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I recently came across an article on the inventor of adaptative equalizer Robert Lucky, In this article : An Oral History by Robert Lucky (http://www.ieeeghn.org/wiki/index.php/Oral-History:Robert_Lucky).

And I have a couple of questions :

  1. Quoting Lucky speaking about DFE :"It was the idea that you could use your own decisions to improve correction ability. When you start transmitting data and the channel is very corrupted the bits you’re getting have a lot of mistakes in them. Nonetheless, you say, “Let’s assume these bits are correct and reconstruct what the perfect signal would have looked like with these bits. Compared to what the signal actually was, we’ll just try to make them look alike with our adaptive filter. ” You're basing this model on corrupted bits on the outside, so that the question is, “Is this ever going to converge ?” " ?

Can some one elaborate on this and expain how the DFE equalizer is assure to converge and provide better equalization than the simple forward filter system ?

  1. My second question is about the design of the filter, I choose to use an MMSE criterion, is that the best alternative ?

  2. My DFE equalizer is for HF ionopheric channels, how can I update the filters coefficients, i.e., based on the channel estimate, what is the exact algorithm ?

Any Hints, thoughts, or ideas will be very appreciated

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I can't provide a lot of theory but I can provide this qualitative description of the problem:

Normally the output of your digital filter sampled every symbol period gives a point on your I/Q plane. Decision feedback usually selects the constellation point closest to this point and modifies your filter to make the output(given the same input) closer to the given constellation point.

If you select the correct point each time then your filter will hopefully converge(approximately) to the set of coefficients which minimize the above distance.

There are some obvious issues with this..

  1. if you aren't selecting the correct constellation point then your filter is converging to the wrong thing. More likely(lets say in a hypothetical AWGN) you will select a point sometimes right, sometimes wrong - hopefully if you select it right more than wrong, the rights will dominate the filter updates.

  2. Your transmitted pulses need to be mostly evenly distributed among the constellation, otherwise I think you could have a DFE converge to something perfect for one pulse train but maybe terrible for the others.

Intuitively you're attempting to invert the frequency response of the channel or reverse the impulse response but this is impossible for finite length equalizers.

I don't know what is currently used but I'm not sure if the standards which use OFDM have DFE or just do channel estimation and lock it down until the next pulse.

Your question about convergence: I don't know, I'm not sure how to analyze DFE, the adaptive filter course I took didn't address it - I have my doubts on how easy a proof is showing it will converge because even proving relatively simple things in adaptive filter theory is really hard(like the convergence pro0f of LMS algorithm was nontrivial even though it's a super simple filter).

I also have seen multiple proofs of algorithms like CMA which either proves it does converge or doesn't converge, or converges for a while and then diverges with a set of assumptions I'm not sure make the proofs relevant to real world applications.

Here's a short MS thesis which talks about some of this:

https://pdfs.semanticscholar.org/106d/7871c03d837a57be6cdf755d4efd44f5e96c.pdf

Basically, bad things can happen.

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