1

Spectral Efficiency of Mixed-ADC Massive MIMO

arXiv:1802.10259v1 [] 28 Feb 2018

Hessam Pirzadeh, Student Member, IEEE, and A. Lee Swindlehurst, Fellow, IEEE

Abstract—We study the spectral efficiency (SE) of a mixedADC massive MIMO system in which K single-antenna users
communicate with a base station (BS) equipped with M antennas
connected to N high-resolution ADCs and M − N one-bit ADCs.
This architecture has been proposed as an approach for realizing
massive MIMO systems with reasonable power consumption.
First, we investigate the effectiveness of mixed-ADC architectures
in overcoming the channel estimation error caused by coarse
quantization. For the channel estimation phase, we study to
what extent one can combat the SE loss by exploiting just
N ≪ M pairs of high-resolution ADCs. We extend the roundrobin training scheme for mixed-ADC systems to include both
high-resolution and one-bit quantized observations. Then, we
analyze the impact of the resulting channel estimation error in the
data detection phase. We consider random high-resolution ADC
assignment and also analyze a simple antenna selection scheme
to increase the SE. Analytical expressions are derived for the
SE for maximum ratio combining (MRC) and numerical results
are presented for zero-forcing (ZF) detection. Performance comparisons are made against systems with uniform ADC resolution
and against mixed-ADC systems without round-robin training
to illustrate under what conditions each approach provides the
greatest benefit.
Index Terms—Massive MIMO, analog-to-digital converter,
mixed-ADC, spectral efficiency.

I. I NTRODUCTION

T

HE seminal work of Marzetta introduced massive MIMO
as a promising architecture for future wireless systems
[2]. In the limit of an infinite number of base station (BS)
antennas, it was shown that massive MIMO can substantially
increase the network capacity. Another key potential of massive MIMO systems which has also made it interesting from
a practical standpoint is its ability of achieving this goal with
inexpensive, low-power components [3], [4]. However, preliminary studies on massive MIMO systems have for the most part
only analyzed its performance under the assumption of perfect
hardware [5], [6]. The impact of hardware imperfections and
nonlinearities on massive MIMO systems has recently been
investigated in [7]-[12]. Although it is well-known that the
dynamic power in
√massive MIMO systems can be scaled down
proportional to M , where M denotes the number of BS
antennas, the static power consumption at the BS will increase
proportionally to M [8]. Hence, considering hardware-aware
design together with power consumption at the BS seems
necessary in realizing practical massive MIMO systems.
This work was supported by the National Science Foundation under Grants
ECCS-1547155 and CCF-1703635, and by a Hans Fischer Senior Fellowship
from the Technische Universität München Institute for Advanced Study.
H. Pirzadeh and A. L. Swindlehurst are with the Center for Pervasive
Communications and Computing, University of California, Irvine, CA 92697
USA (e-mail: hpirzade@uci.edu; swindle@uci.edu).
Portions of this paper have appeared in [1].

Among the various components responsible for power
dissipation at the BS, the contribution of analog-to-digital
converters (ADCs) is known to be dominant [13]. Consequently, the idea of replacing the high-power high-resolution
ADCs with power efficient low-resolution ADCs could be a
viable approach to address power consumption concerns at the
massive MIMO BSs. The impact of utilizing low-resolution
ADCs on the spectral efficiency (SE) and energy consumption
of massive MIMO systems has been considered in [14]-[22]. In
particular, studies on massive MIMO systems with purely onebit ADCs show that the high spatial multiplexing gain owing
to the use of a large number of antennas is still achievable even
with one-bit ADCs [14], [15]. However, many more antennas
with one-bit ADCs (at least 2-2.5 times) are required to attain
the same performance as in the high-resolution ADCs case.
One of the main causes of SE degradation in purely one-bit
massive MIMO systems is the error due to the coarse quantization that occurs during the channel estimation phase. While
at low SNR the loss due to one-bit quantization is only about 2
dB, at higher SNRs performance degrades considerably more
and leads to an error floor [14]. The SE degradation can be reduced by improving the quality of the channel estimation prior
to signal detection. One approach for doing so is to exploit socalled mixed-ADC architectures during the channel estimation
phase, in which a combination of low- and high-resolution
ADCs are used side-by-side. This architecture is depicted in
Fig. 1. Mixed-ADC implementations were introduced in [23],
[24] and their performance was studied from an information
theoretic perspective via generalized mutual information.
The basic premise behind the mixed-ADC architecture is to
achieve the benefits of conventional massive MIMO systems
by just exploiting N ≪ M pairs of high-resolution ADCs.
An SE analysis of mixed-ADC massive MIMO systems with
maximum ratio combining (MRC) detection for Rayleigh and
Rician fading channels was carried out in [25] and [26],
respectively. The SE and energy efficiency of mixed-ADC
systems compared with systems composed of one-bit ADCs
was studied in [27] for MRC detection, and conditions were
derived under which each architecture provided the highest
SE for a given power consumption. The advantage of using a
mixed-ADC architecture in designing Bayes-optimal detectors
for MIMO systems with low-resolution ADCs is reported in
[28]. Although the nonlinearity of the quantization process
increases the complexity of the optimal detectors, it is shown
that adding a small number of high-resolution ADCs to
the system allows for less complex detectors with only a
slight performance degradation. Moreover, the benefit of using
mixed-ADC architectures in massive MIMO relay systems and
cloud-RAN deployments is elaborated in [29], [30].
Most existing work in the mixed-ADC massive MIMO
literature has assumed either perfect channel state information

2

(CSI) or imperfect CSI with “round-robin” training. In the
round-robin training approach [23], [24], [26], the training data
is repeated several times and the high-resolution ADCs are
switched among the RF chains so that every antenna can have
a “clean” snapshot of the pilots for channel estimation. This
obviously requires a larger portion of the coherence interval to
be devoted to training rather than data transmission. More precisely, for M antennas and N pairs of high-resolution ADCs,
M/N pilot signals are required in the single-user scenario
to estimate all M channel coefficients with high-resolution
ADCs. This issue is pointed out in [23] for the single user
scenario and its impact is taken into account. This training
overhead will be exacerbated in the multiuser scenario where
orthogonal pilot sequences should be assigned to the users.
In this case, the training period becomes (M/N )η, where η
represents the length of the pilot sequences (at least as large
as the number of user terminals), which could be prohibitively
large and may leave little room for data transmission. Hence,
it is crucial to account for this fact in any SE analysis of
mixed-ADC massive MIMO systems.
In this paper, we examine the channel estimation performance and the resulting uplink SE of mixed-ADC architectures with and without round-robin training, and compare them
with implementations that employ uniform ADC quantization
across all antennas. The main goals are to determine when,
if at all, the benefits of using the round-robin approach with
ADC/antenna switching outweigh the cost of increasing the
training overhead, and furthermore to examine the question of
whether or not one should employ a mixed-ADC architecture
in the first place. The contributions of the paper can be
summarized as follows.
•

