Читать книгу Spatial Multidimensional Cooperative Transmission Theories And Key Technologies - Lin Bai - Страница 42

In gain combining, the signal z used for detection is a linear combination of all branches.

where W_n denotes the combined weight and W = [W₁, . . . , W_MR]^T. According to the selection of these weights, there are different gain combining methods. It is assumed that the data symbol c is transmitted by the channel and received by M_R antennas. Each antenna is described by the channel h_n = |h_n|e^jϕn (n = 1, . . . , M_R). Suppose they obey the unit variance Rayleigh distribution and all channels are independent. Combining signals from all antennas, the detected variable can be expressed as

where h = [h₁ , . . . , h_MR]^T.

(1) Equal gain combining

The weight of equal gain combining is W_n = 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 is the Gaussian white noise.

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 M_R, and it is greater than the array gain of selective combining. In addition, the diversity gain of equal gain combining is M_R, which is similar to that of the selective combining.

(2) Maximum ratio combining

The selection weight of maximum ratio combining is , and its post-combiner signal

where n′ = h^Hn. Because it maximizes the output SNR ρ_out, this strategy is called maximum ratio combining. And

In the maximum ratio combining diversity scheme, the array gain g_a is always equal to M_R.

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 2M_R 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 M_R.

For other constellations, using maximum likelihood detection,⁵ the error probability is

where and d_min are the nearest neighbor and the minimum separation distance of the constellation, respectively. The analytical solution of the above expression can be obtained by following Eq. (2.106). The upper bound of the bit error rate is usually obtained using the Chernoff bound. Equation (2.108) can also be written as

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 R_ni is the correlation matrix of noise and interference. When there is no interference, R_ni = E{nn^H}. If the noise across the antenna is white noise, then , and the minimum mean square error combining diversity is simplified to a maximum ratio combining diversity with only one coefficient difference.