# What Is the Difference between RLS, LMS and Wiener Filter? When Is One Preferred Over Another?

I'm dealing with a channel equalization problem where the channel is modeled as a WSS process.
I understand LMS utilities a Wiener-like approach, ie it converges to the optimal (wiener) solution.

I understand RLS converges to the normal solution, assuming the statistics of the WSS process are unknown.

I would like to know:

• What class of predictors can Wiener, LMS, and RLS be classified within? Is it correct to define Wiener as 'Optimal (in MSE sense)', LMS as 'Stochastic gradient predictors', and RLS as a 'Linear predictor'? And what class of filters?
• When is one solution preferred to the other and why?
• In particular, when defining the cost function with the MSE, what can we achieve? And when we define it simply with the squared error?
• I prefer LMS to RLS when processing signals in real time, as LMS has lower computational complexity than RLS. RLS converges much faster and has lower MSE. Refer to sections 14.6 and 14.6.1 of the book: Moon, Todd K.; Stirling, Wynn C.; Mathematical_Methods_and_Algorithms_for_Signal_Processing, 2000, Prentice Hall, pp 643-648. Apr 27 '17 at 12:53
• Could you please review my answer? If something missing let me know. Else, could you please mark it?
– Royi
Nov 4 at 7:48

All three are Estimators / Predictors.
All of them try to estimate the coefficients of Linear Filter which minimizes an MMSE Cost Function.

The Wiener filter assumes all data is given and sets the way to calculate the optimal solution.

The LMS and RLS are sequential / on line methods to solve the same problem and given the data is stationary they all will converge to the same solution.

The LMS / RLS can adjust their solution according to data and hence are suitable in cases the model changes in time.

The easiest to implement with the least computing resources needed is the LMS and it is the one you should start with. The LMS works on the current state and the data which comes in.
The RLS, which is more computational intensive, works on all data gathered till now (Weighs it optimally) and basically a sequential way to solve the Wiener Filter.

• There are conditions for the step size in LMS in order to guarantee convergence. RLS solves the matrix equation $$\min ||Ax-b||_2^2$$ in a single step. Each iteration of LMS takes a gradient descent step towards the solution. Jul 26 '19 at 17:29
• I think both methods solve this form of a problem. LMS and RLS deal the case more measurements are added (Added rows to $A$ and elements in $y$) and you want to update the solution easily. Jul 27 '19 at 14:22