Exam1Solutions

# Exam1Solutions - ECEN 661 Exam 1 Solutions Spring 2008...

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ECEN 661 Exam 1 Solutions Spring 2008 Problem 1. (25 points) Zero-mean, white Gaussian noise with a PSD of is passed through a BPF whose transfer function is as described in the figure. (a) Find the variance of the I and Q components of the output bandpass noise. (b) Are the I and Q components of the noise independent? Carefully justify your answer. Solution: Let be the output noise process and its low-pass equivalent. Then . , . . (a) The variance of the I and Q parts is . (b) Since the cross PSD of the I and Q part is zero, its cross-correlation is also zero. Both and are zero-mean Gaussian random process and hence they are uncorrelated and indepen- dent. Half cycle of a sine wave f f c B H f ( ) A N 0 2 n o t ( ) z t ( ) x t ( ) jy t ( ) + = f B 2 --- A 2 N o Φ xx f ( ) Φ n o n o f ( ) N o 2 ----- H f ( ) 2 = Φ xx f ( ) Φ yy f ( ) L.P. Φ n o n o f f c ( ) Φ n o n o f f c + ( ) + { } = = A 2 N o cos 2 π f B ---- = f B 2 < Φ yx f ( ) j L.P. Φ n o n o f f c ( ) Φ n o n o f f c + ( ) { } 0 = = σ 2 Φ xx f ( ) f d 2 A 2 N o cos 2 π f B ---- 0 B 2 df A 2 N o 1 2 π f B -------- cos + f d 0 B 2 A 2 N o B 2 ---------------- = = = = x t ( ) y t ( )

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Problem 2. (25 points) The four waveforms listed below are used to form a 4-ary signalling format.
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