# ps2 - Massachusetts Institute of Technology Department of...

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Massachusetts Institute of Technology Department of Mechanical Engineering 2.160 Identification, Estimation, and Learning Spring 2006 Problem Set No. 2 Out: February 22, 2006 Due: March 1, 2006 Problem 1 A stationary random process X(t) has a mean value of m and an autocorrelation function of the form: R X ( τ ) = σ 2 e τβ Another random process Y(t) is related to X(t) by the deterministic equation: t Y ) = t X a ) + b ( ( where a and b are known constants. (a). What is the autocorrelation function for Y(t) ? (b). What is the cross-correlation function R XY ( )? Problem 2 Two random processes are defined by t X ) = A sin( ω t + θ ) ( t Y ) = B sin( t + ) ( where is a random variable with uniform distribution between 0 and 2 π , and is a 2 ), 0( N and are correlated to each other with a correlation coefficient ρ . Show that the cross- known constant. The A and B coefficients are both normal random variables , )( R XY ( ) = 1 ρσ 2 cos( t ) . 2 Assume A and B are independent of . Problem 3 Wearable medical sensors monitor a patient’s health conditions anytime, anywhere, and continuously. These sensors are expected to revolutionize health care services, including diagnosis and treatment in the home for diabetes, hypertension, and pulmonary disorders. However, the wearable sensors must be robust against disturbances, in particular, the motion of the wearer, since the patients move around for daily activities correlation function R is given by: XY 1

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rather than staying in a hospital. Wearable sensors often cause false signals due to motion
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ps2 - Massachusetts Institute of Technology Department of...

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