New rank detection methods for reduced-rank MIMO systems
Wang and Jing EURASIP Journal on Wireless Communications and
Networking
New rank detection methods for reduced-rank MIMO systems
Qian Wang 0
Yindi Jing 0
0 Department of Electrical and Computer Engineering, University of Alberta , 9107-116 Street, T6G 2V4 Edmonton , Canada
In practical multi-input multi-output (MIMO) systems, the channel matrices often have reduced rank. Reliable detection of the channel rank is essential in achieving the significant gain provided by MIMO configuration. Existing work on MIMO channel rank detection assume a static channel model, so the proposed methods only consider the noise distributions while the distributions of the MIMO channels are not considered. In this paper, we employ a random channel model and propose three threshold-based rank detection methods which take into account the distributions of both the channels and the noises. In our first algorithm, following existing single-threshold rank detection scheme, we rigorously derive an analytical lower bound on the correct rank detection probability and propose a systematic threshold selection method by maximizing the lower bound. Then we propose two new rank detection methods which use multiple thresholds, where each threshold corresponds to one possible rank value. The thresholds are optimized based on the derived lower bounds on the rank detection probability for different channel rank values. The convergence and complexity of the proposed algorithms are analyzed. Simulation results on the correct rank detection probability of the proposed schemes are provided to show their advantage over existing schemes. The mean-squared-error (MSE) and outage probability are also simulated to show the importance of reliable rank detection in MIMO communications.
Rank detection; Reduced-rank MIMO; Threshold optimization
1 Introduction
In the past two decades, a configuration, called
multiinput multi-output (MIMO) system, which utilizes
multiple antennas at both the transmitter and the receiver,
has been extensively studied [1, 2]. MIMO systems have
outstanding performance in increasing the data
throughput and link range over single-antenna systems without
consuming additional bandwidth or transmit power. Early
studies focused on MIMO systems with full-rank
channels. But in many practical propagation environments
such as the number of surrounding scatterers which is
finite and limited, the MIMO channel matrix is likely to
have reduced rank [3, 4], especially when the channel
dimension is large [5]. In [3], the finite scatterer channel
model was described and its capacity was analyzed. In [4],
the model was applied to the large-scale MIMO system,
referred to as massive MIMO. When the channel matrix
has reduced rank, the number of its entries is larger than
its real dimension, and thus designs based on full-rank
channels become inefficient. This motivates the research
on reduced-rank technologies for MIMO systems [6–23].
In [6–14], various reduced-rank filtering technologies
were proposed, where a reduced-rank transformation is
first used on the observed signal vector to obtain a
lowerdimension vector, then a filter is designed to estimate the
desired signal vector. The reduced-rank transformation
can lower the order of the filter and thus require less
computation complexity and shorter training length. If the
covariance matrix of the observed data vector is known,
the optimal reduced-rank design is singular value
decomposition (SVD)-based [13, 14]. But the covariance matrix
estimation needs a very large number of training symbols.
To overcome this drawback, a reduced-rank multistage
Wiener filter design was proposed in [6], which relaxes
the requirement on explicit estimate of the signal
subspace. It was later extended to its adaptive versions in [7, 8]
and applied to MIMO equalization in [9]. Another related
design is the auxiliary-vector filtering algorithm [10].
© 2015 Wang and Jing. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
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Compared with the multistage Wiener filter, it achieves
better performance and does not involve matrix inversion
operation. Joint transformation and filter designs can be
found in [11, 12].
Another important issue for MIMO communications is
the channel rank detection and channel matrix estimation,
since most MIMO transceiver techniques require
channel state information (CSI) at the transmitter side and/or
the receiver side for smart signal processing. For example,
in the MIMO multiplexing transmissions, the transmitter
should align multiple data streams with the eigenspaces
of the channel. To achieve this, precise estimates on the
channel rank and channel matrix are needed. This paper
is concerned with the channel rank detection, which is an
important part of channel estimation. Thus, in what
follows, we explain related literature on channel estimation
and rank detection.
Some existing work on MIMO channel estimation focus
on full-rank MIMO channel matrices with independent
or correlated entries [24–27]. Their channel estimation
schemes are entry-based, where the unknown channel
matrix is parameterized by its entries. For reduced-rank
channel estimation, these schemes will cause performance
degradation. A more natural and efficient approach is
to use SVD-based channel estimation methods [15–23].
It was shown in [16, 17] that the maximum-likelihood
(ML) estimation of the reduced-rank MIMO channel with
Gaussian noise is the truncated SVD method. In truncated
SVD, if the channel rank is known to be r, the MIMO
channel matrix is estimated from the SVD of the received
signal-plus-noise matrix, by keeping the largest r
singular values and their corresponding singular vectors. While
the truncated SVD method traces back to the 1930s [28],
it was rediscovered in [16, 17] for MIMO channel
estimation and further improved to reduce the mean squared
error (MSE) of the estimation [18–22]. One improved
scheme is shrinkage-and-threshold SVD, where the
truncated singular values are further shrunk to remove the
noise effect.
In both truncated SVD and shrinkage-and-threshold
SVD, the channel estimation accuracy largely depends on
the correct truncation of the singular values, which is the
rank detection of the MIMO channel matrix. Thus
correct rank detection can improve the channel estimation
quality, which is crucial for advanced MIMO techniques
such as beamforming and power allocation among data
streams. Furthermore, for a MIMO channel matrix, the
rank is the indicator of how many data streams can be
spatially multiplexed on the channel, and the data streams are
represented by the singular values and the
corresponding singular vectors. Similarly, for a multi-user system
with multiple antennas at the base station, the rank of the
MIMO channel matrix from all users to the base station
determines how many users can be served by the base
station within the same time-frequency bandwidth. Thus,
accurate channel rank detection is an important part of
channel estimation and is essential for MIMO systems.
Various rank detection methods are available in the
literature [17–23]. In [17], a minimum description length
(MDL)-based rank detection was used for MIMO
channels. A threshold is calculated at each instance of the
channel to minimize the MDL. The MDL-based
detection aims at minimizing the MSE. It also requires a large
number of samples to work. In [23], several rank detection
methods were proposed for a real-valued channel matrix.
