VIEWS: 40 PAGES: 34 CATEGORY: Engineering POSTED ON: 9/2/2011 Public Domain
6/2/2010 Outlines 2 Administrative details Signals Systems For examples CHAPTER 1 SIGNALS AND SYSTEMS Lecture 1 SIGNAL Signal Classification 3 4 Signals are functions of independent variables Type of Independent Variable that carry information. For example: Time is often the independent variable. Electrical signals ---voltages and currents in a Example: the electrical activity of the heart circuit recorded with chest electrodes – the Acoustic signals ---audio or speech signals A ti i l di h i l electrocardiogram (ECG or EKG). (analog or digital) Video signals ---intensity variations in an image (e.g. a CAT scan) Biological signals ---sequence of bases in a gene 1 6/2/2010 Independent Variable Dimensionality 5 6 The variables can also be spatial An independent variable can be 1-D (t in the EKG) or 2-D (x, y in an image). For this course: Focus on 1-D for mathematical simplicity but the results can be extended to 2-D or even higher dimensions. Also, we will use a generic In this example, the signal is the intensity as a time t for the independent variable, whether it is time function of the spatial variables x and y. or space. CT SIGNALS 7 8 Continuous-Time (CT) signals: x(t), t — continuous values Discrete Time Discrete-Time (DT) signals: x[n], n— integer values only Most of the signals in the physical world are CT signals, since the time scale is infinitesimally fine, so are the spatial scales. E.g. voltage & current, pressure, temperature, velocity, etc. 2 6/2/2010 DT SIGNALS Many human-made DT Signals 9 10 x[n], n — integer, time varies discretely Ex.#1 Weekly Dow‐ Ex.#2 digital Jones industrial image average Examples of DT signals in nature: Courtesy of Jason Oppenheim. DNA base sequence Used with permission. Population of the nth generation of certain species Why DT? — Can be processed by modern digital computers and digital signal processors (DSPs). CONVERSIONS BETWEEN SIGNAL TYPES Signals Energy and power 11 12 x(t) Continuous-Value Total energy of continuous-time signal x(t) over t Continuous-Time Signal the time interval t1≤ t ≤ t2 (k-1)t kt (k+1)t (k+2)t 2 xt dt Sampling t2 x[n] Continuous-Value n Discrete Time Discrete-Time t1 k-1 k k+1 k+2 Signal Total energy of discrete-time signal x[n] over the Quantizing x[n] Discrete-Value time interval t1≤ t ≤ t2 n Discrete-Time Signal n2 xn k-1 k k+1 k+2 2 Encoding x(t) 111 001 111 011 Discrete-Value n n1 t Continuous-Time Signal (k-1)t kt (k+1)t (k+2)t 3 6/2/2010 13 14 Total energy of continuous-time signal x(t) over The time-average power over an infinite time an infinite time interval interval 1 E lim xt dt xt dt xt T 2 2 T 2 P lim dt T T T 2T T Total energy of discrete-time signal x[n] over an infinite time interval Discrete-time signal x[n] N N 1 xn xn N xn 2 2 2 E lim P lim N n N n N 2 N 1 n TRANSFORMATIONS OF THE AMPLITUDE SCALING INDEPENDENT VARIABLE 15 16 Amplitude scaling x(t) x[n] x(t) A·x(t) x[n] A·x[n] t n Time shift (Time delay or time advance) x(t) x(t-t0) (t) (t t x[n] x[n-n0] ½x(t) (5/4)x[n] Time reversal x(t) x(-t) x[n] x[-n] t n Time scaling x(t) x(At) 4 6/2/2010 TIME SHIFT Time shifting or delay 17 18 x[n] Time-shifting occurs in many real physical systems. x(t) Examples: Listening to someone talking 2m away. Received signal will be delayed, but the delay won’t be t 0 n ti bl noticeable. x(t+t0) x[n-n0] Satellite communication systems (delay can be noticeable if ground stations are not directly