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Forward Concepts of signal and systems arise in wide variety of Signal and Systems fields. Ideas and techniques associated with these concepts play important role in diverse areas of science and technology: Introduction Communication Aeronautics and astronautics Circuit design Seismology Biomedical engineering Control Speech processing Image processing Prepared by Dr Nidal Kamel 1 Prepared by Dr Nidal Kamel 2 Forward Forward Although the different physical natures of the signals that arise in different disciplines, they have two basic features in Automobile itself is a system, the pressure on accelerator common: pedal is the input and the automobile speed is the response (output). Signal are function of one or more independent variable. Contain information about the behavior or nature of some Computer program for electrocardiogram can be viewed as phenomena. the system that has its input from sensors and produce an estimate of heart rate (output). System process particular signal to produce other signal or to follow a desired behavior. Camera receives light and produce photograph. Voltages and currents as a function of time in electrical circuits are Robot arm whose movement are a response to control example of signals. inputs. Circuit is itself an example of a system Prepared by Dr Nidal Kamel 3 Prepared by Dr Nidal Kamel 4 Forward Forward In the many contexts in which signal and systems arise, Another very important class of application in which there are a variety of problem: concepts of signal and systems arise is control. System characterization Control system to regulate chemical processing plant Circuit response, aircraft response, … etc Aircraft autopilot system. Designing systems to process signals in particular way Pilot communicating with air traffic control tower (signal Importance of concepts of signals and systems stems not enhancement). only from the diversity of application, but from the rich Image from deep space or earth-observing satellite (image restoration collection of ideas, analytical tools, and methodologies, or enhancement). developed over the years. Transmitter in communication link. Fourier analysis, Laplace trnasformation, z-trnasformation, … Designing system to extract specific peace of information etc. form the signal. Electrocardiogram, Radar, economic forecast, communication link, … etc Prepared by Dr Nidal Kamel 5 Prepared by Dr Nidal Kamel 6 1 Forward Forward In some phenomena, signal vary continuously with time, Over the last several decades, disciplines of CT and DT whereas with others, evolution is described only at discrete signals and systems have increasingly entwined. points of time. The major impetus for this has come from the dramatic Signals in electrical circuits and mechanical systems are advance in technology. continuous. High-speed digital computer Daily closing stock market average is discrete signal. Integrated circuits Concepts and techniques associated with continuous-time and This made it advantageous to process CT signals by discrete-time signal and systems are conceptually related. representing them by time samples. Prepared by Dr Nidal Kamel 7 Prepared by Dr Nidal Kamel 8 Continuous-Time and Discrete-Time Signals Continuous-Time and Discrete-Time Signals Examples Examples Prepared by Dr Nidal Kamel 9 Prepared by Dr Nidal Kamel 10 Continuous-Time and Discrete-Time Signals Continuous-Time and Discrete-Time Signals Examples Signal Energy and Power In a broad range of phenomena, signals are related to For continuous-time we power and energy in the physical system. use symbol t to denote If v(t) and i(t) are the voltage and current across resistor the independent R, then the instantaneous power is variable. 