ME365chap8_spana

ME365chap8_spana - Spectrum Analysis Spectrum Analysis...

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Unformatted text preview: Spectrum Analysis Spectrum Analysis Spectrum Analysis Spectrum Analysis Signals Periodic Signals Fourier Series Representation of Periodic Signals Frequency Spectra Amplitude and Phase Spectra of Signal KARTIK B. ARIYUR PURDUE UNIVERSITY 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 m m m 1 Spectrum Analysis Signals Signals Signals Signals Linear System G ( j ) Input Signal Output Signal x(t) y(t) Can be characterized by its Frequency Response KARTIK B. ARIYUR PURDUE UNIVERSITY Signals can be categorized as: Periodic Signals Non-Periodic Signals (well defined) Random Signals Would like to characterize signals in the frequency domain ! Frequency Response Function G ( j ) = | G ( j )|e j Arg[ G ( j ) ] 2 Spectrum Analysis Periodic Signals Periodic Signals Periodic Signals Periodic Signals - Fourier Series Fourier Series Fourier Series 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 . ( ) ( ) ( ) [ ] x t A A k t B k t k k k = + + = 1 1 1 2 cos sin KARTIK B. ARIYUR PURDUE UNIVERSITY ( ) ( ) [ ] x t x t T T k k A T = + = = 1 2 1 2 2 : : where and rad /sec and Amplitude of the DC component the k harmonic frequency th A A B k k = = = Fourier Coefficients: 3 Spectrum Analysis Fourier Series Fourier Series Fourier Series Fourier Series Ex: triangle signal with period T sec. ( ) = & = = 2 ) 1 2 cos( ) 1 2 ( 8 ) ( 1 1 2 k k A t k A k A t x 4 4 3 4 4 2 1 Take the first six terms and let A = 10, T = 0.1: x t t t t t ( ) .105 cos( ) . cos( ) . cos( ) .165 cos( ) = + + + 8 20 0 901 3 20 0 324 5 20 7 20 KARTIK B. ARIYUR PURDUE UNIVERSITY-15-10-5 5 10 15 0.05 0.1 0.15 0.2 0.25 Acceleration Time (sec) Summation of Input Signals = = k B A A T 2T t t .1 cos( ) . cos( ) + + 9 20 0 067 11 20 4 Spectrum Analysis 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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ME365chap8_spana - Spectrum Analysis Spectrum Analysis...

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