EECE 301 Signals & Systems Prof. Mark Fowler1.1 Continuous-Time SignalUnit Step Function u(t)Relationship between u(t) & r(t)Example of Time Shift of the Unit Step u(t):The Impulse Function Rectangular Pulse Function: p(t)1/22Note Set #2• What are Continuous-Time Signals???• Reading Assignment: Section 1.1 of Kamen and HeckEECE 301 Signals & SystemsProf. Mark Fowler2/22Ch. 1 IntroC-T Signal ModelFunctions on Real LineD-T Signal ModelFunctions on IntegersSystem PropertiesLTICausalEtcCh. 2 Diff EqsC-T System ModelDifferential EquationsD-T Signal ModelDifference EquationsZero-State ResponseZero-Input ResponseCharacteristic Eq.Ch. 2 ConvolutionC-T System ModelConvolution IntegralD-T System ModelConvolution SumCh. 3: CT Fourier SignalModelsFourier SeriesPeriodic SignalsFourier Transform (CTFT)Non-Periodic SignalsNew System ModelNew SignalModelsCh. 5: CT Fourier SystemModelsFrequency ResponseBased on Fourier TransformNew System ModelCh. 4: DT Fourier SignalModelsDTFT(for “Hand” Analysis)DFT & FFT(for Computer Analysis)New SignalModelPowerful Analysis ToolCh. 6 & 8: Laplace Models for CTSignals & SystemsTransfer FunctionNew System ModelCh. 7: Z Trans.Models for DTSignals & SystemsTransfer FunctionNew SystemModelCh. 5: DT Fourier System ModelsFreq. Response for DTBased on DTFTNew System ModelCourse Flow DiagramThe arrows here show conceptual flow between ideas. Note the parallel structure between the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).3/221.1 Continuous-Time SignalOur first math model for a signal will be a “function of time”Continuous Time (C-T) Signal:A C-T signal is defined on the continuum of time values. That is:f(t) for t ∈ℜReal linef(t)t4/22Unit Step Function u(t)⎩⎨⎧<≥=0,00,1)(tttu. . .u(t)1tNote: A step of height A can be made from Au(t)Step & Ramp FunctionsThese are common textbook signals but are also common test signals, especially in control systems.5/22VsR+–CThe unit step signal can model the act of switching on a DC source…t = 0Vsu(t)R+–CVsu(t)RC–+6/22Unit Ramp Function r(t)Note: A ramp with slope m can be made from: mr(t). . .r(t)1t1Unit slope⎩⎨⎧<≥=0,00,)(ttttr⎩⎨⎧<≥=0,00,)(ttmttmr7/22Relationship between u(t) & r(t)What is ?Depends on t value⇒function of t: f(t)⇒ What is f(t)?-Write unit step as a function of λ-Integrate up to λ= t-How does area change as t changes?i.e., Find Areau(λ)1λλ= tArea = f(t)⇒⇒“Running Integral of step = ramp”∫∞−tduλλ)(∫∞−=tdutfλλ)()()(1)()( trttdutft==⋅==∫∞−λλ∫∞−=tdutrλλ)()(8/22Also note: For we have:Overlooking this, we can roughly say. . .u(t)1. . .r(t)1 t1⎩⎨⎧<>=0,00,1)(ttdttdr⎩⎨⎧<≥=0,00,)(ttttrNot defined at t = 0!dttdrtu)()( =9/22For example… If t0= 2:x(0 – 2) = x(–2) x(1 – 2) = x(–1) At t = 0, x(t – 2) takes the value of x(t) at t = –2Time Shifting SignalsTime shifting is an operation on a signal that shows up in many areas of signals and systems:• Time delays due to propagation of signals─ acoustic signals propagate at the speed of sound─ radio signals propagate at the speed of light• Time delays can be used to “build” complicated signals─ We’ll see this laterAt t = 1, x(t – 2) takes the value of x(t) at t = –1Time Shift: If you know x(t), what does x(t–t0) look like?10/22Example of Time Shift of the Unit Step u(t):. . .u(t)1t1 2 3 4-2 -1. . .u(t-2)1t1 2 3 4-2 -1. . .u(t+0.5)1t1 2 3 4-2 -1General View:x(t ± t0) for t0 > 0“+t0”gives Left shift (Advance)“–t0”gives Right shift (Delay)11/22The Impulse FunctionOne of the most important functions for understanding systems!!Ironically…it does not exist in practice!!⇒ It is a theoretical toolused to understand what is important to know about systems! But… it leads to ideas that are used all the time in practice!!There are three views we’ll take of the delta function:Other Names: Delta Function, Dirac Delta FunctionInfinite heightZero width UnitareaRough View: a pulse with:“A really narrow, really tall pulse that has unit area”12/22Beware of Fig 1.4 in the book… it does not show the real δ(t)…So its vertical axis should NOT be labeled with δ(t) Slightly Less-Rough View:)(1lim)(0tptεεεδ→=)(1tpεεHere we define as:)(1tpεεε12ε2ε−tPulse having… height of 1/ε and width of ε… which therefore has… area of 1 (1 = ε×1/ε)So as ε gets smallerthe pulse gets higher and narrower but always has area of 1…In the limitit “becomes” the delta function13/22Precise Idea:δ(t) is not an ordinary function… It is defined in terms of its behavior inside an integral:The delta function δ(t) is defined as something that satisfies the following two conditions:0any for ,1)(0any for ,0)(>=≠=∫−εδδεεdttttt0δ(t)We show δ(t) on a plot using an arrow…(conveys infinite height and zero width)Caution… this is NOT the vertical axis… it is the delta function!!!14/22The Sifting Property is the most important property of δ(t):0)()()(0000>∀=−∫+−εδεεtttfdttttftf(t)t0f(t0)tt0δ(t- t0)f(t0)Integrating the product of f(t) and δ(t – to) returns a single number… the value of f(t) atthe “location” of the shifted delta function As long as the integral’s limits surround the “location” of the delta… otherwise it returns zero15/22Steps for applying sifting property:Step 1: Find variable of integrationStep 2: Find the argument of δ(•)Step 3: Find the value of the variable of integration that causes the argument of δ(•) to go to zero.Step 4: If value in Step 3 lies inside limits of integration… Take everything that is multiplying δ(•) and evaluate it at the value found in step 3; Otherwise… “return” zero∫+−=−εεδ00)()()(00tttfdttttfStep 1: t Step 2: t –1Step 3: t –1 = 0 ⇒ t = 1Step 4: t = 1 lies in [–4,7] so evaluate… sin(π×1) = sin(π) = 0Example #1:?)1()sin(74=−∫−dtttδπt123)sin( tπ)1( −tδ0)1()sin(74=−∫−dtttδπ16/22Example #2:?)5.2()sin(20=−∫dtttδπ0)5.2()sin(20=−∫dtttδπStep 1: Find variable of integration: tStep 2: Find the argument of δ(•): t –2.5Step 3: Find the value of the variable of integration that causes the argument of δ(•) to go to zero: t – 2.5 = 0 ⇒ t = 2.5Step 4: If value in Step 3 lies inside limits of integration…