•

•

We first present an extension of the round-robin training
approach that incorporates both high-resolution and onebit measurements for the channel estimation. The roundrobin training proposed in [23], [24], [26] based the
channel estimate on only high-resolution observations,
assuming that no data was collected from antennas during
intervals when they were not connected to the highresolution ADCs. In contrast, our extension assumes that
these antennas collect one-bit observations and combine
this data with the high-resolution samples to improve the
channel estimation performance.
We use the Bussgang decompositon [31] to develop a
linear minimum mean-squared error (LMMSE) channel
estimator based on the combined round-robin measurements and we derive a closed-form expression for the
resulting mean-squared error (MSE). We further illustrate the importance of using the Bussgang approach
rather than the simpler additive quantization noise model
in obtaining the most accurate characterization of the
channel estimation performance for round-robin training.
The analysis illustrates that the addition of the one-bit
observations considerably improves performance at low
SNR.
We perform a spectral efficiency analysis of the mixedADC implementation for the MRC and ZF receivers, and
obtain expressions for a lower bound on the SE that takes

•

into account the channel estimation error and the loss of
efficiency due to the round-robin training. We compare
the resulting SE with that achieved by mixed-ADC implementations that do not switch ADCs among the RF
chains, and hence do not use round-robin training. We
also compare against the SE for architectures that do not
mix the ADC resolution across the array, but instead use
uniform resolution with a fixed number of comparators
for different array sizes. We show that, depending on
the SNR, coherence interval, number of high-resolution
ADCs, and the choice of the linear receiver, there are
situations where each of the considered approaches shows
superior performance. In particular, even the round-robin
method with its considerable training overhead can provide the best SE under certain circumstances.
We analyze the possible SE improvement that can be
achieved by using an antenna selection algorithm that
connects the high-resolution ADCs to the subset of antennas with the highest channel gain. We analytically derive
the SE performance of the antenna selection algorithm
for MRC and numerically study its performance for ZF
detection, comparing against the simpler approach of
assigning the high-resolution ADCs to an arbitrary fixed
subset of the RF chains.

In addition to the above contributions, we also discuss
some of the issues related to implementing an ADC switch
or multiplexer in hardware that allows different ADCs to be
assigned to different antennas. We restrict our analysis and
numerical examples to a single-carrier flat-fading scenario, although our methodology can be used in a straightforward way
to extend the results to frequency-selective fading or multiplecarrier signals (e.g., see our prior work in Section III.B of
[14] for the SE analysis of an all-one-bit ADC system for
OFDM and frequency selectivity). The reasons for focusing
on the single-carrier flat-fading case are as follows: (1) the
mixed-ADC assumption already makes the resulting analytical
expressions quite complicated even for the simple flat-fading
case, and it would be more difficult to gain insight into the
problem if the expressions were further complicated; (2) the
original round-robin training idea was proposed in [23] for the
single-carrier flat-fading case, and thus we analyze it under
the same assumptions; (3) the main conclusions of the paper
are based on relative algorithm comparisons for the same set
of assumptions, and we expect our general conclusions to
remain unchanged if frequency rather than flat fading were
considered; and (4) the flat fading case is still of interest in
some applications, for example in a micro-cell setting with
typical path-length differences of 50-100 m, the coherence
bandwidth is between 3-6 MHz, which is not insignificant.
Further assumptions regarding the system model are outlined in the next section. Section III discusses channel estimation using round-robin training, and derives the LMMSE
channel estimator that incorporates both the high-resolution
and one-bit observations. A discussion of hardware and other
practical considerations associated with using a mixed-ADC
system with ADC/antenna switching is presented in Section
IV. Section V then presents the analysis of the spectral

3

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M-N one-bit ADCs
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symbols and changes independently between different intervals. Note that T is a fixed system parameter chosen as the
minimum coherence duration of all users. At the beginning of
each coherence interval, the users send their η-tuple mutually
orthogonal pilot sequences (K ≤ η ≤ T ) to the BS for channel
estimation. Denoting the length of the training phase as ηeff ,
the remaining T − ηeff symbols are dedicated to uplink data
transmission.

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III. T RAINING P HASE
N high-resolution ADCs

Fig. 1. Mixed-ADC architecture.

efficiency for MRC and ZF receivers based on the imperfect
channel state estimates, including an analytical performance
characterization of antenna selection and architectures with
uniform ADC resolution across the array. A number of numerical studies are then presented in Section VI to illustrate
the relative performance of the algorithms considered.
Notation: We use boldface letters to denote vectors, and
capitals to denote matrices. The symbols (.)∗ , (.)T , and (.)H
represent conjugate, transpose, and conjugate transpose, respectively. A circularly-symmetric complex Gaussian (CSCG)
random vector with zero mean and covariance matrix R
is denoted v ∼ CN (0, R). The symbol k.k represents the
Euclidean norm. The K × K identity matrix is denoted by
I K and the expectation operator by E{.}. We use 1N to
denote the N ×1 vector of all ones, and diag{C} the diagonal
matrix formed from the diagonal elements of the square
matrix C. For a complex value, c = cR + jcI , we define
arcsin(c) , arcsin(cR ) + jarcsin(cI ).

In this section, we investigate the linear minimum mean
squared error (LMMSE) channel estimator for different ADC
architectures at the BS. In all scenarios, the pilot sequences
are drawn from an η × K matrix Φ, where the kth column of
Φ, φk , is the kth user’s pilot sequence and ΦH Φ = I K .
Therefore, the M × η received signal at the BS before
quantization becomes
X=

K
X
√

ηpk g k φTk + N ,

(2)

k=1

where N is an M × η matrix with i.i.d. CN (0, σn2 ) elements.
Since the rows of X are mutually independent due to the
assumption of spatially uncorrelated Gaussian channels and
noise, we can analyze them separately. As a result, we will
focus on the mth row of X which is
xTm

K
X
√
ηpk gmk φTk + nTm ,
=

(3)

k=1

where gmk is the mth element of the kth user channel vector,
g k , and nTm is the mth row of N . Since the analysis is not
dependent on m, hereafter we drop this subscript and denote
the received signal at the mth antenna by x.

II. S YSTEM M ODEL
Consider the uplink of a single-cell multi-user MIMO
system consisting of K single-antenna users that send their
signals simultaneously to a BS equipped with M antennas.
Assuming a single-carrier frequency flat channel , the M × 1
signal received at the BS from the K users is given by
r=

K
X
√
pk g k sk + n,

(1)

k=1

where pk represents
√ the average transmission power from the
kth user, g k = βk hk is the channel vector between the
kth user and the BS where βk models geometric attenuation
and shadow fading, and hk ∼ CN (0, I M ) represents the
fast fading and is assumed to be independent of other users’
channel vectors. The symbol
 transmitted by the kth user is
denoted by sk where E |sk |2 = 1 and is drawn from
a CSCG codebook independent of the other users. Finally,
n ∼ CN 0, σn2 I M denotes additive CSCG receiver noise at
the BS.
We consider a block-fading model with coherence bandwidth Wc and coherence time Tc . In this model, each channel
remains constant in a coherence interval of length T = Tc Wc

A. Estimation Using One-Bit Quantized Observations
In this subsection, to have a benchmark for comparison
purposes, we consider the case in which all antennas at the
BS are connected to one-bit ADCs. The received signal xT
after quantization by one-bit ADCs can be written as

(4)
y Tt = Q xT ,
where the element-wise one-bit quantization operation Q(·) replaces each input entry with the quantized value √12 (±1 ± j),
depending on the sign of the real and imaginary parts. According to the Bussgang decomposition [31], the following linear
representation of the quantization can be employed [14]:
r

2 T − 21
T
Q x =
x Dx + q Tt ,
(5)
π

where Dx = diag{Cx } and Cx denotes autocorrelation
matrix of x, which can be calculated as
Cx =

K
X

k=1

ηpk βk φ∗k φTk + σn2 Iη .

