into categories of linear/nonlinear, time- invariant/time-varying, causal/noncausal, memoryless/with-memory, and stable/unstable....

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