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1

Interleaved Training and Training-Based Transmission Design for Hybrid
Massive Antenna Downlink

arXiv:1710.06949v2 [] 11 Mar 2018

Cheng Zhang, Student Member, IEEE, Yindi Jing, Member, IEEE, Yongming Huang, Senior Member, IEEE,
Luxi Yang, Member, IEEE
Abstract—In this paper, we study the beam-based training
design jointly with the transmission design for hybrid massive
antenna single-user (SU) and multiple-user (MU) systems where
outage probability is adopted as the performance measure.
For SU systems, we propose an interleaved training design to
concatenate the feedback and training procedures, thus making
the training length adaptive to the channel realization. Exact
analytical expressions are derived for the average training length
and the outage probability of the proposed interleaved training.
For MU systems, we propose a joint design for the beam-based
interleaved training, beam assignment, and MU data transmissions. Two solutions for the beam assignment are provided with
different complexity-performance tradeoff. Analytical results and
simulations show that for both SU and MU systems, the proposed
joint training and transmission designs achieve the same outage
performance as the traditional full-training scheme but with
significant saving in the training overhead.
Index Terms—Hybrid massive antenna system, outage probability, beam training, beam assignment.

I. I NTRODUCTION
Massive multiple-input-multiple-output (MIMO) is considered to be a promising technique to further increase the
spectrum efficiency of wireless systems, thanks to the high
spatial degrees-of-freedom it can provide [2]–[4]. However,
the conventional full-digital implementation where one full
radio-frequency (RF) chain is installed for each antenna tends
to be impractical due to its high hardware costs and power
consumptions [5], especially for systems targeted at the millimeter wave (mmWave) band [6]–[8]. Recently, enabled by
the cost-effective and low-complexity phase shifters, a hybrid
analog-digital structure has been applied for massive antenna
systems which can effectively reduce the hardware complexity
and costs via a combination of full-dimensional analog RF
processing and low-dimensional baseband processing [9], [10].
It has been shown that this hybrid structure incurs slight
performance loss compared with its full-digital counterpart [9],
when perfect channel state information (CSI) is available.
One crucial practical issue for massive MIMO downlink is
the acquisition of CSI at the base station (BS), especially when
no channel reciprocity can be exploited, e.g., systems with
This work was supported in part by the National Natural Science Foundation
of China under Grants 61720106003, in part by the Research Project of
Jiangsu Province under Grant BE2015156, in part by the Scientific Research
Foundation of Graduate School of Southeast University. Part of this work
has been accepted by IEEE International Conference on Acoustics, Speech
and Signal Processing (ICASSP) 2018 [1]. The guest editor coordinating the
review of this manuscript and approving it for publication was Dr. Christos
Masouros. (Corresponding author: Yongming Huang.)
C. Zhang, Y. Huang and L. Yang are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, P. R.
China (email: zhangcheng1988, huangym, lxyang@seu.edu.cn).
Y. Jing is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada, T6G 1H9 (email:
yindi@ualberta.ca).

frequency-division-duplexing (FDD) [11]. Due to the massive
number of channel coefficients to be estimated [12], traditional
training and channel estimation schemes cause prohibitive
training overhead. Thus, new channel estimation methods have
been proposed for massive MIMO downlink by exploiting
channel statistics [13] or channel sparsity [14]. While these
apply to full-digital systems, they are less effective for the
hybrid structure. Due to the limited number of RF chains
and the phase-only control in the RF processing, it is more
challenging to acquire the channel statistics or align with
statistically dominant directions of the channels accurately and
transmit sensing pilots with high quality [15].
One popular method for the downlink training with hybrid
massive antenna BS is to combine the codebook [16] based
beam training with the traditional MIMO channel estimation
[10], [17], [18]. In this method, a finite codebook is used which
contains all possible analog precoders, called beams. Then the
channel estimation problem is transformed to the estimation
of the beam-domain effective channels. This can reduce the
dimension of the channel estimation problem if the number
of desired beams is limited. The remaining difficulty lies in
finding the desired beams or analog precoders with affordable
training overhead.
Several typical beam-based training schemes have been
proposed for hybrid massive MIMO downlink. For systems
with single RF chain at the BS, which can serve single user
(SU) only, one straightforward method is to exhaustively train
all possible beams in the codebook, then to find the best
beam for transmission. Another typical method is based on
hierarchical search [10], [17], where all possible wide beams
are trained first and the best is selected. Within this selected
best wide beam, the beams with narrower beamwidth are
trained and the best is chosen. The training procedure is
repeated until the optimal beam with acceptable beamwidth
is found. Generally speaking, the hierarchical search scheme
has lower training overhead than the scheme with exhaustive
search. However, since hierarchical search uses wide beams
first, its beam alignment quality is very sensitive to the prebeamforming signal-to-noise-ratio (SNR) [19]. Meanwhile, its
advantage of lower training overhead diminishes as the number
of channel paths increases or when applied to multiple-user
(MU) systems [10], [20].
Beam training procedure for hybrid massive antenna systems with multiple RF chains is analogous to that of the
single RF chain case. The users generally feed back channels
of multiple beams [21], and then the BS selects the best
beam combination, and constructs the corresponding analog
precoder and baseband precoder for data transmission. The
work in [22] studied the beam selection problems with respect to the capacity and signal-to-interference-plus-noise-ratio

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(SINR) maximization. In [23], the tradeoff of transmission
beamwidth and training overhead was studied, where the
beamwidth selection and pair scheduling were jointly designed
to maximize network throughput. In [21], partial beam-based
training was proposed where only a subset of the beams are
trained and the sum-rate loss compared with full training was
studied. Another scheme proposed in literature is based on
tabu search [8] where several initial beams are first chosen and
the training procedure is only executed within the neighbors
obtained by changing the beam of only one RF chain and
fixing the others. The training stops when a local optimal beam
combination is found in terms of mutual information.
For existing beam-based training and corresponding transmission schemes, the basic idea is to obtain the complete
effective CSI for the full or selected partial beam codebook.
Based on the obtained effective CSI values, data transmission
designs are then proposed. In such schemes, the training design
and data transmission design are decoupled. For satisfactory
performance, the size of the training beam codebook needs to
increase linearly with the BS antenna number, leading to heavy
training overhead for massive MIMO systems. The decoupled
nature of existing schemes also imposes limitations on the
tradeoff between training overhead and performance. Further,
only the throughput or diversity gain has been considered
in existing work. In [24], an interleaved training scheme
was proposed for the downlink of SU full-digital massive
antenna systems with independent and identically distributed
(i.i.d.) channels. In this scheme, the BS trains its channels
sequentially and the estimated CSI or indicator is feed back
immediately after each training step. With each feedback, the
BS decides to train another channel or terminate the training
process based on whether an outage occurs. The scheme was
shown to achieve significant reduction in training overhead
with the same outage performance compared to traditional
schemes.
By considering the aforementioned limitations of existing
training schemes and exploiting the interleaved training idea,
in this paper, we study the beam-based training design jointly
with the data transmission design for hybrid massive antenna
systems with single user and multiple users. The outage probability is adopted as the performance measure. We consider
interleaved training designs that are dynamic and adaptive, in
the sense that the length of the training interval depends on the
channel realization and the termination of the training process
depends on previous training results, to achieve favorable
tradeoff between the outage performance and the training
overhead. Compared with the work in [24], our work is
different in the following aspects. First, [24] is on full-digital
massive MIMO systems with i.i.d. channels. Whereas, in this
work, we consider hybrid massive antenna systems and a more
general channel model that incorporates channel correlation
and limited scattering. Our work uses beam-based training
while in [24] the training is conducted for each antenna
sequentially. Second, given the above differences in system,
channel, and transceiver models, the theoretical derivations and
analytical results in our work are largely different. The numbers of resolvable channel paths and RF chains are two new
parameters in our work. They both make the derivations on the

