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 π π π π m m m 1 Spectrum Analysis Signals Signals Signals Signals Linear System G ( j ω ) 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 ω ) = | G ( j ω )|e j Arg[ G ( j ω ) ] 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 ω 1 = 2 π / T . ( ) ( ) ( ) [ ] x t A A k t B k t k k k = + + = ∞ ∑ 1 1 1 2 cos sin ω ω KARTIK B. ARIYUR PURDUE UNIVERSITY ( ) ( ) [ ] x t x t T T k k A T = + = = 1 2 1 2 2 : : ω ω where and rad /sec and Amplitude of the DC component the k harmonic frequency th π π 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 ω π 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 π π π π 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 π π 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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