Should we use Maximal ratio combining and an equalizer together?

Question

I have confusion on MRC(Maximal Ratio Combining) method which I would like to clarify. An example is where I have 1 transmit antenna and 4 receive antennas. The concept of MRC is to combine all the signal from 4 receive antennas such that the combined SNR is increased (given the phase is corrected).

Once the MRC process is completed, are we required to equalize the channel effect with some equalizer (for e.g. ZF or MMSE). Most of the online materials refers MRC as a equalizer, which made me confuse.

If we require to use an equalizer after combining, which channel coefficient should we use ? Because once the received symbols are combines, we just have a vector (with increased SNR). But the channel coefficients are different between 1 tx antenna and each 4 rx antenna.

Answer 1

The MRC technique is applied in the case of flat fading channels. In mathematical terms, this means that the channel coefficient between the Tx antenna and each Rx antenna is a single complex number $h_i$. Rx MRC is implemented by finding the magnitude and phase of this complex number and 'equalizing' it at the Rx. This is spatial equalization which means that the input at each Rx antenna is combined after conjugate weighting by the same channel coefficient $h_i^*$.

Why conjugate? Because the phase cancels out and each antenna signal adds in-phase with each other. For a 2-antenna example and a Tx signal $s$, we have $$ r = \left\{ h^*_1\cdot h_1 + h^*_2 \cdot h_2\right\}\cdot s = \Big\{|h_1|^2+|h_2|^2 \Big\} \cdot s $$

No time domain equalization is needed further. MRC is a special case of beamforming and for the case you have multiple channel taps for each Rx antenna, broadband beamforming is required, e.g., something similar to FIR filters in spatial domain.

Answer 2

Once you receive the MRC output, now you have to go ahead and detect the symbols. Basically, we need to map it back to a constellation. Note that MRC is trying to improve the SNR of the received signal component.

For example assuming $\vec{h} =[1, 2j, -j, 3]^T$, $\vec{n} =[n_1, n_2, n_3, n_4]^T$ is the $ 4\times 1$ noise vector, if we send $x$, and do MRC, we receive $(1+4+1+9)x +(n_1-2n_2j+jn_3+3n_4)$. Normalizing by norm of $\vec{h}$, we get $y =x +(n_1-2n_2j+jn_3+3n_4)/15$. Now, you can find the nearest constellation point to this received point.