102_1_lecture2

# 102_1_lecture2 - UCLA Fall 2010 Systems and Signals Lecture...

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UCLA Fall 2010 Systems and Signals Lecture 2: Signal Characteristics and Models September 29, 2010 EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 1

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Signal Characteristics and Models Operations on the time dependence of a signal Time scaling Time reversal Time shift Combinations Signal characteristics Periodic signals Complex signals Signal Energy and Power EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 2
Time Scaling, Continuous Time A signal x ( t ) is scaled in time by multiplying the time variable by a positive constant b , to produce x ( bt ) . A positive factor of b either expands (0 < b < 1) or compresses ( b > 1) the signal in time. -2 -1 0 1 2 1 2 t x ( t ) b = 1 -2 -1 0 1 2 1 2 t x ( 2 t ) b = 2 -2 -1 0 1 2 1 2 t -3 3 b = 1 / 2 x ( t / 2 ) EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 3

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Time Scaling, Discrete Time The discrete-time sequence x [ n ] is compressed in time by multiplying the index n by an integer k , to produce the time-scaled sequence x [ nk ] . This extracts every k th sample of x [ n ] . Intermediate samples are lost. The sequence is shorter. 2 4 -2 -4 0 1 3 -1 -3 x [ n ] n y [ n ] = x [ 2 n ] 2 -2 0 1 -1 n Called downsampling , or decimation . EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 4
The discrete-time sequence x [ n ] is expanded in time by dividing the index n by an integer m , to produce the time-scaled sequence x [ n/m ] . This speciﬁes every m th sample. The intermediate samples must be synthesized (set to zero, or interpolated). The sequence is longer. 2 -2 0 1 -1 n x [ n ] 2 4 -2 -4 0 1 3 -1 -3 n y [ n ] = x [ n / 2 ] Called upsampling , or interpolation . EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 5

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Time Reversal Continuous time: replace t with - t , time reversed signal is x ( - t ) t x ( t ) t x ( - t ) Discrete time: replace n with - n , time reversed signal is x [ - n ] . t 2 4 -2 -4 0 x [ n ] t x [ - n ] 2 4 -2 -4 0 Same as time scaling, but with b = - 1 . EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 6
Time Shift For a continuous-time signal x ( t ) , and a time t 1 > 0 , Replacing t with t - t 1 gives a delayed signal x ( t - t 1 ) Replacing t with t + t 1 gives an advanced signal x ( t + t 1 ) -2 -1 0 1 2 1 2 t x ( t + 1 ) -2 -1 0 1 2 1 2 t x ( t ) -2 -1 0 1 2 1 2 t x ( t - 1 ) May seem counterintuitive. Think about where t - t 1 is zero. EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 7

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For a discrete time signal x [ n ] , and an integer n 1 > 0 x [ n - n 1 ] is a delayed signal. x [ n + n 1 ] is an advanced signal.
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## This note was uploaded on 04/20/2011 for the course EE 102 taught by Professor Levan during the Fall '08 term at UCLA.

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102_1_lecture2 - UCLA Fall 2010 Systems and Signals Lecture...

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