EECE 3 Signals & Sysems Prof. Mark Fowler Noe Se # Wha are Coninuous-Time Signals??? /6
Coninuous-Time Signal Coninuous Time (C-T) Signal: A C-T signal is defined on he coninuum of ime values. Tha is: f() for Real # Line Time-Varying Volage, Curren, ec. f() Uni Sep Funcion u() u( ),,... u()... Noe: A sep of heigh A can be made from Au() /6
The uni sep signal can model he ac of swiching on a DC source R = V s + C R V s u() + C 3/6
Uni Ramp Funcion r() r( ),, Uni slope r()... Noe: A ramp wih slope m can be made from: mr() mr( ) m,, 4/6
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 = : x( ) = x( ) x( ) = x( ) A =, x( ) akes he value of x() a = A =, x( ) akes he value of x() a = 5/6
Example of Time Shif of he Uni Sep u(): u()... - - 3 4 u(-)... - - 3 4 u(+.5)... General View: x( ± ) for > - - 3 4 + gives Lef shif (Advance) gives Righ shif (Delay) 6/6
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 7/6
Slighly Less-Rough View: Limi of pulse wih widh and heigh / So as ges smaller he pulse ges higher and narrower bu always has area of In he limi as ges smaller i becomes he dela funcion Precise Idea: () is defined by is behavior inside an inegral: The dela funcion () is defined as somehing ha saisfies he following wo condiions: ( ), for any ( ) d, for any 8/6
Showing Dela Funcion on a Plo: We show () on a plo using an arrow (conveys infinie heigh and zero widh) This is he verical axis () Cauion his is NOT he verical axis i is he dela funcion!!! ( - T) A ime shif jus changes where he dela is locaed T 9/6
The Sifing Propery is he mos imporan propery of (): f ( ) ( ) d f ( ) f() f( ) (- ) f( ) Inegraing he produc of f() and ( 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 /6
Seps for applying sifing propery: f ( ) ( ) d f ( ) Example: 7 4 sin( ) ( sin( ) ) d ( ) 3? Sep : Find variable of inegraion Sep : Find he argumen of () Sep 3: Find he value of he variable of inegraion ha causes he argumen of () o go o zero. Sep 4: If value in Sep 3 lies inside limis of inegraion Take everyhing ha is muliplying () and evaluae i a he value found in sep 3; Oherwise reurn zero Sep : Sep : Sep 3: = = Sep 4: = lies in [ 4,7] so evaluae sin() = sin() = 7 4 sin( ) ( ) d /6
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 /6
Power and Energy of Signals Imagine ha signal x() is a volage. If x() drops across resisance R, he insananeous power is p( ) x ( ) R Someimes we don know wha R is here so we normalize his by ignoring he R value: ( ) x ( ) Energy in one period p N Once we have a specific R we can always un-normalize via (In Signals & Sysems we will drop he N subscrip) Recall: power = energy per uni ime ( W = J/s) T de( ) p( ) T de( ) d x ( ) d p N ( ) R de ( ) x ( ) d differenial incremen of energy The Toal Energy x ( ) d = for a periodic signal Noe if x() is no periodic is energy may be finie if i falls off fas enough a is ends 3/6
Recall: power = energy per uni ime Average power over one period = Energy in One Period T P T x T ( ) d Ofen jus called Average Power For periodic signals we use he average power as measure of he size of a signal. The Average Power of pracical periodic signals is finie and non-zero. (Recall ha he oal energy of a periodic signal is infinie.) 4/6
Recangular Pulse Funcion: p () p () -/ / Subscrip specifies he pulse widh We can build a Recangular Pulse from Uni Sep Funcions: p () = u( + /) u( /) u( + /) This is helpful because we will have los of resuls ha apply o he sep funcion -/ u( - /) -/ / = = = 5/6
Building Signals wih Pulses: shifed pulses are used o mahemaically urn oher funcions on and off. g() =.5 + Coninues up forever This Coninues down forever Muliplying By Zero Turns Off g() 3 Delay by p ( -) Widh of Muliplying By One Turns On g() f() = (.5 + )p ( -) Times This Gives This 6/6