# ass1 - Carleton University Department of Systems and...

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SYSC 5504 Department of Systems and Computer Engineering ELG 6154 Principles of Digital Communications Fall 2010/11 Assignment #1 Due on Thursday, October 7, 2010 1. Consider a band-limited zero-mean stationary random process, x ( t ), with power spectral density Φ x ( f )= ± 1i f | f |≤ W 0i f | f | >W . Suppose x ( t ) is sampled at a rate of f s =1 /T to yield a discrete-time process ˜ x n = x ( nT ). (a) Determine the expression for the autocorrelation sequence φ ˜ x ( m )= E x n ˜ x n + m ]. (b) Determine the minimum value of T that results in a white (spectrally ﬂat) sequence. (c) Repeat (b) if the power spectral density of x ( t )is Φ x ( f )= ± 1 −| f | /W if | f |≤ W 0i f | f | >W . 2. For the signals s 0 ( t ), s 1 ( t ), and s 2 ( t ) shown below: (a) Use the Gram-Schmidt orthogonalization procedure to ±nd a set of basis signals. (b) Express s 0 ( t ), s 1 ( t ), and s 2 ( t ) as linear combinations of the basis signals. (c) Show s 0 ( t ), s 1 ( t ), and s 2 ( t ) on a signal space diagram. - 6 t T 3 2 T 3 T A 2 A s 0 ( t ) - 6 t T 3 2 T 3 T A 2 A s 1 ( t ) - 6 t s 2 ( t ) T 3 2 T 3 T A 2 A 2 3 A 2 3. Consider a binary communication system in which 0’s and 1’s are equally likely to be transmitted. Suppose the input to the decision device at the receiver is modelled as R = ± + E b + W, if a 0 was transmitted E b + W, if a 1 was transmitted , where the noise variable, W , is characterized by the Laplacian pdf shown below. If the decision rule is Choose “0” as the estimate of transmitted bit if R> E b

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## This note was uploaded on 01/25/2011 for the course SCE 5201 taught by Professor Huang during the Spring '10 term at Carleton CA.

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ass1 - Carleton University Department of Systems and...

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