1 2 p(t ) = v (t )i (t ) = v (t ) For discrete time we use R symbol n to denote the Total energy expended over time interval t1 ≤ t ≤ t2 is independent variable t2 t2 1 2 ∫t1 p(t )dt = ∫ t1 R v (t )dt Prepared by Dr Nidal Kamel 11 Prepared by Dr Nidal Kamel 12 2 Continuous-Time and Discrete-Time Signals Continuous-Time and Discrete-Time Signals Signal Energy and Power Signal Energy and Power The average power over the time interval t1 ≤ t ≤ t2 is In may systems, we are interested in examining power 1 t2 1 t2 1 2 and energy in signals over infinite time interval. t2 − t1 ∫t1 t2 − t1 ∫t1 R p (t )dt = v ( t )dt In this case, we define the energy of x(t) as Energy of arbitrary complex - continuous signal over ∆ T ∞ E∞ = lim ∫ x(t ) dt = ∫ x( t ) dt 2 2 t1 ≤ t ≤ t2 t2 2 T → ∞ −T −∞ ∫ x(t ) dt t1 E∞ = lim +N ∑ x( t ) 2 dt = +∞ ∑ x (t ) 2 dt N →∞ Energy of discrete-time signal over time interval n =− N n = −∞ n1 ≤ n ≤ n2 is 2 n2 E∞ = ∞ signal has infinite energy ∑ x[n] n =n1 E∞ < ∞ signal has finite energy Prepared by Dr Nidal Kamel 13 Prepared by Dr Nidal Kamel 14 Continuous-Time and Discrete-Time Signals Continuous-Time and Discrete-Time Signals Signal Energy and Power Signal Energy and Power Time-averaged power of x(t) over infinite interval With these definitions we have three classes of signals: ∆ 1. Energy signal: Signal that has finite energy E∞ < ∞ and 1 T ∫ 2 P∞ = lim x(t ) dt thus zero averaged power T →∞ 2T −T ∆ ∆ +N E∞ 1 P∞ = lim =0 ∑N x (t ) dt 2 P∞ = lim T →∞ 2T N →∞ 2 N + 1 n=− 2. Power signal: Signal that has finite average power P∞ and P∞ = ∞ signal has infinite averaged power thus infinite energy E∞ = ∞ P∞ < ∞ signal has finite averaged power 3. Signal for which neither E∞ nor P∞ are finite. Prepared by Dr Nidal Kamel 15 Prepared by Dr Nidal Kamel 16 Transformation of the Independent Variable Transformation of the Independent Variable Time-shift Time-reversal Transformation of the independent variable is a central concept in signal and systems analysis. A simple and very important transformation of the independent variable is the time-shift. Prepared by Dr Nidal Kamel 17 Prepared by Dr Nidal Kamel 18 3 Transformation of Independent Variable Transformation of Independent Variable Time-scaling Example Prepared by Dr Nidal Kamel 19 Prepared by Dr Nidal Kamel 20 Transformation of Independent Variable Transformation of Independent Variable Periodic Signals Periodic Signals Continuous-time signal x(t) has the property that there is a Discrete-time signal x[n] is periodic with period N if positive value of T for which x[n] = x[ n ± N ] x(t ) = x(t ± T ) If signal is periodic with a fundamental period N (or N0), Periodic signal remains unchanged with time-shift of T. then it is also periodic with periods 2N, 3N, 4N,… Periodic signal with T is also periodic with 2T, 3T, 4T T (or T0) is called the fundamental period the signal. Prepared by Dr Nidal Kamel 21 Prepared by Dr Nidal Kamel 22 Transformation of Independent Variable Transformation of Independent Variable Even and Odd Signals Even and Odd Signals In continuous time signal is even if x (−t ) = x (t ) Discrete-time signal is even if x[ − n ] = x[ n ] The signal is odd if x ( −t ) = − x ( t ) x[− n ] = − x[n ] Prepared by Dr Nidal Kamel 23 Prepared by Dr Nidal Kamel 24 4 Exponential and Sinusoidal Signals Transformation of Independent Variable Continuous-Time Complex Exponential and Sinusoidal Even and Odd Signals Signals Any signal can be broken into the Continuous-time complex exponential signal is of the form sum of two signals one is odd and one is even. Ev{x( t )} = 1 2 [x (t ) + x (− t )] x (t ) = Ce at The even signal is where C and a, in general, complex numbers. 1 Ev{x (t )} = [x (t ) + x (− t )] If C and a are real, x(t) is called real exponential signal. 2 The odd signal is For real exponential signal there are two types of behavior. 1 If a is positive then x(t) is a growing exponential. Od {x (t )} = [x (t ) − x (− t )] If a is negative then x(t) is a decaying exponential. 