(6)

4

where Q is an M × η matrix whose mth row is q Tt . The
LMMSE estimate of the channel G = [g 1 , ..., g K ] based on
just one-bit quantized observations (8) is given in the following
theorem.

00 00 00 00 00 00 00 00 00 00 00 00
000000000000

2
σw
=
k

r

π 21
Dx φk
2

(10)


1  2
∗
T
σn + φ̄k Cq t φ̄k .
ηpk

(11)

φ̄k ,

Define the channel estimation error ε , ĝ k − g k . Then we
have
σĝ2k

βk2
=
2
βk + σw
k

and

σε2k

2
σw
β
k k
=
,
2
βk + σw
k

(12)

where σĝ2k and σε2k are the variances of the independent zeromean elements of ĝ k and ε, respectively.
From Theorem 1, it is apparent that in the channel estimation analysis of massive MIMO systems with one-bit ADCs,
the estimation error is directly affected not only by the inner
product of the pilot sequences, but also by their outer product
as well [14]. To get insight into the impact of the one-bit
quantization on the channel estimation, in the next corollary
we adopt the statistics-aware power control policy proposed
in [37]. Apart from its practical advantages, this policy is
especially suitable specially for one-bit ADCs since it avoids
near-far blockage and hence strong interference. Moreover,
this power control approach also leads to simple expressions
and provides analytical convenience for our derivation in
Section VI. Although not the focus of this paper, we note
that in general a massive MIMO system employing a mixedADC architecture will be more resilient than an all one-bit
implementation to the near-far effect and jamming. This is an
interesting topic for further study.

Data Transmission

Antenna set 1
Antenna set 2

00 00 00 00 00 00 00 00 00
0 0 0 0 0 0 0 0 0000000000000000000
000000000

Antenna set 3
Antenna set 4
Antenna set 5

00 00 00 00 00 00 00 00
00000000

Full-resolution
observations

No observation

Wc
Tc

Fig. 2. Transmission protocol for estimation using full-resolution observations.

Corollary 1. For the case in which power control is performed,
i.e., pk = βpk for some fixed value p and for k ∈ K =
{1, · · · , K}, the number of users is equal to the length of
pilot sequences, i.e., η = K, and the pilot matrix satisfies
ΦΦH = IK , we have

(13)
Cx = Kp + σn2 IK = Dx


2
Cq t = 1 −
IK ,
(14)
π
which yields

Theorem 1. The LMMSE estimate of the k-th user channel,
g k , given the one-bit quantized observations Y is [14]
r
1
βk
∗
Yφ̄k ,
(9)
ĝ k =
2
βk + σw
ηp
k
k
where

Training

antennas

In addition, q t represents quantization noise which is uncorrelated with x and its autocorrelation matrix can be derived
based on the arcsine law as [32]
2 −1
2
−1
−1
−1
Cqt = arcsin{Dx 2 Cx Dx 2 } − Dx 2 Cx Dx 2 . (7)
π
π
Much of the existing work on massive MIMO systems with
low-resolution ADCs employs the simple additive quantization
noise model (AQNM) for their analysis [20]-[22], [25]-[30],
[39] which is valid only for low SNRs and does not capture the
correlation among the elements of q t , which turns out to be
of crucial importance in our analysis. Hence, we consider the
Bussgang decomposition instead and will show its effect on
the system performance analysis. Stacking the rows of (5) into
a matrix, the one-bit quantized observation at the BS becomes
r
2
−1
Y=
XDx 2 + Q,
(8)
π

σĝ2k =

2 βk
π 1 + σn2

(15)

Kp

σε2k =



Kp
2
σn

1−

2
π

1+




+ 1 βk

Kp
2
σn

.

(16)

Corrollary 1 states conditions under which Cqt is diagonal.
In addition, it is evident that the channel estimation suffers
from an error floor at high SNRs.
B. Channel Estimation with Few Full Resolution ADCs
Channel estimation with coarse observations suffers from
large errors especially in the high SNR regime. On the
other hand, while estimating all channels using high-resolution
ADCs is desirable, the resulting power consumption burden
makes this approach practically infeasible. This motivates
the use of a mixed-ADC architecture for channel estimation
to eliminate the large estimation error caused by one-bit
quantization while keeping the power consumption penalty
at an acceptable level. In the approach described in [23],
[24], [26] , N ≪ M pairs of high-resolution ADCs are
deployed and switched between different antennas during
different transmission intervals in an approach referred to as
“round-robin” training. In this approach, the M BS antennas
are grouped into M/N sets1 . In the first training sub-interval,
users send their mutually orthogonal pilots to the BS while the
N high-resolution ADC pairs are connected to the first set of
N antennas. After receiving the pilot symbols from all users in
the η-symbol-length training sub-interval, the high-resolution
ADCs are switched to the next set of antennas and so on.
In this manner, after (M/N )η pilot transmissions (M/N subintervals), we can estimate each channel based on observations
with only high-resolution ADCs. This round-robin channel
1 We

assume M/N is an integer throughout the paper.

5

Antenna set 1
Antenna set 2
Antenna set 3
Antenna set 4
Antenna set 5

00 00 00 00 00 00 00 00 00 00 00 00 Training
000000000000
00 00 00 00 00 00 00 00 00
0 0 0 0 0 0 0 0000000000000000000
0 0 0 0 0 0 0 0 00 00 00 00 00 00 00 00 00
0 0 0 0 0 0 0 0 00 00 00 00 00 00 00 00 00
000000000

Data Transmission

antennas

estimation protocol is illustrated in Fig. 2 for a mixed-ADC
system with M/N = 5.
Stacking all N ×η full-resolution observations into an M ×η
matrix, X, the LMMSE estimate of the k-th user channel, g k ,
is [5]
1
1
ĝ k =
Xφ∗k ,
(17)
√
2
σn
ηp
k
1+

σĝ2k =

βk
1+

2
σn
ηpk βk

and

σε2k =

βk
1+

ηpk βk
2
σn

.

(18)

Eq. (18) states that by employing only N pairs of highresolution ADCs and by expending a larger portion of the
coherence interval for channel estimation, the channel can
be estimated with the same precision as that achieved by
conventional high-resolution ADC massive MIMO systems.
However, this comes at the high cost of repeating the training
data M/N times, which can significantly reduce the time
available for data transmission. Indeed, we will see later that in
some cases, a mixed-ADC implementation with round-robin
training achieves a lower SE than a system with all one-bit
ADCs because of the long training interval (even with the
improvements we propose below for the round-robin method).
However, we will also see that there are other situations for
which the mixed-ADC round-robin method provides a large
gain in SE. The primary goal of this paper is to elucidate
under what conditions these and other competing approaches
provide the best performance.
Before analyzing the tradeoff between the gain (lower
channel estimation error) and cost (longer training period) of
the round-robin approach, in the next subsection we propose
channel estimation based on the use of both full-resolution and
one-bit data received by the BS in order to further improve the
performance of the mixed-ADC architecture with round-robin
channel estimation. To our knowledge, this approach has not
been considered in prior work on mixed-ADC massive MIMO.

Full-resolution
observations
One-bit
observations

Wc
Tc

ηpk βk

and the resulting variances of the channel estimate and the
error are given respectively by

00 00 00 00 00 00 00 00
00000000

Fig. 3. Transmission protocol for estimation using full-resolution/one-bit
observations.