In the proposed method, the singular values or
functions of singular values are compared with a threshold for
rank detection. Lower and upper bounds on the threshold
selection were discussed. Josse and Sardy [22] considered
both the MIMO channel rank detection and the
shrinkage of the singular values for the channel estimation.
The rank detection scheme is also threshold-based, where
two threshold selection methods are proposed. In the
first method, the threshold parameter and the shrinkage
parameter are jointly optimized for each channel
realization to minimize a Stein unbiased risk estimate (SURE)
of the MSE of the MIMO channel estimation. In the
second method, the threshold is calculated from the
distribution of the largest singular value of the noise matrix.
In [18–21], for asymptotic MIMO channels where both
dimensions of the channel matrix approach infinity with
a fixed ratio, simple closed-form thresholds were derived
for threshold-based rank detection.
1.1 Summary of our results and distinction to existing work
This paper is on the rank detection of MIMO
channels. We propose a threshold optimization method for the
traditional single-threshold-based rank detection scheme
and two new multiple-threshold-based rank detection
schemes. For the threshold optimization of the traditional
single-threshold scheme, we first derive a closed-form
lower bound on the probability of correct rank
detection based on the a priori channel rank distribution, then
the optimal threshold is decided via the maximization
of the lower bound. For the two new multiple-threshold
schemes, different thresholds are used for different
possible rank values, and each threshold is derived by
maximizing the lower bound on the probability of correct rank
detection when a specific rank value is assumed.
Properties (e.g., well-posedness, convergence, complexity) of
the two new schemes are discussed. Simulation results on
the probability of correct rank detection of the proposed
schemes are shown, and their advantages over existing
schemes are discussed. The MSE and outage
probability are also simulated to show that better rank detection
improves the MSE and outage performance of MIMO
systems.
Our model, problem formulation, and methods differ
from existing ones in the following major aspects. First,
we assume a random channel matrix where the
channel entries follow Rayleigh flat-fading and the distribution
of the channel coefficients is taken into account in the
threshold optimization. On the contrary, in all existing
work, the channels are assumed to be static and the
distribution of the channel matrix is not used in the rank
detection designs [14, 17–23]. Also, in our model, a general
training length and unitary training matrix are considered,
while most existing work (e.g., [17–23]) apply to identity
training matrix only, where the training length equals the
number of transmit antennas. Finally, in this work, we use
the probability of correct rank detection as the
performance measure and optimization objective, while existing
work (except [23]) targeted at minimizing the MSE of the
MIMO channel estimation [17–22].
In what follows, we clarify the major difference of our
channel rank detection and channel estimation problem
to the reduced-rank filter design problem in [6–14]. The
goal of channel estimation is to estimate the channel
matrix itself given a limited training time. In MIMO
communications, channel estimation is usually required in the
transceiver designs to optimize the communication
performance such as outage probability and bit error rate. For
the filter design, the goal is to obtain a precise estimate
of the signals by filtering the observations, and usually
the MSE is used as the design criterion. Regardless of
the channel rank, a reduced-rank filter can be used to
lower the computational complexity and required
training length. Naturally, precise channel rank detection and
channel estimation can be helpful in reduced-rank filter
design, but it is not a necessary step in filter design. Also
the optimal rank for the filter may not be the true rank of
the channel matrix.
The rest of this paper is organized as follows. In
Section 2, the MIMO channel model, the truncated
SVDbased channel estimation, and the rank detection problem
are presented. In Section 3, for the traditional
singlethreshold-based rank detection, we derive a lower bound
on the probability of correct rank detection and propose
to optimize the threshold via the maximization of the
lower bound. Two new rank detection algorithms based
on multiple thresholds corresponding to different
possible rank values are introduced in Section 4, as well as
discussions on their properties. Simulation results on the
probability of correct rank detection, MSE, and outage
probability are shown in Section 5. Section 6 contains the
conclusions.
In this paper, bold upper case letters and bold lower case
letters are used to denote matrices and vectors,
respectively. For a matrix A, its Hermitian, trace, rank, and
determinant are denoted by A∗, tr(A), rank(A), and det(A),
respectively. In is the n×n identity matrix. E(·) denotes the
average operator, and diag {a1, . . . , an} denotes the
diagonal matrix whose diagonal entries starting in the upper
left corner are a1, · · · , an.
2 Reduced-rank MIMO channel and rank detection problem
2.1 Reduced-rank MIMO channel model
A MIMO system with M transmit antennas and N receive
antennas is considered. The channels are assumed to be
flat-fading and block-fading. Denote the M × N channel
matrix between the transmitter and the receiver as H with
its (i, j)-th entry being the channel from the ith transmit
antenna to the jth receive antenna. Define
Denote the rank of the channel matrix as r, i.e.,
rank(H) = r. If r = K , the channel has full rank. If r < K ,
the channel has reduced rank. For a reduced-rank MIMO
channel, the number of degrees of freedom in the channel
matrix is less than its dimension. A rank analysis on
typical propagation environments shows that MIMO channels
often experience rank deficiency [3, 4, 29]. For example, in
[29], the rank distribution of 8 × 8 MIMO channels under
four scenarios, namely generalized typical urban,
generalized bad urban, generalized hill terrain, and generalized
rural area, is reported. For all four scenarios, the
probability that the channel matrix has full rank is 0. Especially for
the scenarios of generalized typical urban and generalized
hill terrain, the rank of the 8 × 8 MIMO channel is always
no higher than 4.
