EECE 3 Signals & Sysems Prof. Mark Fowler Noe Se #2 Wha are Coninuous-Time Signals??? Reading Assignmen: Secion. of Kamen and Heck /22
Course Flow Diagram The arrows here show concepual flow beween ideas. Noe he parallel srucure beween he pink blocks (C-T Freq. Analysis) and he blue blocks (D-T Freq. Analysis). New Signal Models Ch. Inro C-T Signal Model Funcions on Real Line Sysem Properies LTI Causal Ec Ch. 3: CT Fourier Signal Models Fourier Series Periodic Signals Fourier Transform (CTFT) Non-Periodic Signals Ch. 2 Diff Eqs C-T Sysem Model Differenial Equaions D-T Signal Model Difference Equaions Zero-Sae Response Ch. 5: CT Fourier Sysem Models Frequency Response Based on Fourier Transform New Sysem Model Ch. 2 Convoluion C-T Sysem Model Convoluion Inegral Ch. 6 & 8: Laplace Models for CT Signals & Sysems Transfer Funcion New Sysem Model New Sysem Model D-T Signal Model Funcions on Inegers Zero-Inpu Response Characerisic Eq. D-T Sysem Model Convoluion Sum New Signal Model Powerful Analysis Tool Ch. 4: DT Fourier Signal Models DTFT (for Hand Analysis) DFT & FFT (for Compuer Analysis) Ch. 5: DT Fourier Sysem Models Freq. Response for DT Based on DTFT New Sysem Model Ch. 7: Z Trans. Models for DT Signals & Sysems Transfer Funcion New Sysem Model 2/22
. Coninuous-Time Signal Our firs mah model for a signal will be a funcion of ime Coninuous Time (C-T) Signal: A C-T signal is defined on he coninuum of ime values. Tha is: f() for R Real line f() 3/22
Sep & Ramp Funcions These are common exbook signals bu are also common es signals, especially in conrol sysems. Uni Sep Funcion u() u( ) =,, < u()... Noe: A sep of heigh A can be made from Au() 4/22
The uni sep signal can model he ac of swiching on a DC source R = V s + C R V s u() + C + V s u() R C 5/22
Uni Ramp Funcion r() r( ) =,, < Uni slope r()... Noe: A ramp wih slope m can be made from: mr() mr( ) m, =, < 6/22
Relaionship beween u() & r() u(λ) dλ Wha is? Depends on value funcion of : f() f ( ) = u(λ) dλ Wha is f()? -Wrie uni sep as a funcion of λ -Inegrae up o λ = -How does area change as changes? u(λ) λ = Area = f() λ i.e., Find Area f ( ) = u( λ) dλ = = = r( ) r( ) = u(λ) dλ Running Inegral of sep = ramp 7/22
Also noe: For we have:, r( ) =, < dr( ), > = d, < Overlooking his, we can roughly say u( ) = No defined a =! dr( ) d r()... u()... 8/22
Time Shifing Signals Time shifing is an operaion on a signal ha shows up in many areas of signals and sysems: Time delays due o propagaion of signals acousic signals propagae a he speed of sound radio signals propagae a he speed of ligh Time delays can be used o build complicaed signals We ll see his laer Time Shif: If you know x(), wha does x( ) look like? For example If = 2: x( 2) = x( 2) x( 2) = x( ) A =, x( 2) akes he value of x() a = 2 A =, x( 2) akes he value of x() a = 9/22
Example of Time Shif of he Uni Sep u(): u()... -2-2 3 4 u(-2)... -2-2 3 4 u(+.5)... General View: x( ± ) for > -2-2 3 4 + gives Lef shif (Advance) gives Righ shif (Delay) /22
The Impulse Funcion One of he mos imporan funcions for undersanding sysems!! Ironically i does no exis in pracice!! Oher Names: Dela Funcion, Dirac Dela Funcion I is a heoreical ool used o undersand wha is imporan o know abou sysems! Bu i leads o ideas ha are used all he ime in pracice!! There are hree views we ll ake of he dela funcion: Rough View: a pulse wih: Infinie heigh Zero widh Uni area A really narrow, really all pulse ha has uni area /22
Slighly Less-Rough View: δ ( ) = lim pε ( ) ε ε ε p ε ( ) Here we define as: ε ε 2 ε ε 2 p ε ( ) Beware of Fig.4 in he book i does no show he real δ() So is verical axis should NOT be labeled wih δ() Pulse having heigh of /ε and widh of ε which herefore has area of ( = ε /ε) So as ε ges smaller he pulse ges higher and narrower bu always has area of In he limi i becomes he dela funcion 2/22