2 Prepared by Dr Nidal Kamel 25 Prepared by Dr Nidal Kamel 26 Exponential and Sinusoidal Signals Exponential and Sinusoidal Signals Continuous-Time Complex Exponential and Sinusoidal Continuous-Time Complex Exponential and Signals Sinusoidal Signals Important class is obtained when a is purely imaginary. x(t ) = e jωot To verify that this signal is periodic e jωot = e jωo (t +T ) = e jωo t e jωoT = e jωo t The fundamental period of x(t) is 2π To = Prepared by Dr Nidal Kamel 27 ωo 28 Exponential and Sinusoidal Signals Exponential and Sinusoidal Signals Continuous-Time Complex Exponential and Continuous-Time Complex Exponential and Sinusoidal Signals Sinusoidal Signals Signal closely related to periodic complex exponential is If we decrease the sinusoidal signal. magnitude of ωo we slow down the x(t ) = A cos (ωot + φ ) oscillation and therefore e jω0t = cos (ω0 t ) + j sin (ω0t ) increase the period. A j Φ jω0 t A − jΦ − jω0t A cos (ω0t + Φ ) = e e + e e 2 2 { A cos (ω0t + Φ ) = Aℜe e j (ω0 t +Φ ) } { A sin (ω0t + Φ ) = A Im e j (ω0 t + Φ) } Prepared by Dr Nidal Kamel 29 Prepared by Dr Nidal Kamel 30 5 Exponential and Sinusoidal Signals Exponential and Sinusoidal Signals Continuous-Time Complex Exponential and Continuous-Time Complex Exponential and Sinusoidal Signals Sinusoidal Signals General Complex Exponential Discrete time complex Signals exponential signal is defined by Ceat x[ n] = Cα n = Ce β n where α=e β where C = C e jθ and a = r + jw0 then If C and α are real, Ce at = C e jθ e( r + jω0 ) t j (ω0t +θ ) = C e rt e signal is real = C e rt cos (ω0t + θ ) + j C e rt sin (ω0 t + θ ) exponential. Prepared by Dr Nidal Kamel 31 32 Exponential and Sinusoidal Signals Exponential and Sinusoidal Signals Continuous-Time Complex Exponential and Continuous-Time Complex Exponential and Sinusoidal Signals Sinusoidal Signals Important complex General complex signal α >1 exponential signal is C = C e jθ and α = α e jωo then obtained by constraining n n Cα n = C α cos (ωo n + θ ) + j C α sin (ωo n + θ ) β to imaginary. α <1 x[ n] = e jω0 n The signal is closely related sinusoidal signal. By Euler’s relation e jωo n = cos ω0 n + j sin ω0 n and A jΦ jωo n A − j Φ − jωo n Acos ( ωo n + Φ ) = e e + e e 33 Prepared by Dr Nidal Kamel 34 2 2 Exponential and Sinusoidal Signals Exponential and Sinusoidal Signals Continuous-Time Complex Exponential and Continuous-Time Complex Exponential and Sinusoidal Signals Sinusoidal Signals Two basic properties of CT exponential signal ej ωot from its DT counterpart ej ωon : 1. The larger the magnitude of ω0 the higher the frequency is. 2. ej ωot is periodic for any value of ω0 . First property doesn’t hold true for the DT exponential signal, because e j (ω0 + 2π ) n = e j 2π n e jω0 n = e jω0 n Signal doesn’t have a continually increasing rate of oscillation as ω0 increases. Prepared by Dr Nidal Kamel 35 Prepared by Dr Nidal Kamel 36 6 Exponential and Sinusoidal Signals Exponential and Sinusoidal Signals Continuous-Time Complex Exponential and Continuous-Time Complex Exponential and Sinusoidal Signals Sinusoidal Signals Next, is ejωon is periodic for any value of ω0? If x[n] is periodic with fundamental period N, its fundamental In order ejωon is periodic with period N > 0, we must have frequency is 2π/N. jω0 ( n + N ) ) Since the discrete-time exponential signal is periodic if e = e jω0 n or equivelently e jω0 N = 1 ⇒ 2π ω0 ω0 m = ω0 N = 2π m, or= N m 2π N the fundamental frequency of discrete-time exponential signal is Accordingly signal ejωon is periodic if ω0/2π is rational ωo number and not periodic otherwise. m Prepared by Dr Nidal Kamel 37 Prepared by Dr Nidal Kamel 38 Exponential and Sinusoidal Signals Continuous-Time Complex Exponential and Unit Impulse and Step Functions Sinusoidal Signals Discrete-Time Unit Impulse and Unit Step One of the simplest discrete-time is the unit impulse (or unit sample), defined as 0 , n ≠ 0 δ [n ] = 1, n = 0 The second basic discrete- time signal is unit step, 0, n < 0 u[n] = 1, n ≥ 0 Prepared by Dr Nidal Kamel 39 Prepared by Dr Nidal Kamel 40 Unit Impulse and Step Functions Unit Impulse and Step Functions Discrete-Time Unit Impulse and Unit Step Discrete-Time Unit Impulse and Unit Step Relationship between unit Relationship between unit impulse and unit step. impulse and unit step. δ [n ] = u[n ] − u[n − 1] ∞ n u[n] = ∑ δ [n − k ] u[n] = ∑ δ [m ] m = −∞ k =0 Prepared by Dr Nidal Kamel 41 Prepared by Dr Nidal Kamel 42 7 Unit Impulse and Step Functions Unit Impulse and Step Functions Continuous-Time Unit Impulse and Unit Step