Consider a real random variable X with zero mean and variance σ2X . Suppose that wecannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and{Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe information given in the question.)c. Suppose that {Wt : t ∈ R} is white noise with power spectral density value σ2W . Find theoptimal h in this case, and write down an expression for the optimal estimate ˆXT in terms ofT , {Yt : t ∈ [0, T ]}, and SNR := σ2X /σ2W . Comment on how the optimal estimator behavesas T → ∞ or SNR → ∞.
Consider a real random variable X with zero mean and variance σ2X . Suppose that wecannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and{Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe information given in the question.)c. Suppose that {Wt : t ∈ R} is white noise with power spectral density value σ2W . Find theoptimal h in this case, and write down an expression for the optimal estimate ˆXT in terms ofT , {Yt : t ∈ [0, T ]}, and SNR := σ2X /σ2W . Comment on how the optimal estimator behavesas T → ∞ or SNR → ∞.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
Related questions
Question
Consider a real random variable X with zero mean and variance σ2
X . Suppose that we
cannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and
{Wt : t ∈ R} is a WSS process with zero mean andcorrelation function RW , uncorrelated with X.
Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:
ˆXT =
Z T
0
h(T − θ)Yθ dθ,
i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample the
output at time T . We wish to design h to minimize the mean-squared error of the estimate.
a. Use the orthogonality principle to write down a necessary and sufficient condition for the
optimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)
b. Use part a to derive a condition involving the optimal h that has the following form: for all
τ ∈ [0, T ],
a =
Z T
0
h(θ)(b + c(τ − θ)) dθ,
where a and b are constants and c is some function. (You must find a, b, and c in terms of
the information given in the question.)
c. Suppose that {Wt : t ∈ R} is white noise with power spectral density value σ2
W . Find the
optimal h in this case, and write down an expression for the optimal estimate ˆXT in terms of
T , {Yt : t ∈ [0, T ]}, and SNR := σ2
X /σ2
W . Comment on how the optimal estimator behaves
as T → ∞ or SNR → ∞.
X . Suppose that we
cannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and
{Wt : t ∈ R} is a WSS process with zero mean and
Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:
ˆXT =
Z T
0
h(T − θ)Yθ dθ,
i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample the
output at time T . We wish to design h to minimize the mean-squared error of the estimate.
a. Use the orthogonality principle to write down a necessary and sufficient condition for the
optimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)
b. Use part a to derive a condition involving the optimal h that has the following form: for all
τ ∈ [0, T ],
a =
Z T
0
h(θ)(b + c(τ − θ)) dθ,
where a and b are constants and c is some function. (You must find a, b, and c in terms of
the information given in the question.)
c. Suppose that {Wt : t ∈ R} is white noise with power spectral density value σ2
W . Find the
optimal h in this case, and write down an expression for the optimal estimate ˆXT in terms of
T , {Yt : t ∈ [0, T ]}, and SNR := σ2
X /σ2
W . Comment on how the optimal estimator behaves
as T → ∞ or SNR → ∞.
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