The Kalman filter is just one of many
adaptive filtering (or estimation)
algorithms. Despite its elegant derivation and often excellentperformance, the Kalman filter has two drawbacks:
The derivation and hence performance of the Kalman
filter depends on the accuracy of the
a
priori assumptions. The performance can be less than
impressive if the assumptions are erroneous.
The Kalman filter is fairly computationally demanding,
requiring
operations per sample. This can limit the utility
of Kalman filters in high rate real timeapplications.
As a popular alternative to the Kalman filter, we willinvestigate the so-called
least-mean-square (LMS)
adaptive filtering algorithm.
The principle advantages of LMS are
No prior assumptions are made regarding the signal to be
estimated.
Computationally, LMS is very efficient, requiring
per sample.
The price we pay with LMS instead of a Kalman filter is that the
rate of convergence and adaptation to sudden changes is slower forLMS than for the Kalman filter (with correct prior assumptions).
Adaptive filtering applications
Channel/system identification
Channel/system identification
Noise cancellation
Suppression of maternal ECG component in
fetal ECG (
).
Cancelling maternal heartbeat in fetal
electrocardiography (ECG): position of leads.
is an estimate of the maternal ECG signal present
in the abdominal signal (
).
Results of fetal ECG experiment (bandwidth,
3-35Hz; sampling rate, 256Hz): (a)reference input (chestlead); (b)primary input (abdominal lead);
(c)noise-canceller output.
Channel equalization
Channel equalization
Adaptive controller
Adaptive controller
Here, the reference signal is the desired output.
The adaptive controller adjusts the controllergains (filter weights) to keep them appropriate to the
system as it changes over time.
Iterative minimization
Most adaptive filtering alogrithms (LMS
included) are modifications of standard iterative proceduresfor solving minimization problems in a
real-time or
on-line fashion. Therefore, before deriving
the LMS algorithm we will look at iterative methods ofminimizing error criteria such as MSE.
Conider the following set-up:
Linear estimator
Impulse response of the filter:
Vector notation
Where
and
Error signal
Assumptions
are jointly stationary with zero-mean.
Mse
Where
is the variance of
,
is the covariance matrix of
, and
is the cross-covariance between
and
The MSE is quadratic in
which implies the MSE
surface is "bowl" shaped with a unique minimum point (
).
Optimum filter
Minimize MSE:
Notice that we can re-write
as
or
Which shows that the error signal is orthogonal to the input
(by the
orthogonality principle of
minimum MSE estimator).
Steepest descent
Although we can easily determine
by solving the system of equations
Let's look at an iterative procedure for solving this
problem. This will set the stage for our adaptive filteringalgorithm.
We want to minimize the MSE. The idea is
simple. Starting at some initial weight vector
, iteratively adjust the values to decrease the
MSE (
).
In one-dimension
We want to
move
towards the optimal vector
. In order to move in the correct direction, we
must move
downhill or in the direction opposite
to the gradient of the MSE surface at the point
. Thus, a natural and simple adjustment takes the form
Where
is the step size and
tells us how far to move in the negative gradient direction(
).
Generalizing this idea to an iterative strategy, we get
and we can repeatedly update
:
. Hopefully each subsequent
is closer to
. Does the procedure converge? Can we adapt it to
an on-line, real-time, dynamic situation in which thesignals may not be stationary?
Questions & Answers
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
