ANTENNA SELECTION FOR
LARGE-SCALE MIMO SYSTEMS WITH LOW-RESOLUTION ADCS
Jinseok Choi† , Junmo Sung† , Brian L. Evans† , and Alan Gatherer∗

arXiv:1801.09846v1 [] 30 Jan 2018

†

Wireless Networking and Communications Group, The University of Texas at Austin
Email: {jinseokchoi89, junmo.sung}@utexas.edu, bevans@ece.utexas.edu
∗
Systems and Design for Wireless Communications, Huawei, Plano, Texas, USA
Email: alan.gatherer@huawei.com
ABSTRACT

One way to reduce the power consumption in large-scale
multiple-input multiple-output (MIMO) systems is to employ
low-resolution analog-to-digital converters (ADCs). In this
paper, we investigate antenna selection for large-scale MIMO
receivers with low-resolution ADCs, thereby providing more
flexibility in resolution and number of ADCs. To incorporate quantization effects, we generalize an existing objective
function for a greedy capacity-maximization antenna selection approach. The derived objective function offers an opportunity to select an antenna with the best tradeoff between
the additional channel gain and increase in quantization error. Using the generalized objective function, we propose an
antenna selection algorithm based on a conventional antenna
selection algorithm without an increase in overall complexity.
Simulation results show that the proposed algorithm outperforms the conventional algorithm in achievable capacity for
the same number of antennas.
Index Terms— Antenna selection, large-scale MIMO,
low-resolution ADCs, capacity-maximization algorithm.
1. INTRODUCTION
Large-scale MIMO systems have drawn considerable attention as a promising technology for next generation cellular
systems because they offer orders of magnitude improvement
in spectral efficiency [1–3]. Practical challenges such as
hardware cost and power consumption, however, arise due to
the large number of antennas [4]. Since the power consumption of ADCs, PADC , scales exponentially in the number of
quantization bits b, i.e., PADC ∝ 2b [5], high-speed and highresolution ADCs would be the primary power consumers.
Consequently, receivers with low-resolution ADCs (1-3 bits)
have emerged as a possible solution to this problem [6, 7].
Most prior work in low-resolution ADCs has focused on
the case where the number of ADCs is same as the number
of antennas [6–9] without considering the tradeoff between
J. Choi, J. Sung, and B. L. Evans were supported by gift funding from
Huawei Technologies.

the number of quantization bits and the number of ADCs.
Accordingly, a reduced number of low-resolution ADCs
has recently been investigated by using analog processing
with phase shifters [10–12]. Analog processing with phase
shifters, however, is mostly effective in a poor scattering
channel environment with respect to reducing the number of
ADCs [13, 14]. Moreover, phase shifters require additional
hardware cost and power consumption [15].
For large-scale MIMO receivers, analog processing with
phase shifters was compared against analog processing with
switches, i.e., antenna selection, to reduce the number of
ADCs [15]. It was shown in [15] that antenna selection
provides similar spectral efficiencies as analog processing
with phase shifters for equal power consumption in millimeter wave (mmWave) channels. The antenna selection
used in [15] does not exploit the sparsity of mmWave channels, whereas the analog processing with phase shifters does.
Hence, antenna selection is less limited by the channel environment such as low scattering than analog processing
with phase shifters when reducing the number of ADCs. On
that account, antenna selection is more applicable to general
channels for the massive MIMO receiver. Indeed, for channels measured at 2.6 GHz, a great number of ADCs could
be turned off by using antenna selection without a substantial
performance loss [16]. Previously proposed antenna selection
methods [16–19], however, focused on MIMO systems without any quantization errors. Consequently, for low-resolution
ADC receivers, a new antenna selection method that incorporates coarse quantization effect needs to be developed.
In this paper, we investigate antenna selection for largescale MIMO systems with low-resolution ADCs. We derive
the tradeoff between the channel gain from selecting an antenna and its impact on quantization error. In particular, we
derive an antenna selection objective function for greedybased antenna selection, which is a generalized version of
the function in [17]. The objective function measures (i) the
effect of quantization error from previously chosen antennas
in the capacity gain that comes from selecting an additional
antenna, and (ii) the increase in quantization error from the

