Antenna Selection

Antenna selection (ANS) is useful in lessening the RF and baseband implementation complexity. For conventional MIMO systems, the number of antennas is usually relative small and ANS is not a particular concern although proper ANS does save hardware cost

and power consumption. For massive, however, it becomes an important design consid-eration, as the associated number of RF chains reduction can be significant. Moreover, it not always a good policy to use more antennas if the number of antennas used is already large enough; employing more antennas may lead to degraded performance.

An outstanding low-complexity ANS scheme is especially important for massive MIMO systems as it can significantly reduce the hardware requirement of a BS without compromising performance. The solution, however, is challenging to say the least. This can be easily seen by simply counting the number of possible combinations ¹⁰⁰₅₀

≈ 10²⁹ in selecting 50 out of 100 antennas that maximizes the system sum rate.

We will consider first the basic setting of an SU-MIMO system with large-scale BS antenna array and then extend the investigation to an MU-MIMO scenario. Further-more, we focus the ANS study on the DL case, where a massive MIMO BS acts as the transmitter. The UL scenario can be similarly treated with some minor modifications.

2.4.1 ANS for Massive SU-MIMO systems

Consider a DL massive SU-MIMO system with N_T Tx antennas and N_RRx antennas whose received vector is given by (2.38) and N_T  N_R.

In such a system, H is known by the receiver but unknown to the transmitter. The ergodic capacity is given by [21]

R(H) = log₂det

To facilitate subsequent discussion, we need the following definition [23].

Definition 2.4.1. Let f be a function defined as f : U → R⁺. Then f is called monotone if f (S ∪ {a}) − f (S) ≥ 0, ∀a ∈ U , S ⊆ U , a /∈ S, and is called a sub-modular function if f (S ∪ {a}) − f (S) ≥ f (T ∪ {a}) − f (T ), ∀a ∈ U , a /∈ T and S ⊆ T ⊆ U .

A nesting property of R(H) has been derived in [21] when one tries to select one more Rx antenna by a incremental capacity-based selection (CS) algorithm:

R(Hn+1) = R(Hn) + log₂

where Hnis the channel matrix after selecting n Rx antennas and αj,n = h^H_j

 I + _N^P

TH^H_nHn

−1

hj

with h_j being the channel seen by the jth user antenna. (2.28) implies that in an SU-MIMO system, the channel capacity is sub-modular over Rx antenna set {1, 2, . . . , N_R}.

Hence, the main design criterion of SU-MIMO Rx ANS is to reduce the hardware com-plexity which is dominated by the number of radio-frequency (RF) chains. Given the set of selected antennas, one selects, in each step, the J th antenna that maximize kh_J,nk [21]

J = arg max

j kh_j,nk, (2.29)

which gives an important observation that the CS criterion is asymptotically equivalent to the norm-based selection (NS) criterion. Thus, low-complexity NS criterion will be good enough to retain the performance CS can achieve.

On the other hand, [22] and [23] reported that H_n is not sub-modular over the Tx antenna set {1, 2, . . . , N_T}. This means, for a massive MIMO downlink the use of all Tx antennas may not offer better rate. In general, we have two Tx ANS selection criteria, i.e., capacity maximization and feedback overhead reduction for FDD mode.

2.4.2 ANS for Massive MU-MIMO systems

Owing to the results of [24], we know that when M  K, user selection is unneces-sary. Hence, we focus more on the Tx ANS in DL massive MU-MIMO system provided that TDD mode is used.

For an MU-MIMO system with M BS antennas and K single-antenna MSs, M  K, the composite forward link (downlink) channel matrix consists of SSFCs and LSFCs [1]

is given by

G = D

1 2

βH. (2.30)

On the other hand, due to the channel reciprocity [19] in TDD mode, i.e., the forward link and reverse link channel are symmetric, the reverse link (uplink) composite channel matrix is simply the M × K matrix G^T. Therefore, the DL system model is given by [42]

y = Gs + n = GWPx + n (2.31)

where s is the transmitted signal, x ∼ CN (0, I_K) the uncoded data, W the M × K precoding matrix, n ∼ CN (0, IK) the received noise and K × K matrix P diagonal with its kth entry being the square root of power allocated to user k. Thus, the achievable rate

R(H) = log₂det I_K + W^HG^HGWP²

, (2.32)

subject to a total power constraint P

tr WP²W^H

= kWPk²_F ≤ P, (2.33)

can be obtained. It has been proved that, as opposed to SU-MIMO, MU-MIMO using linear precoding techniques, e.g., zero-forcing (ZF) and minimum mean square error (MMSE) precoding, has sub-modular property R(H) over the Tx antenna set [42]. This is because in MU-MIMO, MSs cannot cooperate and no post-detection signal processing is allowed.

Assume decremental transmit antenna selection (TAS) algorithm [42] is used, the capacity loss when getting rid of one more Tx antenna, indexed α, can be computed by using (14) of [42]:

Lemma 2.4.2. Let S and S⁰ be two Tx antenna sets in MU-MIMO system and S = S⁰ \ S = r, where S ⊂ S⁰ ⊆ {1, · · · , M } and |S| = 1. Then, the difference of the sum

rate (throughput loss) between these two sets is given by

, GS is the composite fading channel given the transmit antenna set S, u<sub>`</sub> is the channel vector seen by `th Tx antenna, SNR = _σ^P2, P is the total power constraint, and σ² = 1 is the noise power.

Because R_D(r) ≥ 0, we know that TAS in MU-MIMO system indeed has sub-modularity [23], hence the purpose of TAS is to reduce the hardware complexity and reduce the amount of feedback in FDD mode. Suppose that we want to select M_F  K out of M Tx antennas, the capacity-based selection (CS) algorithm can be summarized in Algorithm 1.

Algorithm 1 Decremental CS algorithm

1: Let S = 1, 2, · · · , M_`;

6: The resulting set S is the desired transmit antenna set.

From Lemma 4 of [42], we obtain

α = arg max

Applying Lemma 3.3.1 to (2.35) and substituting GS = D^1/2_β HS yields

We call (2.36) generalized norm-based selection (GNS), since the form of its selection metric is similar to norm-based selection (NS).

When the serving MSs are not far away from each other, or equivalently, D_β ≈ βI_K, which is in general the case because each RRH only serves the nearby users,

α = arg min

which is just the norm-based selection (NS).

Besides, [1] has shown that in massive MIMO system, ZF precoding is asymptoti-cally equivalent to MF precoding, hence we can get the same result for the case of MF precoding.