Solution to HW 13 - Solution to EE351k fall 2007 hw 13 1....

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Unformatted text preview: Solution to EE351k fall 2007 hw 13 1. RW (t , ) = E[W (t )W (t + )] = E{[ X cos( 2ft ) + Y sin( 2ft )][ X cos( 2f (t + ))) + Y sin( 2f (t + ))]} = E{ X 2 cos( 2ft ) cos( 2f (t + )) + XY [cos( 2ft ) sin( 2f (t + )) + sin( 2ft ) cos( 2f (t + ))] + Y 2 sin( 2ft ) sin( 2f (t + ))} = 2 cos( 2ft ) cos( 2f (t + )) + 0 + 2 sin( 2ft ) sin( 2f (t + )) = 2 cos( 2ft - 2f (t + )) = 2 cos( 2f ) E (W (t )) = E[ X cos(2ft ) + Y sin(2ft )] = E ( X ) cos(2ft ) + E (Y ) sin(2ft ) = 0 A stochastic process is wide-sense stationary if its mean is constant and its autocorrelation depends only on . So W(t) is wss. 2. a. There are at least three approaches we can follow to compute the mean. We can use the definition of the expectation of a random variable X whose PDF is computed in last homework, or use the definition of a function of a random variable W whose PDF is given here, or use the linear property of a expectation operator. We just go the simplest way here: b. E ( X ) = E (t - W ) = t - E (W ) = t - 1 C X (t , ) = cov[ X (t ), X (t + )] = E{[ X (t ) - E ( X (t ))][ X (t + ) - E ( X (t + ))]} = = = = E{ X (t ) X (t + )} - E[ X (t )]E[ X (t + )] - E[ X (t )]E[ X (t + )] + E[ X (t )]E[ X (t + )] E{ X (t ) X (t + )} - E[ X (t )]E[ X (t + )] E{ X (t ) X (t + )} - (t - 1)(t + - 1) E[(t - W )(t + - W )] - (t - 1)(t + - 1) = E[t 2 + t - 2tW - W + W 2 ] - (t - 1)(t + - 1) = t 2 + t - 2t - + 2 - (t - 1)(t + - 1) =1 ...
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This note was uploaded on 04/21/2008 for the course EE 351k taught by Professor Bard during the Fall '07 term at University of Texas at Austin.

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