102_1_lecture3

102_1_lecture3 - UCLA Fall 2010 Systems and Signals Lecture...

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UCLA Fall 2010 Systems and Signals Lecture 3: Signal Models and System Properties October 4, 2010 EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 1
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Agenda Homework 1 due Wednesday in class (10 am) (grace period until 5pm) After class, submit HW to admin. assistant Kimberly Hernandez (Engi IV:56-125AA) Homework 2 will be posted on Wednesday TA Office Hours: F 2pm-4pm (Fang) TA Office Hours: T 11:30am-1:30pm (Karasev) EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 2
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Signal Models and System Properties Today’s topics: Signals Sinusoidal signals Exponential signals Complex exponential signals Unit step and unit ramp Impulse functions and impulse trains Systems What are systems? Linearity Time Invariance EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 3
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Sinusoidal Signals A sinusoidal signal is of the form x ( t ) = cos( ωt + θ ) . where the radian frequency is ω , which has the units of radians/s. Also very commonly written as x ( t ) = A cos(2 πft + θ ) . where f is the frequency in Hertz. We will often refer to ω as the frequency, but it must be kept in mind that it is really the radian frequency , and the frequency is actually f . EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 4
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The (fundamental) period of the sinuoid is T = 1 f = 2 π ω with the units of seconds. The phase or phase angle of the signal is θ , given in radians. t T 2T -2T -T 0 cos ( ω t ) T 2T -2T -T 0 t cos ( ω t - θ ) EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 5
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Complex Sinusoids The Euler relation defines e = cos φ | {z } < ( e ) + j sin φ | {z } = ( e ) . A complex sinusoid is Ae j ( ωt + θ ) = A cos( ωt + θ ) + jA sin( ωt + θ ) . T 2T -2T -T 0 e j ω t Real sinusoid can be represented as the real part of a complex sinusoid <{ Ae j ( ωt + θ ) } = A cos( ωt + θ ) EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 6
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Exponential Signals An exponential signal is given by x ( t ) = e σt If σ < 0 this is exponential decay . If σ > 0 this is exponential growth . -2 -1 0 1 2 1 2 t e - t -2 -1 0 1 2 1 2 t e t EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 7
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Damped or Growing Sinusoids A damped or growing sinusoid is given by x ( t ) = e σt cos( ωt + θ ) Exponential growth ( σ > 0 ) or decay ( σ < 0 ), modulated by a sinusoid. t 0 T 2T -T -2T σ < 0 e σ t cos ( ω t ) σ > 0 t 0 T 2T -T -2T e σ t cos ( ω t ) EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 8
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Complex Exponential Signals A complex exponential signal is given by e ( σ + ) t + = e σt (cos( ωt + θ ) + j sin( ωt + θ )) A exponential growth or decay, modulated by a complex sinusoid. Includes all of the previous signals as special cases. t 0 T 2T -T -2T σ < 0 e ( σ + j ω ) t σ > 0 t 0 T 2T -T -2T e ( σ + j ω ) t EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 9
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Complex Plane Each complex frequency s = σ + corresponds to a position in the complex plane. Left Half Plane Right Half Plane σ σ < 0 σ > 0 j ω Decreasing Signals Increasing Signals EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 10
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Unit Step Functions The unit step function u ( t ) is defined as u ( t ) = 1 , t 0 0 , t < 0 Also known as the Heaviside step function .
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