additional antenna which acts as a penalty. Accordingly, the
tradeoff between the additional channel gain and the increase
of the quantization error in antenna selection can be captured by using the derived objective function. Leveraging the
objective function, we propose a capacity-maximization antenna selection algorithm based on the conventional antenna
selection algorithm without increasing the overall complexity.
Simulation results demonstrate that the proposed algorithm
outperforms the conventional algorithm.
2. SYSTEM MODEL
We consider a multi-user MIMO uplink system in which the
BS with Nr antennas serves Nu users with single antenna.
We assume that the number of the BS antennas is much larger
than the number of users, Nr  Nu . Once the BS receives
user signals, it selects K antennas to use. The selected antennas are connected to RF chains with ADCs. Assuming
a narrowband channel, the received baseband analog signal
r ∈ CNr can be expressed as
√
(1)
r = ρHs + n
where ρ, H, s, and n denotes the transmit power, the Nr ×Nu
channel matrix, the user symbol vector, and the additive white
Gaussian noise (AWGN) vector, respectively. We assume a
zero mean and unit variance for s and n. The ith column of
H corresponds to the channel vector for user i, given as
√
hi = γi gi .
The large scale fading gain γi includes the geometric attenuation and shadow fading, and gi denotes the vector of small
scale fading gains for user i. We use fiH to denote the ith row
of H. We consider that the channel H is known at the BS and
unknown at users. Since the BS selects K antennas and connects the selected antennas to K RF chains, after the antenna
selection, the received analog signal (1) becomes
√
rK = ρHK s + nK
(2)
where K represents the index set of selected antennas with
the cardinality of |K| = K, rK ∈ CK denotes the received
signal vector for the selected antennas, HK ∈ CK×Nu is the
channel matrix for the selected antennas in K, and nK ∈ CK
indicates the corresponding noise vector.
After the antenna selection, each real and imaginary component of the complex output rK(i) is quantized at the pair of
ADCs, where K(i) represents the ith selected antenna. For
analytical tractability, the additive quantization noise model
(AQNM) is used to linearize the quantization process as a
function of quantization bits b. Adopting the AQNM which
provides reasonable accuracy in low and medium SNR ranges
[20], the quantized signal is expressed as


y = Q Re{rK } + jQ Im{rK }
√
(3)
= α ρHK s + αnK + q

Table 1: The Values of β for Quantization Bits b
b
β

1
0.3634

2
0.1175

3
0.03454

4
0.009497

5
0.002499

where Q(·) is the element-wise quantizer function. Here, α
is defined as α = 1 − β and considered to be the quantization gain (α < 1), and β is the normalized mean squared
E[|yi −yqi |2 ]
quantization error β = E[|y
. Assuming a scalar min2
i| ]
imum mean squared error quantizer and Gaussian signaling
for s ∼ CN (0, I) where CN (0, I) represents complex Gaussian distribution with a zero mean vector 0 and the identity
matrix I with proper dimension
for a covariance matrix, β is
√
π 3 −2b
approximated as β ≈ 2 2
for b > 5. Note that b is
the number of quantization bits for each real and imaginary
part. The values of β for b ≤ 5 are shown in Table 1. The
vector q represents the additive quantization noise and is uncorrelated with the quantization input rK . The quantization
noise follows the complex Gaussian distribution with a zero
mean q ∼ CN (0, Rqq ). The covariance matrix Rqq for the
channel HK is given by
Rqq = α(1 − α) diag(ρHK HH
K + I)

(4)

where diag(ρHK HH
K + I) represents the diagonal matrix of
the diagonal entries of ρHK HH
K + I.
3. ANTENNA SELECTION
3.1. Performance Measure
In this section, we examine the key difference of the antenna
selection problem at the receiver with low-resolution ADCs
from the conventional problem, and further propose an antenna selection method based on a capacity-maximization approach. Under the considered system in (3), the capacity for
the given channel matrix HK can be expressed as
−1
C(HK ) = log2 I + ρα2 α2 I + Rqq
HK HH
(5)
K
where Rqq is given in (4). Note that in the system with
low-resolution ADC, the quantization noise covariance matrix Rqq is included in the capacity expression (5) as a penalty
term for each antenna.
Remark 1 Since each diagonal entry of Rqq contains an aggregated channel gains at each selected antenna kfK(i) k2 , the
tradeoff between the channel gain and its influence on quantization error needs to be considered in antenna selection.
3.2. Capacity-Maximization Problem
Using the capacity expression in (5), we formulate the antenna selection problem as follows:

