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Ch2 Continuous-Time Signals

# Ch2 Continuous-Time Signals - Continuous Time Signals Basic...

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Continuous Time Signals Basic Signals – Singularity Functions Transformations of Continuous Time Signals Signal Characteristics Common Signals

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February 10, 2012 Veton K ë puska 2 Continuous-Time Signals Assumptions: Functions ,   x(t),  are of the one independent  variable that typically represents time ,   t . Time  t  can assume all real values:  -∞ < t < ∞, Function x(t) is typically a real function.
Singularity Functions February 10, 2012 Veton K ë puska 3

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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.2 0 0.2 0.4 0.6 0.8 1 u(t) time [sec] Unit Sample Signal Unit Step Function Unit step function  definition: ( 29 < = 0 , 0 0 , 1 t t t u February 10, 2012 Veton K ë puska 4
Unit Step Function Properties Scaling: Unit step function can be  scaled by a real constant K  (positive or negative) Multiplication: Multiplication of any function,  say  x(t) , by a unit step function  u(t) is equivalent to defining the  signal  x(t)  for  t≥0 . February 10, 2012 Veton K ë puska 5 ( 29 ( 29 t Ku t f = ( 29 ( 29 ( 29 0 , t t x t u t x

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Unit Ramp Function Unit Ramp Function is  defined as: February 10, 2012 Veton K ë puska 6 ( 29 < = 0 , 0 0 , t t t t r -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.2 0 0.2 0.4 0.6 0.8 1 r(t) time [sec] Unit Sample Signal
Unit Ramp Function Properties Scaling: Unit step function can be scaled  by a real constant K (positive or  negative) Integral of the unit step function is  equal to the ramp function: Derivative of the unit ramp function is  the unit step function.  February 10, 2012 Veton K ë puska 7 ( 29 ( 29 t Kr t f = Slope of the  straight line ( 29 ( 29 - = t d u t r τ τ ( 29 ( 29 dt t dr t u =

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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.2 0 0.2 0.4 0.6 0.8 1 δ (t) time [sec] Unit Sample Signal Unit Impulse Function Unit Impulse Function,  also know as Dirac  delta function, is  defined as: February 10, 2012 Veton K ë puska 8 ( 29 ( 29 2200 = = = - ε ε τ τ δ δ ε ε & 0 1 0 , 0 0 , d t t t
Unit Impulse Function Properties Scaling: Unit impulse function can be  scaled by a real constant K  (positive or negative) Delta function can be approximated  by a pulse centered at the origin February 10, 2012 Veton K ë puska 9 ( 29 ) ( lim t d t A = δ -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.2 0 0.2 0.4 0.6 0.8 1 δ (t) time [sec] Unit Sample Signal A 2 1 + A 2 1 - A

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Unit Impulse Function Properties Unit impulse function is related  to unit step function: Conversely: February 10, 2012 Veton K ë puska 10 ( 29 ( 29 dt t du t = δ ( 29 ( 29 0 & 2200 = - t t d t u t τ τ δ Proof: 1. t<0 2. t>0 ( 29 ( 29 ( 29 0 since , 0 0 = < 2200 = = - t t d t u t δ τ τ δ ( 29 ( 29 ( 29 0 , 1 since , 0 1 = 2200 = = - - ε τ τ δ τ τ δ ε ε d t d t u t
Time Transformation of Signals February 10, 2012 Veton K ë puska 11

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February 10, 2012 Veton K ë puska 12 Time Reversal: = - - = = 2200 - = ) ( ) ( & ) ( ) ( , ) ( ) ( 0 0 0 0 0 t x t y t x t y t t t x t y 1 2 0 -1 -2 -1 1 2 t 1 2 0 -1 -2 -1 1 2 t y(t)=x(-t) x(t)
February 10, 2012 Veton K ë puska 13 Time Scaling

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Ch2 Continuous-Time Signals - Continuous Time Signals Basic...

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