Filter (estimation)
High school
Consider a quantity that is measured over time, with noisy measurements. In estimation, a filter estimates the current value of the quantity from the measurements made up to now.
Undergraduate
Consider a system with state $x(t)$ and noisy measurements of it. In estimation, a filter computes an estimate of the current state $x(t)$ from the measurements up to time $t$.
Examples
Examples
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A Kalman filter
(definition)
Undergraduate
A Kalman filter is a filter, because it estimates the current state from the measurements up to the current time.
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A particle filter
(definition)
Undergraduate
A particle filter is a filter, because it estimates the current state from noisy measurements up to now, by approximating the state’s distribution with weighted samples.
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An extended Kalman filter
(definition)
Undergraduate
An extended Kalman filter is a filter, because it estimates the current state of a nonlinear system from noisy measurements up to now.
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An unscented Kalman filter
(definition)
Graduate
An unscented Kalman filter is a filter, because it estimates the current state of a nonlinear system from noisy measurements up to now.
Non-examples
Things that may be mistaken for examples.
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A least-squares estimate of a constant parameter from data
High school
A least-squares estimate of a constant parameter from data is not filter, because it estimates a constant parameter, not a changing state.
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A low-pass filter on a sensor signal
High school
A low-pass filter is a filter in the signal-processing sense: it smooths a signal. It is not a filter estimator because it does not estimate an unknown value.
Properties
- The Kalman filter is the optimal (minimum mean-square error) filter for linear systems with Gaussian noise.
Notes
In signal processing, a filter is a system that shapes the frequency content of a signal, such as a low-pass filter. Such a filter is not a state estimator.
Distinctions involving this term
- Types of estimators: When the state of a system is not measured directly, it can be estimated in several ways. An estimator is any rule for estimating an unknown quantity. An observer is a dynamical system whose state estimate converges to the true state. A filter estimates the current state from noisy measurements, and a predictor estimates a future state.