K? =
argmax
C HK .
(6)
K⊂{1,...,Nr }:|K|=K

The large number of antennas Nr makes it almost infeasible
to search over all possible K. Accordingly, to avoid an exhaustive search, we propose a quantization-aware antenna selection algorithm based on the greedy capacity-maximization
approach [17]. The capacity in (5) can be rewritten as
H
C(HK ) = log2 I + ραD−1
K HK HK

(7)

where DK = diag(1 + ρ(1 − α)kfK(i) k2 ) is the diagonal
matrix with 1 + ρ(1 − α)kfK(i) k2 for i = 1, . . . , K at its
diagonal entries.
Using (7), we can express the capacity at the (n + 1)th
selection stage as follows:
C(Hn+1 )

(8)

Algorithm 1 Quantization-Aware Fast Antenna Selection
1) Initialize: T = {1, . . . , Nr } and Q = I.
2) Initialize antenna gain and compute penalty:
cj = kfj k2 and dj = 1 + ρ(1 − α)kfj k2 for j ∈ T .
3) Select antenna J using (10): J = argmaxj∈T cj /dj .
4) Update candidate set: T = T \ {J}.
5) Compute: a = cJ +

 1
dJ − 2
QfJ
ρα

and Q = Q − aaH .

6) Update cj = cj − |fjH a|2 for j ∈ T .
7) Go to step 3 and repeat until select K antennas.

H
= log2 I + ραD−1
n+1 Hn+1 Hn+1
−1
= log2 I + ραHH
n+1 Dn+1 Hn+1


1
−1
H
= log2 I + ρα HH
fK(n+1) fK(n+1)
n Dn Hn +
dK(n+1)

where Hn+1 is the channel matrix that corresponds to the
(n + 1) selected antennas after the (n + 1)th selection stage.
H
Recall that fK(n+1)
denotes the (n + 1)th selected row of H,
and dK(n+1) is the corresponding diagonal entry of Dn+1 .
We can decompose (8) into the capacity after the nth selection and the capacity increase that comes from selecting
the (n + 1)th antenna. To this end, we use the matrix determinant lemma |A + uvH | = |A|(1 + vH A−1 u) as follows:


ρα
C(Hn+1 ) = C(Hn ) + log2 1+
cK(n+1),n
dK(n+1)

error. This corresponds to the intuition in Remark 1. Note
that (10) is the generalized antenna selection objective function of the one in [17]; as the number of quantization bits b
increases, the quantization gain α increases as α → 1, which
leads to dj → 1 and Dn → I.
To propose a quantization-aware antenna selection algorithm, we modify the fast antenna selection algorithm in [17]
by using the derived objective function in (10) without increasing the overall complexity. Unlike the algorithm for perfect quantization, the quantization error term dj needs to be
computed in advance. Then, at the selection step, the proposed algorithm adopts (10) to incorporate the influence of
the quantization error of the previously selected antennas and
the candidate antenna. To compute cj,n in (9), we define

−1
−1
Qn = I+ραHH
D
H
.
n
n
n

where


H
−1
cK(n+1),n = fK(n+1)
I+ραHH
n Dn Hn

−1

fK(n+1) . (9)

To maximize C(Hn+1 ) given the n selected antennas, we
need to find the next antenna j which maximizes cj,n /dj . Accordingly, the antenna selection problem becomes
J = argmax
j

cj,n
.
dj

(10)

Unlike the objective function of the capacity-maximization
algorithm with no quantization error in [17], the derived
objective function cj,n /dj incorporates (i) the effect of the
quantization error from the previously selected n antennas to
the next antenna j in cj,n , and (ii) the additional quantization
error from selecting the antenna j as a penalty for the antenna
j in the form of 1/dj . Since dj is a function of the aggregated
channel gain for the jth antenna, dj = 1 + ρ(1 − α)kfj k2 ,
solving the problem (10) gives the antenna J which offers
the best tradeoff between the channel gain from selecting an
antenna and its influence on the increase of the quantization

(11)

Then, Qn can be updated efficiently by using the matrix
inversion lemma as Qn+1 = Qn − aaH , where a =

dJ −1/2
cJ + ρα
QfJ . Finally, cj,n can be updated as
cj,n+1 = fjH Qn+1 fj = fjH Qn − aa
= cj,n −

H

fj

|fjH a|2 .