Theorem 2. Stacking all N × η full-resolution observations
into an M × η matrix, X, and all (M/N ) − 1 N × η
one-bit quantized observations into M × η matrices, Yt ,
t ∈ T = {1, ..., M/N − 1}, the LMMSE estimate of the k-th
user channel, g k , is


M
r
−1
N
X
1 
∗
w∞k Xφ∗k + w1k
ĝ k =
Yt φ̄k  ,
(19)
ηpk
t=1

where

ηpk
2
σn
ηpk
2 + ςk (pk )
σn

(20)

−1
M
ςk (pk )
N −1
ηpk
1
2 + ςk (pk )
βk + σn

(21)

w∞k =

1
βk

+



w1k =

M
N

ςk (pk ) =
2
σw
k

C̄q t =

−1




2 + M −2 ̺
σw
k
N
k


1
∗
T
σn2 + φ̄k Cq t φ̄k
=
ηpk

(22)

̺k =

(24)

1 T
∗
φ̄ C̄q φ̄
ηpk k t k

2
2 −1
−1
−1
−1
arcsin{D̄x 2 C̄x D̄x 2 } − D̄x 2 C̄x D̄x 2
π
π
C̄x =

K
X

ηpk βk φ∗k φTk

(23)

(25)
(26)

k=1

C. Estimation Using Joint Full-Resolution/One-Bit Observations
While channel estimation performance based on coarsely
quantized observations suffers from large errors in the high
SNR regime, it provides reasonable performance for low
SNRs. Hence, in this subsection we consider joint channel
estimation based on observations from both high-resolution
and one-bit ADCs to further improve the channel estimation
accuracy. Unlike the previous subsection in which the onebit ADCs were not employed, here we incorporate their
coarse observations into the channel estimation procedure. The
protocol for this method is illustrated in Fig. 3 for a mixedADC system with M/N = 5. It can be seen that, in addition to
one set of full-resolution observations for each antenna, there
are (M/N ) − 1 sets of one-bit observations which are also
taken into account for channel estimation. The next theorem
characterizes the performance of this approach.

D̄x = diag{C̄x }.

(27)

This approach yields the following variances for the channel
estimate and the estimation error, respectively:
σĝ2k

=

ηpk
2 + ςk (pk )
σn
βk
ηpk
1
2 + ςk (pk )
βk + σn

σε2k =
Proof. See Appendix A.

1
βk

+

1
.
+ ςk (pk )

ηpk
2
σn

(28)
(29)


Theorem 2 demonstrates the optimal approach for combining the observations from high-resolution and one-bit ADCs.
In addition, this highlights the importance of considering the
correlation among the one-bit observations in the analysis
of mixed-ADC channel estimation, something that could not
be addressed by the widely-used AQNM approach. More

6

precisely, it can be seen that the impact of joint highresolution/one-bit channel estimation is manifested in the variance of the channel estimation error by the term ςk (pk ). To see
this, assume that the correlation among one-bit observations in
different training sub-intervals is ignored (as would be the case
with the AQNM approach). As shown in the appendix, this is
equivalent to setting ̺k = 0 in (24). Under this assumption,
ςk (pk ) becomes

M
N −1
> ςk (pk ),
(30)
ςk0 (pk ) =
2
σw
k
and thus, σε2k > σε2k where σε2k denotes the estimation
0
0
error for ̺k = 0. Consequently, the AQNM model yields an
overly optimistic assessment of the channel estimation error
compared with the more accurate Bussgang analysis. We will
see below that the impact of the AQNM approximation is
significant for mixed-ADC channel estimation.
The next corollary provides insight into the impact of
the system parameters on the joint high-resolution/one-bit
LMMSE estimation.
Corollary 2. For the case in which power control is performed,
i.e., pk = βpk for k ∈ K, the number of users is equal to the
length of pilot sequences, i.e., η = K, and the pilot matrix
satisfies ΦΦH = IK , we have
C̄x = KpIK = D̄x ,
and
C̄q t

1

and σε2k =

where
ς(p) =

2
π σn
2 Kp

+

In addition,
w∞ =

1+

Kp
2
σn
Kp
2 +
σn

ς(p)



M
N −1

π
2 −1

and w1 =

(32)

1
1+

M
N

M
N

Kp
2
σn

−1

−1

Kp
2
σn

βk ,
(33)

.

−1

1+

+ ς(p)

(34)

ς(p)

+ ς(p)

, (35)

where w∞ and w1 denote the weights of the high-resolution
and one-bit observations in the LMMSE estimation, respectively.
Corallary 2 states that in contrast to Theorem 1 where the
correlation among one-bit observations within each training
sub-interval can be eliminated by carefully selecting the system parameters as in Corollary 1, we cannot overcome the
correlation among one-bit observations from different training
sub-intervals. This phenomenon makes the addition of the onebit observations less useful especially in the high SNR regime.
For instance, in the asymptotic case, as the SNR = σp2 goes
n
to infinity, we have
ς −→

π
2

1
,
−1

-10

-15
-20

-15

-10

-5

0

5

2.
Fig. 4. Channel estimation error σε2k /βk versus p/σn

w∞ −→ 1, w1 −→ 0.



2
IK ,
= 1−
π

Kp
2 + ς(p)
σn
βk
+ Kp
2 + ς(p)
σn

-5

(31)

which yields
σĝ2k =

0

(36)

(37)

It is apparent from (36) that in the asymptotic regime ς tends to
a finite value and also is independent of M/N . Moreover, (37)
implies that the optimal approach for high SNRs is to estimate
the channel based solely on the high-resolution observations.
The error for the three channel estimation approaches in
Eqs. (12), (18), and (29) is depicted in Fig. 4 for a case with
M = 100 antennas, K = 10 users, and various numbers of
high-resolution ADCs, N and training lengths η. The label
“Joint” refers to round-robin channel estimation that includes
the one-bit observations as described in the previous section,
”Full resolution” indicates the performance achieved using a
full array of high-resolution ADCs, and “One-bit” refers to
the performance of an all-one-bit architecture. We also plot
the performance predicted for the Joint approach based on the
AQNM analysis, which ignores the correlation among the onebit observations. We see that the AQNM-based analysis yields
an overly optimistic prediction for the channel estimation
error. In particular, unlike AQNM, the more accurate Bussgang
analysis shows that channel estimation with all an one-bit
BS actually outperforms the Joint method for low SNRs, a
critical observation in analyzing whether or not a mixed-ADC
implementation makes sense. However, we see that the mixedADC architecture eventually overcomes the error floor of the
all one-bit system for high SNRs and in such cases can reduce
the estimation error dramatically. Fig. 4 focuses on channel
estimation performance, but does not reflect the full impact
of the round-robin training on the overall system spectral
efficiency, since reducing N increases the amount of training
required by the round-robin method. This will be taken into
account when we analyze the SE in Section V.

