Dear,

Assume H to be the channel matrix of size NxN where N is the number of samples or time instants. The sampling period is Ts = 1/(Nfft * d_f) where Nfft is the number of used subcarriers (with pilots) and d_f is the subcarrier spacing.

Well, H is a circular matrix due to multipath.  

IT IS WELL KNOWN THAT ANY CIRCULAR MATRIX IS DIAGONALISABLE IN AN FFT BASIS, i.e. let W be the IFFT matrix and let X be the vector of information you got. So basically you recieve:

Y = HWX plus some noise. 

When you apply FFT at the receiver (i.e. multiply Y by W^H which is the FFT matrix and is the hermitian of W) 

so W^H * Y = W^H * H * W * X 

Please note that W^H * H * W is a diagonal matrix thus equalisation is much much easier.

BR,

Ahmad

Sure thank you for your great contribution. I know frequency domain equilization is much much better than Time domain one. But the Way to do the equilization in Time domain step by step is what i would like to understand. Thanks again.

Assuming you're right dear Thomas Ketseoglou, how do you make this estimation of the channel without use it for symbol equilization?

We know that generally, we need to make channel estimation for symbol equilization  at receiver side. And in all my useful implementation, I always used Frequency domain to estimate the channel coefficient and also make the symbols equilization. And I saw in this paper [1], the authors propose Time domain Equilization for better demodulation process of their needs.

So I would like to test this other way to estimate the channel coefficient :-)

Best,

[1] https://www.researchgate.net/publication/269328039_Avoiding_spectral_efficiency_loss_in_unipolar_OFDM_for_optical_wireless_communication

Conference Paper Avoiding spectral efficiency loss in unipolar OFDM for optic...

Dear M. F. Sanya,

in OFDM systems, two solutions are possible: either you make data-aided (pilot-aided) frequency estimation and equalization, either time equalization (deconvolution). I understand that you focus of the latter point. Note that in that case, it is rather blind equalization (in fact, it is useless to multiplex pilots in frequency domain and then equalize in time domain). Some papers deal with this problem. I give you four of them as attachment.

In a general way, blind equalization (blind deconvolution) is performed by using a stochastic gradient algorithm, which uses a cost function. One of the most popular is called constant modulus algorithm (CMA).

Regards,

Vincent

more interesting ref

In order to perform time domain(TD) equalization in OFDM you have to put TD equalizer (digital filter with variable coefficients) before FFT unit in the receiver. Moreover, for TD equalization in OFDM you have still two choices:

a) Frequency domain (FD) channel estimation. In this case filter coefficients are determined from the IFFT from FD channel estimate.

b)TD channel estimation. In this case filter must be adaptive (based on LMS, RLS algorithms) and you have to transmit a separate training sequence somewhere. For example, you can replace cyclic prefix with training sequence.

Best wishes

Big thanks to all of your contributions for help me. I got now the main idea. Thanks Vincent Savaux et Arturs Aboltins for papers and explanations. If I have others points, I will come to you again.

Best, to all !

Hi dear People !

I come back to you after trying to understand more my last point about time domain equilization. In fact, I think also it is easy to make the equilization in frequency domain than Time domain. But as I said previously, the author in the paper [1], said that Time domain equilization is good to reduce ICI in OFDM transmission. Therefore with your help, I deduce that it is possible to estimate the channel response in frequency domain and make the equilization in time domain by returning back to time domain using an IFFT. For this need to make the time domain equilization, ones of you proposed to me some solutions (blind equalization, CMA, LMS, RLS algorithms). Sure ! I'm not familiar with these methods and need to code one of them by matlab for my OFDM link transmission. Is it possible to you to help me without more mathematical formula. Thanks.

Best,

[1] https://www.researchgate.net/publication/269328039_Avoiding_spectral_efficiency_loss_in_unipolar_OFDM_for_optical_wireless_communication

Conference Paper Avoiding spectral efficiency loss in unipolar OFDM for optic...

If you use the pilot-assisted approach i.e. estimate the channel in the frequency domain and then IFFT the channel transfer function into a channel impulse response (CIR) for use in time domain equalisation, you would not then need LMS, RLS or CMA. You only need these stochastic gradient methods if you want to estimate and adapt the CIR in the time-domain i.e. blind equalisation.

Do note however that while blind equalisation may well help with ICI, I do not think that using the basic pilot-aided frequency domain channel estimation in the first method would help with ICI.

Read any book on adaptive filtering theory to gain insight on the stochastic gradient methods.

Dear Sanya,

Hi !!

Actually, definitely ICI and ISI will be reduce using IFFT is used in OFDM transmitter , that means convert frequency domain signal to time domain signal and at the receiver FFT is used in the OFDM receiver, that means again convert the time domain signal to frequency. Therefore, time domain estimation will be done & reduce the ICI, ISI with the help of Cyclic Prefix techniques !!

Ok thanks to all your contributions.

Maybe those two 3GPP documents and the work they reference can help: R1-050718, R1-051352 

Thanks

Dear Sanya,

time domain equalization (TEQ) is a technique that is widely adopted in DSL, on Discrete Multitone Modulation (OFDM with bit loading). Equalization in the time domain is discussed in great detail in the book Fundamentals of DSL Technology, chapter 11, or in the lecture notes provided by Prof. John Cioffi, at the attached link, chapter 3.

Best regards,

Igor F.

http://web.stanford.edu/group/cioffi/ee379a/course_reader.html

Thanks for your help dear Igor Antonio Auad Freire.

Thank you Arsia Khan :-) I will check it

i am trying to use a time domain equalizer in an OFDM system but i am not sure if i can use the constant modulus equalizer because the ofdm signal is not constant. Is it possible to use the constant modulus as a time equalizer?