The proposed algorithm, we call quantization-aware fast
antenna selection (QAFAS), is described in Algorithm 1.
Note that the complexity for step 5 and 6 are O(KNu2 ) and
O(KNu Nr ), respectively. Since we assume the large antenna
arrays at the BS (Nr  Nu ), the overall complexity becomes
O(KNu Nr ). Thus, the proposed algorithm does not increase
the overall complexity from the conventional algorithm for
the perfect quantization in [17]. In the following section, we
demonstrate the performance of the proposed algorithm for
the large-scale MIMO systems with coarse quantization.

4. SIMULATION RESULTS
In this section, we evaluate the proposed algorithm and compare the algorithm with the fast antenna selection algorithm
in [17] which shows a comparable performance to the optimal selection case under a perfect quantization assumption.
The Matlab code is available at [21]. We also include a random selection case for a reference in capacity performance.
We assume Rayleigh channel with a zero mean and unit variance. For the large scale fading, we adopt the log-distance
pathloss model [22]. We consider randomly distributed users
over a single cell with a cell radius of 2 km. We assume the
minimum distance between the BS and users to be 100 m.
Considering a 2.4 GHz carrier frequency with 10 MHz bandwidth, we use 8.7 dB lognormal shadowing variance and 5 dB
noise figure with Nr = 128 BS antennas and Nu = 10 users.
Fig. 1 shows the average capacity versus the transmit
power ρ for b = 3 quantization bits in K ∈ {10, 20, 40} antenna selection cases. The proposed QAFAS algorithm outperforms the conventional fast antenna selection method. The
gap between the algorithms increases as the transmit power ρ
increases because the quantization error becomes more dominant than the AWGN as the transmit power increases. Accordingly, the proposed QAFAS algorithm which incorporates the quantization error when selecting antennas improves
the capacity performance more when the quantization error
becomes prevailing. Note that the QAFAS algorithm with
K ∈ {10, 20} achieves the capacity comparable to the random selection case with K ∈ {20, 40} whose number of selected antennas K is twice as many, respectively.
For the different number of quantization bits b, the average capacities for ρ = 5 dBm transmit power in K ∈ {10, 20}
antenna selection cases are shown in Fig. 2. The proposed algorithm provides the highest capacity and improves the capacity compared to the random selection case in the lowresolution regime, whereas the conventional fast antenna selection marginally improves the capacity. Notably, the capacity of the QAFAS algorithm shows relatively larger improvement compared to that of the fast antenna selection algorithm in the low-resolution regime than in the high-resolution
regime, which corresponds to the motivation of this work. In
the low-resolution regime, the QAFAS algorithm with K ∈
{10, 20} shows the comparable capacity to that of random selection with K ∈ {20, 40}, respectively.
5. CONCLUSION
In this paper, we proposed a new antenna selection algorithm
for large-scale MIMO systems with low-resolution ADCs.
For each unselected antenna, the derived objective function measures the increase in capacity, like a conventional
approach. Unlike a conventional approach, the proposed
objective function measures the effect on capacity due to
quantization error from previously chosen antennas and the

Fig. 1: Average capacity for different transmit power with
Nr = 128 BS antennas, Nu = 10 users, b = 3 quantization
bits, and K ∈ {10, 20, 40} selected antennas.

Fig. 2: Average capacity for the different number of quantization bits with Nr = 128 BS antennas, Nu = 10 users, ρ = 5
dBm transmit power, and K ∈ {10, 20} selected antennas.

increase in quantization error if an unselected antenna were
chosen. Since the derived objective function captures the
tradeoff between the additional channel gain and the increase
in quantization error for antenna selection, we used the objective function to propose a capacity-maximization antenna
selection algorithm. The simulation results validated the proposed antenna selection algorithm for low-resolution ADCs.
Therefore, using the proposed algorithm, the antenna selection for large-scale MIMO systems with low-resolution
ADCs offers more flexibility in the resolution and number of
ADCs with higher rates than other selection methods.

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