# Maximal-ratio combining

In telecommunications, maximal-ratio combining (MRC) is a method of diversity combining in which:

1. the signals from each channel are added together,
2. the gain of each channel is made proportional to the rms signal level and inversely proportional to the mean square noise level in that channel.
3. different proportionality constants are used for each channel.

It is also known as ratio-squared combining and predetection combining. Maximal-ratio combining is the optimum combiner for independent AWGN channels.

MRC can restore a signal to its original shape.

## Example: Least Squares estimate in the case of Rx diversity

We consider an example of which the receiver is endowed with N antennas. In this case, the received vector $y$ is

$y=hs+\rho n \,$

(1)

where $n$ is noise vector $n-CN(0,I_{N \times N})$. Following the ML detection criterion the detection procedure may be written as

$\tilde{s}=argmin_{s \in QPSK}|\hat{s}-s|^2,$

(2)

where $\hat{s}$ is the least square solution to the above model.

$\hat{s}=(h^*h)^{-1}h^*y.$

(3)

The least square solution in this case is also known as maximal-ratio-combining (MRC). In the case of N antennas the LS can be written as

$\hat{s}=\frac{h_0^*y_0+h_1^*y_1+...+h_{N-1}^*y_{N-1}}{|h_0|^2+|h_1|^2+...+|h_{N-1}|^2},$

(4)

which means that the signal from each antenna is rotated and weighted according to the phase and strength of the channel, such that the signals from all antennas are combined to yield the maximal ratio between signal and noise terms.