Spatial Multidimensional Cooperative Transmission Theories And Key Technologies. Lin Bai
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In order to overcome the negative impact of fading on the bit error rate, diversity techniques are often employed. The principle of diversity is to provide the receiver with multiple copies of the same transmitted signal, and each replica acts as a diversity branch. If this replication is affected by independent fading conditions, the probability that all branches will be in a fading state at the same time can be greatly reduced. Therefore, diversity stabilizes the link by channel enhancement, which improves the bit error rate performance of the system.
As fading can occur in time, frequency, and space domains, diversity techniques can be used in these domains. For example, time diversity can be obtained by coding and interleaving, and frequency diversity can exploit the time spread of the channel (in the τ domain) by equalization techniques or multi-carrier modulation. Obviously, time and frequency diversity techniques can result in a loss of time or bandwidth due to the introduction of redundancy. Conversely, because multiple antennas are used at one or both ends of the link, space or polarization diversity does not sacrifice time and bandwidth.
2.3.2SIMO system
The SIMO system relies on the number of antennas at the receiving end MR ≥ 2 to realize diversity. If these antennas are sufficiently spaced (such as a wavelength), then when the physical channel exhibits good characteristics, the system will fade independent of the diversity of each branch. Reception diversity can be achieved in two different ways: selective combining and gain combining.
2.3.2.1Reception diversity through selective combining
Among MR received signals, the combiner selects a branch having the largest SNR (or the highest absolute power, bit error rate, etc.) for signal detection. Suppose that MR channels are subject to unit Rayleigh energy independent and identical distribution and the noise level is equal on each antenna. At this time, the selection algorithm compares the instantaneous amplitude of each channel sn(n = 1, . . . , MR) and selects the branch having the largest amplitude smax = max(s1, . . . , sMn}. The probability that smax is below a certain threshold S2 is given by
The distribution corresponding to smax can be obtained by the differentiation of Eq. (2.93).
The average SNR of the combiner output is3
when MR is large, the array gain is approximately
where γ ≈ 0.577 215 66 is the Euler constant.
The diversity obtained by the selective combining can be estimated by calculating the bit error rate using the fading distribution given by Eq. (2.94). For the system using BPSK modulation and owning a two-branch diversity, the bit error rate as a function of the average SNR can be expressed corresponding to each channel ρ4 as follows:
when the SNR is high,
The slope of the bit error rate curve is 2. In general, the diversity gain of the MR-branch selection diversity scheme is equal to MR, which indicates that the selection diversity collects all possible diversity from the channel.
2.3.2.2Reception diversity based on gain combining
In gain combining, the signal z used for detection is a linear combination of all branches.
where Wn denotes the combined weight and W = [W1, . . . , WMR]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 MR antennas. Each antenna is described by the channel hn = |hn|ejϕn (n = 1, . . . , MR). 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 , . . . , 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