A stochastic signal S is amplified by an amplifier that has a stochastic, real-valued gain, A, so that the amplifiers output is Y = AS. Assume that the random variables A and S are independent. It is known that the random gain is either 3 or 7, with P{A = 3} = 0.8. a) Characterize the transformation g: IR→IR that can be applied to the output Y in or- der to estimate the unknown signal S from the data Y such that the mean-square error, E[g(Y) S²], is minimized. b) Find an explicit formula for such optimal estimator g that you characterized in Part a. c) Calculate the mean-square error assuming that S is a {-1, 1, }-valued random variable with P{S=1}=0.6. d) Now someone has used intuition to guess an answer for Part a as 92(Y) = 5Y/31. Ex- plain where this guessed answer may have come from and comment on the optimality of the guessed estimator 92. e) Calculate the mean-square error for the estimator 92 in Part d.
A stochastic signal S is amplified by an amplifier that has a stochastic, real-valued gain, A, so that the amplifiers output is Y = AS. Assume that the random variables A and S are independent. It is known that the random gain is either 3 or 7, with P{A = 3} = 0.8. a) Characterize the transformation g: IR→IR that can be applied to the output Y in or- der to estimate the unknown signal S from the data Y such that the mean-square error, E[g(Y) S²], is minimized. b) Find an explicit formula for such optimal estimator g that you characterized in Part a. c) Calculate the mean-square error assuming that S is a {-1, 1, }-valued random variable with P{S=1}=0.6. d) Now someone has used intuition to guess an answer for Part a as 92(Y) = 5Y/31. Ex- plain where this guessed answer may have come from and comment on the optimality of the guessed estimator 92. e) Calculate the mean-square error for the estimator 92 in Part d.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
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Chapter5: Inner Product Spaces
Section5.CR: Review Exercises
Problem 62CR
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![A stochastic signal S is amplified by an amplifier that has a stochastic, real-valued gain, A, so that
the amplifiers output is Y = AS. Assume that the random variables A and S are independent. It is
known that the random gain is either 3 or 7, with P{A = 3} = 0.8.
a) Characterize the transformation g: IR→IR that can be applied to the output Y in or-
der to estimate the unknown signal S from the data Y such that the mean-square error,
E[lg(Y) S2], is minimized.
b) Find an explicit formula for such optimal estimator g that you characterized in Part a.
c) Calculate the mean-square error assuming that S is a {-1, 1, }-valued random variable with
P{S=1} = 0.6.
d) Now someone has used intuition to guess an answer for Part a as 92(Y) = 5Y/31. Ex-
plain where this guessed answer may have come from and comment on the optimality of the
guessed estimator 92.
e) Calculate the mean-square error for the estimator 92 in Part d.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71f6a4b8-527f-4030-b85a-828002701c6f%2Fa7742464-fcc3-467e-a8b7-7d80a0e10e6b%2Fakbbh17_processed.png&w=3840&q=75)
Transcribed Image Text:A stochastic signal S is amplified by an amplifier that has a stochastic, real-valued gain, A, so that
the amplifiers output is Y = AS. Assume that the random variables A and S are independent. It is
known that the random gain is either 3 or 7, with P{A = 3} = 0.8.
a) Characterize the transformation g: IR→IR that can be applied to the output Y in or-
der to estimate the unknown signal S from the data Y such that the mean-square error,
E[lg(Y) S2], is minimized.
b) Find an explicit formula for such optimal estimator g that you characterized in Part a.
c) Calculate the mean-square error assuming that S is a {-1, 1, }-valued random variable with
P{S=1} = 0.6.
d) Now someone has used intuition to guess an answer for Part a as 92(Y) = 5Y/31. Ex-
plain where this guessed answer may have come from and comment on the optimality of the
guessed estimator 92.
e) Calculate the mean-square error for the estimator 92 in Part d.
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VIEWStep 2: Estimate the unknown signal S from the data Y such that the mean-square error is minimum
VIEWStep 3: Determine an explicit formula for such optimal estimator g that you characterized in Part a
VIEWStep 4: Calculate the mean-square error assuming that S is a {-1, 1, }-valued random variable
VIEWStep 5: Explain where the given answer may have come from and comment on the optimality of this answer
VIEWStep 6: Calculate the mean-square error for the estimator g2 in Part d
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