There are several antenna grouping techniques, which were introduced in [10].

### Sum capacity of sub-channels (Algorithm A1)

In this algorithm, the grouping criterion is the sum capacity of sub-channels [8]. The sum capacity of sub-channels is

{C}_{{S}_{1},{S}_{2},\dots ,{S}_{{N}_{r}}}^{{w}_{1},{w}_{2},\dots ,{w}_{{w}_{{N}_{r}}}}\cong log\left(1+\frac{\mathsf{\text{SNR}}}{{N}_{t}}{\sum}_{i=1}^{{N}_{r}}{w}_{i}^{H}{H}_{i}^{H}{H}_{i}{w}_{i}\right)

(8)

Note that (8) is an approximation, and this algorithm is not optimal even in terms of capacity. To maximize (8), we need to search the sub-channel group that maximizes

{\sum}_{i=1}^{{N}_{r}}{w}_{i}^{H}{H}_{i}^{H}{H}_{i}{w}_{i}

(9)

### Minimum euclidean distance of received constellations (Algorithm A2)

The minimum Euclidean distance of receive constellation is shown [10] as

{d}_{min}^{2}:=\underset{{x}_{i},{x}_{j}\in X,{x}_{i}\ne {x}_{j}}{min}\frac{{\u2225{H}_{AG}\left({x}_{i}-{x}_{j}\right)\u2225}^{2}}{{N}_{r}}

(10)

where *X* is the set of all possible transmitted vector **x**_{
AG
}. We consider all possible effective channel **H**_{
AG
}'s in (5). We calculate the minimum Euclidean distance of receive constellation for every possible **H**_{
AG
}, and find the optimal sub-channel **H**_{
i
}'s and the optimal **W**_{
AG
}that maximize (10).

### Minimum singular value of effective channel (Algorithm A3)

A MIMO channel can be decomposed into multiple SISO channels by SVD, and the received SNR is proportional to the squared singular value of a channel. The BER performance is thus dominated by the minimum singular value. We find the minimum singular value of each **H**_{
AG
}, and pick the best **H**_{
i
}'s and **W**_{
AG
}which maximize the minimum singular value of **H**_{
AG
}.

### Effective channel capacity (Algorithm A4)

Unlike Algorithm A1, this does not consider the sum capacity of sub-channels but overall channel capacity itself. As in other algorithms, for every possible effective channel **H**_{
AG
}, we calculate channel capacity

C=logdet\left({I}_{{N}_{r}}+\frac{1}{{N}_{r}\underset{n}{\overset{2}{\sigma}}}{{H}_{AG}}^{H}\cdot {H}_{AG}\right).

(11)

We can then select the grouping and the precoding matrix which maximize (11).

### Based on normalized instantaneous channel correlation matrix (Algorithm A5)

Transmit antennas which are highly correlated are grouped together and transmit antennas which are less correlated are separately grouped in this algorithm. Let us define a normalized instantaneous channel correlation matrix (NICCM) as

{\left[R\right]}_{ij}=\frac{1}{\u2225{h}_{i}\u2225\cdot \u2225{h}_{j}\u2225}\cdot {\left[{H}^{H}H\right]}_{ij}

(12)

R=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill {r}_{12}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {r}_{1{N}_{t}}\hfill \\ \hfill {r}_{12}^{*}\hfill & \hfill 1\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {r}_{2{N}_{t}}\hfill \\ \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill {r}_{1{N}_{t}}^{*}\hfill & \hfill {r}_{2{N}_{t}}^{*}\hfill & \hfill \dots \hfill & \hfill 1\hfill \end{array}\right]

(13)

In (13), if the amplitude of *r*_{13} is large, then it means that the first and the third columns of **H** are more correlated than the other pairs. This can be interpreted as the correlation between the transmit antennas 1 and 3 is large.

Using this concept, we can devise a simple antenna grouping algorithm. For simplicity's sake, assume *N*_{
t
}is 4 and *N*_{
r
}is 2. In a 4*×* 2 system, **R**_{4× 2}is written as

{R}_{4\times 2}=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill A\hfill & \hfill B\hfill & \hfill C\hfill \\ \hfill {A}^{*}\hfill & \hfill 1\hfill & \hfill D\hfill & \hfill E\hfill \\ \hfill {B}^{*}\hfill & \hfill {D}^{*}\hfill & \hfill 1\hfill & \hfill F\hfill \\ \hfill {C}^{*}\hfill & \hfill {E}^{*}\hfill & \hfill {F}^{*}\hfill & \hfill 1\hfill \end{array}\right]

(14)

For simplicity, we consider only the antenna grouping where the size of each group is 2, which we call (2, 2) grouping. The possible antenna grouping cases are (1,2//3,4), (1,3//2,4) and (1,4//2,3). We then compare (*|A|* + *|F|*), (*|B|* + *|E|*), and (*|C|* + *|D|*). If (*|A|* + *|F|*) is the largest, it means that the correlation between transmit antennas 1 and 2 and between transmit antennas 3 and 4 is larger than the others so we group (1, 2) and (3, 4) together which are denoted by (1,2//3,4). Similarly, if (*|B|*+*|E|*) is the largest, then we use the grouping of (1,3//2,4). If (*|C|*+*|D|*) is the maximum, then we use the grouping of (1,4//2,3). The advantage of this algorithm is that it reduces the search complexity significantly. This antenna grouping algorithm can be extended to any MIMO system where *N*_{
t
}is an integer multiple of *N*_{
r
}. The BER performance of this algorithm is compared to other criteria in [9], and it is very close to others.