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Unformatted text preview: EE 562a Homework Set 4 Due Monday 26 February 2007 1 (The following well-defined problems come from different sources, and the notation used may vary. Don’t let that bother you!) 1. Let x ( u ) and y ( u ) be random vectors that are related to each other by the equation y ( u ) = Gx ( u ) , where G and the mean vector and covariance matrix of x ( u ) are given by G = 6 2 3 0 1 6 2 3 0 1 6 2 0 0 1 6 m x = K x = I . (a) What is the LMMSE estimator of x ( u, 3) x ( u, 4) from the observation x ( u, 1) x ( u, 2) ? (b) What is the LMMSE estimator of y ( u ) based on the observation of x ( u, 1) x ( u, 2) ? (c) Now let’s consider a more general problem. Let ˆ w ( u ) be a LMMSE estimate of w ( u ) that has been constructed from an observation of v ( u ). Suppose that z ( u ) = Hw ( u ). How is the LMMSE estimate ˆ z ( u ) of z ( u ) from an observation of v ( u ), related to ˆ w ( u )? (d) You can use the results of (c) to check your solution to (b). Indicate how v ( u ), w ( u ), and z ( u ) can be equated with the components of the problem in (a) and (b) in order to accomplish this check....
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- Spring '07
- Probability theory, LMMSE, random vectors, LMMSE estimate