7

IV. P RACTICAL C ONSIDERATIONS
The improvement in channel estimation performance provided by the round-robin training clearly comes at the expense
of a significantly increased training overhead. For example,
consider a simple worst-case example with a 400 Hz Doppler
spread in a narrowband channel of 400 kHz bandwidth; in
this case, the coherence time is roughly 1000 symbols. For
higher bandwidths or smaller cells with lower mobility, the
coherence time can easily approach 10,000 symbols or more.
A mixed-ADC array of 128 antennas with 16 high-resolution
ADCs would require repeating the pilots 8 times, which for 20
users would amount to 160 symbols, or 16% of the coherence
time when T = 1000 symbols. This is a relatively high price
to pay, and as we will see later, in many instances the resulting
loss in SE cannot be offset by the improved channel estimate.
However, we will also see that on the other hand, there are
other situations where the opposite is true, where the roundrobin method leads to significant gains in SE even taking the
training overhead into account.
Besides the extra training overhead, the round-robin method
has the disadvantage of requiring extra RF switching or
multiplexing hardware prior to the ADCs, as shown in Fig. 1.
It is unlikely that a single large M × M multiplexer would be
used for this purpose, since complete flexibility in assigning
a given high-resolution ADC to any possible antenna is not
needed. A more likely architecture would employ a bank of
smaller multiplexers that allows one high-resolution ADC to
be switched among a smaller subarray of antennas, ensuring
that each RF chain has access to high-resolution training data
during one of the round-robin intervals. Such an approach
is similar to the simplified “subarray switching” schemes
proposed for antenna selection in massive MIMO [33]-[35]. In
an interesting earlier example, a large 108 × 108 multiplexer
chipset for a local area network application was developed
in [36], composed of several 36 × 36 differential crosspoint
ASIC switches that consume less than 100 mW each, with a
bandwidth of 140 MHz and a 0 dB insertion loss.
In the 20 years since [36], RF switch technology has advanced considerably. For the example discussed above involving a 128-element array with 16 high-resolution ADCs and 112
one-bit ADCs, the multiplexing could be achieved using 16
8×8 analog switches arranged in parallel. Consider the Analog
Devices ADV3228 8 × 8 crosspoint switch as an example
of an off-the-shelf component for such an architecture2. The
ADV3228 has a 750 MHz bandwidth, a switching time of 15
ns, and a power consumption of 500 mW, which is similar
to that of an 8-bit ADC (for example, see Texas Instruments’
ADC08B200 8-bit 200 MS/s ADC3 ). Since the switches can
be implemented at a lower intermediate frequency prior to the
I-Q demodulation, only one per subarray is required, and thus
the total power consumption of the switches would be less
than half that of the ADCs.
Note that for the vast majority of the coherence time, the
switch is idle. To accommodate the round-robin training, the
2 See

http://www.analog.com/en/products/switches-multiplexers/bufferedanalog-crosspoint-switches/adv3228.html#product-overview for product
details.
3 http://www.ti.com/product/ADC08B200/technicaldocuments.

switches only need to be operated M
N −1 times, once for every
repetition of the training data. This reduces the actual power
consumption to below the specification, and further reduces the
impact of the additional training. Short guard intervals would
need to be inserted between the training intervals to account
for the switching transients, but these will typically not impact
the SE. For the example discussed above with 128 antennas
and 8 switches, 7 switching events are required for a total
switching time of 105 ns, which is insignificant compared to
the coherence time of 2.5 ms at a 400 Hz Doppler.
The insertion loss of the analog switches would also have
to be taken into account in an actual implementation, since
this will directly reduce the overall SNR of the received
signals. Harmonic interference due to nonlinearities in the
switch are likely not an issue; for example, the specifications
for a Texas Instruments switch (LMH6583) similar to the
ADV3228 indicate that the power of the second and third
harmonic distortions were -76 dBc. Furthermore, it has been
shown that the use of signal combining with a massive antenna
array provides significant robustness to such nonlinearities and
other hardware imperfections [7]-[12].
V. S PECTRAL E FFICIENCY
Although channel estimation with a mixed-ADC architecture using round-robin training can substantially improve
the channel estimation accuracy, it requires a longer training
interval and, therefore, leaves less room for data transmission
in each coherence interval. More precisely, (M/N )η symbol
transmissions are required for round-robin channel estimation
which could be large when the number of high-resolution
ADCs, N , is small4 . Despite losing a portion of the coherence interval for channel estimation due to the mixed-ADC
architecture, the improvement in the signal-to-quantizationinterference-and-noise ratio (SQINR) can be significant owing
to more accurate channel estimation, and thus a higher rate
would be expected during this shorter data transmission period.
In this section, we study this system performance trade-off in
terms of spectral efficiency for the three mentioned channel
estimation approaches.
In the data transmission phase, all users simultaneously send
their data symbols to the BS. To begin, assume the antennas
are ordered so that the last N antennas are connected to highresolution ADCs in this phase. A more thoughtful assignment
of the high-resolution ADCs will be considered below. From
equation (1), and based on the Bussgang decomposition, the
received signal at the BS after one-bit quantization is
#
"q
 
2 − 21
q̄
D̄
0
π
(38)
r+ d
yd =
0
0
IN
| {z }
qd

D̄ = diag{Cr }

(39)

4 Note that in designing a mixed-ADC system with round-robin channel
training, one should consider the ratio M/N in scaling the system instead
of just increasing the number of antennas M . In particular, increasing the
number of BS antennas requires increasing of the high-resolution ADCs, N ,
as well.

8

Cr =

K
X

2
pk ḡ k ḡ H
k + σn IM−N ,

(40)

k=1

where ḡ k denotes the M − N elements of g k corresponding
to the M − N one-bit ADCs and q̄ d is the (M − N ) × 1
quantization noise in the data transmission phase. It is apparent
that the covariance matrix in (40) is not diagonal which makes
analytical tractability difficult. However, by adopting statisticsaware power control [37], i.e., pk = βpk , and assuming that
the number of users is relatively large (typical for massive
MIMO systems), channel hardening occurs [14], and (40) can
be approximated as

Cr ∼
(41)
= Kp + σn2 IM−N = D̄.
As a result, according to the arcsine law (see (7)), the covariance matrix of the quantization noise in the data transmission
phase becomes Cq̄d ∼
= (1 − 2/π)IM−N and (38) simplifies to
!
K
X
√
∼A
y =
phk sk + n + q
(42)
d

d

k=1



αIM−N
A=
0


0
,
IN

q
1
where α , π2 (Kp+σ
2 .
n)
For data detection, the BS selects a linear receiver W ∈
CM×K as a function of the channel estimate. Note that the
quantization model considered in (4) and (5) does not preserve
the power of the input of the quantizer since the power of the
output is forced to be 1. Thus we premultiply the received
signal as follows to offset this effect:
ŷ d = A−1 y d .

(43)

By employing the linear detector W, the resulting signal at
the BS is
ŝ = W H ŷ d .
(44)
Thus, the kth element of ŝ is
ŝk =

K
√ H
√ X
pw k hk sk + p
wH
k hi si
i=1,i6=k

H −1
+ wH
q d , (45)
k n + wk A

where w k is the kth column of W . We assume the BS
treats w H
k hk as the gain of the desired signal and the other
terms of (45) as Gaussian noise when decoding the signal5 .
Consequently, we can use the classical bounding technique of
[37] to derive an approximation for the ergodic achievable SE
at the kth user as
Sk = R (SQINRk ) ,

(46)

where the effective SQINRk is defined by (47) at the top of the
next page, and R (θ) , (1 − ηeff /T ) log2 (1 + θ) where ηeff
represents the training duration which is η and (M/N ) η for
the pure one-bit and mixed-ADC architectures, respectively.
5 Note that in general, the quantization noise is not Gaussian. However, to
derive a lower bound for the SE, we assume it is Gaussian with covariance
Cq d .

