Systems Lecture 6 PDF

Systems lecture 6 1  System A system is formally defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals. Input signal Output signal System...

Systems lecture 6 1  System A system is formally defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals. Input signal Output signal System 2  Examples of Systems In a communication system, the input signal could be a speech signal or computer data, the system itself is made up of the combination of a transmitter, channel and receiver and the output signal is a an estimate of the information contained in the original message.                System 3  Examples of Systems In an automatic speaker recognition system, the input signal is a speech (voice) signal, the system is a computer, and the output signal is the identity of the speaker. 4  Systems Viewed as Interconnections of Operation In mathematical term, a system may be viewed as an interconnection of operation that transforms an input signal into an output signal with properties different from those of the input signal. Let the overall operator H denote the action of a system. Then, the application of a continuous-time signal x(t) to the input of the system yields the output signal. y(t)=H{x(t)} x(t) y(t) H 5  Systems Viewed as Interconnections of Operation Correspondingly, for the discrete-time case, we may write y[n]=H{x[n]} x[n] y[n] H 6  Systems Viewed as Interconnections of Operation Consider a discrete-time system whose output signal y[n] is the average of the three most recent values of the input signal x[n]; that is 1 y[n]  x[n]  x[n  1]  x[n  2] 3 Such a system is referred to as a moving average system, for two reasons: y[n] is the average of the sample values x[n], x[n-1] and x[n- 2]. The value of y[n] changes as n moves along the discrete time axis. 7  Systems Viewed as Interconnections of Operation Problem Formulate the operator H for this system, hence develop a block diagram representation for it. Solution Let the operator Sk denote a system that shifts the input x[n] by k time units to produce an output equal to x[n-k]. 8  Systems Viewed as Interconnections of Operation Solution-Continued 1 H can be written as,  H  1 S  S 2 3  Two different (but equivalent) implementations of the moving- average system: (a) cascade form of implementation and (b) parallel form of implementation. 9  Systems Viewed as Interconnections of Operation Problem Express the operator that describes the input-output relation 1 y[n]  x[n  1]  x[n]  x[n  1] 3 In terms of the time-shift operator S. Solution 1 1  H  S 1  S 1 3  10  Properties of system Memory A system is said to possess memory if its output signal depends on past or future values of the input signal. In contrast, a system is said to be memoryless if its output depends only on the present value of the input signal. The delay system, y[n]=x[n-1] must retain or store the preceding value of the input. 11  Properties of system Memory The system described by the input-output relation, 1 y[n]  x[n]  x[n  1]  x[n  2] 3 has memory, since the value of the output signal y[n] at time n depends on the present and on two past values of the input signal x[n]. 12  Properties of system Memory In contrast, a signal described by the input-output relation 2 y[n]  x [n] is memoryless, since the value of the output signal y[n] at time n depends only on the present value of the input signal. 13  Properties of system Memory (Problem) How far does the memory of the moving –average system described by the input-output relation extend into the past? 1 y[n]  x[n]  x[n  2]  x[n  4] 3 Answer: Four time units 14  Properties of system Memory (Problem) The input-output relation of a semiconductor diode is represented by i (t ) a0  a1v(t )  a2 v 2 (t )  a3v 3 (t ) ... where, v(t) is the applied voltage, i(t) is the current flowing through the diode, and a0, a1, a2, a3….. Are constants. Does the diode have memory? Answer: No 15  Properties of system Memory (Problem) The input-output relation of a capacitor is described by, t 1 v(t )  i ( )d C  What is the extend of the capacitor’s memory? Answer: The capacitor’s memory extends from time t back to the infinite past. 16  Properties of system Causality A system is said to be causal if the present value of the output signal depends only on the present or past values of the input signal. For example, the moving-average system described by 1 y[n]  x[n]  x[n  1]  x[n  2] 3 is causal. 17  Properties of system Causality Some other examples of causal systems are given below: 𝑦[n]= 𝑥[n − 3] + 3𝑥[n] y[n]=nx[n] y[n]=x[n-1]+x[n] 18 Properties of system 19  Properties of system 1. 𝑦[n] = 𝑥2[n] + 𝑥[n − 3] 2. 𝑦[n] = 𝑥[3 − n] + 𝑥[n − 2] 3. 𝑦[𝑛] = 𝑥[2𝑛] 4. 𝑦[𝑛] = sin[𝑥[𝑛]] 20  Properties of system Causality of the given system can be determined by considering different values of n as follows- n = 0 → 𝑦 = 𝑥2 + 𝑥[−3] n = −2 → 𝑦[−2] = 𝑥2[−2] + 𝑥[−5] n = 2 → 𝑦 = 𝑥2 + 𝑥[−1] Hence, for all the values of n, the output depends only on the present and past values of the input. Thus, the given system is a causal system. 21  Properties of system Causality of the given system can be determined by considering different values of n as follows- n = 0 → 𝑦 = 𝑥 + 𝑥[−2] n= −1 → 𝑦[−1] = 𝑥 + 𝑥[−3] n = 1 → 𝑦 = 𝑥 + 𝑥[−1] Hence it is clear that , for some values of n, the output depends only on the future values of the input. Thus, the given system is a non-causal system. 22  Properties of system 3. 