{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HW-12-solution

# HW-12-solution - HW-12 1 In the complex notation there is a...

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

HW-12 1. In the complex notation there is a clever device for finding the time average of a product. Suppose ) cos( ) , ( a t r k A t r f and ) cos( ) , ( b t r k B t r g . Show that *) ~ ~ Re( ) 2 / 1 ( g f fg , where f ~ and g ~ are the complex of f and g and the star denotes complex conjugation. Solution: First, let us perform a direct calculation fgdt T fg T 0 1 where 2 T is the periodicity. ) cos( 2 )] cos( 0 [ 2 )] cos( ) 2 2 [cos( 2 ) cos( ) cos( 1 1 0 0 0 b a b a b a b a T b a T T AB T T AB dt t r k T AB dt t r k B t r k A T fgdt T fg Second, let us calculate *) ~ ~ Re( g f . ) exp( ~ ), exp( ~ b a t r k B g t r k A f . Then ) cos( )] ( exp[ Re )] ( exp[ )] ( exp[ Re *) ~ ~ Re( b b b b b b AB i AB t r k i B t r k i A g f .

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}