Fund Signals ECE 3500
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ECE 3500 Chapter 1 Introduction to Signals and Systems 1 Chapter 1 Introduction to Signals and Systems In this chapter we introduce the most basic concepts of signals and systems We introduce the classification of signals as well as the classification of systems We also introduce some useful signals that are frequently used in the study of systems Moreover elementary signal operations are discussed Finally to pave the way for the next chapter a simple example of a linear time invariant continuoustime system is presented The content of illis Cll pl l is related to materials in pages Sl72 and 7981 of Law s book ECE 3500 Chapter 1 Introduction to Signals and Systems Signals A signal as the term implies is a set of information or data In general a signal is a function of one or more variables Examples Speech Speech is an audio signal that varies with time This is an example of onedimensional signals as the signal is a function of only one variable time The time here varies continuously The speech is thus a continuouslime signal Although it is not generally true when there is only one independent variable we always refer to it as time and use the letter I for it when it is continuoustime and the letter k when it is discretetime Can you mention other examples of onedimensional signals DowJones Averages DowJones averages are typically recorded at the end of each working day This is an example of discretetime signals Other examples Pictures A Picture is a twodimensional signal that depends on the spatial coordinates x and y coordinates of the picture plane A picture in general is continuous in x and y directions and the strengthintensity of colors at each point Other examples Television Images This is an example of threedimensional signals Our discussion in this class is limited to onedimensional signals only ECE 3500 Chapter 1 Introduction to Signals and Systems 3 Systems Signals are often processed by systems to modify them or to extract some desired information from them Inputs gt System gtOutputs Fig 11 Examples A noisy speech may be cleared by passing it through a proper filter Input here is the noisy speech filter is the system and output is the noise free less noisy speech An antiaircraft gun operator may want to know the future location of a hostile moving target Past locations of the target may be used to predict its future location A system may be made up of physical components such as electrical circuits or mechanical components or may be an algorithm implemented on a microcontroller or a computer ECE 3500 Chapter 1 Introduction to Signals and Systems 11 Classification of Signals There are several classes of signals Those that are of interest to us are I Continuouslime and discretelime signals An example of a continuoustime signal Fig 12 An example of a discretetime signal ECE 3500 Chapter 1 Introduction to Signals and Systems 2 Analog and digital signals 39 Analog signals are continuous both in time and amplitude such as Fig 12 Digital signals are discrete both in time and amplitude eg Fig 14 A signal which is discrete in time but continuous in amplitude is called sampled or discretetime signal eg Fig 13 Fig 14 ECE 3500 Chapter 1 Introduction to Signals and Systems 6 3 Periodic and aperiodic signals A signal which gets the same magnitude after every T 0 seconds is said to be periodic with the period of To eg Fig 15 f0 flflTo W V Av T 0 A V V Fig 15 A signal is aperiodic if it is nonperiodic eg Fig 12 4 Energy and power signals An energy signal t satisfies the condition 0 lt E lt 00 Where E is the total energy given by E fl ft I2 ah ECE 3500 Chapter 1 Introduction to Signals and Systems 7 39 A power signal t satis es the condition 0 lt P lt 00 where P is the average power given by l P 139 if z 2 dz T130 Tf Examples 1 f I ell is an energy signal since f ft 2 dz z wethz 1 lt oo 2 f0 511197127 is not an energy signal since f N 2 dz 00 However it is a power signal since 1 2 1 39 7 t dt m mn 2 5 Deterministic and probabilistic signals A deterministic signal can be described as a function of time A random signal can only be characterized by its statistics Examples The signals given in the above examples are both 2 at deterministic s1gnals while sm2nfot and 31 7 5 where is a random phase which varies in every selection of signals are random signals Electronics noise is also a random signal ECE 3500 Chapter 1 Introduction to Signals and Systems 12 Some Useful Signal Operations 1 Time shifting f0 Aft T A T 7 gt AftT T t Fig 16 ECE 3500 Chapter 1 Introduction to Signals and Systems 2 Time Scaling f0 T1 I T2 xf2t V T22 kft2 A TlZ A 2T1 Fig 17 22 V ECE 3500 Chapter 1 Introduction to Signals and Systems 10 3 Time Inversion Time Reversal f0 L 1 T2 W fl T T V 2 1 Fig 18 13 Some Useful Signal Models 1 Unit Step Function ul I 1 I20 u 0 lt0 A ul V Fig 19 ECE 3500 Chapter 1 Introduction to Signals and Systems 11 We can use unit step function to express truncated waveforms For example the function 6quot I20 fl 0 rlt0 may be written as ff e39mu for all values of I from 0 to 00 Example 11 Give expressions for the following waveforms in terms of the unit step function A gl 1 z 1 V g0 1 r l 1 2 V A gl 1 z 05 0 5 r ECE 3500 Chapter 1 Introduction to Signals and Systems Example 12 Plot the function ft tut 2t 2ut 2 t 4ut 4 Example 13 Give an expression for the