A typical reduced-rank channel model for MIMO
system is the finite scatterers/dimensional channel [3, 4],
where the number of (clusters of ) scatterers is finite. The
rank of the channel matrix is not only constrained by
the number of transmit and receive antennas but also
constrained by the number of scatterers. In the finite
scatterers model in [3], under the assumptions that both the
transmit and receive elements are isotropic and
uncoupled and the signal bandwidth is narrow compared with
the overall channel bandwidth, the MIMO channel matrix
can be described as the product of a full-rank steering
matrix and a propagation matrix. The propagation matrix
models independent fast fading, geometric attenuation,
and shadow fading. Following the model in [3] and the
rank factorization in [14, 29], we assume that the MIMO
channel matrix has the following decomposition:
where A is an M × r full-rank matrix and B is a r × N
fullrank rectangular unitary matrix. In this work, we focus on
Rayleigh fading by assuming that entries of A are
independent and identically distributed (i.i.d.) and follow
circularly symmetric complex Gaussian (CSCG) distribution,
Hˆentry =
(S∗S)−1S∗Y.
The entry-based estimation leads to a full-rank matrix.
When r = K , i.e., the channel has full rank, the
SVDbased estimation and entry-based estimation are
equivalent. When r < K , i.e., the channel has reduced rank, the
entry-based estimation will contain subspaces due to the
noise effect only and thus have a worse performance.
Therefore, the rank detection is an essential problem for
the SVD-based estimation. Wrong rank detection will lead
to channel estimation error. In addition, it can degrade
the performance of the MIMO communications. If the
detected rank is smaller than r, some singular values and
the corresponding singular spaces existing in the
channel matrix may be detected as the noise effect only, and
the subspaces will be lost in the estimated MIMO
channel. On the other hand, if the detected rank is larger
than r, some singular values and the corresponding
singular spaces which do not exist in the channel matrix
but appear in Y˜ because of the noise disturbance may be
detected as part of the channel matrix. In MIMO
communications, information and power will be allocated to such
subspaces, which cause loss of information and wasting of
power since the subspaces do not exist in the channel. Our
problem of this paper is to detect the channel rank from
the received signal Y, or the transformed received signal Y˜.
It is noteworthy that the rank detection does not require
extra training since the same observations can be used for
both rank detection and channel estimation.
S∗Y,
Y˜ = H + W˜ ,
with zero mean and unit-variance, i.e., aij ∼ CN (0, 1),
where aij is the (i, j)-th entry of A. It can thus be shown
straightforwardly that each entry of H has CSCG
distribution and hi,j ∼ CN (0, bj 2F ), where bj is the jth column
of B. It is noteworthy that the rank detection schemes
proposed in this paper are not constrained to Rayleigh
distribution and can be extended to more general
channel fading models. The schemes are also not constrained
to the unitary B case and can be extended to any general
deterministic B matrix.
2.2 Training model, SVD-based channel estimation, and rank detection problem
To estimate the channel matrix, a training process is
needed. Denote the length of the training period as T and
average transmit power used for each training time slot
as P. During the training period, the transmitter sends
√PT /MS, where S is the T × M pilot matrix. For the
observability of the channel rank detection model with
respect to all possible rank values, we assume that T ≥
M, which guarantees that the number of independent
equations in the training equation is no less than the
number of independent unknown coefficients in the channel
matrix. We further assume that S is unitary, i.e., S∗S = IM,
which means that the pilot vector sent from each
transmit antenna is orthogonal to each other and has the same
energy. Denote the T × N matrix received at the receiver
as Y. We have
Y =
where W is the T × N noise matrix. Entries of the noise
matrix are assumed to be i.i.d. CSCG random variables
with zero mean and unit-variance. The pilot and the noise
are assumed to be independent to the channel matrix,
which applies to most practical systems. It is noteworthy
that the channel model and training model also apply to
multi-user massive MIMO systems with M single-antenna
users and N base station antennas, where channel training
is conducted in the uplink [5].
The channel estimation problem is to estimate H from
the observation Y. To do this, we first transform the
training equation in (4) to obtain a more direct
relationship between the channel and the (transformed) received
signal. Define
which is the M × N transformed received signal matrix.
By left-multiplying both sides of (4) with
PMT S∗, we have
Since S∗ is an M × T unitary matrix, W˜ is an M × N
random matrix. Entries of W˜ can be shown to be i.i.d. CSCG
random variables with zero mean and their variances are
M/(PT ). From (6), Y˜ is a noisy observation of the channel
matrix H with white Gaussian noises.
Let Y˜ = Pdiag{σ1, · · · , σK }Q∗ be the SVD of Y˜, where P
and Q are M×K and K ×N unitary matrices and σi’s are in
non-increasing order, i.e., σ1 ≥ · · · ≥ σK ≥ 0. If the rank
of H is known to be r, the ML estimation of H has been
proved to be the truncated SVD of Y˜ given as follows [28]:
˜
W
S∗W.
In (8), an estimation with rank r is obtained by
keeping the subspaces with respect to the r strongest singular
values of Y˜. Subspaces with respect to the K − r smallest
singular values are seen as the noise effect and are ignored.
This process guarantees that the estimator has the same
rank with the real channel.
On the other hand, if the channel rank is unknown or
the channel has full rank, an entry-based ML estimation
can be obtained as [24]
To help presenting our results, we first introduce the
following definitions. Define the K × K matrix F(1)(μ) and
the r × r matrix F(r)(μ), whose (i, j)-th entries are:
where γ (k, u) and (k, u) are the lower and upper
incomplete gamma functions [30], respectively.
The following proposition on the probability of correct
rank detection under the condition that the rank of the
MIMO channel matrix is r (for 1 ≤ r ≤ K ) is derived.
Proposition 1. If the rank of H is r, the probability of
correct rank detection of Algorithm 1 with threshold th has
the following lower bound:
C1=
i=1
[(M − i)! (r − i)! ]−1 , C2=
[(L − i)! (K − i)!]−1 .
i=1
3 Threshold selection for single-threshold-based rank detection
We can detect the rank of H from the singular values
of Y˜. Threshold-based algorithm appears to be a
natural and common strategy [23], where the rank of H is
detected as the number of singular values of Y˜ that are
larger than the threshold. With this scheme, singular
values of Y˜ that are smaller than the threshold are seen as
the effect of the noise only; while singular values of Y˜ that
are larger than the threshold are seen as the effect of
nonzero component of the channel matrix with small noise
disturbance.
