Maximal-ratio combining

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

${\displaystyle y=hs+\rho n\,}$

(1)

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

${\displaystyle {\tilde {s}}=argmin_{s\in {\mathcal {M}}}|{\hat {s}}-s|^{2},}$

(2)

where ${\displaystyle {\mathcal {M}}}$ is the considered constellation of ${\displaystyle s}$ and ${\displaystyle {\hat {s}}}$ is the least square solution to the above model.

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

(3)

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

${\displaystyle {\hat {s}}={\frac {h_{0}^{*}y_{0}+h_{1}^{*}y_{1}+...+h_{N-1}^{*}y_{N-1}}{|h_{0}|^{2}+|h_{1}|^{2}+...+|h_{N-1}|^{2}}},}$

(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 maximum ratio between signal and noise terms.

• Siae Microelettronica patent