A. MRC Detection
1) Random Mixed-ADC Detection: In this subsection, we
consider the case in which the high-resolution ADCs are
connected to an arbitrary set of N antennas. Denoting the
estimate of the channel by Ĥ = [ĥ1 , ..., ĥK ], setting W = Ĥ,
and following the same reasoning as in [14], the SE of the
mixed-ADC architecture with MRC detection can be derived
as


2
pM
σ
ĥ

(48)
SkMRC = R 
2
 ,
1−
(
)
N
pK + σn2 + α2π 1 − M

where the channel estimate variance σĥ2 = σĝ2k /βk depends on
the estimation approach as denoted in (12), (18), and (28).
From (48), it can be observed that the gain of exploiting the
mixed-ADC architecture is manifested in the SE expressions
by two factors, channel estimation improvement by a factor of
σĥ2 , and quantization noise reduction by a factor of 1 − N/M .
2) Mixed-ADC Detection with Antenna Selection: Having
an accurate channel estimate can help us to employ the
N high-resolution ADCs in an intelligent manner to further
improve the performance of the mixed-ADC architecture. A
careful look at the SQINR expression in (47) reveals that
the effect of one-bit quantization on the SE is manifested by
the last term of the denominator. Hence, one can maximize
the SE by minimizing this term through smart use of the N
high-resolution ADCs. We refer to this approach as MixedADC with Antenna Selection. We consider an antenna selection
scheme suggested by the SQINR expression in (47). In this
approach, the N high-resolution ADCs are connected to the
antennas corresponding to rows of Ĥ with the largest energy,
2
PK
i.e. k=1 ĥmk . Besides numerical evaluation in Section VI,
in Theorem 3 we derive a bound for the SE achieved by MRC
detection with antenna selection.
Theorem 3. The spectral efficiency of the mixed-ADC system
with antenna selection and an MRC receiver is lower bounded
by


2
pM σĥ


S̄kMRC = R 
  , (49)

1−
( π2 ) PM−N
2
pK + σn + MKα2
m=1 χm
where χm is defined at the top of the next page, and FA
denotes the Lauricella function of type A [45].
Proof. See Appendix B.



The lower bound (49) explicitly reflects the benefit of
antenna selection in the data transmission phase. By comparing (49) with (48), it is evident that antenna P
selection has
M −N
χm
improved the SE by replacing 1 − N/M by m=1
. In
MK
Section VI we illustrate how antenna selection improves SE
for different SNRs. Note that Theorem 3 assumes the ability
to make an arbitrary assignment of the high-resolution ADCs
to different RF chains, which may not be possible if the ADC
multiplexing is implemented by a bank of subarray switches.
In the numerical results presented later, we show that this does
not lead to a significant degradation in performance.

9

SQINRk =

χm =

p

PK

i=1

E

n

wH
k hi

2


p E wH
k hk

o


− p E wH
k hk

2

2


+ σn2 E {kwk k2 } + α−2 E wH
k Cq d w k

(47)



M−m
X
M −m
M!
(Γ(K))−m−ℓ K 1−m−ℓ Γ (1 + K (m + ℓ))
(−1)−ℓ
(m − 1) ! (M − m) !
ℓ
ℓ=0

(m+ℓ−1)

× FA

(1 + K (m + ℓ) ; K, · · · , K; K + 1, · · · , K + 1; −1, · · · , −1) (50)

B. ZF Detection
In this section, we study the SE of the mixed-ADC architecture with ZF detection. To design a mixed-ADC adapted ZF
detector, we re-write the last two terms of the denominator of
(47) as follows:



H
2
−2
H
(51)
wk σn IM + α Cqd w k = W Cneff W ,
kk

where Cneff = σn2 IM +α−2 Cqd . Accordingly, the ZF detector
for the mixed-ADC architecture can be written as
−1

H −1
.
(52)
Ĥ
C
Ĥ
W = C−1
Ĥ
neff
neff

Plugging (52) into (47) yields (53) at the top of the next page.
Similar to the MRC case, the SQINR in (53) suggests the
same antenna selection approach for ZF detection. In general,
calculating the expected values in (53) is not tractable neither
for arbitrary-antenna mixed-ADC detection nor mixed-ADC
with antenna selection. Hence, we numerically evaluate the
performance of mixed-ADC with ZF detection in the next
section.

C. Massive MIMO with Uniform ADC Resolution
Contrary to the mixed-ADC architecture where the ADC
comparators are concentrated in a few antennas, uniformly
spreading the comparators over the array is an alternative
approach [19], [20], [21], [41], [44]. In this subsection, we provide the SE expressions for such systems. These expressions
will be used in the next section for performance comparisons
with the mixed-ADC architecture.
The SE for the case of all one-bit ADCs was derived in
[14] using the Bussgang decomposition. For ADC resolutions
of 2 bits or higher, the AQNM model is sufficiently accurate.
Using AQNM and following the same reasoning as in [21],
[41], [44], the SE of a massive MIMO system with uniform
resolution ADCs can be derived as
S̃kMRC =



pM σ̃ĥ2
 


R
0)
2 + K + σ2
pK + σn2 + (1−α
p
σ̃
n
α2
ĥ
0

(54)

S̃kZF =


R



pK 1 − σ̃ĥ2



p (M − K) σ̃ĥ2

(M−K)σ̃2
ĥ
+ σn2 +
E wH
k C0 w k
α2




,

(55)

for MRC and ZF detection, respectively. In (54) and (55),
σ̃ĥ2 =

α20 ηp

+

α20 σn2

α20 ηp
,
+ α0 (1 − α0 ) (pK + σn2 )

(56)

α0 is a scalar depending on the ADC resolution and can
be found in Table
 I of [21], wk is the kth column of
−1

, and C0 denotes the covariance matrix
W = Ĥ ĤH Ĥ
of the quantization noise based
on the AQNM model [21]. The

detailed calculation of E w H
C
0 w k in (55) is provided in
k
[44] which we do not include here for the sake of brevity.
VI. N UMERICAL R ESULTS

By substituting from (12), (18), and (28) into (48), (49),
and (53), we can evaluate the performance of mixed-ADC
architectures for different system settings. We consider a
system with M = 100 antennas at the BS, and K = 10 users.
Also, we assume the power control approach of [37] is used,
so that pk βk = p for all k. We also assume that an optimal
resource allocation has been performed [41], [42] such that
the training length, ηeff , transmission power during the training
phase, pt , and data transmission phase, pd are optimized under
a power constraint ηeff pt + (T − ηeff )pd = Pave T . In the
following figures, the SNR is defined as SNR , Pave /σn2 .
Fig. 5 illustrates the optimal weights for combining
high-resolution and one-bit observations for the joint highresolution/one-bit LMMSE channel estimation. Interestingly, it
can be seen that when M/N is large, the one-bit observations
are emphasized in the low SNR regime relative to the highresolution observations. In addition, in contrast to the weights
for the high-resolution observations, which rise monotonically
with increasing SNR, the weight for the one-bit observations
grows at first and then decreases to zero.
To study the performance improvement due to joint channel estimation and antenna selection in mixed-ADC massive
MIMO, the sum SE for the MRC and ZF detectors for a system
with coherence interval T = 400 symbols and N = 20 highresolution ADCs is depicted in Fig. 6 and Fig. 7, respectively.