below the satellite) Radar systems -t0 t 0 n n0 Transmitted signal Ag(t) t0 ,n0 > 0 Received signal Bg(t-to), with B<A, due to attenuation. Example : Time shifting TIME REVERSAL 19 20 x(t) x[n] s(t) s(t) 1 1 01 t 01 t 0 t 0 n x(t) = s(t-2) x[-n] x(t) = s(t+1) x(-t) 1 1 0 1 2 3 t -1 0 1 t Shifted to the right or delayed Shifted to the left or advanced in time t by -1 sec. 0 0 n by 2 sec. 5 6/2/2010 TIME SCALING Example 21 22 x(t) Examples: Given signal x(t) x(t) 1 Playing an audio -1 t tape at a faster or t 1 x(2t) slower speed. 0 1 2 (a) Sketch x(t+1) -½ t ½ (b) Sketch x(-t+1) x(t/2) (c) Sketch x(3t/2) (d) Sketch x(3t/2+1) t -2 2 Example (2) Example (3) 23 24 x(t) x(t) 1 1 t t 1 2 1 2 x(t+1) x(3t/2) 1 1 (a) (c) t t -1 0 1 2 2/3 4/3 0 x(-t+1) 1 x(3t/2+1) 1 t (b) -1 0 1 (d) -2/3 0 2/3 t 6 6/2/2010 TYPES OF SIGNALS Signal Properties 25 26 C i Continuous Discrete Di Periodic signals (analog) ( g) (digital) ( g ) Even and odd signals Exponential and sinusoidal signals Step and pulse signals N Non- Non- N Periodic Periodic periodic p periodic p Analysis Fourier Fourier Discrete Discrete-Time Tool Series Transform Fourier Fourier Series Transform & Z-Transform PERIODIC SIGNALS PERIODIC SIGNALS (2) 27 28 An important class of signals is the class of Periodic signals x(t) periodic signals. A periodic signal is a CT: x(t) = x(t + T) continuous time signal x(t), that has the property x(t) = x(t+T) -2T -T 0 T 2T t x[n] = x[n+N] Ex. 60-Hz power line, computer clock, etc. DT: x[n] = x[n + N] x(n) n 7 6/2/2010 PERIODIC SIGNALS (3) EXAMPLE 29 30 2 Given cost if t 0 x t cos(t+2π) = cos(t) sin t if t 0 sin(t+2π) = sin(t) Is signal periodic? Are both periodic with period 2π EXAMPLE (2) EVEN AND ODD SIGNALS 31 32 An even signal is identical to its time reversed signal, i.e. it can be reflected in the origin and is x(t) equal to the original: x(-t) = x(t) [ ] [ ] x[-n] = x[n] 0 t Signal x(t) is not periodic because x(t) does not An odd signal is identical to its negated, time recur at the time origin. reversed signal, i.e. it is equal to the negative reflected signal x(t) x(-t) = -x(t) x[-n] = -x[n] 0 t 8 6/2/2010 EVEN AND ODD SIGNALS (2) EXAMPLE (2) 33 34 Any signals can be expressed as a sum of Even DT unit-step function. Even signal 1 ,n 0 ,n 0 x even n 2 and Odd signals. That is 1 ,n 0 1 ,n 0 1 2 x n ,n 0 0 ,n 0 ½ dd(t), x(t) = xeven(t) + xodd(t) 1 -3 -2 -1 3 2 1 0 1 2 3 n Where + -3 -2 -1 0 1 2 3 n = 1 2 ,n 0 Odd signal x odd n 0 ,n 0 xeven(t) = [x(t) + x(-t)]/2 1 ,n 0 ½ 2 xodd(t) = [x(t) - x(-t)]/2 -3 -2 -1 0 1 2 3 n -½ EXPONENTIAL & SINUSOIDAL Example SIGNALS 35 36 Compute the even part of the function Exponential and sinusoidal signals are x[n] = ancos[πfn] characteristic of real-world signals and also from a basis (a building block) for other signals. A generic complex exponential signal is of the xn x n a n cosfn a n cos fn form: xe n 2 2 a n cosfn a n cosfn x(t) = Ceat 2 a n a n cos fn where C and a are, in general, complex numbers. 