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20092ee132A_1_hwk5_sol

20092ee132A_1_hwk5_sol - EE132A Spring 2009 Prof John...

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EE132A, Spring 2009 Communication Systems Prof. John Villasenor Handout# 16 TAs: Pooya Monajemi and Erica Han Homework 5 Solutions 1. Signal Space (a) The space spanned by this set of signal has a dimensionality of 2 because the basis functions consist of two orthonormal signals. (See part (b)). (b) Using the trigonometric identity cos( ) cos cos sin sin A B A B A B + = , the four signals can be written as: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 2 cos 2 cos 2 sin 2 4 2 2 cos 2 sin 2 2 3 2 cos 2 cos 2 sin 2 4 2 2 cos 2 cos 2 c c c c c c c c c c E E E s t f t f t f t T T T E E s t f t f t T T E E E s t f t f t f t T T T E E s t f t f t T T π π π π π π π π π π π π π π = + = = + = − = + = − = + = − From above, the signals can be represented as a linear combination of the following two orthonormal signals. 1 2 2 cos(2 ), 0 ( ) 0, elsewhere 2 sin(2 ), 0 ( ) 0, elsewhere c c f t t T t T f t t T t T π ϕ π ϕ = = Note: The basis function has unit energy.
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