10

SQINRZF
k =

p

 
−1


−1  
−1  
−1
−2
−1
−1
2
H
H
H
H
Ĥ Cneff Ĥ
Ĥ Cneff Ĥ Ĥ Cneff Ĥ
pK 1 − σĥ E
+E
Ĥ Cneff Ĥ
kk

1

100

0.9

90

0.8

80

0.7

70

0.6

60

0.5

50

0.4

40

0.3

30

0.2

20

0.1

10

0
-20

-15

-10

-5

0

5

10

15

20

Fig. 5. Weights used in the LMMSE channel estimator for high-resolution
and one-bit observations.

30

25

20

15

10

5

0
-20

-15

-10

-5

0

5

10

15

20

Fig. 6. Sum SE for MRC detection versus SNR for M = 100, N = 20, and
T = 400.

In these and subsequent figures, “Joint with AS” indicates
that the channel estimation was performed with both onebit and high-resolution ADCs and that antenna selection (AS)
was used for data detection, “Joint without AS” represents
the same case without antenna selection, “Joint Subarray AS”
means that the antenna selection only occurred within each
M/N -element subarray (one high-resolution ADC assigned
to the strongest channel within each subarray), and “Not

0
-20

-15

-10

(53)

kk

-5

0

5

10

15

20

Fig. 7. Sum SE for ZF detection versus SNR for M = 100, N = 20, and
T = 400.

Joint without AS” represents the case in which channel is
estimated based on only high-resolution observations and no
antenna selection is employed. For both MRC and ZF, it can
be seen that antenna selection slightly improves the SE for
high SNRs, where the channel estimation is most accurate.
At low SNR, we see that joint channel estimation provides
a gain from the use of one-bit ADCs, which provide useful
information at these SNRs. We also see that the constrained
AS required when the switching is only performed within
subarrays provides nearly identical performance to the case
where arbitrary AS is allowed.
Note that the main reason for the small gain for antenna
selection is due to the fact that, with multiple users, selecting
a given antenna does not benefit all users simultaneously, and
the strong users responsible for a given antenna being selected
will in general be different for different antennas. Thus, the
improvement due to increased signal-to-noise ratio for some
users is somewhat offset by the fact that other users may
experience a lower SNR on those same antennas. We would
see a much larger benefit for antenna selection if only a single
user were present.
Figs. 8 and 9 provide a comparison among a mixed-ADC
massive MIMO system with joint channel estimation and
antenna selection, an all-one-bit architecture (“One-bit”), and a
mixed-ADC without round-robin training for which the highresolution ADCs are connected to a fixed set of antennas
without ADC switching or antenna selection (“Non-roundrobin”) [27]. Since mixed-ADC channel estimation improves
the channel estimation accuracy by expending a larger portion

11

100

30
20

50
10
0
-20

-15

-10

-5

0

5

10

15

20

0
-20

-15

-10

-5

0

5

10

15

20

-15

-10

-5

0

5

10

15

20

100

30
20

50
10
0
-20

-15

-10

-5

0

5

10

15

20

Fig. 8. Sum SE for MRC detection versus SNR for M = 100, N = 20, 10,
and T = 400, 1000.

of the coherence interval for training, its benefit is directly
related to the length of the coherence interval. For MRC
detection, when T = 400, the mixed-ADC architecture performs better than the all-one-bit architecture for N = 20,
but when N = 10 the all-one-bit architecture is better due
to the larger training overhead incurred when N is smaller.
However, for T = 1000, mixed-ADC outperforms the allone-bit architecture at high SNRs for both N = 10, 20,
while the all-one-bit case is still better for N = 10 at low
SNRs. Round-robin training provides better SE performance
at high SNR when N = 20 compared to the case without
antenna switching, especially for the larger coherence interval.
However, for other cases, the round-robin training overhead
significantly reduces the SE, especially for N = 10 and the
shorter coherence interval.
For ZF detection, we see that the mixed-ADC architectures
can provide very large gains in SE compared to the one-bit
case at high SNRs, regardless of T . For low SNRs, there is
little to no improvement. These cases still do not show a
significant benefit for round-robin training compared with a
fixed ADC assignment; only when N = 20 and T = 1000 do
we see a slight improvement.
For N = 20, Figs. 10 and 11 show how the coherence
coherence interval T impacts the effectiveness of the mixedADC architecture for MRC and ZF detectors, respectively.
For mixed-ADC MRC detection, it is apparent that the best
choice among the three architectures (all one-bit, mixed-ADC
with and without round-robin training) depends on the SNR
operating point and the length of the coherence interval. The
advantage of round-robin training becomes apparent for long
coherence intervals, where the increased training length has a
smaller impact. The gain for round-robin training is greatest
at higher SNRs. For shorter coherence intervals, mixed ADC
with fixed antenna/ADC assignments provides the best SE,
with the largest gains again coming at higher SNRs. For this

0
-20

Fig. 9. Sum SE for ZF detection versus SNR for M = 100, N = 20, 10,
and T = 400, 1000.

30
28
26
24
22
20
18
16
14
12
500

1000

1500

2000

Fig. 10. Sum SE for MRC detection versus T for M = 100, N = 20, and
SN R = −10, 0, 10 dB.

value of N , the all-one-bit system generally has the lowest
SE, although the difference is not large for MRC.
The next example investigates the impact of distributing the
resolution (i.e., the comparators of the ADCs) across the array
with different numbers of antennas. If we assume that the
“high-resolution” ADCs consist of 5 bits [43], a mixed-ADC
architecture with N = 20 high-resolution and M − N = 80
one-bit ADCs will have 180 total comparators. Figs. 12 and 13
illustrate the SE achieved by distributing the 180 comparators
across arrays of different length for MRC and ZF detection,
respectively. In these figures, “Joint with AS” and “Non-roundrobin” refer to mixed-ADC architectures with N = 20 5-bit
ADCs and M −N = 80 one-bit ADCs, “One-bit” corresponds
to M = 180 antennas with one-bit ADCs, and “Multi-bit”

12

70

100

60

50

50
0
-20

-15

-10

-5

0

5

10

15

20

-15

-10

-5

0

5

10

15

20

40
100
30
50
20

0
-20

10
500

1000

1500

2000

Fig. 11. Sum SE for ZF detection versus T for M = 100, N = 20, and
SN R = −10, 0, 10 dB.

Fig. 13. Sum SE for ZF detection versus SNR for 180 comparators and
T = 400, 1000.

40
30

35

20
10
0
-20

30
-15

-10

-5

0

5

10

15

20
25

40
30

20

20
10
0
-20

15
-15

-10

-5

0

5

10

15

20
10
10

20

30

40

50

60

70

80

90

100

Fig. 12. Sum SE for MRC detection versus SNR for 180 comparators and
T = 400, 1000.
Fig. 14. Sum SE for MRC detection versus N for SN R = −10, 0, 10 dB
and T = 1000.

indicates a system with either M = 90 2-bit ADCs or M = 60
3-bit ADCs. As we see in the figures, it can be inferred that
for MRC detection, which is interference limited, it is better to
have a larger number of antennas with lower-resolution ADCs
instead of equipping the BS with fewer antennas and high
resolution ADCs. This is consistent with the results of [30],
[39], and is due to the fact that a larger number of antennas
helps the system to more effectively cancel the interference.
On the other hand, for ZF detection which is noise limited,
the use of high-resolution ADCs avoids additional quantization
noise imposed by the low-resolution ADCs, and is more
beneficial than having a larger number of antennas with lowresolution ADCs at high SNR.
Finally, Figs. 14 and 15 show the impact of the number of

high-resolution ADCs in a mixed-ADC system with M = 100
antennas, K = 10 users, and various numbers N of highresolution ADCs, where N = 100 denotes the all-highresolution system. It is apparent that with a large enough
coherence interval and a sufficient number of high-resolution
ADCs, the mixed-ADC implementation with joint round-robin
channel estimation and antenna selection outperforms the
all-one-bit architecture and mixed-ADC without round-robin
training. The gains are greatest when ZF detection is used and
the SNR is high, but such gains must be weighed against the
increased power consumption and hardware complexity.