2 9 6/2/2010 COMPLEX EXPONENTIAL REAL EXPONENTIAL SIGNALS SIGNALS 37 38 C and a are real Consider when a is purely imaginary: x(t) = Ceat x(t) = Ceat xt e j0t Exponential growth E ti l th E ti l d Exponential decay An important property of this signal is that it is x(t) x(t) periodic j 0 t j 0 t T e e a<0 The fundamental period T0 is x(t) a>0 2 C T0 C 2 0 T0 t t 0 t SINUSOIDAL SIGNAL 39 40 By Euler’s relationship, this can be expressed as: x(t) = Acos(0t+) e j0t cos0t j sin 0t x(t) 2 Acos A T e j 0 t , e j 0 same fundamental period t T : fundamental time period : fundamental frequency 10 6/2/2010 Example Cont. 41 42 x(t) = 2cos(2t +/3) By Euler's relation x(t) = Acos(0t+ ) e j0t cos0t j sin 0t A = 2, 0 = 2 = /3 Acos = 2cos(/3) =1 Sinusoidal signal can be written in terms of periodic complex exponential. x(t) A j j 0 t A j j t Period 2 2 A cos0t e e e e 2 2 2 2 T 2 1 2 A A j 0t e j 0t t e -2 2 2 Cont. Cont. 43 44 If we decrease the A A magnitude of ω0, we A cos0t e j 0t e j 0t slow down the rate of 2 2 oscillation and therefore increase th f i A cos0t e Ae j 0t the period. Exactly the opposite effects A sin t mAe j 0t occur if we increase 0 the magnitude of ω0. 2 T0 0 11 6/2/2010 Cont. Cont. 46 45 Fundamental t Total energy and average power over one 1 1 x(t) = cos(t) period T0 T0 frequency 2 e dt 1 dt T0 -10 -5 0 5 10 j0 t -1 E period t 0 0 1 () (2 ) x(t) = cos(2t) 1 -10 -5 0 5 10 Pperiod E period -1 T t 1 Finite average power x(t) = cos(3t) T 1 j 0 t 2 e -10 -5 0 5 10 0T0 = 2πk P lim dt 1 -1 k = 0, 1, 2,… T 2T T Cont. Cont. 47 48 A necessary condition for a complex exponential Harmonically related set of complex to ejωt be periodic with period T0 is exponentials is a set of periodic exponentials with fundamental frequencies that are all e joT0 1 multiples of a signal positive frequency ω0. With implies that k t e jk t 0 k 0,1,2,...... ωT0 = 2k , k = 0, ±1, ±2,... The fundamental period We can see that ω must be multiple of ω0. We 2 T define the set of Harmonically related complex 0 exponentials k0 k 12 6/2/2010 Example 49 50 Plot signal x(t) = 2ej2t |x(t)| = 2 x(t) = 2t R e[x(t)] 2 x(t) = 2ej2t = rejt = 2cos(2t) + j2sin(2t) 2 2 2 1 1 1 t t t r=2 2, 1 1 -2 Re [x(t)] = 2cos(2t) Im Im [x(t)] = 2sin(2t) 2 Very Difficult too plot mag[x(t)] = r = 2 1 x(t) = tan-1[sin(2t)/cos(2t)] = 2t Re -2 COMPLEX EXPONENTIAL Example SIGNALS 51 52 Plot the magnitude of the signal Consider complex exponential signal: xt e j 2t e j 3t Solution x(t) = Ceat xt e j 2.5t e 0.5t e j 0.5t 2e j 2.5t cos0.5t Where C and a are Wh d C = |C|ej xt 2 cos0.5t a = r + jω0 13 6/2/2010 53 54 xt Ce at C e j e r j0 t C e rt e j 0t Ce at C e rt cos 0 t j C e rt sin 0t For r = 0, the real and imaginary parts of complex exponential are sinusoidal, For r > 0, they correspond to sinusoidal signals multiplied by a growing exponential, and for r < 0 they correspond to sinusoidal signals multiplied by a decaying exponential. Discrete-Time Complex Exponential and Sinusoidal Signals 55 56 Complex exponential signal or sequence If C and α are real || > 1, > 0 x[n] = Cn x[n] = Cn Where C and α are complex numbers x[n] = Ceβn || < 1, > 0 Where = eβ 14 6/2/2010 57 58 || > 1, < 0 If β is purely imaginary xn e j0 n Euler s By Euler's relation || < 1, < 0 e j0 n cos 0 n j sin 0 n and A j j 0 n A j j 0 n A cos0 n e e e e 2 2 59 60 Example of x [n]=Acos(0n+) 15 6/2/2010 COMPLEX EXPONENTIAL SIGNALS 61 62 General Complex Exponential Signals C n C e n cos0 n j C e n sin 0 n x[n] = Cn Where C and are C = |C|ej e j 0 Periodicity Properties of Discrete- Time Complex exponentials 63 64 Two properties of continuous-time counterpart e j 0 t Two properties of discrete-time counterpart e j0 n The larger the magnitude of ω0 the higher is the First property rate of oscillation in the signal The signal with frequency ω0 is identical to the signals j 0 t with frequencies ω0 ± 2π, ω0 ± 4π and so on. Therefore we e is periodic for any value of ω0 need only consider a frequency interval of length 2π . e j 0 2 n e j 2n e j0 