Let th be the threshold. Recall that σi’s are the singular
values of Y˜ in non-increasing order. The
single-thresholdbased rank detection scheme, denoted as RD1, can be
represented as follows:
If no singular value is larger than th, i.e., σ1 < th,
the rank detection result is set to be 1 since the rank of
a MIMO channel cannot be 0. The algorithm is given in
Algorithm 1.
This single-threshold-based rank detection idea is not
new and was proposed and used in [17–23]. But the major
challenge of using this scheme for rank detection lies in
the selection of the threshold th. Appropriate selection of
the th value is crucial to the detected result.
In this section, we first derive a lower bound on the
probability of correct detection in Section 3.1, then
propose a systematic method for the threshold selection
based on maximizing the lower bound in Section 3.2, and
finally discuss the difference of the proposed method with
existing ones in Section 3.3.
3.1 Derivation of a lower bound on the conditional probability of correct rank detection
To find a systematic way of optimizing the threshold,
we first derive a lower bound on the probability of
correct detection conditioned on an arbitrary rank value of
the MIMO channel matrix. The lower bound takes into
consideration the system dimensions (e.g., T, M, and N ),
training power P, and the distributions of the channel
coefficients and the noises. It will be used in the threshold
optimization in later sections.
Proof 1. Recall that σi’s are the singular values of Y˜ in
non-increasing order, i.e., σ1 ≥ · · · ≥ σr ≥ · · · ≥ σK ≥ 0.
Let λi and γi be the singular values of H and W˜,
respectively, both in non-increasing order, i.e., λ1 ≥ · · · ≥ λr ≥ 0
and γ1 ≥ · · · ≥ γr ≥ · · · ≥ γK ≥ 0. Since rank(H) = r, we
have λr+1 = · · · = λK = 0.
We will first show that when λr ≥ 2 th ≥ 2γ1, our
algorithms will detect the rank of H as r, which is the
correct detection. According to [23], from (6), we have for all
i = 1, · · · , K ,
By noticing that λr+1 = 0, from (15) with i = r + 1, we
have σr+1 ≤ γ1. Thus when λr ≥ 2 th ≥ 2γ1, we have
σr+1 ≤ γ1 ≤ th. Also from (15) with i = r, σr ≥ λr −
γ1 ≥ 2 th − γ1 ≥ th. By noticing that σi’s are in
nonincreasing order, we can conclude that the rank detection
result of Algorithm 1 is r, which is the correct detection.
Thus, a lower bound on the probability of correct
detection is obtained as follows,
P(correct detection|rank(H) = r)
≥ P (λr ≥ 2 th ≥ 2γ1|rank(H) = r)
= P (λr ≥ 2 th & γ1 ≤ th|rank(H) = r)
= P (λr ≥ 2 th|rank(H) = r) P(γ1 ≤ th),
P (λr ≥ 2 th|rank(H) = r) = 1 − Fωr 4 t2h
= C1 · det F(r) 4 t2h
Next, we calculate P(γ1 ≤ th). Recall that entries of W˜
are i.i.d. following CN (0, M/PT ). Thus, (PT /M) W˜W˜ ∗ is
an M × M central Wishart matrix with degree N. The CDF
of its largest eigenvalue is known to be [30]
Given Algorithm 1 and threshold th, the overall
probability of correct rank detection can be lower bounded by
φ ( th). The derivations are as follows.
By using (18) and (20) in (16), the lower bound in (13) is
obtained.
3.2 Threshold optimization for Algorithm 1
Assume that the a priori probability mass function of the
channel rank, P(rank(H) = r) for r = 1, · · · , K , is known.
Define
r=1
P(rank(H) = r).
P(γ1 ≤ th) = C2 det F(1) PT 2
M th
P(correct detection|rank(H) = r)P(rank(H) = r)
r=1
where the last step is because that γ1, the largest eigenvalue
of W˜ , is independent of both λr and the rank of H.
Recall that our channel is modeled as H = AB, where
the M × r matrix A has independent entries following
CN (0, 1) and B is a r × N unitary matrix. Then HH∗ =
ABB∗A=AA∗, which is an M × M central Wishart matrix
with degree r. The singular values of H are the square roots
of the eigenvalues of HH∗. The cumulative density function
(CDF) of the smallest non-zero eigenvalue of HH∗ is known
as follows [30]:
r=1
× P(rank(H) = r),
The optimization problem in (24) is one-dimensional
and can be optimally solved via exhaustive grid search. But
there is a natural tradeoff between precision and
computational complexity. For low computational complexity, in
solving (24), we can find a zero point of d ln φ ( th)/d th
via bisection method and use it as the threshold. This
low-complexity method can result in sub-optimality when
d ln φ ( th)/d th has multiple zero points.
In this section, for the traditional
single-thresholdbased rank detection, we rigorously derived an analytical
lower bound on the correct rank detection
probability, based on which a systematic threshold optimization
scheme that maximizes this lower bound is proposed. The
derived optimal threshold is adaptive to the number of
transmit and receive antennas of the MIMO channel, the
training length and power, and the distributions of the
channel coefficients and the noise. However, it is
independent of the instantaneous channel values or singular
values of the channel matrix. Thus the threshold
optimization can be conducted off-line, which largely reduces
the delay of channel rank detection in real applications.
It is noteworthy that the proposed method is not
limited to Rayleigh fading channel model but can be extended
to other distributions. Our lower bound calculation and
threshold optimization need the distributions of the
singular values of the channel matrix. For other channel
fading models, even if no closed-form expressions for the
distributions of the singular values are available,
numerical estimations of the singular value distributions can be
obtained via simulation, and our method can still be used.
Especially as our methods can be conducted off-line, the
computation complexity is not an issue in the real-time
channel rank detection and channel estimation within a
coherence interval of the channel.