following waveform ECE 3500 Chapter 1 Introduction to Signals and Systems 13 2 The Unit Impulse Function 6t This is the most important function in the study of signals and systems It is a narrow tall pulse of Width zero and height infinity which satis es the equation f 6Idt 1 Note that the above statement implies that 50 0 fort 0 Fig 110 depicts a unit impulse and its approximation 50 s a O l 8 I L I L 3 2 Fig 110 Multiplication of a function by an impulse We note that I5I 05I 11 and in general I5IT T5IT 12 ECE 3500 Chapter 1 Introduction to Signals and Systems 14 Sampling propem of the unit impulse function From 11 it follows that f r6 0dr 0f 6 0dr lt0gt Similarly using 12 we get f z5z Tdt Tf so Tdr T This result is known as sampling or sifting property of the unit impulse function Example 13 Simplify the following expressions a t2t16t N b t3 t46r 1 c Z JjMx Z Example 14 Evaluate the following integrals a amewt b jt315t 5dt 5 0 2 0 2 0 5a 2dw 1 c f ECE 3500 Chapter 1 Introduction to Signals and Systems 15 14 Classification of systems Systems may be classified broadly in the following categories i Linear and nonlinear systems Constantparameter and timevaryingparameter systems Instantaneous memoryless and dynamic with memory systems 4 Causal and noncausal systems 5 Lumpedparameter and distributedparameter systems 6 Continuoustime and discretetime systems 7 Analog and digital systems WN The main emphasis in this class is on linear continuoustime and linear discretetime systems In general a system may be de ned by a mathematical function such as yf Tff Where l is the system input and yl is the system output TI denotes the function which is performed by the system on the input t to generate the output yl ECE 3500 Chapter 1 Introduction to Signals and Systems 16 Linear systems A system TI is linear if y1Tx1 y2Tx2 y1y2 Tx1x2 and y1Tx1ll gt k1y1lTk1x1ll The first property is known as additivity and the second one homogeneity Combining the two properties into one we get k1y1l kzyz I Tk1x1 kzxz y1l Tx1l y2 Tlx2 where k and k are arbitrary constants In the above x1tx2ty1t and y2tcan in general be vectors This corresponds to a multipleinput multipleoutput system In this class our discussion is limited to the cases of singleinput singleoutput systems ie where x1lx2ly1landeU are scalars ECE 3500 Chapter 1 Introduction to Signals and Systems 17 Example 15 Show that the system described by the equation dy 3 I I dz y f is linear Solution For the inputs f1l and f2l we have d g 3y1z 10 and 3yzr 122 Multiplying the first equation by k1 the second equation by kg and adding them yields k1y1r k2y2ltrgt1 3k1y1ltrgt mm mm mm which shows that the response of the system to the input f0 klfl t k2f2t is W k1y1t k2y2t thus the system is linear Using the same argument one can say that a system described by a differential equation of the form dny dn ly dm lf a a t b b n dtn n l dtn1 0y m m l dtm1 a bofltrgt is a linear system The coefficients al and b can be constants or functions of time Thus the system may be timeinvariant or time varying ECE 3500 Chapter 1 Introduction to Signals and Systems 18 Example 15 Show that the following systems are linear thwwm WFWH Wkwmmwmwmmmwmwmm timevarying coef cient ECE 3500 Chapter 1 Introduction to Signals and Systems 19 T ime invariant and limevarying systems Systems Whose parameters do not change with time are time invariant systems For such a system if the input is delayed by T seconds output also will be delayed by T seconds Systems Whose parameters change with time are timevarying systems Example 16 For systems in example 15 which of the systems are time invariant and which ones are timevarying ECE 3500 Chapter 1 Introduction to Signals and Systems 20 15 Circuits as example of linear systems Consider the simple RC circuit of Fig 111 R TC Fig 111 If the capacitor voltage is known at time I 0 and ft is given for t 2 0 the output yl is given by 1 yt vC 0 Rft E frdr 13 where vc 0 is the capacitor voltage at time I 0 From this result we may note that the output yl consists of two components zeroinput component and zerostale component where the latter refers to the case where vc00 This result is true in general for all linear systems That is in general we have Total response zeroinput response zerostate response This is known as the decomposition property of linear systems Here the principle of superposition is applicable to zeroinput and zerostate components This follows from the linearity of the system with respect to both components ECE 3500 Chapter 1 Introduction to Signals and Systems 21 Example 17 Assuming that Vc 0 0 show that the circuit shown in Fig 111 is linear That is if the system output to inputs f1t and f20 are y1t and y2t respectively the system output to Cl110 k2f20 is Cd11 k2y20 Example 18 Consider the RLC circuit shown in Fig 112 Find the equation that relates f I and yt and see whether the system is i linear or nonlinear ii timeinvariant or timevarying Note that yt is the loop current in the circuit L10H R1Q f0 W CZSF Fig 112 ECE 3500 Chapter 1 Introduction to Signals and Systems 22 Example 19 Show that the system described by equation I 32 1yl 2r 1fz is linear but is timevarying Example 110 Show that the system described by equation mg 3yt f0 is nonlinear Example 111 The response of a LTIC system to f1t is equotut and to f2t is e 4 ut Find the response of the system to a 3 f1t 4 b 5f1l 1 4f2l 5