rphw4solf - Stochastic Processes nctuee07f Homework 4...

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Stochastic Processes nctuee07f Homework 4 Solutions 1. (a) The covariance matrix K of the interference-plus-noise vector i + z is E [( i + z )( i + z ) T ] = N X i =2 h i h T i + σ 2 I . (b) The optimal detector in terms of minimum error probability for X 1 if there were only 1 user in the system is given by the MAP criterion, and can be obtained as h T 1 y ˆ X 1 =1 ˆ X 1 = - 1 0 . The average probability of error P e for user 1 is P e = P [ h T 1 y < 0 | X 1 = 1] P [ X 1 = 1] + P [ h T 1 y > 0 | X 1 = - 1] P [ X 1 = - 1] = P [ h T 1 y < 0 | X 1 = 1] = P h T 1 z < - || h 1 || 2 + X i 6 =1 h T 1 h i X i · = E P h T 1 z < - || h 1 || 2 + X i 6 =1 h T 1 h i X i · { X i ∈ {- 1 , 1 } ,i 6 = 1 } = X { X i ∈{- 1 , 1 } ,i 6 =1 } Q ˆ || h 1 || 2 + i 6 =1 h T 1 h i X i σ || h 1 || ! 2 1 - N . (c) The LMMSE estimate of X 1 is ˆ X 1 lm ( y ) = E [ X 1 ] + K X 1 y K - 1 y ( y - E [ y ]) , where K X 1 y = E [ X 1 y T ] = E [ X 1 ( X 1 · h 1 + i + z ) T ] = h T 1 K y = E [ yy T ] = N X i =1 h i h T i + σ 2 I , where I is the M × M identity matrix. Note that K y is nonsingular since x T K y x > 0 for all nonzero vector x . Thus, ˆ X 1 lm ( y ) = h T 1 ˆ N X i =1 h i h T i + σ 2 I ! - 1 y . Let u T = h T 1 N i =1 h i h T i + σ 2 I · - 1 . Then the decision rule is u T y ˆ X 1 =1 ˆ X 1 = - 1 0 .

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(d) The optimum combining ﬁlter w op is the vector w that maximizes the signal to interference-plus-noise ratio (SINR) deﬁned by SINR = E [ | w H s | 2 E [ | w H ( i + z ) | 2 = w H h 1 h H 1 w w H Kw . As we did in HW#3, the maximum SINR is achieved when w = K - 1 h 1 = ˆ N X i =2 h i h T i + σ 2 I ! -
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This note was uploaded on 11/28/2010 for the course EE 301 taught by Professor Gfung during the Winter '10 term at National Chiao Tung University.

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rphw4solf - Stochastic Processes nctuee07f Homework 4...

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