generalizedharmoonicanalysis

generalizedharmoonicanalysis - Generalized Harmonic...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Page1 / 2

generalizedharmoonicanalysis - Generalized Harmonic...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online