the transmitter may not have knowledge of the MISO system channel. Since the channel characteristics can be estimated at the receiving end, while the channel information needs to be fed back from the receiver to the transmitter at the transmitting end, there are basically two ways to obtain direct transmit diversity.
•When the transmitter has complete channel knowledge, beamforming can be achieved by various optimization metrics such as SNR and SINR to obtain diversity and array gain.
•When the transmitter has no channel information, the so-called space–time coding pre-processing can be used to obtain the diversity gain, but the array gain cannot be obtained.
In the following, different beamformers will be evaluated and several indirect transmit diversity techniques will be discussed, which can convert space diversity into time or frequency diversity.
2.3.3.1Transmit diversity formed by matched beamforming
This beamforming technique is also known as transmitting maximum ratio combining and it assumes that the transmitter knows all the information about the channel. In order to use diversity, signal c is appropriately weighted before being transmitted to each antenna. At the receiving end, the signal can be expressed as
where h = [h1 , . . . , hMR] denotes the MISO channel vector and W is the weight vector. And the weight vector that maximizes the received SNR is
where the denominator guarantees that the average total transmitting power remains unchanged and is equal to ES. This vector makes the transmission in the direction of the matched channel and is therefore also called the matched beamforming or conventional beamforming. Similarly, for reception maximum ratio combining, the average output SNR is ρout = MTρ, and therefore, the array gain is equal to the number of transmitting antennas MT. If the bit error rate has the following upper bound at a high SNR, the diversity gain is also equal to MR.
Therefore, the matched beamformer exhibits the same performance as the reception maximum ratio combining. It requires knowledge of the complete information of the transmitting channel, which means there is feedback from the receiver in the time duplex system. If frequency duplexing is adopted, the interchangeability of the upper and lower channels is no longer guaranteed, and the understanding of the channel information at the transmitting end is greatly reduced. In addition, the matched beamformer is optimal in the absence of interfering signals but cannot cancel the interference.
Similar to the aforementioned augmented selection algorithm for SIMO systems, the matched beamformer can be combined with the selective combining algorithm. In the beamformer, the transmitter selects
2.3.3.2Space–time coded transmit diversity
The beamforming technique described previously requires channel information for the transmitter to obtain optimal weights. Conversely, Alamouti proposes a particularly simple but original diversity approach for the two transmit antenna systems, called the Alamouti algorithm, which does not require information on the transmit channel. Considering that in the first symbol period, two symbols c1 and c2 are simultaneously transmitted from antenna 1 and antenna 2, and then two symbols –
The symbol received in the second symbol period is
where each symbol is divided by
It can be seen that the two symbols are extended on two antennas over two symbol periods. Therefore, Heff represents a space–time channel. Adding the matched filter
where n′ satisfies
It shows that the Alamouti algorithm cannot provide array gain due to a lack of information about the transmitting channel (note E{||h||2} = MT = 2).
However, for independent and identically distributed Rayleigh channels, the average bit error rate of the above problem has the following upper bound at high SNR.
It means that despite the lack of transmit channel information, the diversity gain is equal to MT = 2, which is the same as the transmit maximum ratio combining. From a global perspective, the Alamouti algorithm has a lower performance than the transmit or receive maximum ratio combining due to its zero array gain.
2.3.3.3Indirect transmit diversity
The technique of obtaining space diversity by combining or space–time coding described above belongs to the direct transmit diversity technique. By using well-known SISO techniques, converting space diversity to time or frequency diversity can also be realized.
Assuming MT = 2, the phase shift is achieved by delaying the signal on the second transmit branch by one symbol period or by selecting the appropriate frequency shift.
