I am having difficulty in trying to understand the working mechanism of an equalizer. Please correct me where wrong.
Mechanism of an Equalizer - The purpose of an equalizer is to reduce intersymbol interference due to noise and fading effect. A linear equalizer is a filter that can undo these channel effects. When the channel coefficients $w$ are unknown, we perform blind equalization. In this scenario, we first estimate $w$ and then apply an equalizer.
The output of an equalizer is a delayed version of the transmitted signal. This is what I do not understand. We need to find what is the input to the system, but the transmitted signal is not the input. What I mean to say, let the channel model be a single input, single output moving average of order 2, MA(2): $y(n) = a_1*x(n-1) + a_2*x(n-2) + x(n) + \eta(n)$ where $x$ is the input data (unknown) and $\eta$ is an additive zero mean white Gaussian measurement noise; $n$ is the sample, $*$ is multiplication of the coefficient with the data. If I do, out_eq = conv(w,y) (In Matlab) where $w$ are the estimates and $y$ is the received signal that is transmitted, then how come the input $x$ is obtained through this operation? Another way is not to use conv() and use an FIR filter because I read that linear equalizers are FIR filter. I can understand the part where we do $out_eq = y./w' = x$ but how is this similar to the convolution or using FIR filter?
Why Equalizers are linear filter? - posted as a new Question
Suggestion on what equalizer I should use for my case- I have a Rayleigh fading channel that is modeled as FIR filter. Let the impulse response / channel be $h$. Then the transmitted signal $tx(n) = h.*x(n) + AWGN$. I wish to apply Expectation Maximization and Kalman filter to estimate the $h$. Then, how do I equalize and which equalizer to apply and why?
UPDATE :
UPDATE -
Part (1) - Mechanism of equalizer
Reference - Book Blind Equalization & System Identification, batch processing Algorithm, performance & applications, by Chong-Yung Chi, Springer.
Chapter 4 defines blind linear equalization for Single input single output (SISO). Suppose that $u[n]$ is the source signal of interest and is distroted by SIS LTI system $h[n]$ via the data model:
$y[n] = x[n] + w[n]$ where $x[n] = h[n]*u[n]$ is the noise free signal and $w[n]$ is the additive measurement noise. The goal of designing an equalizer $v[n]$ is such that the output of the equalizer given by $e[n] = v[n]*y[n]$ approximates $u[n]$ as well as possible. $e[n]$ is called the equalized signal. The book says that we can do blind equalization through the following steps (a) estimation of $h[n]$ by means of blind system identification algorithm (b) design of a non-blind equalizer with these estimated parameters for the retreival of $u[n]$.
Confusion on Terminology problem and mechanism : I was under the impression that the output of the equalizer is the source input to the system, as in case of an ideal equalizer we perform the operation out_equalizer = received_signal./channel_coefficients = h[n]*u[n] = y./hand looking at this equation, out_equalizer = u.
In other cases when we Constant Modulus ALgorithm, we use convolution or the FIR filter as
filter(estimates_channel_coefficients,1,received_signal)
But, according to the book, the output of the equalizer is not = received_signal./channel_coefficients. How will the operation $e[n]$ give me the input $u[n]$? Can you please explain how the equation for $e[n]$ is the output of the equalizer that gives the input $u[n]$.
Confusion - What is the output of the Equalizer- is it the transmitted signal (noise-free) or the input to the system?Your answer says that the output of the equalizer is the transmitted signal but the book says it should be the input to the system. But the transmitted signal is a noise free signal which is the convolution of the channel coefficients and the input source. The received signal is the input of your equalizer - Understood
you need to find the transmitted signal. It could be considered the input of your channel (the channel can be modeled as a FIR filter). - Can you elaborate on this a bit more? My understanding is say if the impulse response of the channel is modeled as a FIR filter, then how is the transmitted signal = input of this channel? Considering the MA(2) that I have put in the question, the transmitted signal is $\mathbf{y}$ but it is not the input to the channel. The input to the channel is $x[n]$.
PART (2) - Choosing an equalizer for my case
If the Rayleigh fading channel is a one tap FIR filter, then the ideal equalizer (dividing the recieved signal by estimated $h$) would work fine, based on your answer.
But , if it is an $m$ tap filter, then I should resort to some other techniques as you have said. Would another FIR filter as an equalizer be applicable? I know of Zero-forcing, MMSE techniques but was thinking of using the linear filter as most books say that equalizers are mostly FIR filters.
