hw3 - ECE 562 Fall 2011 September 22, 2011 HOMEWORK...

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Unformatted text preview: ECE 562 Fall 2011 September 22, 2011 HOMEWORK ASSIGNMENT 3 Due Date: October 4, 2011 (in class) 1. For a complex random vector Y , show the following: = ( I + Q ) + j ( QI IQ ) and = ( I Q ) + j ( QI + IQ ) 2. Consider the signal s ( t ) = [sin( t ) + j cos( t )]11 t 1 . Suppose this signal is corrupted by complex WGN w ( t ) with PSD N = 2 to form the received signal r ( t ) = s ( t ) + w ( t ) . Further suppose we form the random variable Z as: Z = integraldisplay 1 r ( t ) sin( t ) dt (a) Find P { Z I > 1 } . (b) Find P { Z I + Z Q > 1 } . (c) Find P ( { Z I > 1 } { Z Q > 1 } ) 3. Unequal Priors. Describe how the optimal decision regions for BPSK signaling get modified when the priors on the messages are not the same, say 1 = 2 / 3 and 2 = 1 / 3. 4. P e for PAM. Compute (the exact) P e as a function of E s for 4-ary PAM. Note that P e ,m is not the same for all m in this case....
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hw3 - ECE 562 Fall 2011 September 22, 2011 HOMEWORK...

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