n e j0 n j0 n e dose not have a continually increasing rate of oscillation as ω0 is increased in magnitude. We increase the rate of oscillation when ω0 increase from 0 to π and We decrease the rate of oscillation when ω0 increase from π to 2π. 16 6/2/2010 65 66 Second property There must be an integer of m Concerns the periodicity of the discrete-time complex exponential 0N = 2m Signal e j0 n to be periodic with period N > 0 2 0 or j 0 n j 0 n N N m e e Or e j 0 N 1 0N must be a multiple of 2 Comparison of the signals ejωot and ejωon 67 68 Comparison of the signals ejωot and ejωon 17 6/2/2010 DISCRETE –TIME UNIT IMPLUSE 70 The discrete unit impulse(or unit sample) signal 69 is defined: 0 n 0 x[ n ] [ n ] The Unit Impulse and 1 n 0 Unit Step Functions DISCRETE –TIME STEP DISCRETE – TIME UNIT SIGNALS IMPLUSE AND STEP SIGNALS 71 72 The discrete unit step signal is defined: Note that the unit impulse is the first difference (derivative) of 0 n 0 the step signal x[n] u[ n] 1 n 0 [n] u[n] u[n 1] n<0 Similarly, the unit step is the running sum (integral) of the unit impulse. n un m n>0 m 18 6/2/2010 73 74 n Unit impulse sequence can be used to sample un m the value of a signal at n = 0, since [n] is m nonzero(and equal to 1) only for n = 0 n<0 Change m to k = n – m x[n] [n] = x[0] [n] 0 un n k k A unit impulse [n-n0] at n = n0 n k n>0 k 0 x[n] [n-n0] = x[n0] [n-n0] CT UNIT STEP FUNCTIONS 75 76 The continuous unit step signal is defined: t du t x(t ) u (t ) ( )d t dt 0 t 0 x(t ) u (t ) 1 t 0 u(t) (t) 1 1/ 0 t 0 t 19 6/2/2010 The CT Unit Step and CT Unit CT UNIT IMPLUSE FUNCTIONS Impulse 77 78 (t) is a short pulse, of duration and with unit As approaches zero, x(t) approaches a CT area for any value of . As 0, (t) becomes narrower and higher, maintaining its unit area. unit step and x(t) approaches a CT unit impulse x(t) x'(t) t lim t 1 0 1/ (t) k (t) 1 k -/2 /2 t -/2 0 /2 t The CT unit step is the integral of the CT unit 0 t 0 t impulse and the CT unit impulse is the Note: It is not the shape of the function that is generalized derivative of the CT unit step important in the limit, but it’s the area. The impulse t u(t ) ( )d du t has an area of one. t dt Relationship between u(t) and (t) SAMPLING PROPERTY 79 80 Interval of integration Interval of integration The continuous unit impulse signal is defined: () (t-) (t) 0 t t 2 t 0 t 0 1 t 2 t -/2 /2 Interval of integration t Interval of integration Let x(t) be finite and continuous at t = 0. () (t-) (t) (t) 1/ 1/ x(t) x(0) 0 t 0 t t u t d u t t d - /2 t - /2 t /2 /2 0 20 6/2/2010 SAMPLING PROPERTY Properties of the CT Impulse 81 82 The area under the product of the two functions is The sampling (also called sifting) property xt t t dt xt 2 1 xt dt A 0 0 2 The sampling property “extracts” the value of a As the width of x(t) approaches zero x(t) x(0) zero, function at a point. 2 1 1 The scaling property lim A x0 limdt x0 lim x0 0 1 0 2 0 at t0 t t0 a The CT unit impulse is implicitly defined by This property makes the impulse different from ordinary function x0 t xt dt Properties of the CT Impulse Example (continued) 83 84 The equivalence property Consider the discontinuous signal x(t) xt A t t0 A xt0 t t0 x(t)A(t-t0) A(t-t0) x(t) A/ Ax(t0)/ x(t0) t0 t t0 t t0- /2 t0+ /2 t0- /2 t0+ /2 Calculate and graph the derivative of this signal 21 6/2/2010 85 86 SYSTEM Continuous‐Time and Discrete‐Time Systems SYSTEMS SYSTEM (2) 87 88 For the most part, our view of systems will be from Eg. Voltage buffer an