# Recursive Least Square Adaptive Linear Equalizer

For the adaptive filter to work properly, a desired signal d(n) needs to be provided. The output from the equalizer y(n) is subtracted from d(n) to produce an error signal, which is used to adjust the filter weights.

1. The adaptive filter is located on the receiver side, so how to obtain the desired signal, and use it at the receiver? When using a training sequence, the desired sequence can easily be located at the receiver. What will be the situation with other unknown sequences?

2. What is the difference between least mean square and recursive least squares adaptive linear filters?

To answer (1) the adaptive equalizer without a training sequence (blind equalization) can be used based on the decisions of the received sequence. This specifically is called a "decision directed equalizer". Of course it can not work in very low SNR conditions, where a training sequence would be required. A typical approach is to have the training sequence first to remove channel ISI and establish a higher SNR signal. From this point a decision directed approach can continue to track and maintain the equalized signal as the channel varies with time. (If I recall correctly, a decision directed approach can typically converge if the bit error rate is better than 1E-3 -- but this exact figure I am not confident in).

I have heard (but have no experience with) that the most commonly used adaptive algorithm for blind channel equalization is the Constant Modulus Algorithm. Perhaps someone more knowledgeable about this approach can add an additional response with more comments and details about CMA (or better if I cannot Google a satisfactory explanation I can post it as a question myself).

To answer (2), the LMS (Least Mean Square, also referred to as the Gradient Algorithm) and the RLS (Recursive Least Squares algorithm) are the two most common algorithms for recursively minimizing mean square error in adaptive linear equalizers (in contrast to non-linear Decision Feedback Equalizers, which are not to be confused with "Decision-Directed" mentioned above, would be preferred in the case of frequency selective fading).

The LMS algorithm is well known, easy to implement and computationally cheap (requires $2M+1$ multiply operations, where M is the number of equalizer coefficients). It uses a Stochastic Gradient Descent Rule, so may converge slowly.

The RLS algorithm in comparison is computationally more expensive (requires $2.5M^2+4.5M$ multiply operations), but has faster convergence and good tracking.

For details on the LMS and RLS algorithms (and likely where I got the metrics quoted above), see Wireless Communications by Rappaport which I have notated in the plots below. Notice the computational difference between the two algorithms:

Also see this post here where I give the code and detailed intuitive explanation for the determination of the LMS equalizer coefficients using the Wiener-Hopf equation: Compensating Loudspeaker frequency response in an audio signal. In this post I used a known copy of the signal to determine the channel impulse response in the static case (not a recursive solution). However as noted above this could also be done blind with the decisions of what we think was transmitted if the SNR was high enough to make a sufficient number of correct decisions.

you can find additional detailed explanations on Wikipedia: https://en.wikipedia.org/wiki/Recursive_least_squares_filter

• 1. I am not asking about blind equalization. I am asking about how the “decision directed mode” works after the training mode. In the training mode, I understand that the desired signal is the actual input signal, and the error is calculated as the difference between the desired signal and the estimated signal from the equalizer. In the “decision directed mode” how the desired signal is obtained?? It is not of course the actual input signal. @Dan Boschen – user24907 Aug 14 '18 at 13:44
• 2. I read in the reference that the rapid convergence of the Recursive Least Square (RLS) equalizer relies on error measures expressed in terms of a time average of the actual received signal instead of a statistical average. What does this mean? @Dan Boschen – user24907 Aug 14 '18 at 13:44
• The decision directed mode is indeed the input signal. A simple example is equiprobable BPSK, where you “decide” 1 or 0 based on the hard limit of the input signal. I am referring to blind equalization as equalization without a training sequence such as this case, where instead it is “decision directed”. Does that clear it up for you? – Dan Boschen Aug 14 '18 at 13:58
• As far as your second question if you google “Ensemble vs Time Average” I think you will find very good explanations (easier and more detailed than what I can write in a couple sentences here) – Dan Boschen Aug 14 '18 at 14:04