Thank you.
Answer
Ok, there is some misconceptions in your question. I strongly recommend you to read a little more about the topics, but I will try to help you a little. My answers and some comments:
...linear equalizer is a filter that can undo these channel effects. When the channel coefficients w are unknown, we perform blind equalization. In this scenario, we first estimate w and then apply an equalizer.
Yes, the linear equalizer is a filter that can undo the effects caused by the channel. When you don't know the channel coefficients, you can perform blind equalization, but it is not estimate the channel coefficients and them apply equalization. Those are things completely different.
When you perform blind equalization, you don't necessary need to know the channel coefficients. There are equalizers that need the channel information and equalizer that don't. and this has nothing to do with blind equalization.
The act of estimate the channel coefficients is called Channel Estimation, and there are a lot of techniques on how to do that. It is used when you have an equalizer that need the information of the channel coefficients, but it is unknown.
To perform an equalization without knowing the channel coefficients, or even to perform channel estimation, you need something called training sequence. It is a sequence previous defined, that both transmitter and receiver know, and the receiver use this information to estimate the channel or to perform equalization. If you have an equalizer that doesn't need a training sequence to perform the equalization, you have a blind equalizer. The linear equalizer is not a blind equalizer because you need the channel estimation, and the channel estimator uses training sequence to estimate the channel.
The output of an equalizer is a delayed version of the transmitted signal. > This is what I do not understand. We need to find what is the input to the system, but the transmitted signal is not the input.
You are right when you say that the output equalizer is a delayed version of the transmitted signal. What you didn'1t understand yet is that the transmitted signal is different from the received signal. The transmitted signal doesn't have the channel effects yet, and the received signal do. The received signal is the input of your equalizer. and no, you don't need to find the input of your equalizer, you need to find the transmitted signal. It could be considered the input of your channel (the channel can be modeled as a FIR filter).
You can only use the equation outeq = y./w′ if you have a flat channel. If your channel is not flat, this equation will not work. A flat channel is a channel that only has a multiplicative effect in the transmitted signal. Imagine that you will perform a convolution, but the filter that represents your channel has only one coefficient. That's why you can just divide things in the receiver to perform equalization. But be careful, if you are not using OFDM (a robust technique against frequency selective channels) or if your channel has more than one coefficient, your channel will not be flat (it is more complex than this, but it is the main idea).
Then, how do I equalize and which equalizer to apply and why?
There are a lot of ways of doing that. I suggest you to start from the basic. Read a little about Zero Forcing equalizers, MSE equalizers and Maximum Likelihood Sequence Estimation. without more details, it is difficult to tell you which one to use. What you want to achieve? What is your goal? Each one has its advantages and disadvantages. Read a little bit about them and you will be able to determine which one you need.
UPDATE:
Look at the figure to understand better:
In this figure, you can see that the transmitted signal is the input of the channel filter (a FIR filter). The output of this filter is the received signal, which is the input of the equalizer. I have just drawn the main blocks to show the concept of equalization. Of course you can add the channel estimation block and other things inside the receiver block.
The coefficients of your equalizer are not the same coefficients of your channel filter. The impulse response of your equalizer must be in such a way that it corrects the effects imposed by the channel.
When you project an equalizer, you want the output of you equalizer to be an estimation of the input of your channel, that in the figure I called the transmitted signal.
But , if it is an m tap filter, then I should resort to some other techniques as you have said. Would another FIR filter as an equalizer be applicable? I know of Zero-forcing, MMSE techniques but was thinking of using the linear filter as most books say that equalizers are mostly FIR filters.
But ZF and MMSE equalization are techniques to find the taps of a linear FIR filter with as many taps as you want. Once you have found the taps through one of this techniques, you can just use
filter(equalizer_coefficients, 1, received_signal)
You use the ZF and MMSE to find the equalizer_coefficients. Once you have it, you just need to perform the filtering. The only problem is that these techniques are not blind techniques because they use training sequences to find the taps. If you want a blind technique, you can search for CMA equalization, for example, but the technique that you should use will depend directly of the nature of your signal. Is a modulated signal? What kind of modulation you have?
You said that you are using Kalman Filter and expectation to estimate your channel impulse response. Once you have it, you can easily use the ZF technique to find the equalizer coefficients, for example. And your equalizer surely will be a linear FIR filter.
Let me know if I could make it more clear.
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