Types of equalizers

Question

I can't get my mind around the different types of equalizers and the difference between some of them. I've been looking at different books unsuccessfully.

So far, I have Wikipedia's list and this post.

Let me know what's wrong with this list:

• Linear (ZF, MMSE)
• Non-linear (MLSE, DFE)
• Blind (could you name a few? I've seen something about Bussgang methods based on LMS but that's it)
• Adaptive (LMS, SG, RLS)

Now, my questions are:

1. The only difference between linear and non-linear is that the latter has some sort of feedback?
2. It's obvious a linear equalizer can't be non-linear and viceversa but, can a blind equalizer be Linear and/or non-linear?
3. Same about adaptive, can it be linear and/or non-linear? If I understand correctly, it could only be non-linear.
4. Do adaptive equalizers always need a training sequence? That means an equalizer can't be adaptive and blind at the same time?

Thanks!

Answer 1

1. Yes. But of course there are other differences when you consider what linear vs nonlinear means. Things like stability, rate of convergence, complexity (implementation and processing), etc.
2. Yes. If you have feedback in your equalizer algorithm then it is nonlinear and you can do this for either decision aided or blind equalization techniques.
3. No. Adaptive implies there is feedback so by definition this is a non-linear operation
4. No. You can you implement adaptive equalizers for decision directed or blind.

Answer 2

The additive white Gaussian noise (AWGN) ISI channel model is given by:

\begin{equation}u_{k}=\sum_{j=0}^{L} f_{j} c_{k-j}+n_{k}\end{equation}

where $n$ is an AWGN, $c$ is the output of the matched filter and $u$ the output of the channel impaired by awgn and ISI. The resulting ISI channel has to be equalized for reliable detection. In real life those are example of the value taken by L:GSM with L = 4, Wifi with L = 16, ADSL with L = 100. we assume linear memoryless modulations such as PAM, PSK, and QAM.

There are many different equalization schemes:

1. Maximum–Likelihood Sequence Estimation (MLSE) like Viterbi Equalizer: The MLSE decision is then the sequence of symbols $\left\{c_{\mathrm{m}}\right\}$ minimizing this distance \begin{equation}\hat{c}_{m}=\arg \min _{c_{m}} \sum_{m}\left| u_{m}-\left.\sum_{j=0}^{L} f_{j} c_{m-j}\right|^{2}\right.\end{equation}

2. Linear Equalization (LE): linear or transverse filter equalizer structures adapt tap weights by using the LMS, RLS, or CMA adaptive algorithm. When using these equalizer structures, the number of samples per symbol determines whether symbols are processed using whole or fractional symbol spacing. you can find some information about linear equalizer here ( non linear equalizer vs linear equalizer )

3. Decision–Feedback Equalization (DFE) A decision feedback equalizer (DFE) is a filter that uses feedback of detected symbols to produce an estimate of the channel output. This add more complexity to the Equalizer structure and allow to better combat ISI. The DFE is fed with detected symbols and produces an output which typically is subtracted from the output of the linear equalizer

4. Turbo-Equalizer: applies turbo decoding while treating the channel as a convolutional code.
5. Wiener Equalization, is based on the WienerHopf equation https://en.m.wikipedia.org/wiki/Wiener_filter .

If we focus on time-domain equalization, you need to remember that there are a lot of types optimization criteria:

-Zero Forcing: The ZF filter tries to force the interference level to zero at all costs. Most often, this causes noise enhancement.

-MMSE equalizer: designs the filter to minimize $E[|e|2]$, where $e$ is the error signal, which is the filter output minus the transmitted signal.

-CMA: Constant Modulus Algorithm