2

average training length and the outage probability considerably
more complicated compared to the case in [24]. Our work also
provides more comprehensive analytical performance results
along with abundant new insights. Third, this work considers
both SU and MU systems. Especially for MU systems, new
design issues appear, such as the beam assignment and the
local CSI at each user. Although the same basic interleaved and
adaptive training idea is used, the implementation of this idea
for MU systems is far from immediate applications and the
algorithm has large difference to the one for the SU case. The
distinct contributions of this work are summarized as follows.
•

•

•

•

For SU massive antenna systems with arbitrary number of
RF chains, we propose a beam-based interleaved training
scheme and the corresponding joint data transmission
design. The average training length and the outage probability of the proposed scheme are studied, where exact
analytical expressions are derived.
For MU massive antenna systems with arbitrary number
of RF chains, we propose a joint beam-based interleaved training and data transmission design. Two beam
assignment solutions, i.e., exhaustive search and maxmin assignment, are proposed with different complexityperformance tradeoff. Compared to exhaustive search,
the low-complexity max-min method induces negligible
increment in the average training length and small degradation in the outage performance. Due to its advantage
in complexity, the max-min method is more desirable for
the proposed MU scheme.
Analytical results and simulations show that for both
SU and MU systems, the proposed training and joint
transmission designs achieve the same outage probability
as the traditional full-training scheme but with significant
saving in the training overhead.
Based on the analytical results and simulations, useful
insights are obtained on the performance of several
special but typical scenarios, e.g., small channel angle
spread (AS) or limited scattering, and also on the effect
of important system parameters, e.g., the BS antenna
number, the RF chain number, the channel path number
or AS, and the rate requirement, on the average training
length and the outage performance.

Specifically, for the average training length of the proposed
SU scheme, our analysis reveals the following. 1) For channels
with limited scattering, the average training length is no
longer a constant as in the case of i.i.d. channels, but linearly
increases with respect to the BS antenna number. However, it
decreases linearly with increasing channel paths. Meanwhile,
fewer RF chains or higher rate requirement has negligible
effect on the average training length. 2) For channels with
non-negligible AS, the average training length is a constant
dependent on the AS, the RF chain number, and the rate requirement. The constant increases for higher rate requirement
while the increasing slope decreases with more RF chains.
Moreover, smaller AS (larger channel correlation) reduces the
increasing speed of the average training length with higher
rate requirement. For the outage probability of the proposed
scheme, the following major new insights are achieved. 1)

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For channels with limited scattering and single RF chain, the
outage probability scales as the reciprocal of the BS antenna
number to the power of the channel path number. 2) For
channels with non-negligible AS and single RF chain, the
outage probability decreases exponentially with respect to the
BS antenna number. 3) More RF chains can further decrease
the outage probability for both kinds of channels.
Notation: 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 conjugate transpose, transpose,
and trace are denoted by AH , AT and tr{A}, respectively.
For a vector a, its conjugate counterpart is a∗ . E[·] is the
mean operator and Pr[·] indicates the probability. The notation
a = O (b) means that a and b have the same scaling with
respect to a parameter given in the context. kak denotes the
2-norm of a Rand kAkF denotes the Frobenius norm of A.
x
Υ (s, x) = 0 ts−1 e−tRdt is the lower incomplete gamma
∞
function and Γ (s, x) = x ts−1 e−t dt is the upper incomplete
gamma function. X 2 (k) denotes the chi-squared distribution
with k degrees of freedom. CN (0, Σ) denotes the circularly
symmetric complex Gaussian distribution where the mean
vector is 0 and the covariance matrix is Σ.
II. S YSTEM M ODEL AND P ROBLEM S TATEMENT
A. System Model
Consider the downlink of a single-cell1 massive antenna
system with hybrid beamforming structure. The BS employs
Nt ≫ 1 antennas with NRF ∈ [1, Nt ) RF chains and serves
U single-antenna users. Since for effective communications,
each user requires a distinct beam, U ≤ NRF is assumed. Let
H = [h1 , ..., hU ] ∈ CNt ×U be the downlink channel matrix.
1) Channel Model: We consider the typical uniform array, e.g., uniform linear array or uniform planar array, at
the BS and high dimension. The beamspace representation
of the channel matrix becomes a natural choice [15], [25]
where the antenna space and beamspace are related through
a spatial discrete Fourier transform (DFT). Denote the DFT
matrix as D ∈ CNt ×Nt where the ith column is di =
[1, e−j2π(i−1)/Nt , ..., e−j2π(i−1)(Nt −1)/Nt ]T , ∀i. Assume that
there are L ∈ [1, Nt ] distinguishable scatterers or paths [15]
in User u’s channel ∀u, and define the set of their direction
indices as Iu = {Iu,1 , ..., Iu,L }. The channel vector of User
u can be written as
hu = Dh̄u = [d1 , ..., dNt ][h̄u,1 , ..., h̄u,Nt ]T ,

(1)

where h̄u,i ∼ CN (0, 1/L) for i ∈ Iu and h̄u,i = 0 for i ∈
/
Iu . This channel model can be understood as an asymptotic
approximation of the geometric channel model in [26] which
has been widely used for the mmWave band [10], [27] with
discretized angle distribution of the channel paths.
Specifically, we assume that different users have independent path directions and gains, and the L-combination
(Iu,1 , · · · , Iu,L ) follows discrete uniform distribution with
1 This work only considers single-cell systems without inter-cell interference. The main reason for the simplification is to focus on interleaved training
design and fundamental performance behavior. The multi-cell case is left for
future work.

3

each element on [1, Nt ]. One justification is given as follows.
Generally, for the geometric channel model, the angles of
different paths are independent following uniform distribution
[10] and no two paths’ continuous angles are the same. The
beamspace representation equivalently divides the angle space
into Nt uniform sections [15]. When Nt is large enough,
no two paths are in the same section. As the variances of
h̄u,i , i∈Iu are set to be the same, the average power difference
among different paths is not embodied in this channel model.
Further, it is assumed that all users have the same L. When
L = Nt , our channel model becomes the i.i.d. one [24]. When
L = 1, it reduces to the single-path one [21].
While generally speaking, the number of distinguishable
channel paths L is arbitrary in our work, two typical scenarios
are of special interest, corresponding to different scaling with
respect to Nt . The first typical scaling for L is that it is a
constant with respect to Nt , i.e., L = O(1). This corresponds
to channels with extremely small AS where having more
antennas does not result in more distinguishable paths. One
application is the outdoor environment with a few dominant
clusters [15]. Another typical scaling is when L linearly
increases with Nt , i.e., L = cNt with c ∈ (0, 1] being a
constant. It corresponds to channels with non-negligible AS
where c is the value of the AS. Since the spatial resolution
increases linearly with Nt , it is reasonable to assume that
the number of distinguishable path increases linearly with
Nt . One application is the indoor environment with a large
amount of reflections [15]. Similarly, two typical scalings for
NRF are NRF = O(1) and NRF = c̄Nt with c̄ ∈ (0, 1]
being a constant. The former case is more interesting from
the perspective of low hardware costs.
2) Hybrid Precoding and Outage Probability: The hybrid
structure at the BS calls for an analog RF precoding followed
with a baseband precoding.
The analog precoder FRF ∈ CNt ×Ls is realized by phase
shifters for low hardware complexity, where Ls is the number
of beams used for the transmission and Ls ≤ NRF . All
elements of FRF have the same constant norm. Without loss
of generality, we assume k[FRF ]i,j k2 = 1/Nt , ∀i, j. The
codebook-based beamforming scheme is used in this work,
where columns of FRF are chosen from a codebook of
vectors FRF [10], [17]. Naturally, with the channel model
in (1),
is used [28], where FRF =
√
√ the DFT codebook
{d∗1 / Nt , · · · , d∗Nt / Nt }. Each element in the codebook
is also called a beam and there are Nt beams in total.
With a given analog beamforming matrix FRF , the effective
channel√matrix √
for the baseband is HT FRF . More specifically,
T ∗
hu di / Nt = Nt h̄u,i is the effective channel of User u on
Beam i. If h̄u,i 6= 0, Beam i is a non-zero beam for User u.
The next to discuss is the baseband precoding and the outage
probability. In what follows, we consider the SU case and the
MU case separately due to their fundamental difference.
For the SU case (i.e., U = 1) where the BS chooses Ls
beams
for analog precoding, the transmitted signal vector is
√
P FRF fBB s and the transceiver equation can be written as
y=

