generalizedharmoonicanalysis

# generalizedharmoonicanalysis - Generalized Harmonic...

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Unformatted text preview: Generalized Harmonic Analysis Generalized Harmonic Analysis (see Davenport and Root, Introduction to the Theory of Random Signals and Noise) Parseval's Theorem: 1 2 - f (t ) 2 dt = - F ( ) 2 d (1) For random variable f(t), consider a modification to Parseval's theorem given by averaging both sides to give lim 1 T T T /2 -T / 2 f (t ) 2 dt = f (t ) 2 = lim 2 2 F ( ) d T - (2) Recall the Fourier Transform is defined (here) as: 1 -it F ( ) = f (t )e dt 2 - giving 2 1 2 0 F ( ) = 2 T 2 or 1 2 2 2 2 - it (3) - f (t )e dt (4) - f (t ) e -it dt = lim 1 T 2 T /2 T /2 -T / 2 -T / 2 f (t ) f (t )e - it e -it dt (5) = lim 1 T 2 T /2 T /2 -T / 2 -T / 2 f (t ) f (t ) e -it e -it dt (6) Recall the autocorrelation R(t-t'), or for a stationary random variable, R( ) f (t ) f (t ) = R (t - t ) = R ( ) Focus on the integral in eqn (6) for a few minutes: T /2 T /2 I= -T / 2 -T / 2 R(t - t )e - i ( t - t ) dtdt Breaking this into regions with a change of variable = t - t T /2 T /2 I= -T / 2 -T / 2 - i ( t - t ) dtdt = R ( )e -i R(t - t )e 0 T T / 2 - -T / 2 - i dt d + R( )e -T 0 T /2 - -T / 2 dt d t' T/2 -T/2 T/2 t -T/2 which simplifies to I = R( )e -i (T - )d + R ( )e -i (T + ) d 0 T 0 -T -T t' T Using the fact that R ( ) is symmetric in , T I= resulting in -T R( )e 2 - i (T - )d T 2 F ( ) = lim 1 T 2 -T - i R( )e (1 - 1 )d = T 2 - R( )e - i d ...
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## This note was uploaded on 09/16/2011 for the course ME 563 taught by Professor Staff during the Spring '11 term at Auburn University.

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generalizedharmoonicanalysis - Generalized Harmonic...

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