𝑦[𝑛] = 𝑥[3𝑛] 𝑛 = 0 → 𝑦 = 𝑥 𝑛 = −1 → 𝑦[−1] = 𝑥[−3] 𝑛 = 1 → 𝑦 = 𝑥 𝑛 = 2 → 𝑦 = 𝑥 23 Properties of system 24 Properties of system 25  Properties of system Invertibility Then the output signal of the second system is defined by H inv H I 26  Properties of system Invertibility A system is not invertible unless distinct inputs applied to the system produce distinct outputs. That is, there must be a one-to- one mapping between input and output signals for a system to be invertible. 27  Properties of system An example of an invertible discrete-time system is 28  Properties of system Invertibility An example of an invertible discrete-time system is 29  Properties of system Invertibility An example of a noninvertible system is y[n] 0 The system that produces the zero output sequence for any input sequence. 30  Properties of system Invertibility Show that a square-law system described by the input-output relation is not invertible. The square law system violates a necessary condition for inverse system, namely, that distinct input must produce distinct outputs. Here the distinct inputs x[n] and –x[n] produce the same output y[n]. 31  Properties of system Invertibility (Problem) We cannot determine the sign of the input from knowledge of the output. 32  Properties of system Time Invariance A system is said to be time invariant if a time shift in the input signal results in an identical time shift in the output signal. Otherwise the system is said to be time variant. Thus if y[n] is the output of a discrete-time invariant system when x[n] in the input then y[n-n0] is the output when x[n-n0] is applied. 33  Properties of system Time Invariance To check whether a system is time invariant, we must determine whether the time invariance property holds for any input and any time shift n0. 34  Properties of system Time Invariance The output of the system H 35  Properties of system Time Invariance The system is time invariant if 36  Properties of system Time Invariance Consider the continuous time system defined by Determine whether the system is time-invariant or not. 37  Properties of system Time Invariance Let x1 be an arbitrary input, then 38  Properties of system Time Invariance Then, 39  Properties of system Linearity A system is said to be linear in terms of system input x[n] and the system output y[n] if it satisfies the following two properties Additive property Homogeneity property 40  Properties of system Linearity 41  Properties of system Linearity 42  Properties of system Linearity 43  Properties of system Linearity The two properties defining a linear system can be combined into a single statement: ax1[n]  bx2 [n] ay1[n]  by2 [n] where, a and b are constants. 44  Properties of system Linearity Furthermore, it is straightforward to show from the definition of linearity that if xk[n], k=1, 2, 3, …., then the response to a linear combination of these inputs given by, x[n]  ak xk [n] a1 x1[n]  a2 x2 [n]  a3 x3[n] ... k is, y[n]  ak yk [n] a1 y1[n]  a2 y2 [n]  a3 y3[n] ... k This very important fact is known as the superposition property, which holds for linear systems. 45  Properties of system Linearity Let the operator H represents a discrete-time system. Let the signal applied to the system input be defined by the weighted sum 46  Properties of system Linearity 47  Properties of system Linearity 48  Properties of system Linearity 49  Properties of system Linearity 50  Properties of system Linearity The linearity property of a system. (a) The combined operation of amplitude scaling and summation precedes the operator H for multiple inputs. (b) The operator H precedes amplitude scaling for each input; the resulting outputs are summed to produce the overall output y[n]. If these two configurations produce the same output y[n], the operator H is linear. (a) (b) 51  Properties of system Linearity When a system violates either the additive property or the property of homogeneity, the system is said to be nonlinear. 52  Properties of system Linearity 53  Properties of system Linearity Let the input signal x[n] be expressed as the weighted sum. N x[n]  ai xi [n] i 1 54  Properties of system Linearity The resulting output signal of the system is, N N y[n] n ai xi [n]  ai nxi [n] i 1 i 1 55  Properties of system Linearity N N y[n]  ai yi [n]  ai nxi [n] i 1 i 1 Since the two configurations produce the same output y(t), the operator H is linear. 56  Properties of system Linearity Consider a discrete-time system described by the input- output relation y[n] ex[n] Check the system for linearity. 57  Properties of system Linearity y[n] ex[n] x[n] r[n]  jI [n] where, r[n] real part of x[n] I [n] imaginary of x[n] x1[n] r1[n]  jI1[n] x2 [n] r2 [n]  jI 2 [n] y1[n] H x1[n] r1[n] y2 [n] H x2 [n] r2 [n] y3 [n] a1r1[n]  a2 r2 [n]  j[r1[n] r 2 [n]} here, a1 a2  j 58  Properties of system Linearity 59  Properties of system Linearity 60  Properties of system Linearity 61  Properties of system Linearity y4 [n] H x1[n]  x2 [n] 2 x1[n]  x2 [n]  3 Since y3 [n]  y4 [n], the system is not linear 62  Properties of system Linearity 0 0.x[n] 0. y[n] 0 This is known as “zero-in/zero-out” property. 63  Properties of system Linearity Consider a discrete-time system described by y[n]=2x[n]+3. Here it can be seen that y[n]=3 if x[n]=0, so the system violates the zero-in/zero-out property of linear systems. Hence the system is not linear. 64  Properties of system Linearity Show that the moving average system described by is linear. 65  Properties of system Linearity 1 y1[n]   x1[n]  x1[n  1]  x1[n  2] 3 1 y2 [n]   x2 [n]  x2 [n  1]  x2 [n  2] 3 1 1 y3[n]  a1 x1[n]  a2 x2 [n]  a1 x1[n  1]  a2 x2 [n  1] 3 3 1  a1 x1[n  2]  a2 x2 [n  2] 3 66  Properties of system Linearity y4 [n] H a1 x1[n]  a2 x2 [n] 1 1  a1 x1[n]  a2 x2 [n]  a1 x1[n  1]  a2 x2 [n  1] 3 3 1  a1 x1[n  2]  a2 x2 [n  2] 3 Since y3 [n]  y4 [n], the system is linear 67 End 68