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Unformatted text preview: ECE 562 Fall 2011 November17, 2011 SOLUTIONS TO HOMEWORK ASSIGNMENT 6 1. MF Maximizes SNR. We showed in the class that the SNR for the kth symbol (ignoring the interference from other symbols) when a linear equalizer c k is used is given by: SNR k = E s  c † k h k  2 N k c k k 2 Using the CauchyShwarz inequality, show that SNR k is maximized by the matched filter equalizer c k, MF = h k . Ans: By the CauchyShwarz inequality SNR k = E s  c † k h k  2 N k c k k 2 ≤ E s N k c k k 2 k h k k 2 k c k k 2 = E s N k h k k 2 This maximum value is obtained when c k is a scaled version of h k . In particular, we can choose c k = h k . 2. MSE of linear equalizers. The MSE for the kth symbol when a linear equalizer c k is used was defined in class to be: MSE k = E h  c † k Z s m k  2 i Using the fact that Z = h k s m k + X j 6 = k h j s m j + W show that MSE k = E s  c † k h k 1  2 + E s X j 6 = k  c † k h j  2 + N k c k k 2 Ans: Expanding the definition of MSE gives MSE k = E s m k c † k Z 2 = E s m k N X j =1 c † k h j s m j c † k w 2 = E s m k c † k h k 1 X j 6 = k c † k h j s m j c † k w 2 = E " c † k h k 1 2  s m k  2 + X j 6 = k c † k h j 2  s m j  2 +  c † k w  2 # = E s c † k h k 1 2 + E s X j 6 = k c † k h j 2 + N k c k k 2 where the second to last line follows from the fact that the symbols are independent and zero mean, and independent from the noise....
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 Fall '09

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