Continuous-Time Unit Impulse and Unit Step Continuous-time step function is defined in similar way Continuous-time unit impulse function is related to unit to discrete-time counterpart step in a manner analogous to discrete time. 0, t < 0 t u(t) = u (t ) = ∫ δ (τ )dτ δ [ n] = u[n ] − u[ n − 1] 1, t > 0 −∞ Thus n du(t ) u[n ] = ∑ δ [ m] δ (t ) = m =−∞ dt Unit step is discontinuous at t = 0. Prepared by Dr Nidal Kamel 43 Prepared by Dr Nidal Kamel 44 Unit Impulse and Step Functions Unit Impulse and Step Functions Continuous-Time Unit Impulse and Unit Step Continuous-Time Unit Impulse and Unit Step u(t) is discontinuous at δ∆(t) is a short pulse of duration t = 0, thus not ∆ and unit area. differentiable. As ∆→0, δ∆(t) becomes Consider u∆(t), as shown narrower and higher, maintaining in the figure its unit area. Formally u(t) is the limit Its limiting form is of u∆(t) as ∆ → 0, and δ(t ) = lim δ∆ (t ) ∆→ 0 du (t ) δ ∆ (t ) = ∆ Since δ(t) has, no duration but dt unit area, we adopt the graphical Prepared by Dr Nidal Kamel 45 notation, shown the Figure. 46 Unit Impulse and Step Functions Unit Impulse and Step Functions Continuous-Time Unit Impulse and Unit Step Continuous-Time Unit Impulse and Unit Step More generally, a scaled CT-impulse has very impulse kδ(t) has an area important sampling of k, thus property. x(t ) = x (t )δ ∆ (t ) t ≈ x (0)δ ∆ (t ) ∫−∞ kδ(τ )dτ = kδ(t ) Since δ(t) is the limit as ∆→0 of δ∆(t), it follows that The height of the arrow is x(t )δ (t ) = x(0)δ (t ) proportional to the area of By the same argument, we impulse. have x(t )δ (t − t 0 ) = x(t 0 )δ (t − t 0 ) Prepared by Dr Nidal Kamel 47 Prepared by Dr Nidal Kamel 48 8 Continuous-Time and Discrete-Time Systems Continuous-Time and Discrete-Time Systems Simple Examples of Systems Continuous-time (CT) x (t ) → y ( t ) systems, transform v s ( t ) − vc ( t ) i (t ) = continuous input signals R into continuous outputs. dvc (t ) i (t ) = C x[ n] → y[ n ] dt x (t ) → y (t ) dvc (t ) 1 1 + vc ( t ) = v s (t ) dt RC RC Discrete-time (DT) systems, transform discrete inputs into discrete outputs. x[ n] → y[ n] Prepared by Dr Nidal Kamel 49 Prepared by Dr Nidal Kamel 50 Continuous-Time and Discrete-Time Systems Continuous-Time and Discrete-Time Systems Interconnection of Systems Interconnection of Systems Prepared by Dr Nidal Kamel 51 Prepared by Dr Nidal Kamel 52 Basic System Properties Basic System Properties System with and without Memory System with and without Memory The system is memoryless if its output for each value of Voltage across capacitor in RC circuit is an example of CT the independent variable at any given time is dependent system with memory: on the input at only the same time. 1 t y [n ] = 2 x[ n ] + x 2 [n ] v( t ) = C ∫ i(τ )dτ −∞ y ( t ) = Rx ( t ) Examples of DT system with memory: Concept of memory corresponds to the presence of a mechanism in the system that stores information about n input at times other than the current time. y[ n] = ∑ x[k] k = −∞ capacitor in RC circuit y[ n] = x[ n − 1] + x[ n] kinetic energy in automobile y[ n] = x[ n + 1] + x[n ] + x[ n − 1] memory and registers in digital computers. Prepared by Dr Nidal Kamel 53 Prepared by Dr Nidal Kamel 54 9 Basic System Properties Basic System Properties Invertibility and Inverse Systems Causality A system is invertible System is causal if the output at any time depends on if distinct inputs lead to values of the input at only the present and past times. distinct output. Voltage across capacitor in RC circuit ( running integral). The system is Velocity of automobile as a function of fuel flow (driver action). invertible if the inverse Causal system is also called nonanticipative system. system exists. n Cascading the inverse y[n] = ∑ x[k ] k = −∞ (Causal) system with the y[n] = x[ n − 1] + x[ n] (Causal) original system, yields y[n] = x[ n + 1] + x[n] + x[ n − 1] (Noncausal) output equal to the y[n] = x[-n] (?) input. 