In a receiver with OFDM demodulator using post FFT combining with one transmitter antenna and two receiving antennas, I see that MRC coefficients usually are defined as
C_1(K) = H*_1[k] / ( |H_1[k]|^2 + |H_2[k]|^2)
C_2(K) = H*_2[k] / ( |H_1[k]|^2 + |H_2[k]|^2),
where k shows the subcarrier number. The above coefficents basically create an "equalized" combined signal as H[K] = C_1(k) H_1[k] + C_2(k) H_2[k].
But what I would like to do is a sequential implementation of a diversity combiner and MMSE equalizer, where the coefficients of the equalizer are defined as
W[k] = H*[k] / (|H[k]|^2 + noise variance), where H[k] is the combined channel defined as
H[K] = C_1(k) H_1[k] + C_2(k) H_2[k]
Does anyone know what the coefficients of C_1(K) and C_2(K) should be in this case?
It is hard to answer precisely since you haven't stated how your received signals are modeled. But if there is only a signal plus thermal noise, then the MRC and MMSE combining schemes are equal up to a scaling factor.
If you want to sequential implementation, then you can first apply MRC and then apply what you call the MMSE equalizer as a second step.
So to answer you question: C_1(K) and C_2(K) should be the same as you have written for MRC.
Thanks, Professor Björnson,
I'm modeling the received signal as:
R[k] = I_k H[k] + Z[k],
where H[k] is the combined channel (combining of H_1[k] and H_2[k]), and Z[k] is the noise corresponding to the combined channel.
The C_1(K) and C_2(K) as I have written for MRC will provide an equalized combined channel, so I think it is not reasonable to have an MMSE equalizer after an equalized signal, and this is why I want to change the MRC coefficients for the sequential implementation of maximum ratio combining and MMSE equalizer.
I do not understand the reason why do you want to apply :
1) firstly, an MRC combiner with coefficients C1 and C2 that provides a single output per subcarrier number ;
2) secondly a single-channel equalizer that is applied on each subcarrier.
This does not sound to be an optimal approach !!
In my opinion, an MMSE equalizer W[k] applied to the 2 inputs R1[k] and R2[k] and providing a single output Z[k] should be sufficient to follow the optimal approach !
I do not see the advantage of deriving such a sequential approach MRC + equalizer !
Pascal Scalart There are many resources that can help you to understand in an OFDM scenario, the combining need to be implemented in the subcarrier level, the following is an example
Article Implementation of joint Pre-FFT adaptive array antenna and P...
When you are saying "optimal" you need to clarify optimal in which sense if you mean in terms of complexity, there are some solutions for that, again referring to the above article, however, firstly my question is not about optimal implementation, and secondly, depending on the application you might want to have a complex implementation for better performance in terms of bit-error-rate, that is a trade-off and another story.
Please note that the coefficients that I mentioned are C_1(K) and C_2(K), not "C1 and C2" that you have mentioned, there is a huge difference between these two sets.
If you know about any reference that implemented/simulated post FFT combining in OFDM scenario with non-subcarrier level coefficients for MRC, sharing that can be helpful.
Farah Arabian I understand that the MRC and equalizer is applied on each subcarrier (and therefore the coefficients C1 and C2 are in fact C1[k] and C2[k]) but what I meant was that I don't see the interest of cascading a MRC combiner with a multi-channel equalizer.
Instead a single multi-channel equalizer operating on the subcarriers should be sufficient to obtain the optimal solution.
For example, if we consider that the optimization criterion is MMSE, then the optimal MIMO (thus multi-channel) equalizer is given by :
Wmmse = inv(Ryy)*P*H
where
- Ryy represents the cross-correlation matrix of the multi-channel vector received after FFT, i.e. y=[y1 y2] with yi = [yi(1) yi(2) ... yi(nfft)] the vector obtained after FFT on the ith channel (here in your case we have i=1,2)
- P is the diagonal matrix of power loading at transmitter
- H is the MIMO channel matrix expressed between the transmitted and received subcarriers.
In the previous mathematical relation, there is no need of a MRC when minimizing the MMSE criterion because the MRC is automatically included into the Wmmse equalizer !
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