Lecture 1: Signals & Systems Concepts(1) Systems, signals, mathematical models. Continuous-time and discrete-time signals and systems. Energy and power signals. Linear systems. Examples, introduction to MatlabSpecific Objectives:Introduce, using examples： what is a signal what is a systemhow are they relatedEE-2027 SaS, L11Recommended Reading MaterialSignals and Systems, Oppenheim & Willsky, Chapter 1MatlabMany other introductory sources available. EE-2027 SaS, L12What is a Signal?A signal is a pattern of variation of some formSignals are variables that carry informationExamples of signal include:Electrical signalsVoltages and currents in a circuitAcoustic signalsAcoustic pressure (sound) over timeMechanical signalsVelocity of a car over timeVideo signalsIntensity level of a pixel (camera, video) over timeEE-2027 SaS, L13How is a Signal Represented?Mathematically, signals are represented as a function of one or more independent variables.For instance a black & white video signal intensity(亮度) is dependent on x, y coordinates and time t f(x,y,t)On this course, we shall be exclusively concerned with signals that are a function of a single variable: timetf(t)EE-2027 SaS, L14Example: Signals in an Electrical CircuitThe signals vc and vs are patterns of variation over timeNote, we could also have considered the voltage across the resistor or the current as signals+-ivcvsRC Step (signal) vs at t=1 RC = 1 First order (exponential指数) response for vcvs, vctEE-2027 SaS, L15Continuous & Discrete-Time SignalsContinuous-Time SignalsMost signals in the real world are continuous time, as the scale is infinitesimally无无穷穷小小 fine.Eg voltage, velocity, Denote by x(t), where the time interval间间隔隔 may be bounded (finite) or infiniteDiscrete-Time SignalsSome real world and many digital signals are discrete time, as they are sampledE.g. pixels, daily stock price (anything that a digital computer processes)Denote by xn, where n is an integer value that varies discretelySampled continuous signal xn =x(nk) k is sample timex(t)txnnEE-2027 SaS, L16Many human-made Signals are DTWhy DT? Can be processed by modern digital computersand digital signal processors (DSPs).EE-2027 SaS, L17Mandrill ExampleUnblurred Image & No NoiseEE-2027 SaS, L18Mandrill ExampleBlurred Image (bad focus)EE-2027 SaS, L19Mandrill ExampleUnblurred Image 0.1% Noise (too dark)EE-2027 SaS, L110Signal Properties On this course, we shall be particularly interested in signals with certain properties:Periodic signals: a signal is periodic if it repeats itself after a fixed period T, i.e. x(t) = x(t+T) for all t. A sin(t) signal is periodic.Even and odd signals: a signal is even if x(-t) = x(t) (i.e. it can be reflected in the axis at zero). A signal is odd if x(-t) = -x(t). Examples are cos(t) and sin(t) signals, respectively.Exponential and sinusoidal signals: a signal is (real) exponential if it can be represented as x(t) = Ceat. A signal is (complex) exponential if it can be represented in the same form but C and a are complex numbers.Step and pulse signals: A pulse signal is one which is nearly completely zero, apart from a short spike, d(t). A step signal is zero up to a certain time, and then a constant value after that time, u(t).These properties define a large class of tractable(易处理的), useful signals and will be further considered in the coming lecturesEE-2027 SaS, L111What is a System?Systems process input signals to produce output signalsExamples:A circuit involving a capacitor can be viewed as a system that transforms the source voltage (signal) to the voltage (signal) across the capacitorA CD player takes the signal on the CD and transforms it into a signal sent to the loud speakerA communication system is generally composed of three sub-systems, the transmitter, the channel and the receiver. The channel typically attenuates and adds noise to the transmitted signal which must be processed by the receiverEE-2027 SaS, L112How is a System Represented?A system takes a signal as an input and transforms it into another signalIn a very broad sense, a system can be represented as the ratio of the output signal over the input signalThat way, when we “multiply the system by the input signal, we get the output signalThis concept will be firmed up in the coming weeksSystemInput signalx(t)Output signaly(t)EE-2027 SaS, L113Example: An Electrical Circuit System+-ivcvsRCvs, vctEE-2027 SaS, L114Continuous & Discrete-Time Mathematical Models of SystemsContinuous-Time SystemsMost continuous time systems represent how continuous signals are transformed via differential equations.E.g. circuit, car velocityDiscrete-Time SystemsMost