√ T
√
P h FRF fBB s + n = P h̄T DFRF fBB s + n, (2)

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where P can be shown to be the short-term total transmit
power, h is the channel vector from the BS to the user,
fBB is the baseband beamformer, s denotes the data symbol with unit power, and n is the additive noise following
CN (0, 1). For a fixed h, with perfect effective channel vector,
i.e., h̄T DFRF , at the user side, the channel capacity is
log2 (1+P kh̄T DFRF fBB k2 ) bps/Hz. For a given transmission
rate Rth , an outage event occurs if
2Rth − 1
,
(3)
P
where α is called the target normalized received SNR. Thus,
for random h, the outage probability is
kh̄T DFRF fBB k2 ≤ α ,

out(FRF , fBB ) = Pr(kh̄T DFRF fBB k2 ≤ α).

(4)

If further the effective channel vector is perfectly known at the
BS, fBB can be designed to match the effective channel vector,
i.e., fBB = (h̄T DFRF )H /kh̄T DFRF k, which is optimal in
the sense of minimizing the outage probability. In this case,
the outage probability becomes Pr(kh̄T DFRF k2 ≤ α) which
is dependent on FRF .
If more than Ls non-zero beams are available, beam selection is needed. Define the set of indices of the available
non-zero beams as A = {a1 , ..., aj }. The optimal beamselection is to find the strongest Ls ones within the set.
By ordering the magnitudes of the effective channels as
kh̄s1 k ≥ kh̄s2 k ≥ ... ≥ kh̄sLs k ≥ · · · ≥ kh̄sj k. The set of
indices of the selected beams is S = {s1 , ..., sLs }. Thus the
beamforming matrices are:

 ∗
d∗sLs
ds1
,
FRF = √ , ..., √
Nt
Nt
[h̄s1 , ..., h̄sLs ]H
fBB = q
.
(5)
kh̄s1 k2 + ... + kh̄sLs k2

P
Ls
2
.
The outage probability reduces to Pr
k
≤
α/N
k
h̄
t
s
i
i=1
For the MU case, we assume that the BS uses U out of
the NRF RF chains to serve the U users, i.e., Ls = U . The
received signal vector at the users can be presented as
√
y = P HT FRF FBB s + n,
(6)
where s ∼ CU×1 contains the U independent data symbols
satisfying E[ssH ] = (1/U )IU ; and n ∼ CN (0, I) is the
additive noise vector. For the shot-term power normalization
H
at the BS, we set tr{FRF FBB FH
BB FRF } = U .
Without loss of generality, assume that Beams n1 , · · · , nU
are selected
to serve Users
1, · · · , U respectively. Thus FRF =
√
√
[d∗n1 / Nt , ..., d∗nU / Nt ]. The effective channel matrix Ĥ =
HT FRF is therefore


h̄ 1 · · · h̄1,nU
p  1,n

..
..
..
Ĥ=[h̄1 , ..., h̄U ]T DFRF = Nt 
 .(7)
.
.
.
h̄U,n1 · · · h̄U,nU
One of the most widely used baseband precodings is the zeroforcing (ZF) scheme [22], [27]: FBB = λĤH (ĤĤH )−1 ,
where
q
(8)
λ = U/kFRF ĤH (ĤĤH )−1 k2F .

4

With ZF baseband precoding, the user-interference is cancelled
and the received SNRs of all users are the same, which are
SNRMU = (P/U )λ2 .

(9)

For a given target per-user transmission rate Rth , an outage
event occurs for User u if SNRMU ≤ (P/U )ᾱ, where
2Rth − 1
U
(10)
P
is the target normalized per-user received SNR. The outage
probability for the system with U users is thus

out(FRF , FBB ) = Pr λ2 ≤ ᾱ .
(11)
ᾱ ,

B. Beam-Based Training and Existing Schemes
To implement hybrid precoding, including both the beam
selection/assignment and the baseband matching/ZF, CSI is
needed at the BS, thus downlink training and CSI feedback
must be conducted. Instead of all entries in H, for the hybrid
massive MIMO system under codebook-based beamforming,
√
the BS only needs values of the effective channels Nt h̄u,i ’s.
Thus beam-based training is a more economical choice than
traditional MIMO training [12]. In what follows, existing
beam-based training schemes are briefly reviewed.
For SU systems, the typical beam-based training scheme operates as follows. For each channel realization, the BS sequentially transmits along the Nt beams for the user to estimate the
corresponding effective channels. The effective channel values
are then sent back to the BS. Another type of beam-based
training scheme uses hierarchical search [10], which generally
has smaller training length. But this advantage diminishes as
the path number L increases or the pre-beamforming SNR
decreases, along with performance degradation [19]. Specifically, for SU systems with Nt BS antennas, NRF RF chains,
and L channel paths, the training length of the hierarchical
scheme is THT−SU = M L2 logM (Nt /L), where M ≥ 2 is
the partition number for the beamwidth, e.g., M = 2 means
bisection [10, Section V-B-1]. Note that the training length
is independent of NRF since the ideal hierarchical codebook
is assumed. Meanwhile, the mechanism of the hierarchical
scheme needs L ≤ Nt /M < Nt /e0.5 = Nt /1.65 for plausible
results. It can be shown that the training length increases with
L for the aforementioned range of L. As for the tabu search
based training [8], it is sensitive to the initialization step and
stops when a local optimal beam combination is found. Thus
it has worse outage performance.
For MU systems, beam-based training has been studied in
[21], [27], [28] with similar procedure to the SU case reviewed
above since all users can conduct channel estimation at the
same time when the BS sends a pilot along one beam. But in
[21], a more general scheme was proposed where only Lt out
of Nt beams are selected for training. The value of Lt can
be used to leverage the tradeoff between the training overhead
and the performance. Larger Lt means longer training period,
less time for data transmission, and better transmission quality;
while smaller Lt leads to the opposite. For the special case of
Lt = Nt , the scheme becomes the full training case in [28].

