are independent. Combining signals from all antennas, the detected variable can be expressed as
where h = [h1 , . . . , hMR]T.
(1) Equal gain combining
The weight of equal gain combining is Wn = e−jϕn, indicating that the signals from different antennas are in phase and can be added together. This approach requires the combiner to have a complete knowledge of the known signal phase. And the post-combiner signal of Eq. (2.100) becomes
where
When the channel is a Rayleigh distribution, the mean value of the output SNR can be obtained.
It can be seen that the array gain increases linearly with MR, and it is greater than the array gain of selective combining. In addition, the diversity gain of equal gain combining is MR, which is similar to that of the selective combining.
(2) Maximum ratio combining
The selection weight of maximum ratio combining is
where n′ = hHn. Because it maximizes the output SNR ρout, this strategy is called maximum ratio combining. And
In the maximum ratio combining diversity scheme, the array gain ga is always equal to MR.
Consider the case of transmitting with BPSK modulation. It is well known that when u = ||h||2 and different channels are independently distributed Rayleigh channels, u obeys 2MR degrees of freedom χ2 distribution.
The bit error rate can be given by
when the SNR is large, the above equation becomes
It can be seen that the diversity gain is still MR.
For other constellations, using maximum likelihood detection,5 the error probability is
where
Since u is a χ2 variable, the average upper bound above is
when the SNR is large, Eq. (2.110) is simplified to
Similar to the case of BPSK, the diversity gain is equal to the number of receiving branches in an independent and identically distributed Rayleigh channel.
(3) Minimum mean square error combining
When the noise is spatially correlated or non-Gaussian interference occurs, the maximum ratio combining is no longer optimal. In this case, the minimum mean-squared error combining is an optimal gain combining, from which the weight is obtained by minimizing the mean square error between the transmitted symbol c and the combiner output z, namely
And it is easy to get the optimal weight vector
where Rni is the correlation matrix of noise and interference. When there is no interference, Rni = E{nnH}. If the noise across the antenna is white noise, then
2.3.2.3Reception diversity through hybrid selection combining or gain combining
A hybrid approach combines the selection algorithm with the maximum ratio combining. At each moment, the receiver first selects
Obviously, it can be concluded that the average SNR of the combiner output is the sum of the two items. The first item corresponds to the maximum ratio combining
Similarly, for selective combining (
2.3.3MISO system
The MISO system utilizes MT transmitting antennas with pre-processing or precoding to perform diversity at the transmitting end. And the obvious difference from reception