Notice that in the proof of Proposition 1, we can loose
our condition for the lower bound λr ≥ 2 th ≥ 2γ1 to λr ≥
2 th ≥ 2γr without affecting the validity of the proof. With
this change, another lower bounds on the conditional and
overall probabilities of correct rank detection, denoted as
φ˜r( th) and φ˜ ( th), can be obtained as follows:
r=1
Since γr < γ1, it can be shown straightforwardly that
φ˜r( th) > φr( th) and φ˜ ( th) > φ ( th), which means
that the new lower bounds are tighter, and we may obtain
a better threshold by maximizing φ˜ ( th). However, the
expression for P(γr ≤ th) is much more complex than
P(γ1 ≤ th). Notice that it is the rth largest singular value
of W˜ , not the smallest singular value since W˜ has full rank
with probability 1. Thus the calculation of the derivative
of φ˜ ( th) is more involved, and the maximization of φ˜ ( th)
has higher computational complexity. Our simulations
show that the use of φ˜ ( th) improves the performance but
the improvement is moderate. Thus, balancing the
performance and computation complexity, we choose φ ( th) for
our algorithms.
3.3 Difference to existing single-threshold-based rank detection schemes
In this subsection, we explain the difference of our work
with existing results on threshold-based rank detection in
[19, 20, 22, 23].
First, the research in [19, 20, 22, 23] are for real-valued
channel matrix and real-valued noise matrix. Also, they
consider the special case of T = M and S = Im. Our work
applies for complex-valued channel matrix and
complexvalued noise matrix, a general training length T where
T ≥ M, and unitary T × M pilot S. In what follows,
we explain existing threshold selections and calibrate the
results to our model and notation.
In [23], lower and upper bounds on the threshold
selection were provided. No specific threshold value or
optimization method was given.
In [19] and [20], the channel rank detection and
channel estimation problem were considered jointly for the
asymptotic case, where the channel matrix dimensions
approach to infinity but with a fixed ratio. The threshold
for the rank detection was chosen to minimize the MSE
of the truncated SVD channel estimation. The rank
detection follows the traditional single-threshold algorithm in
Algorithm 1. After calibrating their results to our model,
the threshold proposed in [20] is
th,[20] =
and the threshold proposed in [19] is
th,[19] =
In [22], the single-threshold Algorithm 1 was used and
the threshold selection was based on the minimization
of the SURE of the MSE of the channel estimation. Two
methods were proposed. The first method needs
numerical threshold optimization, and the optimization
problem changes with the instantaneous channel realization.
Thus, the threshold optimization needs to be repeated
for every coherence interval, impairing its practicality and
efficiency. As in the second method, an analytical
threshold was proposed. After calibrated to our model, the
threshold is
1 −
where F−1 (x) represents the inverse function of the CDF
W˜ 1
of the largest singular value of the noise matrix W˜ in (7).
In all three aforementioned works, the proposed
threshold selection methods depend on the distribution of the
noise matrix only, where [19, 20] used the asymptotic
behavior of the singular values and singular vectors of
the noisy observation matrix when the dimensions of the
channel matrix approach infinity; and Josse and Sardy [22]
used the distribution of the largest singular value of the
noise matrix. It was assumed in their work that the MIMO
channel matrix is deterministic and their results cannot
take advantage of the distribution of the channel matrix.
On the contrary, we adopt a random channel model, and
our threshold selection takes into account both the
distribution of the channel coefficients and the distribution of
the noises.
4 Improved multiple-threshold rank detection methods
To use the rank detection scheme in Algorithm 1 with
the proposed threshold in the previous section, the a
priori probabilities of the channel rank need to be known.
However, for some wireless communication systems, the
channel rank distribution may not be precisely known due
to the mobility and complexity of the signal propagation
environment. In this case, a rank detection algorithm that
does not rely on the channel rank distribution is required.
In addition, the lower bound on the probability of correct
rank detection in (21) provides the average rank
detection performance over all possible rank values. It may not
be sharp enough for one channel realization with a
specific rank value. Thus, in this section, we propose two
improved rank detection algorithms which do not need
the rank distribution.
4.1 Rank detection algorithm with multiple thresholds
Instead of using only a single threshold for the rank
detection as in Algorithm 1, we propose to use K thresholds
t∗h,1, t∗h,2, · · · , t∗h,K , each corresponding to one of the K
possible rank values, 1, · · · , K . These thresholds are
optimized by maximizing the lower bound on the probability
of correct rank detection conditioned on the channel rank
value, i.e.,
where φi( ) is defined in (13).
t∗h,i serves as the rank detection threshold when the
channel rank is i, and t∗h,i+1 serves as the rank
detection threshold when the channel rank is i + 1. Recall that
σ1, · · · , σK are ordered singular values of Y˜. Thus, it is
natural to detect the channel rank as i when both the
following two conditions are satisfied C1) σi ≥ t∗h,i and C2)
σi+1 < t∗h,i+1. To help the presentation, we define the
following set:
I = {i|σi ≥ t∗h,i & σi+1 < t∗h,i+1, i = 1, · · · , K − 1},
which is the set of rank detection values that satisfy
the two conditions. Since it is possible that I has 2 or
more elements, for the uniqueness of the detection result,
we detect the rank as the largest index that satisfies the
two conditions. In other words, the detection rule, called
RD2, can be represented as follows:
RD2: r = mi∈aIx {i}.
To guarantee the existence of a rank detection result, we
also need to consider the case that there is no element in
I, i.e., I = ∅. The following claim is proved.
Claim 1. For two sequences of real numbers a1, · · · , aK
and b1, · · · , bK . If there exists no integer i (for 1 ≤ i ≤ K −
1) such that ai ≥ bi and ai+1 < bi+1, one of the following
two cases must be true:
1. there exists an integer D, 1 ≤ D ≤ K, such that
ai < bi for i < D, and ai ≥ bi for i ≥ D
2. ai < bi for all i.
Proof 2. Assume that there exists no integer i (for 1 ≤
i ≤ K − 1) such that ai ≥ bi and ai+1 < bi+1. We consider
the two cases aK < bK and aK ≥ bK separately.
When aK < bK , if there exists an i such that ai ≥
bi, let imax be the largest i satisfying ai ≥ bi. We have
aimax ≥ bimax and aimax +1 < bimax +1 , which contradicts the
assumption. Thus ai < bi for all i.
