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# SIGNALS AND SYSTEMS EE 421G

UK

GPA 3.81

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This 12 page Class Notes was uploaded by Adaline Pollich on Friday October 23, 2015. The Class Notes belongs to EE 421G at University of Kentucky taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/228319/ee-421g-university-of-kentucky in Electrical Engineering at University of Kentucky.

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Date Created: 10/23/15

11 Signals and Systems Signals convey information Systems respond to or process information Engineers desire mathematical models for signals and systems in order to solve design problems efficiently and thoroughly Signal encoded information data a dynamic or change in some quantity that has meaning In most cases this is modeled as a function of time or space Noise irrelevant data variability in a quantity that has no meaning or significance In most cases this is modeled as a random variable System a mapping between a set of inputs to a set of outputs an entity that processes signals received at its input and produces another set of signals at the output a process that responds to actions or events at its input by generating actions or events at its output Most physical systems are approximately modeled by differential equations or convolution integrals 12 Classi cations of Systems 1 Linear vs Nonlinear For all linear systems superposition holds for inputoutput relationships Denote a general system in the following manner ym System H operates on inputx to produce output y The system H is linear if and only if iff for any inputoutput pair y1Hx1 and y2 Hx2 the following statement is also true aly1 a2y2 Ha1x1a2x2 where a1 and 12 are constants 13 Determine whether or not each system described below is linear Assume inputs and outputs are functions of time denoted byx and y and constants are denoted k O dzx dx 0 y x dlz dt 0 ykx10 o yk1xk2x2 o xd2xx y c112 2 Constant arameter timeinvariant vs timevar in s stems 14 A system whose model parameters change with time is considered time varying Note that the output and input will be varying with time for both constant parameter and timevarying systems Systems where outputs differ only by a time shift when the same inputs are applied at corresponding time shifts is a constant parameter system A system is a constant parameter system iff for any input output pair yI Hxf the following statement is also true yl 239 Hxl 7 for all 1 Determine whether the systems below are time varying or time invariant o yl kxl 10 0 yl cos2721xl 15 3 Instantaneous memor less vs d namic with memor s stems For an instantaneous system the present output value depends only on the present input value In a dynamic system the present output value depends on the present and past input values Dynamic systems usually contain some type of energy storage elements The response of a dynamic system results from two components the initial condition and the input The state of the system refers to the information needed along with the present input to determine the present output Zeroinput response system output due only to system state or initial condition Zerostate response system output due only to the input of the system In general Determine which systems are instantaneous and which are dynamic o y klx kzxz o y g x dt 4 Causal vs Noncausal 16 Systems where the output depends only on the present andor past values of the input are referred to as causal Note that for a causal system the output cannot depend on future input values Systems where the output depends on future input value is referred to as a noncausal system 5 LinpedParaineter vs DistributedParaILeter Svstems In most real systems the interactions between the signal energy and the system elements happen continuously over space ie resistance over a wire In modeling these systems the interaction can be considered to occur at one point in space This is referred to as a lumpedparameter model This is a reasonable model when the dimensions of the elements are small with respect to the energy wavelength When this is not done ie transmission lines the model is referred to as a distributed parameter system 17 6 ContinuousTime vs DiscreteTime System The input and output for a discretetime system is de ned only at discrete points in time yn Hxn for n e 2 1 0 1 2 If the inputs and outputs are defined over a continuum of time values then the system is a continuoustime system yt Hxt for l E 0oo 7 Analog vs Digital Systems A system whose input and output values take on only a set of discrete values is referred to as a digital system If the values of the input and output can take on a continuum of values then the system is an analog system Analog Signal Elements of a Digital Signal Processing System Discretetime Digital Signal 96611 AW n X Signal Quantizer Coder xan T n T Processed Processed Digital Analog Signal Signal Computing Interpolator Hardware and smoothing ya I 18 19 Differential Equation Models for Current and Voltage Systems m Ca acitors it c p dz v jjam dil 1 t Ind ctors vt L It vz39dz39 u 0 dt 0 LL lt gt Resistors vl Ril gt 1390 T vl J 11 vl gt i0 V0 Example find the inputoutput equation relating input iS to output v0 2 CL 1 R RR A 0 Va 5 m V Differential Equation Models for Position and Force Systems Translational Systems Consider motion output and force input in one direction denoted by yt and ft respectively M21580quot f0 M f Linear Spring Stiffness K fl Kyt Linear Dashpot damping coefficient B f I Byl Rotational Systems Consider an angular position output and torque input denoted by 9t and T t respectively Rotational Mass J T l J9 Torsional Spring K TZ K6 Torsional Dashpot B Tl B Electromechanical Systems For a DC motor consider the an angular position output and current input denoted by 6t and it respectively Motor Constant KT T I KT il 112 Example consider a torsional spring with stiffness K2 ntmrad fastened to the rotor of an armature controlled DC motor with motor constant K 5 ntmA and the rotational mass is J 5 ntmrads2 The friction coefficient for the spinning rotor is B 05 ntm2radsec Find the equation that relates the rotor position to the armature current Assume the polarity of the motor is such that a positive current moves the angular position is a positive direction Describe the motion of the rotor for a step input going from 0 to 2 A Ans K J 3639 K6 6t 5 eXp Z 205cos2l 00125 sin2t In Matlab gtgt t 02pi20100 gtgt sig 5 exp t205cos2t 0125sin2t gtgt plottsig gtgt title39rotor response39 0 gtgtxlabel39seconds39 03 gtgtylabel39radians39 07 0 60 seconds

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