{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

EE351kHW_15SolnFall2009

# EE351kHW_15SolnFall2009 - Solution to HW 15 Due 30 November...

This preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: Solution to HW # 15, Due 30 November 2009 Problem 1 (a) E[ X (t )] = âˆ« A cos(Ï‰0t + Î¸ ) âˆ’Ï€ Ï€ 1 A Ï€ [sin(Ï‰0t + Î¸ )]Î¸ = âˆ’Ï€ = 0 dÎ¸ = 2Ï€ 2Ï€ A2 1 dÎ¸ = 2Ï€ 4Ï€ (b) E[ X 2 (t )] = âˆ« A2 cos 2 (Ï‰0t + Î¸ ) âˆ’Ï€ Ï€ âˆ«âˆ’Ï€ (1 + cos(2Ï‰0t + 2Î¸ ))dÎ¸ = 1 dÎ¸ 2Ï€ Ï€ A2 A2 2Ï€ = 4Ï€ 2 (c) RX (Ï„ ) = E[ X (t ) X (t + Ï„ )] = âˆ« A2 cos(Ï‰0t + Î¸ ) cos(Ï‰0 (t + Ï„ ) + Î¸ ) âˆ’Ï€ Ï€ = A2 4Ï€ âˆ«âˆ’Ï€ (cos(âˆ’Ï‰0Ï„ ) + cos(2Ï‰0t + Ï‰0Ï„ + 2Î¸ ))dÎ¸ = Ï€ A2 cos(Ï‰0Ï„ ) 2 Problem 2 (a) < X (t ) >= 1 T0 âˆ« T0 0 A cos(Ï‰0t + Î¸ )dt = A [sin(Ï‰0t + Î¸ )]T=0 0 = 0 t T0 A2 2T0 A âˆ«0 (1 + cos(2Ï‰0t + 2Î¸ ))dt = 2T0 T0 = T0 (b) < X 2 (t ) >= (c) 1 T0 âˆ« T0 0 A2 cos 2 (Ï‰0t + Î¸ )dt = 2 A2 2 RX (Ï„ ) =< X (t ) X (t + Ï„ ) >= A2 = 2T0 1 T0 âˆ« T0 0 A2 cos(Ï‰0t + Î¸ ) cos(Ï‰0 (t + Ï„ ) + Î¸ )dt âˆ« T0 0 A2 A2 (cos(âˆ’Ï‰0Ï„ ) + cos(2Ï‰0t + Ï‰0Ï„ + 2Î¸ ) )dt = T0 cos(Ï‰0Ï„ ) = cos(Ï‰0Ï„ ) 2T0 2 (d) From the results above E[ X (t )] =< X (t ) > E[ X 2 (t )] =< X 2 (t ) > E[ X (t ) X (t + Ï„ )] =< X (t ) X (t + Ï„ ) > ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online