discrete time systems represent how discrete signals are transformed via difference equationsE.g. bank account, discrete car velocity systemFirst order differential equationsFirst order difference equationsEE-2027 SaS, L115Properties of a SystemOn this course, we shall be particularly interested in signals with certain properties:Causal: a system is causal if the output at a time, only depends on input values up to that time.Linear: a system is linear if the output of the scaled sum of two input signals is the equivalent scaled sum of outputsTime-invariance: a system is time invariant if the systems output is the same, given the same input signal, regardless of time.These properties define a large class of tractable, useful systems and will be further considered in the coming lecturesEE-2027 SaS, L116How Are Signal & Systems Related (i)?How to design a system to process a signal in particular ways?Assume a signal is represented asx(t) = d(t) + n(t)Design a system to remove the unknown “noise component n(t), so that y(t) d(t)System?x(t) = d(t) + n(t)y(t) d(t)EE-2027 SaS, L117How Are Signal & Systems Related (ii)?How to design a system to extract specific pieces of information from signalsEstimate the heart rate from an electrocardiogramEstimate economic indicators (bear, bull) from stock market valuesAssume a signal is represented asx(t) = g(d(t)Design a system to “invert the transformation g(), so that y(t) = d(t)System?x(t) = g(d(t)y(t) = d(t) = g-1(x(t)EE-2027 SaS, L118Lecture 1: SummarySignals and systems are pervasive in modern engineering courses:Electrical circuitsPhysical models and control systemsDigital media (music, voice, photos, video)In studying the general properties of signals and systems, you can:Design systems to remove noise/enhance measurement from audio and picture/video dataInvestigate stability of physical structuresControl the performance mechanical and electrical devices This will be the foundation for studying systems and signals as a generic subject on this course.EE-2027 SaS, L119Introduction to MatlabSimulink is a package that runs inside the Matlab environment.Matlab (Matrix Laboratory) is a dynamic, interpreted, environment for matrix/vector analysisUser can build programs (in .m files or at command line) C/Java-like syntaxIdeal environment for programming and analysing discrete (indexed) signals and systemsEE-2027 SaS, L120Basic Matlab Operations % This is a comment, it starts with a “% y = 5*3 + 22;% simple arithmetic x = 1 2 4 5 6;% create the vector “x x1 = x.2;% square each element in x E = sum(abs(x).2);% Calculate signal energy P = E/length(x);% Calculate av signal power x2 = x(1:3);% Select first 3 elements in x z = 1+i;% Create a complex number a = real(z);% Pick off real part b = imag(z);% Pick off imaginary part plot(x);% Plot the vector as a signal t = 0:0.1:100;% Generate sampled time x3=exp(-t).*cos(t);% Generate a discrete signal plot(t, x3, x);% Plot points EE-2027 SaS, L121Other Matlab Programming StructuresLoopsfor i=1:100 sum = sum+i;endGoes round the for loop 100 times, starting at i=1 and finishing at i=100i=1;while i=100 sum = sum+i; i = i+1;endSimilar, but uses a while loop instead of a for loopDecisionsif i=5 a = i*2;else a = i*4; endExecutes whichever branch is appropriate depending on testswitch icase 5 a = i*2;otherwise a = i*4;endSimilar, but uses a switchEE-2027 SaS, L122Matlab Help!These slides have provided a rapid introduction to MatlabMastering Matlab 6, Prentice Hall, Introduction to Matlab (on-line)Lots of help availableType help in the command window or help operator. This displays the help associated with the specified operator/functionType lookfor topic to search for Matlab commands that are related to the specified topicType helpdesk in the command window or select help on the pull down menu. This allows you to access several, well-written programming tutorials. newsgroupLearning to program (Matlab) is a “bums on seats activity. There is no substitute for practice, making mistakes, understanding conceptsEE-2027 SaS, L123Using the Matlab DebuggerBecause Matlab is an interpreted language, there is no compile type syntax checking and the likelihood of a run-time error is higherRun-time debugging can helpUse the debug and breakpoints pull-down menus to determine where to stop program and inspect variablesStep over lines/step into functions to evaluate what happensEE-2027 SaS, L124Lecture 1: ExercisesRead SaS OW, Chapter 1. This contains most of the material in the first three lectures, a bit of pre-reading will be extremely useful!In lecture 2, well be looking at signals in more depthEE-2027 SaS, L125