à 2.1 General Definitions and Classifications

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1 DOUBLE CLICK on Right Brackets to expand the ections 2.ystems à 2.1 General Definitions and Classifications A ystem is a transformation of an input signal x into an output signal y. Every there is a relationship of cause and effect between an input signal (the cause) and an output signal (the effect), we can define a system, and it is represented as in Figure Figure 2.1.1: A Discrete Time ystem In this book we will be focusing on discrete systems, where the input signal is a discrete sequence x@nd and the output signal is a discrete sequence y@nd. Very similar considerations can be made when the signals are defined in continuous and an extensive literature exists on the matter. Example. The bank where you keep your money. Call x@nd the total amount of money you deposit and withdraw during the n-th month, let y@nd be the balance at the end of the n-th month, and p the yearly interest rate in percent. Then the bank constitutes a system which takes your money x@nd as input, and gives you a balance y@nd as output. This system can be described by the recursion with p y@nd = y@n 1D + p y@n 1D +x@nd being the monthly interest rate (this is an approximate value for illustration purposes only). Example. In many applications we want to filter out noise from a signal. Although we will be seeing techniques to design filters, if we do not know anything about signal processing the first thing we try is to do some sort of averaging, to smooth the signal and attenuate the disturbances. For example, if we average over ten points we can relate the input and output signals as y@nd = 10 1 Hx@nD +x@n 1D x@n 9DL

2 2 Unit1.nb This difference equation describes the operation of the system, and the relationship between the input signal and the output signal In this section, to streamline the notation, we indicate the input and the output sequences of a system as follows x@nd y@nd ystems are classified according to a number of properties we list below. a) Linearity. Consider a system and any two input output pairs x y x y Then the system is linear if and only if, for any two constant a 1, a 2 we can write This is shown in Figure a 1 x +a 2 x a 1 y + a 2 y x [ ] y [ n] 1 1 n x [ ] y [ n] 2 2 n x [ n] = a x [ n] + a x [ n] y [ n] = a1 y1[ n] + a2 y 2[ n] Figure 2.1.2: Linearity Property In other words the output is the superposition of the corresponding outputs. Example. Consider the system defined by the input output relation Is the system linear? y@nd = x@nd +x@n 1D In order to answer this question, let x and x be two distinct inputs, and y y be the respective ouputs, ie x y x y

3 Unit1.nb 3 Then an input x@nd = a 1 x +a 2 x yields an output which can be rearranged as y@nd = Ha 1 x +a 2 x + Ha 1 x 1D +a 2 x 1DL y@nd = a 1 Hx +x 1DL + a 2 Hx +x 1DL The terms between round brackets are the output signals y and y and therefore we can say that y@nd = a 1 y +a 2 y which shows superposition and therefore the given system is linear. Example. Consider the system defined by the input output relation Then applying superposition consider the input Therefore the respective output becomes After simple algebra we can say that, in general y@nd = 2 Hx@nDL 2 x@nd = a 1 x +a 2 x y@nd = 2 Ha 1 x +a 2 x 2 y@nd a 1 2 Hx 2 +a 2 2 Hx 2 where the terms on the right hand side represents the outputs due to x and x respectively. ince superposition does not hold, the system is non linear. b) Time Invariance. Let be a system with an input output pair Then the system is invariant if and only if x@nd y@nd x@n DD y@n DD In other words the system is invariant it its characteristic does not change with. As a consequence if we shift the input signal by an amount D, the output is also shifted by same amount D. This is shown in Figure

4 4 Unit1.nb x[ n D ] y[ n D] D D Figure 2.1.4: Time Invariance Example. Consider the system defined by the input output relation y@nd = x@nd +x@n 1D Is the system invariant? Consider any input x@nd and the output y@nd, ie x@nd y@nd Now consider the same input, delayed by L, and call y the respective output defined as y =x@n LD +x@n L 1D Now the question is whether y = y@n LD. From the above expression it is easy to answer is affirmative, and therefore x@n LD y@n LD for all input signals x@nd. As a consequence the system is invariant. Example. Consider the system defined by input output relation y@nd = 2nx@nD Then, to determine whether the system is invariant or not, consider any input x@nd and the respective response y@nd. If the input now is shifted in, the output becomes y = 2nx@n LD We immediately see that this output is different from the shifted output y@n LD given by y@n LD = 2 Hn LLx@n LD Therefore the system is not invariant (ie it is varying). c) Bounded Input Bounded Output (BIBO) tability. A system is BIBO stable if and only if whenever the input signal is bounded as»x@nd» A for all n, the output is also bounded as»y@nd» B for all n. This has to be true for all input signals. If this is violated by an input

5 Unit1.nb 5 output pair, the system is not stable. This is shown in Figure Example Consider the system defined by y@nd = x@nd +x@n 1D Is the system BIBO stable? We have to check. Let x@nd be an input which is bounded as Then the output signal y@nd is bounded as»x@nd» A for all n.»y@nd»»x@nd» +»x@n 1D» 2 A where we use the fact that»x@nd +x@n 1D»»x@nD» +»x@n 1D». Therefore for this system a bounded input yields a bounded output, and therefore the system is BIBO stable. Example. Consider the system y@nd = nx@nd. Then we can see that the bounded input x@nd = 1 for all n, yields the ouput y@nd = n for all n. Clearly this output is not bounded, sonce it goes to infinity as n ± and therefore the system is not stable. Bounded Input Bounded Output Figure 2.1.5: Bounded Input Bounded Output (BIBO) tability d) Causality. A system is causal if the output y@nd at any is independent on any value x@n +md of the input signal, with m > 0. In other words in a causal system the effect follows the cause, as in Figure Example. Consider the system defined by the input output relation y@nd = 2x@n 1D. Then this system is causal since the ouput y@nd at any n depends on past values of the input signal only. Therefore the system is causal. Example. Consider the system defined by the input output relation y@nd = 2x@n +1D +x@n 1D. Then this system is non causal since the ouput y@nd at any n depends on some future values of the input signal. Figure 2.1.6: Causal ystem

6 6 Unit1.nb Homework Problems: Problem 2.1 ee if you are following. Click HERE for a Question (not graded) Animations

EE Homework 5 - Solutions

EE Homework 5 - Solutions EE054 - Homework 5 - Solutions 1. We know the general result that the -transform of α n 1 u[n] is with 1 α 1 ROC α < < and the -transform of α n 1 u[ n 1] is 1 α 1 with ROC 0 < α. Using this result, the