lec08_spana

lec08_spana - 3 j GHP j 3 j 1 R2 1 j GBP j R1 1 j 1 2 j 1 1...

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  1 3 3 j j j G HP      1 1 1 2 2 1 1 j j j R R j G BP 1 1 3 1 1 3 3 1 3 3 1 1 1 1 1 1 1 3 1 1 j j C R j j R C j R C j C j R C j R L
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ME365 Spectrum Analysis Slide 1 Spectrum Analysis Signals Periodic Signals Fourier Series Representation of Periodic Signals Frequency Spectra Amplitude and Phase Spectra of Signals Signal Through Systems - a Frequency Spectrum Perspective Non-periodic Signals - Fourier Transform Random Signals - Power Spectral Density sin( ) sin( )cos( ) cos( )sin( ) cos( ) sin( ) sin( ) cos( ) cos( )cos( ) sin( )sin( ) sin( ) ( ) cos( ) A B A B A B A A A A A B A B A B A A A A   ; sin 2 cos 2 ; cos
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ME365 Spectrum Analysis Slide 2 Signals Signals can be categorized as: Periodic Signals Non-Periodic Signals (well defined) Random Signals Would like to characterize signals in the frequency domain ! Linear System G ( j ) Input Signal Output Signal x(t) y(t) Can be characterized by its Frequency Response Function G ( j ) = | G ( j )|e j Arg[ G ( j ) ] x(t )=∑x i (t) y(t)=∑ y i (t)         sin cos 2 1 0 k k k k k t B t A A t x       ) ( sin | ) ( | ) ( cos | ) ( | ) 0 ( 2 1 0 k k k k k k k k k j G t j G B j G t j G A j G A t y
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ME365 Spectrum Analysis Slide 3 Periodic Signals - Fourier Series Any periodic function x ( t ), of period T , can be represented by an infinite series of sine and cosine functions of integer multiples of its fundamental frequency 1 = 2 / T .               T A k k k k k T T t x t x t k B t k A A t x th 2 1 2 1 1 1 1 0 frequency harmonic k the : component DC of Amplitude : and rad/sec 2 and where sin cos 2 0 : ts Coefficien Fourier T T k T T k T T dt t k t x B dt t k t x A dt t x A 0 1 2 0 1 2 0 2 0 ) sin( ) ( ) cos( ) ( ) (
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ME365 Spectrum Analysis Slide 4 Fourier Series Ex: triangle signal with period T sec. Summation of Input Signals -10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.05 0.1 0.15 0.2 Time (sec) Acceleration x t A k A k t A B k k k ( ) ( ) cos( ) 8 2 1 0 0 2 1 1 0   A T 2T Take the first six terms and let A = 10, T = 0.1: x t t t t t t t ( ) .105cos( ) . cos( ) . cos( ) .165cos( ) .1cos( ) . cos( )           8 20 0 901 3 20 0 324 5 20 0 7 20 0 9 20 0 067 11 20 x(t)
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ME365 Spectrum Analysis Slide 5 Pointers for Calculating the Fourier Coefficients: A 0 /2 represents the average of the signal x ( t ). It contains the “DC” (zero frequency) component of the signal.
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lec08_spana - 3 j GHP j 3 j 1 R2 1 j GBP j R1 1 j 1 2 j 1 1...

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