1 t y(t) = C −∞∫ x(τ ) dτ (Causal) Prepared by Dr Nidal Kamel 55 y(t) = 2.6 x(t Prepared by Dr Nidal Kamel + 1) (Noncausal) 56 Basic System Properties Basic System Properties Stability Stability Informally, a stable system Consider applying force is the one in which small f(t)=F to the automobile, inputs lead to responses with the vehicle initially that do not diverge. at rest. In different words, The velocity will increase, unstable system is the one but not without bound. in which responses grow The bound when the without bound in response frictional forces exactly to small inputs. balance the applied force, i.e. ρ 1 F V = F ⇒V = Prepared by Dr Nidal Kamel 57 m m ρPrepared by Dr Nidal Kamel 58 Basic System Properties Basic System Properties Stability Time Invariance A useful strategy to test the stability of a system is to A system is time invariant if the behavior and look for a specific bounded input that leads to characteristics are fixed over time. unbounded output. RC circuit with constant values for R and C over time. To Check the stability of the system S1 : y (t ) = tx(t ) we constantly loaded auto’s trunk. consider a simple bounded input like unit step ( x(t) = u(t) ). In systems language, system is time invariant if time shift in Accordingly this system is unstable input signal results in identical time shift in the output. To check the stability of the system S 2 : y (t ) = e x (t ) we consider abounded input by B. Thus –B < x(t) < B for all t. Then In DT systems, we express this as, if y[n] is the output when x[n] is y(t) must satisfy the input then y[n-n0] is the output when x[n-n0] is applied. e − B < y (t ) < e B In CT systems, we express this as, if y(t) is the output when x(t) is the input then y(t-t0) is the output when x(t-t0) is applied. Accordingly, this system is stable. Prepared by Dr Nidal Kamel 59 Prepared by Dr Nidal Kamel 60 10 Basic System Properties Basic System Properties Time Invariance/Example Time Invariance Check whether the system y(t) = sin[x(t)] is time invariant or Check whether the system y[n] = n x[n] is time invariant not. or not. We must determine whether the time-invariance property holds for any input and any time shift t0. Consider input signal x1[n] = δ[n] which yields output y1[n]=0 (since nδ[n] = 0) . Let x1(t) be an arbitrary input, and let y1(t) = sin[x1(t)]. However the input x2[n] = δ[n-1] yields output Then consider second input obtained by shifting x1(t) in time: y2[n] = nδ[n-1]= δ[n-1], which is not a shifted version of y1(t). x2(t) = x1(t-t0). Thus, the system is time-variant. The output corresponding to this input is y2(t) = sin[x1(t-t0)]. Since y2(t)= y1(t-t0) the system is time invariant. Prepared by Dr Nidal Kamel 61 Prepared by Dr Nidal Kamel 62 Basic System Properties Basic System Properties Linearity Linearity/Example Linear system, in continuous or discrete time, is a system Consider a system S whose input and output are related as that possesses the property of superposition. y (t ) = tx(t ) More precisely, let y1(t) be the response of CT system to determine whether the system is linear or not. input x1(t) and y2(t) to input x2(t) Then the system is let x1(t) and x2(t) be a two arbitrary inputs with their outputs linear if: y1(t) = t x1(t) and y2(t) = t x2(t), respectively. The response to x1(t) + x2(t) is y1(t) + y2(t) [additive property]. The response to ax1(t) is ay1(t), where a is any complex constant. Let x3(t) a linear combination of the two inputs [scaling property] x3 (t ) = ax1 (t ) + bx2 (t ) The two properties can be combined into single statement If x3(t) is the input of S, then the output is CT: ax1(t) + bx2(t)+… → ay1(t) + by2(t)+…. y3 (t ) = tx3 (t ) DT: ax1[n]+ bx2[n]+… → ay1[n] + by2[n]+…. = t ( ax1 (t ) + bx2 (t ) ) = atx1 (t ) + btx2 (t ) Prepared by Dr Nidal Kamel 63 Prepared by Dr Nidal Kamel 64 = ay1 (t ) + by2 (t ) 11

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posted: | 4/15/2012 |

language: | English |

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This article show the introduction of signal and system in our life. How the concept of Signal and System will be applied in our life such as communication, circuit design, control, speech processing, image processing and others.

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