input-output perspective: A system responds to applied input signals, and its response is described in terms of one or more output signals x(t) CT system y(t) x(t) System y(t) x[n] DT system y[n] 22 6/2/2010 EXAMPLES OF SYSTEMS (2) EXAMPLES OF SYSTEMS (3) 89 90 RLC circuit – an electrical system By KVL, we have Fine the relationship of input voltage vs and output vs t vR t vc t Rit vc t voltage vc in RC circuit The current in this circuit is dvc t it C dt Replace i(t) dvc t vs t RC vc t dt or dvc t 1 1 vc t vs t dt RC RC EXAMPLES OF SYSTEMS (4) EXAMPLES OF SYSTEMS (5) 91 92 RLC circuit – an electrical system A shock absorber – a mechanical system Force Balance: f - input, v – output dv t v t 1 t i t C v d dyt f t M Bvt K vt t dt R L dt capacitanceresistance inductance This equation looks quite familiar 23 6/2/2010 EXAMPLES OF SYSTEMS (6) EXAMPLES OF SYSTEMS (7) 93 94 Observation: different systems could be A robot car – a close-loop system described by the same input/output relations EXAMPLES OF SYSTEMS (8) EXAMPLES OF SYSTEMS (9) 95 96 A thermal system Cooling Fin in Steady State Observations Independent variable can be something other than time, such as space. Such systems may, more naturally, have t = distance along rod boundary conditions, rather than “initial” y(t) = Fin temperature as function of position conditions. x(t) = Surrounding temperature along the fin 24 6/2/2010 EXAMPLES OF SYSTEMS (10) EXAMPLES OF SYSTEMS (11) 97 98 A rudimentary “edge” detector Financial System y[n] = x[n + 1] – 2x[n] + x[n -1] Bank account = {x[n + 1] – x[n]}- {x[n] – x[n -1]} y[n] – balance at nth month = “Second difference” Second difference x[n] – deposit at nth month This system detects changes in signal slope r – monthly interest (a) x[n] = x[n] y[n] = 0 Then (b) x[n] = nu[n] y[n] y[n] = (1+ r)y[n-1] + x[n] SYSTEM SYSTEM INTERCONNECTIOINS INTERCONNECTIOINS(2) 99 100 An important concept is that of interconnecting Signal flow (Block) diagram systems System 1 System 2 Cascade To build more complex systems by interconnecting simpler subsystems To modify response of a system System 1 Parallel System 2 Feedback System 1 System 2 25 6/2/2010 SYSTEM Example : Circuit system INTERCONNECTIOINS(3) 101 102 Feedback interconnection Example System 1 System 2 System 4 Input System 3 Output Example: A Communication System 103 A communication system has an information 104 signal plus noise signals This is an example of a system that consists of an interconnection of smaller systems. SYSTEM PROPERTIES Cell phones are based on such systems Noise Noise Noise Noisy Informatio Transmitter Channel Receiver Informatio n Signal n Signal 26 6/2/2010 System properties System properties (cont.) 105 106 WHY? Systems with and without Memory Memoryless system is the system that its output Important practical/physical implications. We for each value of the independent variable at a can make many important predictions of the given time is dependent on the input at only that t b h i ith t having to d system behaviors without h i t do any time. same ti mathematical derivations Example of memoryless systems y[n] = (2x[n]−x2[n])2 They allow us to develop powerful tools (transforms, more on this later) for analysis and y(t) = Rx(t) design System properties (cont.) System properties (cont.) 107 108 Example of memory systems Invertibility and Inverse Systems n A system is said to be invertible if distinct inputs yn xk lead to distinct outputs. It the system is k invertible, then an inverse system exists that, y yn xn 1 when cascaded with the original system, yields t 1 an output w[n] equal to the input x[n] to the first y t x d C system. 