# Explain the Adaptive Part of Adaptive Algorithms - Kalman Filter and Least Mean Square / Constant Modulus

General questions:

• Is the Kalman filter (they have used Unscented Kalman Filter) adaptive or not? Is the Unscented Kalman Filter used in the paper an adaptive algorithm?
• Adaptive algorithms such as Constant Modulus and Least Squares are adaptive. Why? What is being adapted ? Based on my understanding, the step size is adapted and the weights as well. But I am not sure for the case of Kalman filters -- is the Kalman Gain getting adapted? Adapted to what?

Adaptive Filters are called "Adaptive" when they can adapt to changes in data.
In the filters you mentioned above, which are part of the Linear Filters family the property means their coefficients are changing over time.

Linear Filters are basically weighing and summing the data.
For instance, given no prior information on data you may want to have exact weight for any data given.

Yet in most Signal Processing use cases we'd like to (Or might want to) give higher weight for each data sample according to its properties such as SNR, How good it fit the model, etc...

In the classic Kalman Filter the coefficients are set according to a model matrix - $P$.
For Kalman filter this matrix stands for the Covariance of the estimation.
It changes according to 2 other matrices which are properties of our model and data - $Q$ - The model confidence level matrix, $R$ the data confidence model (For Gaussian data it is basically the SNR).

Since the algorithm support the cases those are changing over time it is called adaptive.

For instance, in the case of tracking a target using RADAR the algorithm can change the matrix $R$ according to the SNR of the measurement.

The Recursive Least Squares have similar properties (Though it depends only on one factor, something similar to the matrix $R$ of the Kalman Filter).
One could even see the RLS as a private case of the Kalman Filter (Or the Kalman Filter as a generalization).

There are many extension for the Kalman Filter for many cases.
The UKF (Unscented Kalman Filter) you mentioned is built to handle Non Linear cases of the Model.
But adaptive concept stays the same, the weights are changing according to properties of the Data and the Model.

• So why is a classic Kalman Filter adaptive if all the covariances and gain can be computed offline? It doesn't matter what the data is – Stanley Pawlukiewicz Jul 29 '17 at 22:28
• You got it right. If you assume $R$ and $Q$ are constant it means the Kalman filter doesn't change its properties due to data change. It is Pseudo Adaptive in the meaning isn't constant (As the Covariance of the states improves with integration of data). – Royi Jul 30 '17 at 13:18
• @StanleyPawlukiewicz, Pay attention that if the model is correct, namely the process is stationary, being constant while the statistical properties of the signal are constant, is the right and logical thing to do. – Royi Jul 30 '17 at 17:53
• The notion that the measurement noise, process noise , state transition matrix, measurement matrix, and deterministic inputs need to be constant or stationary for offline calculation is erroneous. The terms must be known but they don't have to be static. The Kalman Filter is optimal when these objects are known and the gain can be computed offline, without any interaction with the data. What is an optimal algorithm supposed to adapt to? – Stanley Pawlukiewicz Jul 31 '17 at 16:33
• @RiaGeorge, Kalman Filter is an AR Filter in its form. But if you calculate the equivalent FIR (Or approximate, as the FIR is infinite) you'd see its coefficients are adaptively changing according to the data properties you feed the Kalman Filter. This holds for any type of Kalman Filter (Linear, Extended, Unscented and Cubature for that matter). – Royi Aug 4 '17 at 8:31

An adaptive filter is one that updates its coefficients or parameters as a function of the input signal. Kalman filters are adaptive filters. On each time step, they update the estimate of the states they are tracking as well as the estimate of the covariance of these states.

• The LF is an optimal state estimator. By your definition, any estimator would be adaptive. I disagree – Stanley Pawlukiewicz Jul 27 '17 at 19:43
• The State Covariance, State Prediction Covariance, Innovations Covariance, Filter Gain, and Updated State Covariance can all be calculated offline without any data passing through the filter – Stanley Pawlukiewicz Jul 27 '17 at 20:04
• You can have order adaptive filters too, which isn't quite covered in this answer. – Batman Jul 27 '17 at 20:39
• @StanleyPawlukiewicz the Kalman Filter can be modified (Godard 1974) ) to work in adaptive transversal filter structures [Haykin, Adaptive filter theory, ch.6, 1986] Therefore they can implement adaptive systems. The estimated state becomes the filter coefficients. Yet in its pure classical mode it does not seem like an adaptive system as you have declared by the independence of Covariance and Gain matrices from the input data. – Fat32 Jul 27 '17 at 20:51
• The offline calculation is well documented, in Bar Shaloam, Anderson and Moore, Kailath, .., when a Kalman Filter is optimal, it is not adaptive. – Stanley Pawlukiewicz Jul 27 '17 at 21:41