# lecture2n - Stanford University Winter 2009-2010 Signal...

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Stanford University Winter 2009-2010 Signal Processing and Linear Systems I Lecture 2: Signal Characteristics and Models January 4, 2010 EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 1 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 Signals sizes Signal Energy and Power EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 2

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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 ) EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 3 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 . EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 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 . EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 5 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 . EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 6

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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.
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