13

A PPENDIX
100

A. Proof of Theorem 2

90

From (2), the observations from the high-resolution ADCs
can be written as
r
1
v(0) =
Xφ∗k = g k + ñ(0),
(57)
ηpk

80
70
60

σ2

where ñ(0) ∼ CN (0, ηpnk I M ). In addition, from (8), the
observations from the one-bit ADCs become
r
1
∗
v(t) =
Yt φ¯k = g k + ñ(t) + q̃(t), t ∈ T , (58)
ηpk

50
40
30

σ2

20
10
10

20

30

40

50

60

70

80

90

100

Fig. 15. Sum SE for ZF detection versus N for SN R = −10, 0, 10 dB and
T = 1000.

VII. C ONCLUSION
We studied the spectral efficiency of mixed-ADC massive
MIMO systems with either MRC or ZF detection. We showed
that properly accounting for the impact of the quantized
receivers using the Bussgang decomposition is important for
obtaining an accurate analysis of the SE. We introduced a joint
channel estimation approach to leverage both high-resolution
ADCs and one-bit ADCs and our analytical and numerical
results confirmed the benefit of joint channel estimation for
low SNRs.
Mixed-ADC detection with MRC and ZF detectors and
antenna selection were also studied. Analytical expressions
were derived for MRC detection and a numerical performance
analysis was performed for ZF detection. It was shown that
antenna selection provides a slight advantage for high SNRs
while this advantage tends to disappear for low SNRs.
We showed that the SNR, the number of high-resolution
ADCs and the length of the coherence interval play a pivotal
role in determining the performance of mixed-ADC systems.
We showed that, in general, mixed-ADC architectures will
have the greatest benefit compared to implementations with all
low-resolution ADCs when ZF detection is used and the SNR
is relatively high. In such cases, the gain of the mixed-ADC
approach can be substantial. Gains are also possible for MRC,
but they are not as significant, and require larger numbers of
high-resolution ADCs to see a benefit compared with the ZF
case. The more complicated mixed-ADC approach based on
ADC switching and round-robin training can achieve the best
performance in some cases, particularly when the coherence
interval is long and more high-resolution ADCs are available
to reduce the number of training interval repetitions. Otherwise, a mixed-ADC implementation without ADC switching
and extra training is preferred.

where ñ(t) ∼ CN (0, ηpnk I M ) is independent of ñ(t′ ) for
q
1
¯∗
t 6= t′ , and q̃(t) =
ηpk Q(t)φk . Since the elements of
v(t) are independent, we can estimate the mth channel gmk
separately. Therefore, stacking all the observations in a vector,
we can write


  

vm (0)
ñm (0)
1


  .. 

..
..


 .

.
.


  

 vm (t)  = 1 gmk + 
.
ñ
(t)
+
q̃
(t)
m
m


  



 .

..
.
.
.






.
.
.
M
M
M
1
vm ( N − 1)
ñm ( N − 1) + q̃m ( N − 1)
|
|
{z
} |{z}
{z
}
1M

v

u

N

(59)
As a result, the LMMSE estimation of the mth channel
coefficient for the kth user is [40]
−1

1
1TM C−1
+ 1TM C−1
1
M
(60)
ĝmk =
u
u v.
N
N
N
βk
In Eq. (60), Cu denotes the covariance matrix of u which is
a block diagonal matrix of the form

 σ2
n
0
...
0
ηpk
" 2
#
 0
2
σn
σw
. . . ̺k 


0
k
= ηpk
,
(61)
Cu =  .
..
.. 
..
 ..
0 S
.
.
. 
0

̺k

...

2
σw
k

where

∗

̺k = E{(ñm (t) + q̃m (t)) (ñm (t′ ) + q̃m (t′ )) }, t 6= t′ , (62)
can be easily calculated with the aid of the Bussgang decomposition and the arcsine law as in (24). Substituting (61) into
(60), we have
ĝmk =



−1
ηpk
1
+ 2 + 1TM −1 S−1 1 M −1
N
N
βk
σn
#
"
σn2
×
1TM −1 S−1 v.
N
ηpk

(63)

To calculate the inverse of the matrix S, we re-write it as

2
S = σw
− ̺k I M −1 + ̺k 1 M −1 1TM −1 ,
(64)
k
N

N

N

14

where γ(., .) denotes the incomplete Gamma function. From
[47], the integral (70) can be calculated in closed form for
Gamma random variables as

and use Woodbury’s matrix identity:
S−1 =

2
σw
k

1
IM −
− ̺k N −1

1
2 −̺
σw
k
k

which yields

2

( M − 1)
1
+ 2N
̺k
σwk − ̺k

1TM −1 S−1 =
N

T

1 M −1 S
N

−1

1
2 +
σw
k

1 M −1 =
N

M
N

2
σw
k

!−1

T

1 M −1 1 M −1 , (65)
N

N

 1TM ,
− 2 ̺k N −1

M
N −1 
.
+ M
N − 2 ̺k

(66)
(67)

Substituting (66) and (67) into (63) completes the proof.

E{E(m) } = σĥ2 χm .

(72)

This is in contrast to the unordered case where E{Em } =
Kσĥ2 . As a result
 2 M−N
oo 
n n H
2 σĥ X
= 1−
min E ĥk Cq d ĥk
χm . (73)
π K m=1

The remaining terms in (47) can be calculated similar to
the case where the high-resolution ADCs are connected to
arbitrary antennas. Plugging these terms and (73) into (47)
and some algebraic manipulation results in (49).
R EFERENCES

B. Proof of Theorem 3
Denote the energy of the mth row, m ∈ M = {1, ..., M },
of Ĥ by Em , i.e.,
Em ,

K
X

2

ĥmk .

(68)

k=1

To do antenna selection, we must connect the N highresolution ADCs to the antennas corresponding to the largest
Em . Suppose that the indices of the N antennas to which the
high-resolution ADCs are connected are contained in the set
N . Hence, we have
K
X

k=1

o
n H
E ĥk Cq d ĥk =


o 
n H
2 X
E{Em }. (69)
KE ĥk Cqd ĥk = 1 −
π
M\N

Eq. (69) provides a criterion for connecting the N highresolution ADCs in the data transmission phase. In fact, it
states that, for the MRC receiver, the expected value in (69)
will be minimized if the high-resolution ADCs are connected
to the antennas corresponding to the largest Em . Denote E(m)
as the mth smallest value of Em , i.e.,
E(1) ≤ E(2) ≤ · · · ≤ E(M) .
Hence, E(m) is the mth order statistic, and assuming that the
E(m) are statistically independent and identically distributed,
we have [46]
E{E(m) } =

Z ∞
M −1
m−1
M−m
M
x [F (x)]
[1 − F (x)]
dF (x),
m−1
−∞
(70)
where x is the realization of E(m) and F (x) is the cumulative distribution function of Em . For the case that we have
considered, where the channel coefficients are i.i.d. Rayleigh
distributed, the Em are independent Gamma random variables
with
!
x
F (x) = γ
,K ,
(71)
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