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It should be noted that while the training procedure for the
MU case is similar to that for the SU case, the effective CSI
feedback and the beam assignment at the BS are different.
Specifically, for an MU system with Lt beams being trained,
assume that there are ju non-zero beams among the trained
beams for User u, ∀u. Then User u feeds back to the BS the
effective channels of the ju non-zero beams along with their
indices. And the BS finds the beam assignment based on the
CSI feedback. The work in [22] considered the magnitude of
the path and the SINR, while the sum-rate maximization was
used in [21], [27].
C. Motivations of This Work
This work is on beam-based training design for SU and MU
massive MIMO systems with hybrid structure. The object is to
propose beam-based training schemes and corresponding SU
and MU transmission schemes with reduced training length,
without sacrificing the performance compared with the full
training case. The training length, or the number of symbol
transmissions required in the training period, is a crucial
measure for training quality since it affects both the available
data transmission time and the beamforming/precoding gain
during transmission. In existing beam-based training schemes,
the training length is fixed regardless of the channel realization
and further the effective CSI feedback is separated from
the training procedure. Thus we refer such designs as noninterleaved training (NIT). The combination of the NIT and
data transmission for SU and MU systems are referred to as
NIT-SU and NIT-MU transmission schemes, respectively. For
our object, interleaved training idea is used, where for each
channel realization, the training length is adaptive to the channel realization. Further, the effective CSI or indicator feedback
is concatenated with the pilot transmissions to monitor the
training status and guide the action of the next symbol period.
Naturally, for interleaved schemes, the training and the data
transmission need to be designed jointly.
In addition, in this work, outage probability is used as
the major performance measure, which is different to most
existing work where the sum-rate [27] and SINR [22] are
used. While outage probability is a useful practical quality-ofservice measure for wireless systems, its adoption in massive
MIMO is very limited [29] due to the high challenge in the
analysis. With outage probability as the performance measure
and the aim of reducing training length, in the proposed
interleaved schemes, the basic idea is to stop training when the
obtained effective CSI is enough to support the targeted rate
to avoid an outage. Other than training designs, quantitative
analysis will be conducted on the outage performance of
the proposed schemes for useful insights in hybrid massive
antenna system design.
III. I NTERLEAVED T RAINING FOR SU S YSTEMS
P ERFORMANCE A NALYSIS

AND

This section is on the SU system, where interleaved beambased training and the corresponding SU transmission are
proposed. Further, outage probability performance is analyzed
as well as the average training length of the proposed scheme.

5

A. Proposed Interleaved Training and SU Transmission
Recall that the object of interleaved training is to save
training time while still having the best outage probability
performance. Thus, instead of training all beams and finding
the best combination as in NIT, the training should stop right
after enough beams have been trained to avoid outage. Since
the set of L non-zero beams for the user I is random with
uniform distribution on the set {1, · · · , Nt }, and the channel
coefficients along the non-zero beams are i.i.d., the priorities of
the training for all beams are the same. Therefore, the natural
order is used for beam training, i.e., the BS trains from the
first beam to the Nt -th beam sequentially. The training stops
when the outage can be avoided based on the already trained
beams or no more beam is available for training.
Let B contain the indices of the non-zero beams that have
been trained. Let LB , min(NRF , |B|) be the maximum number of non-zero beams that can be used for data transmissions
given the number of RF chains and the number of known nonzero beams. Let S contain the indices of the LB known nonzero beams with the largest norms. The proposed interleaved
training and the corresponding SU transmission scheme are
shown in Algorithm 1.
Algorithm 1 Proposed Interleaved Training and Corresponding SU Transmission (IT-SU) scheme
1: B = ∅;
2: for i = 1, ..., Nt do
3:
The BS trains the ith beam; The user estimates h̄i ;
4:
If kh̄i k > 0, B = B ∪ {i} and the user finds S, which
contains the indices of the LB P
non-zero beams with the
2
largest norms, then
calculates
l∈S kh̄l k ;
P
2
5:
if kh̄i k = 0 or l∈S kh̄l k ≤ α/Nt then
6:
The user feeds back “0”; Continue;
7:
else
8:
The user feeds back h̄l , for all l ∈ S along with their
indices;
9:
The BS constructs FRF and fBB as in (5) and
conducts data transmission; Break;
10:
end if
11: end for
In the proposed scheme, at the ith training interval where
i ≤ Nt , the BS sends a pilot
√ for the user to estimate the ith
beam value: h̄i (the scalar Nt is omitted for brief notation)2.
If it is a non-zero beam (i.e., kh̄i k > 0), the user compares
the received SNR provided by the LB strongest beams known
from the finished i training intervals with the target normalized
received SNR α to see if an outage
P event will happen given the
obtained CSI. If kh̄i k = 0 or l∈S kh̄l k2 ≤ α/Nt and i < Nt ,
the already trained beams cannot provide a beam combination
to avoid outage. Thus the user feeds back the indicator “0”
to request the BS to continue training the next beam. For
the special case of i = Nt , all beams have been trained
2 In this work, the channel estimation is assumed to be error-free. For
massive antenna systems, the post-beamforming SNR during the training
phase is usually high, leading to small channel estimation error. Meanwhile,
this simplification has little effect on the structure of the proposed scheme,
although it may cause degradation in the outage performance.

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andPan outage is unavoidable with any beam combination.
2
If
l∈S kh̄l k > α/Nt , enough beams have been trained
to avoid outage. Thus the user feeds back the LB non-zero
effective channels h̄l , l ∈ S along with their indices, and the
BS aligns the LB beams with FRF and matches the effective
channel vector with fBB as in (5) to conduct data transmission.
Since the training and feedback are interleaved in the proposed
scheme, we name it interleaved training based SU transmission
(IT-SU) scheme.
B. Average Training Length Analysis
This subsection studies the average training length of the
IT-SU scheme. Since for different channel realizations, the
number of beams being trained in our proposed IT-SU scheme
can vary due to the randomness in the path profile and path
gains, we study the average training length measured in the
number of training intervals per channel realization where the
average is over channel distribution.
Before showing the analytical result, we first discuss the
effect of NRF on the average training length. Intuitively, for
any given channel realization and at any step of the training
process, larger NRF means that the same or more beam
combinations are available based on the already trained beams.
Thus the same or larger received SNR can be provided, which
results in the same or an earlier termination of the training
period. Therefore, with other parameters fixed, the average
training length is a non-increasing function of NRF , i.e.,
larger NRF helps reduce the average training length. On the
other hand, since there are at most L non-zero paths in the
channel for each channel realization, having a larger NRF
than L cannot provide better beam combination for any given
already trained beams compared with that of when NRF = L.
Therefore, with other parameters fixed, the average training
length of the IT-SU scheme is a non-increasing function of
NRF for NRF ≤ L, then keeps unchanged for NRF ≥ L.
The average training length for NRF = L is a lower bound
for a general NRF value and the average training length for
NRF = 1 is an upper bound.
With the above discussion, in the following analysis, we
only consider the scenario of 1 ≤ NRF ≤ L. Define


  
i−1
Nt − i
Nt
(12)
ξ(i, j) ,
/
L
j
L−j−1

and the target normalized received SNR α, the average training
length of the proposed IT-SU scheme is
TIT-SU = Nt −
where

l

(1)
bi

(j − NRF

, min{i − 1, L − 1}
, max{0, L − 1 − Nt + i},
for i = 2, · · · , Nt − 1.

