(a) An estimate of ryz [m] is obtained as 1 fyz[m] = √ (y[m] * x*[−m]) · N What is the mean of îyx [m]? Is it unbiased? (b) The cross power spectral density Âyz (e¹w) is estimated as the Fourier transform of the cross-correlation sequence fyz[m]. Provide an expression for µr(e³“) in terms of the Fourier Transform of the observed sequences. (c) What is the asymptotic mean and variance of µx(ejw) ? No need for a proof. Use your experience with the Periodogram, (d) Provide a Bartlett type estimate for the cross power spectral density Ryz(e³w), and comment on its variance
(a) An estimate of ryz [m] is obtained as 1 fyz[m] = √ (y[m] * x*[−m]) · N What is the mean of îyx [m]? Is it unbiased? (b) The cross power spectral density Âyz (e¹w) is estimated as the Fourier transform of the cross-correlation sequence fyz[m]. Provide an expression for µr(e³“) in terms of the Fourier Transform of the observed sequences. (c) What is the asymptotic mean and variance of µx(ejw) ? No need for a proof. Use your experience with the Periodogram, (d) Provide a Bartlett type estimate for the cross power spectral density Ryz(e³w), and comment on its variance
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Let x[n] and y[n] be zero mean, jointly circular Gaussian wide sense stationary pro-
cesses with cross power spectral density Rya (eju) = E ryæ[m]e-jum, where ryæ[m] =
E[y[n + m]x*[n]. One realization of a[n],0 < n < N – 1, and y[n], 0 <n< N – 1, is
observed.
(a) An estimate of rya m] is obtained as
1
fya [m] = (y[m] * x*[-m])
N
What is the mean of îyæ [m]? Is it unbiased?
(b) The cross power spectral density Rya (eJw) is estimated as the Fourier transform of
the cross-correlation sequence îyæ [m]. Provide an expression for Ryæ(ejw) in terms
of the Fourier Transform of the observed sequences.
(c) What is the asymptotic mean and variance of Rya (ejw) ? No need for a proof.
Use your experience with the Periodogram,
(d) Provide a Bartlett type estimate for the cross power spectral density Rya (eJw),
and comment on its variance](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe68f0066-db31-42e2-87ae-f5acbabd5711%2Fc1e2b66d-8c01-4132-9242-47be28b929f0%2Fjk94enc_processed.png&w=3840&q=75)
Transcribed Image Text:Let x[n] and y[n] be zero mean, jointly circular Gaussian wide sense stationary pro-
cesses with cross power spectral density Rya (eju) = E ryæ[m]e-jum, where ryæ[m] =
E[y[n + m]x*[n]. One realization of a[n],0 < n < N – 1, and y[n], 0 <n< N – 1, is
observed.
(a) An estimate of rya m] is obtained as
1
fya [m] = (y[m] * x*[-m])
N
What is the mean of îyæ [m]? Is it unbiased?
(b) The cross power spectral density Rya (eJw) is estimated as the Fourier transform of
the cross-correlation sequence îyæ [m]. Provide an expression for Ryæ(ejw) in terms
of the Fourier Transform of the observed sequences.
(c) What is the asymptotic mean and variance of Rya (ejw) ? No need for a proof.
Use your experience with the Periodogram,
(d) Provide a Bartlett type estimate for the cross power spectral density Rya (eJw),
and comment on its variance
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