27 6/2/2010 109 110 Example of Inverse system Example of noninvertible discrete-time systems y[n] = 0 This system produces the zero output q y p sequence for any input sequence The accumulator y(t) = x2(t) We cannot determine the sign of input from the knowledge of the output. System properties (cont.) System properties (cont.) 111 112 A system is causal Mathematically (in CT): A system x(t) y(t) is if the output does not anticipate future values of the causal if input, i.e., if the output at any time depends only on values of the input up to that time. when x1(t) y1(t) () () x2(t) y2(t) () () All real-time physical systems are causal, because and time only moves forward, Effect occurs after cause. (imagine if you own a noncausal system whose output x1(t) = x2(t) for all t < t0 depends on tomorrow’s stock price.) Then All memoryless systems are causal, since the output y1(t) = y2(t) for all t < t0 responds only to the current value of the input. 28 6/2/2010 System properties (cont.) System properties (cont.) 113 114 Causal or noncausal Stability system is one in which small inputs y(t) = x2(t – 1) lead to responses that do not diverge y(5) depend on x(4) Causal y(t) = x(t + 1) Stable system y(5) = x(6), y depends on future Noncausal y[n] = x(– n) y[5] = x[-5] ok But y[-5] = x[5] Noncausal Unstable system y[n] = (½)n+1x3[n - 1] y[5] depend on x[4] Causal System properties (cont.) 115 116 TIME-INVARIANCE (TI) : A system is time invariant if a time shift in input A system is time invariant if the behavior and signal result in an identical time shift in the output characteristics of the system are fixed overtime. signal. Informally, a system is time-invariant (TI) if its behavior does not depend on what time it is. Example discrete time If y[n] is the output of a discrete-time time invariant - RC circuit is time invariant if the resistance and system when x[n] is the input, then y[n-n0] is the capacitance values R and C constant over time: We would output when x[n-n0] is applied. expect to get the same results form an experiment with this circuit today as we would if we ran the identical experiment Mathematically (in DT): A system x[n] y[n] is tomorrow. TI if for any input x[n] and any time shift n0, - On the other hand, if the values of R and C are changed or fluctuate over time: We would expect the results of our If x[n] y[n] experiment to depend on the time a which we run it. then x[n-n0] y[n-n0]. 29 6/2/2010 System properties (cont.) System properties (cont.) 117 118 In continuous time with y(t) the output TIME-INVARIANCE: TIME SHIFT WAVEFORM corresponding to the input x(t), a time - invariant Replace argument t by t – t0 where t0 is amount system will have y(t-t0) as the output when x(t-t0) of shift is the input. x[n] x[n n x[n-n0] Mathematically (in CT): A system x(t) y(t) is TI if for any input x(t) and any time shift n0, If x(t) y(t) then x(t-t0) y(t-t0) n n 1 2 3 4 n0 n0+4 System properties (cont.) Example 119 120 If the input to a TI System is periodic, then the output is periodic with the same period. “Proof”: Suppose x(t + T) = x(t) pp ( ) () and x(t) → y(t) Then by TI x(t + T) → y(t + T). y2(t) y1(t-2) ↑ ↑ System is not time-invariance These are So these must be the the same same output, i.e., y(t) = input! y(t + T). 