The following theorem has been proved.
Theorem 1: For the hybrid SU massive antenna system with
Nt BS antennas, the L-path channel, NRF ≤ L RF chains,

i=1

(Nt − i)Pi ,

L −Lα
e Nt ,
Nt

(13)

(14)

for i = 2, ..., Nt − 1,
 (2)


bP

i
−Lα j
−Lα


Nt
Nt

1
−
e
if NRF = 1,
ξ(i,
j)e



(1)

j=b

i

(2)

min(NRF
−Lα
P−1,bi )
j Nt
/j!
ξ(i, j)( Lα
(15)
Pi =
Nt ) e

(1)

j=bi



(2)



bP

i

(2)
(1)


+
if 1 <NRF ≤ L,
+P
ξ(i,
j)
P

j
j

(1)

(−1) j!LNRF
− l)!(NRF − 1)!(NRF − 2)!l!
for j = 0, · · · , L − 1, l ≤ j − NRF ,
(2)
bi

N
t −1
X

P1 =

for i = 2, · · · , Nt − 1 and j < i, which is the probability of a
path being aligned by the ith beam and j paths being aligned
by the first i − 1 beams. Define
βj,l ,

6

j=max(NRF,bi )

(1)

(2)

and Pj and Pj are shown in (16) and (17) at the top of
next page.
Proof: See Appendix A.
Theorem 1 provides an analytical expression on the average
training length of the proposed IT-SU scheme. Other than
the two special functions Υ and Γ, it is in closed-form. The
two functions are well studied and their values can be easily
obtained. The Pi given in (14) and (15) is the probability
that the training length is i. Together, P1 , · · · , PNt form the
probability mass function (PMF) of the training length. From
(13), it can be easily concluded that the average training length
of the proposed scheme is always less than Nt since Pi ≥ 0,
∀i.
The result in Theorem 1 is general and applies for arbitrary
values of NRF , L, Nt . But due to the complicated format, it is
hard to see insightful behaviours of the average training length
directly. In what follows, several typical scenarios for massive
antenna systems are considered.
1) The Channel with Finite L, i.e., L = O(1): We first
consider the channels for the massive antenna system with
a finite number of paths. That is, L is a finite value while
Nt → ∞. The asymptotic result on the average training length
of the proposed IT-SU scheme is provided for both the special
case of single RF chain and the general case of multiple RF
chains.
Lemma 1: For the hybrid massive antenna system with
Nt ≫ 1 BS antennas, finite constant number of channel paths
L, and target normalized received SNR α, when the number
of RF chains is one, i.e., NRF = 1, or is the same as the
path number, i.e., NRF = L, the average training length of
the proposed IT-SU scheme can be written as follows:
Nt
+ O(1).
(18)
L+1
Proof: See Appendix B.
The result in Lemma 1 shows that for the two special
NRF values, the average training length of the IT-SU scheme
increases linearly with Nt , but the slope decreases linearly
with L, the number of channel paths. The traditional NIT-SU
TIT-SU =

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(1)
Pj

,



1 − NRF
L

NRF

Lα
j−N
NRF
XRF
X−2 NRF − 2 (−1)m
e− Nt
βj,l
NRF − 1
(l + 1)NRF
m
m=0

l=0

 m+1
t= Lα(l+1)
N N
t
Υ (NRF − 1 − m, t) + Γ (NRF , t) t=0 t RF
.
×
m+1

,



(16)

Lα
j−N
NRF
XRF
X−2 NRF − 2
−e− Nt
1
βj,l
2
m
(NRF − 1)
(NRF − 1) (l + 1)NRF
m
m=0
l=0
 m−n+1
0
n t
m  
Υ (NRF − 1 − m, t) + Γ (NRF − n, t) Lα(l+1)
X
m
−Lα(l + 1)
Nt NRF
×
.
n
N
m
−
n
+
1
t
n=0

1 − NRF
L

NRF

scheme with full training has a fixed training length Nt . Thus
the proposed IT-SU scheme has significant saving in training
time as Nt is very large. For example, when NRF = L = 1,
we have TIT-SU ≈ Nt /2, meaning that the IT-SU scheme
reduces the average training length by half. It is noteworthy
that this gain in time is obtained with no cost in outage
probability performance (more details will be explained in
the next subsection). Moreover, the average training length
is independent of the threshold α.
From the discussion at the beginning of this subsection, we
know that for any value of NRF , the average training length is
lower bounded by its value for NRF = L and upper bounded
by its value for NRF = 1. Thus the analytical results for the
two special cases in Lemma 1 lead to the following corollary.
Corollary 1: For the hybrid massive antenna system with
Nt ≫ 1 BS antennas, finite constant number of channel paths
L, and target normalized received SNR α, the average training
length of the proposed IT-SU scheme can be written as (18)
for any number of RF chains NRF .
2) The Channel with Linearly Increasing L, i.e., L =
O(Nt ): The next typical scenario to consider is that the
number of channel paths has a linear scaling with the number
of BS antennas. That is, L = cNt while Nt → ∞ for
a constant but arbitrary c. For this case, due to the more
complicated form of Pi than that of the finite L case, simple
expression of the average training length is hard to obtain.
Two special cases are analyzed in what follows.
For the special case where NRF = 1 and c = 1, i.e., single
(2)
(1)
RF chain and i.i.d. channels, we have bi = bi = i − 1 for
i = 2, ..., Nt − 1. Thus, from (14) and (15),
Pi = e−α (1 − e−α )i−1
for i = 1, ..., Nt − 1. This is the same as the result of the
interleaved antenna selection scheme for full-digital massive
antenna systems with i.i.d. channels in [24]. The same asymptotic upper bound on the average training length for large Nt
can be obtained as:
TIT-SU = eα (1 − (1 − e−α )Nt ) → eα when Nt → ∞.
This bound is only dependent on the threshold α.

(17)

For another special case where NRF = L and c = 1,
from (14) and (15), we have Pi = e−α αi−1 /(i − 1)! for
i = 1, ..., Nt − 1. This is in accordance with that of the
interleaved Scheme D for full-digital massive antenna systems
with i.i.d. channels in [24, Theorem 2]. The corresponding
upper bound on the average training length is 1 + α, which
again is only dependent on the threshold α.
For the general case of 0 < c < 1, the expression of Pi in
(15) can be used along with numerical evaluations for further
studies. Fig. 1 shows the average training length TIT-SU with
respect to Nt for different parameter values. The following
observations are obtained from the plot.

25

c=0.1, N RF=1
c=0.1, N
c=0.2, N

RF
RF

=L
=1

c=0.2, N RF=L

20

Average training length

(2)
Pj

7

c=1, N RF=L
=4
=8

15

10

5
20

40

60

80

100

120

140

160

180

Antenna number

Fig. 1. Average training length of the IT-SU scheme for α = 4, 8.

•

•

For any NRF value, TIT-SU asymptotically approaches a
constant upper bound that is independent of Nt .
TIT-SU is an increasing function of the threshold α. Thus
the advantage of the proposed scheme over NIT-SU
degrades for larger α. The advantage degradation with
increasing α is most severe when c = 1 and NRF = 1,
but for larger NRF or smaller c, it is considerably slower.

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•

When NRF = L, TIT-SU is a decreasing function of c.
When NRF = 1, depending on the value of α, TIT-SU may
not be a monotonic function of c. This can be explained
by the two opposite effects of increasing c: the increase
in multi-path diversity and the decrease in average path
power. For NRF = L, there are enough beams to be
used to compensate for the second effect, thus larger c
tends to decrease TIT-SU via higher multi-path diversity.
For NRF = 1, the second effect is more dominant, thus
larger c tends to increase TIT-SU due to the path power
loss, especially for large threshold α.