When aK ≥ bK , if there exists an i such that ai < bi, let
imax be the largest i satisfying ai < bi. Then, we will have
ai < bi for all i ≤ imax, based on the same reasoning as
above. In this case, D = imax + 1. When there does not exist
an i such that ai < bi, meaning ai ≥ bi for all i, D = 1.
Based on Claim 1, when I = ∅, we have either σi < ∗
th,i
for i < D and σi ≥ t∗h,i for i ≥ D (1 ≤ D ≤ K ), in
which case the rank detection result should be K ; or σi <
t∗h,i for all i = 1, · · · , K , in which case the rank detection
result should be 1, which is the lowest possible rank for
the random Rayleigh fading channel matrix H.
Given these discussions, our second rank
detection scheme with multiple thresholds is described in
Algorithm 2.
4.2 Iterative rank detection algorithm with multiple thresholds
In our third rank detection algorithm, we iteratively use
the K thresholds t∗h,1, · · · , t∗h,K defined in (25) to refine
our rank detection threshold and detection result. In each
iteration, single threshold-based rank detection is
performed and the threshold value is set using the rank
detection result of the previous iteration. The iteration
ends when the rank detection result of the current
iteration is the same as the result of the previous one. More
specifically, first, a rank detection result r1 is initialized,
e.g., r1 = 1, then t∗h,r1 is used for the threshold of the next
iteration. The new rank detection result is found using
Algorithm 1, that is, the rank is detected as the maximum
index i such that σi ≥ t∗h,r1 or 1 if all singular values of
Y˜ are smaller than t∗h,r1 . This new rank detection result is
denoted as r2. If r2 = r1, t∗h,r2 is used as the threshold for
the next iteration, and a new rank detection result can be
obtained. The scheme is described in Algorithm 3. Please
note that lines 3–8 of Algorithm 3 are the same as those of
Algorithm 1.
t∗h,2 ≥ · · · ≥
t∗h,K , Algorithm 3
Proof 3. We prove the convergence by contradiction.
Assume that the algorithm does not converge. From the
algorithm, r2 is the new rank detection result when using
threshold t∗h,r1 . For any initial value for r1, if r2 = r1,
Algorithm 3 converges and the rank detection result is r1,
which causes a contradiction. Next we consider the cases
r2 > r1 and r2 < r1 separately.
Case 1: r1 < r2. Recall that r2 is the new rank detection
result when using threshold t∗h,r1 . Thus, we have either
• Case A: r2 = K and σK ≥ t∗h,r1; or
• Case B: σr2 ≥ t∗h,r1 and σr2+1 < t∗h,r1.
Since t∗h,i’s are in non-increasing order, for Case A, we
have σK ≥ t∗h,r1 ≥ t∗h,K . Thus the new rank detection
result is K, i.e., r3 = K . Then r3 = r2 and Algorithm 3
terminates, which contradicts the assumption. For Case B,
since r1 < r2, we have σr2 ≥ t∗h,r1 ≥ t∗h,r2 . Thus the
next rank detection result cannot be smaller than r2, i.e.,
r2 ≤ r3. If r2 = r3, Algorithm 3 terminates, which
contradicts the assumption. Thus r2 < r3. The same situation
happens for the next iterations. So, if Algorithm 3 does not
converge, we will find an infinite strictly increasing integer
sequence r1 < r2 < r3 < · · · . This contradicts the fact that
rm’s are in the range of [ 1, K ].
Case 2: r1 > r2. Similarly, we have either Case A (r2 = 1
and σ1 < t∗h,r1 ) or Case B listed above. For Case A, we have
σ1 < t∗h,r1 ≤ t∗h,1. Thus the new rank detection result is 1,
i.e., r3 = 1 and Algorithm 3 terminates, which contradicts
the assumption. For Case B, since r1 > r2, we have σr2+1 <
t∗h,r1 ≤ t∗h,r2 . Thus the next rank detection result cannot
be larger than r2, i.e., r3 ≤ r2. If r2 = r3, Algorithm 3
terminates, which contradicts the assumption. Thus r2 < r3.
The same situation happens for the next iterations. So,
if Algorithm 3 does not converge, we will find an infinite
strictly decreasing integer sequence r1 > r2 > r3 > · · · .
This contradicts the fact that rm’s are in the range
of [ 1, K ].
Claim 2 shows that when the thresholds
corresponding to the K rank values are in non-increasing order,
Algorithm 3 is guaranteed to converge. Also, from the
proof of Claim 2, we can see that the convergence is
guaranteed within K iterations. Our limited simulation
results indicate that the algorithm converges very fast
(within 2–3 iterations). Intuitively, as t∗h,i is the
threshold when the channel rank is i and the singular
values are in a non-increasing order, it is natural to have
t∗h,1, t∗h,2, · · · , t∗h,K in a non-increasing order. However,
we cannot prove this analytically. When violation of the
ordering happens on one threshold, we can simply reset
the threshold to be the average of the one before and the
one after to fix the ordering problem and have guaranteed
convergence.1
Although the convergence of Algorithm 3 is
guaranteed, we cannot not guarantee the uniqueness of the rank
detection solution with respect to different initial values
for r1. In other words, for different initial rank values,
the algorithm may converge to different solutions. An
example is as follows. Assume that σ1 > t∗h,1 > σ2 >
t∗h,2 > · · · > σK > t∗h,K . Then the final rank
detection result of Algorithm 3 will equal to r1 for any initial r1
value.
4.3 Discussion on complexity
The computational complexity of Algorithms 2 and 3 is
composed of two parts: the calculations of t∗h,1, · · · , t∗h,K
and the rank detection part.
The optimization of t∗h,i only depends on the
dimensions of the channel matrix M and N, the training time
T, and the training power P. It is independent of the
channel realization of each coherence interval. Thus the
optimization can be conducted off-line. Further, the
following lemma is proved which can be used to reduce the
computational complexity of the optimization.
