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problem_set6 - EE 511 Problem Set 6 Due on 5 November 2007...

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EE 511 Problem Set 6 Due on 5 November 2007 1. White noise of power spectral density N 0 / 2 is filtered using an ideal low pass filter of bandwidth B . What is the variance of the output noise process? 2. Suppose X t is a Wiener process defined for t 0, i.e., a Gaussian process with m X ( t ) = mt for t 0, where m is a constant, and R X ( t, s ) = σ 2 min ( t, s ) + m 2 ts for t, s 0. Define a process Y t = X t + D - X t for t 0, where D is a fixed positive number. a) Find m Y ( t ) and R Y ( t, s ). b) Show that Y t is stationary and find S Y ( f ). 3. Let X t be a zero-mean stationary Gaussian process with auto-correlation function R X ( τ ). This process is applied to a square-law device defined by Y t = X 2 t . a) Show that E [ Y t ] = R X (0). b) Show that the auto-covariance function of Y t , C Y ( τ ) = 2 R 2 X ( τ ). 4. A stationary Gaussian process X t with zero-mean and power spectral density S X ( f ) is applied to a linear filter with impulse response as shown in Figure 1. A sample Y is taken of the random process at the filter output at time T . a) Determine the mean
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