C. Outage Performance Analysis
In this subsection, the outage probability of the proposed
IT-SU scheme is analyzed.
Theorem 2: For the hybrid SU massive antenna system with
Nt BS antennas, the L-path channel, NRF ≤ L RF chains and
the target normalized received SNR α, the outage probability
of the IT-SU scheme is
out(IT-SU) =

 

 Υ N , αL L−N


XRF
RF Nt
L
NRF +l−1 L−NRF

+
(−1)
NRF
(NRF − 1)!
l
l=1


 
αL
NRF −1

−1− N l
Nt
RF
−1
NRF
e 
 − B(l) ,
(19)
×
l
l
−1 −
NRF

where

B(l) ,


P−2
NRF
m=0



0

1
m!



l
− NRF

m



NRF ≥ 2
Υ m + 1, αL
Nt

.
otherwise
(20)

Proof: See Appendix C.
Theorem 2 provides an analytical expression for the outage
probability of the proposed IT-SU scheme. The expression is
in closed-form other than the special function Υ. Although
the effect of NRF on the outage performance of the ITSU scheme is implicit in (19), from its derivations, we have
out(IT-SU) = Pr(x ≤ α/Nt ) where x is the sum of the largest
NRF elements in {kh̄i k2 , i ∈ I}. Apparently, for any channel
realization, the value of x increases as NRF increases from
1 to L. Therefore, for any given finite α, larger NRF means
smaller outage probability.
1) Single RF Chain Analysis: To obtain further insights in
the effect of Nt , L and α on the outage performance, we
consider the special case with NRF = 1 in what follows.
Lemma 2: For the hybrid massive antenna system with Nt
BS antennas, the L-path channel, single RF chain, and the
target normalized received SNR α, the outage probability of
the IT-SU scheme is as follows:
out(IT-SU) = (1 − e
Proof: See Appendix D.

−αL
Nt

)L .

(21)

8

It can be seen from (21) that for arbitrary values of α, P
and arbitrary scaling of L with respect to Nt between constant
and linear, i.e., L = O(NtrL ) for rL ∈ [0, 1], we have
lim out(IT-SU) = 0.

Nt →∞

This means that for the SU massive antenna system with single
RF chain, arbitrarily small outage probability can be obtained
for any desired date rate and any fixed power consumption P
as long as Nt is large enough. This shows the advantage of
having massive antenna array at the BS.
Specifically, for finite L, i.e., L = O(1), we have
L



αL
−(L+1)
+ O Nt
,
out(IT-SU) =
Nt

meaning that the outage probability scales as O Nt−L for
large Nt . For linearly increasing L where L = cNt , we have
out(IT-SU) = (1 − e−αc )cNt ,

meaning that the outage probability decreases exponentially
with respect to Nt . For i.i.d. channels where c = 1, the outage
probability of the IT-SU scheme reduces to (1 − e−α )Nt . This
is the same as that of the antenna selection scheme in the
full-digital massive antenna systems with i.i.d. channels [24].
2) Multiple RF Chain Analysis: For the case of multiple
RF chain where 1 < NRF ≤ L, since larger NRF results in
smaller outage probability, for both finite L and L = cNt ,
it can be concluded that arbitrarily small outage probability
can also be achieved for an arbitrary date rate with any fixed
power consumption P when Nt is large enough.
3) Comparison with NIT Schemes: In Section III-B, the
proposed IT-SU scheme was compared with the NIT-SU
scheme with full training in terms of training length. Here,
we give the outage probability comparison. As utilized in
the proof of Theorem 2, for the IT-SU scheme, an outage
happens only when all beams have been trained and no beam
combination can satisfy the target SNR requirement. This is
the same as that of the NIT-SU scheme with full training, thus
the two schemes have the same outage performance.
Another possible non-interleaved scheme is to have partial
training with a fixed training length of Lt < Nt [21]. It can be
seen easily that as Lt decreases, the outage probability of the
partial non-interleaved scheme increases. Thus, the proposed
IT-SU scheme is superior in terms of outage probability
compared with the NIT-SU scheme with the same training
length. Numerical validation will be given in the simulation
section.
IV. I NTERLEAVED T RAINING

FOR

MU T RANSMISSION

This section is on the more general and complicated MU
systems, where the joint beam-based interleaved training and
the corresponding MU transmission is proposed. Compared to
SU systems where the optimal transmission is the maximumratio combining of the best trained beams, the beam assignment problem is a challenging but crucial part of the MU transmission. In what follows, we first study the beam assignment
issue. Subsequently, the joint beam-based interleaved training
and MU transmission is proposed.

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A. Feasible Beam Assignment and MU Beam Assignment
Methods
For the hybrid massive antenna BS to serve multiple users
with a fixed beam codebook, a typical idea is to assign a
beam to each user. But the beam assignment problem is far
from trivial and is a dominant factor of the performance. We
first introduce the definition of feasible beam assignment, then
propose MU beam assignment methods.
Definition 1: For the hybrid massive antenna downlink
serving U users with codebook-based beam transmission, a
beam assignment is an ordered U -tuple, (n1 , . . . , . . . nU ),
where ni is the index of the beam assigned for User i. A
feasible beam assignment is a beam assignment where the
resulting effective channel matrix as given in (7) has full rank.
In other words, a beam assignment is feasible if ZF baseband can be conducted with no singularity, thus the received
SNR, denoted as SNRMU for the MU transmission, or λ
in (9) is non-zero. If an infeasible beam assignment is used
for the MU transmission, ZF baseband precoding cannot be
conducted. Even if other baseband precoding, e.g., regularized
ZF, is used, the received SINR will be very small due to
the high interference and outage occurs. Thus feasible beam
assignment is a necessary condition to avoid outage. Two cases
that can cause infeasible beam assignment are 1) all beams
(n1 , . . . , . . . nU ) are zero-beams for any user thus the effective
channel matrix has a row with all zeros, and 2) one beam is
assigned to more than one user thus two identical columns
appear in the effective channel matrix. On the other hand,
depending on the effective channel values and interference
level, a feasible beam assignment may or may not be able
to avoid outage.
A straightforward and optimal beam assignment method is
the exhaustive search. Denote the set of known (e.g., already
trained) non-zero beam indices for User u as Bu . Define
B , ∪U
u=1 Bu . By searching over all possible feasible beam
assignments over B and finding the one with the maximum λ,
the optimal beam assignment is obtained. The complexity of
the exhaustive search is however O(|B|U ), which is unaffordable for large |B| and/or U .
Thus for practice implementation, beam assignment methods with affordable complexity are needed. To serve this
purpose, we transform the SNR maximization problem for the
beam assignment to the problem of maximizing the minimum
effective channel gain among the users, i.e.,
arg

max

n1 ,··· ,nU ∈B

min{|h̄u,nu |}.
u

(22)

Then by drawing lessons from the extended optimal relay
selection (ORS) method for MU relay networks in [30], the
following beam assignment algorithm is proposed. First, the
original ORS algorithm in [31] is used to maximize the
minimum effective channel gain. Suppose that the minimum
gain is with User i and Beam j. Then we delete User i and
Beam j and apply the original ORS scheme again to the
remaining users and beams. This procedure is repeated until
all users find their beams. It has been shown in [30], [31] that
this scheme not only achieves an optimal solution for (22),
but also achieves the unique optimal solution that maximizes

