Please solve all part in details Briefly answer the following questions. Describe the difference between Kalman filter, Wiener filter, and sequential LMMSE estimator. When does Kalman filter converge to Wiener filter? Sequential LMMSE estimator? Consider the vector LMMSE estimator for the Bayesian linear model, where the model for observations is given by x = H theta + w. When does the LMMSE estimator for this model become BLUE? Show and justify using equations. What is the advantage of method of moments estimator over other classical estimators? Comment on its optimality properties. Consider again the linear model x = H theta + w. For this linear model, the LSE, BLUE, MLE, and MYUE can all be described using the following estimator for uncorrelated observations: Describe the probabilistic assumption for the noise vector w for all the four estimators (LSE. BLUE, MLE, MVUE), and comment on differences. For the linear model x = H theta + w with correlated observations, use the whitening transformation technique to show that the MVUE estimator is given by theta = (H^TC^-1H)^- 1H^TC^-1x. Solution Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone. The filter is named after Rudolf E. Kálmán, one of the primary developers of its theory. The Kalman filter has numerous applications in technology. A common application is for guidance, navigation and control of vehicles, particularly aircraft and spacecraft. Furthermore, the Kalman filter is a widely applied concept in time series analysis used in fields such as signal processing and econometrics. Kalman filters also are one of the main topics in the field of robotic motion planning and control, and they are sometimes included in trajectory optimization. In signal processing, the Wiener filter is a filter used to produce an estimate of a desired or target random process by linear time-invariant (LTI) filtering of an observed noisy process, assuming known stationary signal and noise spectra, and additive noise..

1. Please solve all part in details Briefly answer the following questions. Describe the difference
between Kalman filter, Wiener filter, and sequential LMMSE estimator. When does Kalman
filter converge to Wiener filter? Sequential LMMSE estimator? Consider the vector LMMSE
estimator for the Bayesian linear model, where the model for observations is given by x = H
theta + w. When does the LMMSE estimator for this model become BLUE? Show and justify
using equations. What is the advantage of method of moments estimator over other classical
estimators? Comment on its optimality properties. Consider again the linear model x = H theta +
w. For this linear model, the LSE, BLUE, MLE, and MYUE can all be described using the
following estimator for uncorrelated observations: Describe the probabilistic assumption for the
noise vector w for all the four estimators (LSE. BLUE, MLE, MVUE), and comment on
differences. For the linear model x = H theta + w with correlated observations, use the whitening
transformation technique to show that the MVUE estimator is given by theta = (H^TC^-1H)^-
1H^TC^-1x.
Solution
Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a
series of measurements observed over time, containing statistical noise and other inaccuracies,
and produces estimates of unknown variables that tend to be more precise than those based on a
single measurement alone. The filter is named after Rudolf E. Kálmán, one of the primary
developers of its theory.
The Kalman filter has numerous applications in technology. A common application is for
guidance, navigation and control of vehicles, particularly aircraft and spacecraft. Furthermore,
the Kalman filter is a widely applied concept in time series analysis used in fields such as signal
processing and econometrics. Kalman filters also are one of the main topics in the field of
robotic motion planning and control, and they are sometimes included in trajectory optimization.
In signal processing, the Wiener filter is a filter used to produce an estimate of a desired or target
random process by linear time-invariant (LTI) filtering of an observed noisy process, assuming
known stationary signal and noise spectra, and additive noise.