Proof 4. For the Hermitian matrices F(1)(μ) and
F(r)(μ), we can show that all their leading principal minors
are positive when μ > 0 from the definitions in (11), (12),
and the CDFs in (17), (19). Thus the two matrices are
positive definite and det F(1) PMT t2h and det F(r) 4 t2h
are log-concave functions since the determinant of a
positive definite matrix is log-concave [31]. Based on [31], the
product of log-concave functions is also log-concave. This
ends the proof.
Notice that t∗h,r is the maximum point of ln φr( th).
With the log-concavity of φr( th), we can find t∗h,r by
finding the unique zero point of d ln φr( th)/d th, using
bisection method. The calculations are as follows. From
(13) and the definitions in (11) and (12), we have
= tr F(r) 4 t2h
−1
−1
/d th. The (i, j)-th entries of Dr and D1
[Dr]i,j = −8 the−4 t2h 4 t2h M−r+i+j−2 ,
[D1]i,j = 2
i+j−2
Next, we analyze the complexity of the rank detection
part. For Algorithm 2, the total number of comparisons in
the worst scenario is 2K , thus the complexity is O(K ). For
Algorithm 3, in the worst case, the number of iterations
is K ; and for each iteration, at most K + 1
comparisons are needed. The overall number of comparisons is
K (K + 1). Thus the complexity is O(K 2). Notice that
K = min{M, N }. Even for massive MIMO systems with N
base station antennas and M single-antenna users, where
N is large (e.g., hundreds), the complexities of the two
proposed rank detection algorithms are linear and quadratic
in the number of users, respectively.
5 Simulation results
5.1 Simulation on the probability of correct rank detection
In this section, simulation results are shown for
Algorithm 1 with our proposed threshold optimization
in Section 3, and the two new rank detection algorithms
with multiple thresholds, Algorithm 2 and Algorithm 3,
proposed in Section 4. We simulate the probability of
correct rank detection for different parameters, such as the
average training power P, the training length T, and the
numbers of transmitter and receiver antennas M and N.
In our simulations, channel coefficients are generated as
Rayleigh fading following the model in (3). S is
generated as a random T × M unitary matrix following the
isotropic distribution. For comparison, we also show the
rank detection accuracy of Algorithm 1 with the
threshold values proposed in [19, 20, 22]. While, the results in
[19, 20, 22] are for T = M and S = IM, we extend them for
a general T ≥ M and unitary S as explained in Section 3.3.
We first consider an 8 × 8 MIMO system, where M =
N = 8. This configuration is typical in 4G standards (e.g.,
LTE-advance, WiMAX Release 2) [32]. Figure 1 shows
the probabilities of correct rank detection for different
average training powers, where T = M = 8. The
channel rank is randomly generated with uniform distribution,
i.e., P(rank(H) = i) = 1/8 for i = 1, · · · , 8. We can
observe from this figure that for the single-threshold
algorithm, Algorithm 1, the proposed threshold has about the
same performance as the threshold in [22] for the whole
training power range and are much better than the ones
in [19, 20]. The proposed multiple-threshold algorithms,
Algorithm 2 and Algorithm 3, achieve considerably higher
detection rate than Algorithm 1 when the training power
is higher than 6.5 dB. Algorithm 2 is slightly better than
Algorithm 3 at high training power, but is slightly worse at
low training power.
To better understand the performance of the schemes
for different channel ranks, for the same network, in
Figs. 2 and 3, we show the probabilities of correct rank
detection for low-rank channel matrix where the rank can
be 1, 2, or 3 with equal probability and for high-rank
channel matrix where the rank can be 4, 5, or 6 with
equal probability.2 For the low-rank case, Algorithm 1
with the proposed threshold outperforms the proposed
Algorithms 2 and 3. Compared with existing work, the
proposed threshold has better performance than [22] at
high training power but worse performance at low
training power. It is significantly better than results in [19, 20]
for most training power values. For the high-rank case,
among the three proposed schemes, Algorithm 2 has
the best performance, followed by Algorithm 3. They
achieve significantly higher detection rate than schemes in
[19, 20], especially for low training power. Compared with
[22], the proposed schemes are largely better for small and
medium P and slightly worse for large P.
In Fig. 4, for the same 8 × 8 MIMO system with
uniform rank value from 1 to 8, we simulate the probability of
correct rank detection for different training lengths. The
figure shows that the proposed Algorithms 2 and 3 are
significantly better than Algorithm 1. For Algorithm 1, the
proposed threshold is slightly worse than the one in [22],
but better than those in [19, 20]. The figure also shows that
the proposed Algorithm 2 achieves the highest correct
detection probability.
In Figs. 5 and 6, we show the rank detection
performance for different numbers of transmit antennas while
the number of receive antennas is fixed as 8 and different
numbers of receive antennas while the number of
transmit antennas is fixed as 8. The channel rank is random and
uniformly distributed from 1 to K. The training length is
T = 12. For Fig. 5, the average transmit power of each
transmit antenna is set to be 1 dB, thus, P/M = 1 dB, or
equivalently, P = 100.1M = (1 + 10 log10 M) dB. This
is for fair comparison among systems with different M
values. For Fig. 6, we set P = 10 dB. From the figures,
we can see that our proposed Algorithms 2 and 3 always
achieve significantly higher detection rate than existing
Fig. 1 Probability of correct rank detection of 8 × 8 MIMO systems for different average training power. The rank of the channel matrix is uniformly
distributed between 1 and 8
ones. Algorithm 1 with the proposed threshold is also
superior to existing ones in general except for large M
values in Fig. 5, where it is slightly worse than [22].
In Fig. 7, we consider a 10 × 100 uplink massive MIMO
system. The receiver is a base station where 100
antennas are deployed. The transmitters are 10 single-antenna
users or one user with 10 antennas. The training length
is T = 20. The channel rank is assumed to be uniformly
distributed over {5,6,7,8,9}3. The probabilities of correct
rank detection for different average training powers are
shown. We can see that the proposed schemes outperform
all existing ones for the whole range of the training power.