9

the uth minimum channel gain conditioned on the previous
1st to the (u − 1)th minimum channel gains for all u. Further,
the worst case complexity of this scheme is O(U 2 |B|2 ), much
less than that of the exhaustive search. This beam assignment
is referred to as the max-min assignment. With respect to
the outage probability performance, it is suboptimal. But the
method targets at maximizing the diagonal elements of the
effective channel matrix, which in general is beneficial to ZF
transmission. Simulation results exhibited in Section V show
that its outage performance loss is small compared with the
exhaustive search especially for channels with small L.
B. Joint Beam-Based Interleaved Training and MU Transmission Design
Similar to the SU case, the main goal of the interleaved
training and joint MU data transmission scheme (referred to
as the IT-MU scheme) is to save training time while preserving
the outage probability performance. The fundamental idea is to
conduct the training of each beam sequentially and terminate
right after enough beams have been trained to avoid outage.
However, different from the SU case, a big challenge for the
MU case is that each user does not know the effective channels
of other users since user cooperation is not considered. Thus
the users are not able to decide whether to terminate the
training interval given an SNR threshold. Our solution for this
is to make users feed back their acquired non-zero effective
channels during training and let the BS to make the decision.
Other differences of the MU scheme to the SU one include
the initial training steps, the beam assignment problem, and
the termination condition for the training interval. These will
be studied in details in the explanation of the scheme that
follows.
The detailed steps for the proposed IT-MU scheme is given
in Algorithm 2. At the beginning of this scheme, the first U
beams in the codebook are trained and every user estimates
the corresponding effective channels and constructs its set of
non-zero beam indices Bu . Then the non-zero beam values and
their indices are fed back to the BS (with this information, the
BS also knows Bu , ∀u). While for the SU case the beams are
trained one by one, U beams need to be trained initially for
the MU case since at least U beams are needed for a feasible
beam assignment.
After this initial training stage, if for any user, no non-zero
beam is found (in which case the user feeds back “0”) or
|B| < U where B is the union of non-zero beam indices of
all users, the training of the next beam starts. Otherwise, the
BS finds a beam assignment on B with either the exhaustive
search or the max-min method given in Section IV-A. If the
beam assignment is feasible and can avoid outage, training
terminates and data transmission starts with this beam assignment and the corresponding ZF baseband precoding as shown
in Section II-A2. Otherwise, the BS starts the training of the
next beam. When the new ith beam has been trained, each user
again estimates the corresponding effective channel. If it is a
zero-beam, the user feeds back “0”; otherwise, it feeds back
the effective channel value. If this ith beam is a zero-beam for
all users or any user still has no non-zero beam or the updated

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Algorithm 2 The Joint Beam-Based Interleaved Training and
MU Transmission (IT-MU) Scheme.
1: The BS trains the 1st to U th beams. User u, ∀u estimates
the corresponding effective channels h̄u,1 , ..., h̄u,U and
constructs its set of non-zero beam indices Bu ;
2: If |Bu | = 0, User u, ∀u feeds back “0”. Otherwise, User
u feeds back the non-zero effective channel values along
with their beam indices;
3: if any user’s feedback is “0” or |B| < U then
4:
set os = 1 and goto Step 13;
5: else
6:
The BS finds a beam assignment on B;
7:
if the beam assignment is not feasible or the resulting
received SNR is below the outage threshold then
8:
set os = 1 and goto Step 13;
9:
else
10:
set os = 0 and goto Step 27;
11:
end if
12: end if
13: for i = U + 1, ..., Nt do
14:
The BS trains the ith beam; User u, ∀u estimates the
corresponding effective channel h̄u,i ;
15:
For all u, if kh̄u,i k = 0, User u feeds back “0”; else
User u feeds back the value h̄u,i and let Bu = Bu ∪ {i}
and B = B ∪ {i};
16:
if all users’ feedbacks are “0” or |Bu | = 0 for any u
or |B| < U then
17:
set os = 1 and continue;
18:
else
19:
The BS finds a beam assignment on B;
20:
if the beam assignment is not feasible or the resulting
received SNR is below the outage threshold then
21:
set os = 1 and continue;
22:
else
23:
set os = 0 and goto Step 27;
24:
end if
25:
end if
26: end for
27: if os = 0 then
28:
The BS uses the found beam assignment to construct
FRF and the ZF FBB for MU transmission;
29: end if
|B| is still less than U , the BS starts the training of the next
beam if an un-trained beam is available. Otherwise, the BS
finds a beam assignment on B with either the exhaustive search
or the max-min method. If the beam assignment is feasible and
can avoid outage, training terminates and transmission starts.
Otherwise, the BS starts the training of the next beam if an
un-trained beam is available. The procedure continues until a
beam assignment that can avoid outage is found or there is no
new beam for training.
C. Discussion on Average Training Length and Outage Performance
For the IT-MU scheme, the minimum possible training
length is U and the maximum possible training length is Nt .

10

Similar to the IT-SU scheme, it is reasonably expected that the
IT-MU scheme has a smaller average training length than the
NIT-MU scheme with full training. Meanwhile, since complete
effective CSI is available for the BS if necessary in the IT-MU
scheme, it achieves the same outage performance. Moreover,
the outage probability of the IT-MU scheme is smaller than
that of the NIT-MU scheme with partial training at the same
training length.
V. N UMERICAL R ESULTS AND D ISCUSSIONS
In this section, simulation results are shown to verify
the analytical results in this paper. Meanwhile, properties
of the proposed interleaved training and joint transmission
schemes are demonstrated. We also make comparison with
non-interleaved schemes.
In Fig. 2, the average training length of the IT-SU scheme in
Algorithm 1 is shown for NRF = 1 and α = 4, 83 . First, it can
be seen that the derived average training length in Theorem
1 well matches the simulation. Second, for L = 1, 3, 6 and
α = 4, the average training length increases linearly with
Nt with slope about 0.50, 0.26 and 0.13, respectively. These
match the theoretical results in Lemma 1 where the slope is
1/(L + 1) = 0.5, 0.25, 0.14, respectively. Further, the dashed
line without marker is the line of TIT-SU = Nt /4, which is the
asymptotic average training length for L = 3 in (18). Third,
for L = cNt where c = 0.1, 0.2, the average training lengths
approach to constants as Nt increases. While for c = 1, the
asymptotic constant upper bound is less explicit since Nt is
not large enough to reveal the asymptotic bound eα . When
c = 1 and α = 8, the average training length is almost the
same as the NIT-SU scheme with full training (dotted line) due
to the high SNR requirement and limited simulation range of
Nt . Finally, the average training length increases with α (much
less significant for finite L and small c) which is in accordance
with the comments on Lemma 1 and Fig. 1.
In Fig. 3, the average training length of the IT-SU scheme is
studied for NRF > 1 and α = 4, 8. Again, the results in Theorem 1 have tight match with the simulation. For L = 3 and
NRF = 3, the average training length increases linearly with
Nt with ratio about 1/(L+1) = 0.25 for α = 4, 8. Meanwhile,
the average training length has negligible reduction compared
with that of NRF = 1 in Fig. 2, e.g., 34.5 for NRF = 1 and
33.8 for NRF = 3 with Nt = 110 and α = 8. These validate
Lemma 1. Further for L = cNt where c = 0.1, 1, the average
training lengths are upper bounded by different constants as
Nt grows. The effect of c, α and NRF on the upper bound
has been studied in Section III-B2, which can be referred to
directly. Finally, the result for L = NRF = Nt , i.e., full-digital
massive MIMO with i.i.d. channels matches the theoretical
result 1 + α. Both Figs. 2 and 3 show that compared with
the NIT-SU scheme with full training (represented by dotted
lines), the IT-SU scheme achieves huge reduction in training
length with the same outage performance.
3 For P = 0 dB, α = 4, 8 mean that R
th = 2.3219, 3.1699 bps/Hz
respectively which represent the low rate scenario. For P = 10 dB, α = 4, 8
mean that Rth = 5.3576, 6.3399 bps/Hz respectively which represent the
intermediate to high rate scenario.

