Ch2 Continuous-Time Signals

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

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Unformatted text preview: Continuous Time Signals Basic Signals Singularity Functions Transformations of Continuous Time Signals Signal Characteristics Common Signals February 10, 2012 Veton Kpuska 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 Kpuska 3-1-0.8-0.6-0.4-0.2 0.2 0.4 0.6 0.8 1-0.2 0.2 0.4 0.6 0.8 1 u(t) time [sec] Unit Sample Signal Unit Step Function Unit step function definition: ( 29 < = , , 1 t t t u February 10, 2012 Veton Kpuska 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 t0 . February 10, 2012 Veton Kpuska 5 ( 29 ( 29 t Ku t f = ( 29 ( 29 ( 29 , t t x t u t x Unit Ramp Function Unit Ramp Function is defined as: February 10, 2012 Veton Kpuska 6 ( 29 < = , , t t t t r-1-0.8-0.6-0.4-0.2 0.2 0.4 0.6 0.8 1-0.2 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 Kpuska 7 ( 29 ( 29 t Kr t f = Slope of the straight line ( 29 ( 29 -= t d u t r ( 29 ( 29 dt t dr t u =-1-0.8-0.6-0.4-0.2 0.2 0.4 0.6 0.8 1-0.2 0.2 0.4 0.6 0.8 1 (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 Kpuska 8 ( 29 ( 29 2200 = = = - & 1 , , 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 Kpuska 9 ( 29 ) ( lim t d t A = -1-0.8-0.6-0.4-0.2 0.2 0.4 0.6 0.8 1-0.2 0.2 0.4 0.6 0.8 1 (t) time [sec] Unit Sample Signal A 2 1 + A 2 1-A Unit Impulse Function Properties Unit impulse function is related to unit step function: Conversely: February 10, 2012 Veton Kpuska 10 ( 29 ( 29 dt t du t = ( 29 ( 29 & 2200 = -t t d t u t Proof: 1. t<0 2. t>0 ( 29 ( 29 ( 29 since , = < 2200 = = -t t d t u t ( 29 ( 29 ( 29 , 1 since , 1 = 2200 = = -- d t d t u t Time Transformation of Signals February 10, 2012 Veton Kpuska 11 February 10, 2012...
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This note was uploaded on 02/10/2012 for the course ECE 3111 taught by Professor Earles during the Fall '09 term at FIT.

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

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