In this section, we consider various spatial diversity techniques aimed at reducing the error probability.

Receive diversity

Consider the SIMO channel depicted in Figure 2.

Figure 2: SIMO channel
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Let N be the number of receive antennas. The signal received in antenna i is given by

(1.15)

where h_i and n_i are the fading and noise, respectively, as experienced by antenna i. We assume the fading is independent, which is the case, provided the antennas are sufficiently spaced from each other.

Consider the following weighted combination of the antennas' inputs

(1.16)

where the α_i's are some deterministic numbers. The SNR of the above channel is given by

(1.17)

It is straightforward to verify (applying the Cauchy-Schwartz inequality) that by setting

(1.18)

the SNR is maximized. The weighted combination of the antennas' inputs with the above α_i's is referred to as maximal ratio combining. Substituting (1.18) into (1.17), the maximal SNR is given by

(1.19)

Thus, the error probability obtained by the ML receiver when applied to the MRC output satisfies

(1.20)

Let

(1.21)

then

(1.22)

z_i's are statistically independent, Rayleigh distributed random variables. Thus, their joint density is simply given by the product of their individual densities

(1.23)

Averaging (1.22) with respect to (1.23) yields

(1.24)

As we can see, by using N receive antennas we have managed to substantially reduce the error probability. Note that in order to perform MRC, the receiver has to know the fading, or, in other words, the receiver has to have access to the channel state information (CSI). This is usually done by sending some known signal through the channel, called pilot, and measuring the channel’s response. Cleary, such a procedure does not allow for having perfect CSI, but rather approximate CSI. However, empirical results indicate that using MRC with approximate CSI, instead of perfect CSI, slightly deteriorates performance. In general, the performance of spatial diversity techniques is measured using two terms: diversity order

(1.25)

and antenna gain

(1.26)

MRC achieves diversity order of N and antenna gain of N. If we draw a curve of the error probability as a function of the SNR on the logarithmic axis, the diversity order is the slope of the curve, and the antenna gain is the left-hand horizontal shift of the curve with respect to the curve

In some cases, it is not practical to have multiple antennas at the receiver. Consider for example handled devices: their small form factor does not allow for the positioning of multiple antennas that are spaced far enough from each other. Once the antennas are close, the fading seen by them is not independent, and then the error probability can not be made small as indicated by (1.24). Can we achieve the performance of MRC but with multiple antennas at the transmitter? The answer is yes. Essentially, the reason that MRC works is that it increases the SNR. By applying transmit diversity we can also increase the SNR and in turn decrease the error probability.

Transmit diversity

Consider the MISO channel depicted in Figure 3. Let M be the number of transmit antennas. The received signal is given by

(1.27)

where h_j is the fading corresponding to transmit antenna j, and x_j is the symbol sent through antenna j. Again, we assume that the fading is independent. Suppose that we transmit

(1.28)

Figure 3: MISO channel
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where w_j's are some weighting factors satisfying

(1.29)

The above constraint ensures we are not increasing the transmission power. Substituting (1.28) into (1.27) we have

(1.30)

The SNR of the above channel is given by

(1.31)

As before, we would like to maximize the SNR. Setting

(1.32)

the SNR is maximized to the value

(1.33)

Following the exact same steps as in the case of the MRC, we readily obtain

(1.34)

The procedure described in equation (1.28) with the optimal weighting factors of (1.32) is referred to as transmit beamforming. It is so named because the signal x is being formed before being transmitted. Transmit beamforming achieves a diversity order of M and an antenna gain of M, the same as MRC with M receive antennas. However, note that for transmit beamforming, the transmitter must have the CSI. This presents us with a bit of a problem, since in order for the transmitter to have the CSI, the receiver must send it to the transmitter, unavoidably reducing the throughput. Can we achieve transmit diversity without having to provide the transmitter with the CSI? Yes, we can, using Alamouti’s scheme.

Alamouti’s scheme consists of two transmit antennas and one receive antenna. It achieves the error probability

(1.35)

while only requiring CSI to be at the receiver. It does so by employing transmission and reception mechanisms stretched across space and time. Alamouti’s scheme achieves a diversity order of 2 and an antenna gain of 1, as opposed to an antenna gain of 2 for MRC 1x2 and transmit beamforming 2x1. This means that MRC 1x2 and transmit beamforming 2x1 outperform Alamouti’s scheme by 3db (the SNR term in (1.35) is divided by 4 and not 2 as for MRC and transmit beamforming). However, as explained earlier, MRC needs the antennas to be sufficiently spaced, and transmit beamforming needs to know the CSI at the transmitter.

Transmit/receive diversity

Consider the MIMO channel depicted in Figure 4. Let M and N be the number of transmit and receive antennas, respectively. The received signal at antenna i is given by

(1.36)

h_ij is the fading corresponding to the path from transmit antenna j to receive antenna i.

Figure 4: MIMO channel
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As before, we assume the fading is independent. n_i is the noise corresponding to receive antenna i. Let

(1.37)

(1.38)

then

(1.39)

We now describe the procedure of transmit/receive beamforming. The transmitter sends

(1.40)

where v is a vector of size Mx 1 satisfying

(1.41)

For vectors and matrices "*" denotes the Hermitian conjugate, i.e., the vector, or matrix, is first transposed and then complex conjugated, entry by entry. The received signal is then

(1.42)

The receiver multiplies the received signal with a Nx 1sized vector u, creating the channel

(1.43)

The above channel SNR is given by

(1.44)

How should one choose v and u such that the SNR is maximized? The maximizing vectors are derived from the singular value decomposition (SVD) of H [4], and the maximal SNR satisfies

(1.45)

where

(1.46)

The error probability is then bounded by

(1.47)

Averaging the error probability with respect to the fading yields

(1.48)

Thus, for transmit/receive beamforming we have a diversity order of MN, referred to as full diversity. The antenna gain on the other hand satisfies

(1.49)

Transmit/receive beamforming requires CSI at the receiver as well as in the transmitter. For a 2 x 2 setting, transmit/receive diversity can also be achieved by using 2 x 2 Alamouti-based scheme (obtained by an extension of the Alamouti’s scheme) which achieves a diversity order of 4, an antenna gain of 2, and requires CSI only at the receiver. In Table 1, we summarize the antenna gain and diversity order for the different channel configurations.