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# hw3 - ECE 562 Fall 2011 HOMEWORK ASSIGNMENT 3 Due Date...

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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: Σ = (Σ I + Σ Q ) + j QI Σ IQ ) and ˇ Σ = (Σ I Σ Q ) + j QI + Σ IQ ) 2. Consider the signal s ( t ) = [sin( πt ) + j cos( πt )]11 0 t 1 . Suppose this signal is corrupted by complex WGN w ( t ) with PSD N 0 = 2 to form the received signal r ( t ) = s ( t ) + w ( t ) . Further suppose we form the random variable Z as: Z = integraldisplay 1 0 r ( t ) sin( πt ) 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 π 1 = 2 / 3 and π 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. 5. Performance of MPSK. (a) Using the Intelligent Union Bound, show that the symbol error probability for MPSK signaling in AWGN is bounded by P e 2 Q parenleftBigg radicalbigg 2 E s N 0 sin π M

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hw3 - ECE 562 Fall 2011 HOMEWORK ASSIGNMENT 3 Due Date...

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