Precise Idea: δ() is no an ordinary funcion I is defined in erms of is behavior inside an inegral: The dela funcion δ() is defined as somehing ha saisfies he following wo condiions: δ ( ) =, for any ε ε δ ( ) d =, for any ε > We show δ() on a plo using an arrow (conveys infinie heigh and zero widh) δ() Cauion his is NOT he verical axis i is he dela funcion!!! 3/22
The Sifing Propery is he mos imporan propery of δ(): + ε ε f ( ) δ ( ) d = f ( ) ε > f() f( ) δ(- ) f( ) Inegraing he produc of f() and δ( o ) reurns a single number he value of f() a he locaion of he shifed dela funcion As long as he inegral s limis surround he locaion of he dela oherwise i reurns zero 4/22
Seps for applying sifing propery: + ε ε f ( ) δ ( ) d = f ( ) Example #: 7 4 sin( π) δ ( sin( π) ) d δ ( ) 2 3 =? Sep : Find variable of inegraion Sep 2: Find he argumen of δ( ) Sep 3: Find he value of he variable of inegraion ha causes he argumen of δ( ) o go o zero. Sep 4: If value in Sep 3 lies inside limis of inegraion Take everyhing ha is muliplying δ( ) and evaluae i a he value found in sep 3; Oherwise reurn zero Sep : Sep 2: Sep 3: = = Sep 4: = lies in [ 4,7] so evaluae sin(π ) = sin(π) = 7 4 sin( π) δ ( ) d = 5/22
2 Example #2: sin( π) δ ( 2.5) d =? Sep : Find variable of inegraion: Sep 2: Find he argumen of δ( ): 2.5 Sep 3: Find he value of he variable of inegraion ha causes he argumen of δ( ) o go o zero: 2.5 = = 2.5 Sep 4: If value in Sep 3 lies inside limis of inegraion No! Oherwise reurn zero sin( π) δ ( 2.5) 2 sin( π) δ ( 2.5) d = 2 3 Range of Inegraion Does NOT include dela funcion 6/22
7 2 Example #3: sin( ω)( 3) δ (3 + 4) d =? 4 Sep : Find variable of inegraion: τ Sep 2: Find he argumen of δ( ): τ + 4 Sep 3: Find he value of he variable of inegraion ha causes he argumen of δ( ) o go o zero: τ = 4 Sep 4: If value in Sep 3 lies inside limis of inegraion Yes! Take everyhing ha is muliplying δ( ): (/3)sin(ωτ/3)(τ/3 3) 2 and evaluae i a he value found in sep 3: (/3)sin( 4/3ω)( 4/3 3) 2 = 6.26sin( 4/3ω) Because of his handle slighly differenly! Sep : Change variables: le τ = 3 dτ = 3d limis: τ L = 3(-4) τ L = 3(7) 2 2 sin( ωτ / 3)( τ / 3 3) δ ( τ + 4) dτ =? 3 2 7 4 ( ) 2 sin( ω)( 3) δ (3 + 4) d = 6.26sin 4 / 3ω 7/22
One Relaionship Beween δ() & u() δ ( λ) dλ = u( ) For < : he inegrand = inegral = for < δ(λ) λ = < Range of Inegraion λ Defines he uni sep funcion For > : we inegrae over he dela inegral = for > δ(λ) λ = > λ Range of Inegraion 8/22
Anoher Relaionship Beween δ() & u() d δ ( ) = u( ) d Derivaive = u() Derivaive =... Derivaive = ( Engineer Thinking ) Our view of he dela funcion having infinie heigh bu zero widh maches his inerpreaion of he values of he derivaive of he uni sep funcion!! 9/22
Periodic Signals Periodic signals are imporan because many human-made signals are periodic. Mos es signals used in esing circuis are periodic signals (e.g., sine waves, square waves, ec.) A Coninuous-Time signal x() is periodic wih period T if: x( + T) = x() x()...... T x() x( + T) Fundamenal period = smalles such T When we say Period we almos always mean Fundamenal Period 2/22
Recangular Pulse Funcion: p τ () p τ () -τ/2 τ/2 Subscrip specifies he pulse widh We can build a Recangular Pulse from Uni Sep Funcions: p τ () = u( + τ/2) u( τ/2) u( + τ/2) This is helpful because we will have los of resuls ha apply o he sep funcion -τ/2 u( - τ/2) -τ/2 τ/2 = = = 2/22
Building Signals wih Pulses: shifed pulses are used o urn oher funcions on and off. This allows us o mahemaically describe complicaed funcions in erms of simpler funcions. 2 Coninues up forever g() =.5 + This Coninues down forever Muliplying By Zero Turns Off g() 3 2 2 Delay by p 2 ( -) 2 Widh of 2 Muliplying By One Turns On g() f() = (.5 + )p 2 ( -) Times This Gives This 2 22/22