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 Hf () A N 0 2 n o t zt xt jy t + = f B 2 --- A 2 N o Φ xx f Φ n o n o f N o 2 ----- 2 = Φ f Φ yy f L.P. Φ n o n o ff c Φ n o n o c + + {} == A 2 N o cos 2 π f B ----   = fB 2 < Φ yx f j L.P. Φ n o n o c Φ n o n o c + 0 σ 2 Φ 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 ---------------- = = yt

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Problem 2. (25 points) The four waveforms listed below are used to form a 4-ary signalling format. , , , , , . (a) Find a set of orthonormal basis functions such that each waveform can be written as a linear combination of the set of basis functions.
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## This note was uploaded on 02/21/2012 for the course ECEN 611 taught by Professor Miller during the Spring '08 term at Texas A&M.

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

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