Fig. 2 Probability of correct rank detection of 8 × 8 MIMO systems for different average training power. The rank of the channel matrix is uniform
between 1 and 3
Fig. 3 Probability of correct rank detection of 8 × 8 MIMO systems for different average training power. The rank of the channel matrix is uniform
between 4 and 6
The figure also shows that Algorithm 3 is slightly better
than Algorithm 2 at low and medium training power, but
slightly worse at high training power.
5.2 Simulation on MSE and outage probability
In this section, to show the effect of rank detection on the
channel estimation and the MIMO communication
performance via simulations on the MSE of the truncated
SVD channel estimation and the outage probability of
the MIMO multiplexing communications. The truncated
SVD channel estimation given the rank detection result
has been shown in (8) in Section 2.2. The MSE is defined
as
1 E{ Hˆ − H 2F },
Fig. 4 Probability of correct rank detection of 8 × 8 MIMO systems for different training lengths T. The rank of the channel matrix is uniformly
distributed between 1 and 8
which is the MSE per channel coefficient. Traditionally,
the average in (29) is over the noises. But here it is
over both the noises and the channels, since the channel
coefficients are also random variables. In the conducted
Monte-Carlo simulation, a distinct channel realization is
used for each iteration.
In Fig. 8, we show the MSE for an 8 × 8 MIMO
system with different rank detection methods. The rank of
the channel matrix is uniformly distributed between 1 and
8. The training length is T = 8. The simulated train
ing power range is from 8 to 18 dB, which is the medium
and high power range. Previous simulations show that our
algorithms achieve higher rank detection probability in
this training power range. From the figure, we can see
that our algorithms achieve lower MSE than the existing
algorithms. For comparison, we also simulated the MSE
Fig. 7 Probability of correct rank detection of 10 × 100 massive MIMO system for different average training power, with high channel rank values
for SVD-based estimation with perfect rank detection. We
can see that it has the lowest MSE, which helps verify
that better rank detection will improve the SVD-based
channel estimation. The figure also shows that the
SVDbased estimation achieves lower MSE than entry-based
estimation.
For the outage probability, MIMO multiplexing
transmission is adopted, where multiple data streams are sent
through different channel eigenspaces. First we rewrite
the channel estimate in (8) as follows
where rˆ is the rank detection result, Uˆ rˆ is the M × rˆ
matrix composed of the first rˆ columns of P, and Vˆ r
ˆ
is the M × rˆ matrix composed of the first rˆ columns
of Q. Since the rank detection result is rˆ, only rˆ users
Fig. 8 MSE of 8 × 8 MIMO systems for different average training powers. The rank of the channel matrix is uniformly distributed between 1 and 8
y˜ = y Vˆrˆ =
where w˜ is the equivalent noise vector. Define
Z = Uˆ∗HVˆ rˆ. The signal-to-noise-plus-interference ratio
r
ˆ
(SINR) of the ith data stream can be represented as
SINRi =
Pr |zi,i|2
ˆ
r
ˆ
j=1,j=i |zj,i|2
where σn2 is the noise variance. In the following
simulations, we assume that the noises have unit variance, i.e.,
σn2 = 1.
For fair comparison of different rank detection schemes,
the system outage probability is calculated as follows. We
assume that there are K data streams/symbols in total. If
the rank detection result is rˆ, only rˆ symbols are
transmitted and the communications of the rest symbols are
seen to be in outage. For example, for a 8 × 8 channel,
if the rank is detected as 5, the three symbols that are
not served are in outage. This setup will make the
outage probability value appear to be high especially when
the channel rank is low. But it is fair since it avoids the
preference of always having a lower rank detection.4 For
the rˆ symbols that are actually transmitted, a symbol is
in outage if its SINR is below the pre-defined threshold.
In Fig. 9, the outage probabilities for an 8 × 8 MIMO
system are shown for different training powers. The rank
of the channel matrix is uniformly distributed between 1
and 8. The training length is T = 8. The total transmit
power is fixed and set as 8 × 10 dB. The SINR threshold is
3 dB. We compare our schemes with the cases of perfect
channel estimation, entry-based estimation, and
SVDbased estimation with perfect rank detection. The figure
shows that perfect rank detection achieves the lowest
outage probabilities at medium and high training power
ranges, which verifies that better rank detection will
improve the outage performance. It is also shown in the
figure that the proposed multiple-threshold algorithms
achieve almost the same outage probability as perfect
rank detection and are better than other rank detection
algorithms.
6 Conclusions
In this paper, we proposed novel threshold-based rank
detection algorithms for reduced-rank MIMO systems.
Different from previous work, we consider a MIMO
system with a random channel matrix model, a general
training length, and a unitary training matrix. Lower
bounds on the probability of correct rank detection were
derived using the distribution of the channel matrix and
noise matrix, based on which the rank detection
thresholds can be optimized. In addition to the traditional
single-threshold detection algorithm, we further
proposed two low-complexity multiple-threshold algorithms.
Fig. 9 Outage probability of 8 × 8 MIMO systems for different average training powers. The rank of the channel matrix is uniformly distributed
between 1 and 8
Compared with the existing schemes, our proposed
schemes can achieve higher rank detection rate for various
scenarios. Our simulation results also show that better
rank detection can improve the channel estimation quality
and the outage performance of the MIMO system.
1In simulations, the thresholds we numerically obtain
from (25) are in non-increasing order, except when the
MIMO channel dimension gets large. When the values of
M and N are large, violation of the non-increasing order
occasionally happens due to the limited precision of
computer calculation.
2The cases where the channel rank is 7 and 8 are not
considered, since based on the the experimental results
on several scenarios in [33], the rank of the 8 × 8 channel
matrix is always less than 7.
3For very small rank values, e.g., 1,2,3,4, our schemes
are inferior to existing ones.
4For example, for a 8 × 8 MIMO system whose channel
rank is uniformly distributed between 1 and 8, under this
setup, a lower bound on the outage probability will be
i7=0 i/8/8 = 43.75 %. Another way to show the
comparison is to draw the simulated outage probability
values subtracted by the lower bound. But either way, the
same comparison result can be obtained.
Competing interests
The authors declare that they have no competing interests.
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