30 6/2/2010 Example (2) Example (3) 121 122 Consider the continuous-time system y1(t) = sin|x1(t)| Time shift y(t) = sin|x(t)| y1(t-t0) = sin|x1(t-t0)| Let Issystem time invariant? x2(t) = x1(t-t0) y2(t) = sin|x2(t)| = sin|x1(t-t0)| Check system is time invariant Determine whether the time-invariance property holds for any input and any time shift t0 Because y2(t) = y1(t-t0), Then this system is time invariant Example (4) TIME-INVARIANT OR VARYING ? 123 124 Consider the discrete-time system y(t) = x2(t+1) y[n] = nx[n] TI y1[n] = nx1[n] y[n] = (½)n+1x3[n – 1] 1 time shifting y1[n-n0] = (n-n0)x1[n-n0] Time-varying (NOT time-invariant) x2[n] = x1[n-n0] y2[n] = nx2[n] = nx1[n-n0] y2[n] y1[n-n0] Time-Varying system 31 6/2/2010 System properties (cont.) System Linearity 125 126 LINEAR AND NONLINEAR SYSTEMS The most important property that a system possesses is linearity Many systems are nonlinear. It means allows any system response to be Ex.Economic system. analysed as the sum of simpler responses policies, labor, Input: fiscal and monetary policies labor ( (convolution) ) resources, etc Simplistically, it can be imagined as a line Output: GDP, inflation, etc. Specifically, a linear system must satisfy the two properties: System behavior is very unpredictable because it Additive: the response to x1(t)+x2(t) is y1(t) + y2(t) is highly nonlinear. Scaling: the response to ax1(t) is ay1(t) where aC However, we focus exclusively on linear systems. Combined: ax1(t)+bx2(t) ay1(t) + by2(t) Linear systems can be analyzed accurately. PROPERTIES OF LINEAR LINEARITY SYSTEMS 127 128 A (CT) system is linear if it has the superposition Superposition property: If xk[n] yk [n] If x1(t) → y1(t) and x2 (t) → y2 (t) then ax1(t) + bx2(t) → ay1(t) + by2(t) Then a x n a y n k k k k k k For linear system, zero input → zero output “Proof” 0 = 0x[n] → 0y[n] = 0 32 6/2/2010 Example 129 130 an + x(t) y(t) an + Ex#1 Linear y(t) = 3x(t) why? + + Prove ax1(t) + bx2(t) ay1(t) + by2(t) linear xn(t) xn(t) Find ax1(t) + bx2(t) = ? a1 x1(t) y1(t) = 3x1(t) a1 Equivalent x2(t) y2(t) = 3x2(t) x1(t) x1(t) x3(t) = ax1(t) + bx2(t) Linear (a) (b) y3(t) = 3x3(t) = 3ax1(t) + 3bx2(t) * If the system is linear outputs y(t) in (a) and (b) Find ay1(t) + by2(t) = ? are equal. ay1(t) + by2(t) = 3ax1(t) + 3bx2(t) * 131 132 Ex#2: Is this system linear : y(t) = 3x(t)+2, Ex#3 : Consider a system whose input x[n] and Prove ax1(t) + bx2(t) ay1(t) + by2(t) linear output y[n] are related by y[n] = e{x[n]}. Determine whether or not the system is linear. x1(t) y1(t) = 3x1(t) + 2 x2(t) y2(t) = 3x2(t) + 2 x1[n] = r1[n] + js1[n] y1[n]= e{x1[n]} = r1[n] [ ] [ ] j [ ] [ ] { [ ]} [ ] x3(t) = ax1(t) + bx2(t) x2[n] = r2[n] + js2[n] y2[n]= e{x2[n]} = r2[n] x3[n] = r3[n] + js3[n] y3[n]= e{x3[n]} = r3[n] y3(t) = 3x3(t) + 2 = 3ax1(t) + 3bx2(t) + 2 * ay1(t)+by2(t) = a[3x1(t) + 2] + b[3x2(t) + 2] x3[n] = ax1[n] + bx2[n] = ar1[n] +jas1[n]+br2[n]+ jbs2[n] not linear = a3x1(t) + 2a + 3bx2(t) + 2b * y3[n] = ar1[n] + br2[n] (1) 33 6/2/2010 133 134 ay1[n] = ar1[n] But If a = j by2[n] = br2[n] x3[n] = jx1[n] + bx2[n] = jr1[n] - s1[n]+br2[n]+ jbs2[n] ay1[n] + by2[n] = ar1[n] + br2[n] (2) y3[n] = e{x3[n]} = -s1[n] + br2[n] (*) (1) = (2) It should be Linear jy1[n] = jr1[n] by2[n] = br2[n] jy1[n] + by2[n] = jr1[n] + br2[n] (**) (*) (**) This system is not linear Example DT LTI System 135 136 Is this system linear : y(t) = 3x2(t) x1[n] y1[n] 2 1 1 1 -1 0 1 0 1 x2[n] 2x1[n-1] x1[n-2] Not linear 3 4 2 2 2 2 3 1 1 0 1 2 -1 0 1 2 1 2 3 y2[n] 2y1[n-1] y1[n-2] 2 2 1 1 3 1 2 1 2 2 3 -1 34