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60

50

70

L=1
L=3
L=6
L=0.1N t

60

L=N t

Average training length

Average training length

24
23

L=0.2N t

L=0.2N t
40

L=1
L=3
L=6
L=0.1N t

11

Theoretical
Simulated

30

20

50

L=N t

22

Theoretical
Simulated

21

40

20
70

80

90

100

110

30

20
10
10
0
30

40

50

60

70

80

90

100

110

30

40

50

60

Antenna number

70

80

90

100

110

Antenna number

Fig. 2. Average training length of the IT-SU scheme with NRF = 1 for α = 4 (left) and α = 8 (right). The dotted line shows the training length of the
NIT-SU scheme with full training.

40

L=0.1N t , N RF=3

35

L=N t , N RF=0.1N t

Average training length

Theoretical
Simulated

20

15

5

60

70

80

90

100

110

Antenna number

L=N t , N RF=Nt
Theoretical
Simulated

15

5

50

L=N t , N RF=0.1N t

20

10

40

L=0.1N t , N RF=0.1N t

25

10

0
30

L=0.1N t , N RF=3

30

L=N t , N RF=Nt

25

L=3, NRF=3

35

L=0.1N t , N RF=0.1N t

30

Average training length

40

L=3, NRF=3

0
30

40

50

60

70

80

90

100

110

Antenna number

Fig. 3. Average training length of the IT-SU scheme with NRF > 1 for α = 4 (left) and α = 8 (right). The dotted line shows the training length of the
NIT-SU scheme with full training.

In Fig. 4, comparison on the average training length is
shown for the proposed scheme, the non-interleaved scheme
with full training, and the hierarchical scheme with M = 3
where NRF = 1 and α = 8. Note that for the considered
setting, M = 3 results in the minimum training length thus
is the most favourable for the hierarchical scheme. The figure
shows that 1) when L = 1, the hierarchical scheme has the
lowest average training length, 2) for L = 2, the IT-SU scheme
has lower training length than the hierarchical scheme when
Nt < 120 and larger training length when Nt ≥ 120, 3) for
L = 3, the IT-SU scheme has lower training length than the
hierarchical scheme for all simulated values of Nt in the range
[30, 180], 4) for L = 0.1Nt , the training length of the proposed
IT-SU scheme is lower and bounded by a constant while the
hierarchical scheme experiences fast increase in the training

length with larger Nt .
In Fig. 5, the outage performance of the IT-SU scheme is
compared with that of the NIT-SU scheme with full training
and partial training at the same training length (by setting Lt
to be the same as the average training length of the IT-SU
scheme). The cases of L = 1, 3, 0.1Nt, NRF = 1, 3, 0.1Nt
and α = 4 are studied. It can be seen that 1) the theoretical
outage probabilities of the IT-SU scheme in Theorem 2 match
the simulated values well; 2) the outage probability of the ITSU scheme is the same as that of the NIT-SU scheme with
full training, and significantly lower than that of the NIT-SU
scheme with partial training; 3) the outage probability of the
IT-SU scheme diminishes fast as Nt grows; 4) by increasing
NRF from 1 to 3 for L = 3 or from 3 to 0.1Nt for L = 0.1Nt ,
the outage probability of the IT-SU scheme decreases. These
validate the discussions in Section III-C.

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100
90

NIT-SU, full training
Hierarchical
IT-SU
L=1
L=2
L=3
L=0.1N t

Average training length

80
70
60
50
40
30
20
10
0
40

60

80

100

120

140

160

180

Antenna Number

Fig. 4. Average training length comparison of the IT-SU scheme, the NIT-SU
scheme with full training, and the hierarchical scheme. NRF = 1, α = 8,
and M = 3.

10 0

L=1, NRF=1

Outage probability

10

L=3, NRF=1

-1

L=3, NRF=3
L=0.1N t , N RF=3
L=0.1N t , N RF=0.1N t
IT-SU, theoretical
IT-SU, simulated
NIT-SU, full training
NIT-SU, partial training

10 -2

40

50

60

70

80

90

100

very positive result for the proposed IT-SU scheme designed
with the outage performance goal. It shows that interleaved
and adaptive training design in general benefits the system
performance compared to non-interleaved training.
Figs. 7 and 8 show the average training length and outage
performance of the IT-MU scheme in Algorithm 2 respectively
where NRF = U = 3 and ᾱ = 6. Both the exhaustive
search and the max-min method are considered for the beam
assignment in the IT-MU scheme. It can be seen that for
L = 1, 3, the average training lengths of the IT-MU scheme
have linear increase with Nt , where the slopes are approximately 0.75 and 0.38, respectively. Compared with the NITMU scheme with full training, where the training length equals
Nt , the reduction in training length of the proposed scheme is
significant, and larger L results in bigger reduction. Second,
when L = 0.1Nt , the training length of the IT-MU scheme
approaches a constant, which equals 23.9, as Nt grows. The
IT-MU scheme has the same outage performance as that of the
NIT-MU scheme with full training which is much better than
that of the NIT-MU scheme with the same training length
(partial training). Lastly, by replacing the exhaustive search
with the max-min method for beam assignment, the outage
performance of the IT-MU scheme has some small degradation
for L = 3 and the degradation diminishes for L = 1. When
L = 0.1Nt , the outage performance degradation due to the
sub-optimal beam assignment is more visible. On the other
hand, the increment of average training length due to the use of
this sub-optimal assignment method is negligible. Considering
the lower complexity of the max-min method, its application
in the IT-MU scheme is more desirable.
VI. C ONCLUSIONS

10 -3

10 -4
30

12

110

Antenna number

Fig. 5. Comparison of outage probability between the IT-SU scheme and the
NIT-SU scheme.

In Fig. 6 we show the ergodic rate of the proposed ITSU scheme with slight modification4 and the NIT-SU scheme
for α = 4, 8 and P = 10 dB. It can be seen that the ITSU scheme has lower ergodic rate compared with the NITSU scheme with full training. Compared with the NIT-SU
scheme with the same average training length (partial training),
the IT-SU scheme achieves higher ergodic rate. This is a
4 For the rate comparison, we change the condition in Line 5 of Algorithm
P
1 to “kh̄i k = 0 or l∈S kh̄l k2 ≤ α/Nt and i < Nt ”. The only difference
is for the case that all beams have been trained and an outage is still
unavoidable. Previously, no transmission is conducted since outage is not
avoidable, while with the change, the user feeds back the channel coefficients
of the min(NRF , L) known non-zero beams with the largest norms and
the BS uses this information for hybrid beamforming. This change has no
effect on the outage performance of the proposed scheme but is sensible
when considering the rate performance.

For the hybrid massive antenna systems, we studied the
beam-based training and joint beamforming designs for SU
and MU transmissions with outage probability as the performance measure. For SU systems, via concatenating the
feedback with the training, an interleaved training scheme
was proposed whose training length is adaptive to channel
realizations. Then, exact analytical expressions were derived
for the average training length and outage probability of the
proposed scheme. For MU systems, we proposed a joint
interleaved training and transmission design, which contains
two new techniques compared to the single-user case: having
the BS control the training process due to the limited local CSI
at the users and feasible beam assignment methods. Analytical
and simulated results show that the proposed training and joint
transmission designs achieve the same performance as the traditional full-training scheme while save the training overhead
significantly. Meanwhile, useful insights were obtained on
the training length and outage probability of typical network
scenarios and on the effect of important system parameters,
e.g., the BS antenna number, the RF chain number, the channel
path number or angle spread, and the rate requirement.
Many future directions can be envisioned on interleaved
training designs for hybrid massive MIMO systems. One possible extension is to consider systems with channel estimation
error and analyze how it affects the system performance. An-

10

10

9

9

Ergodic Rate (bps/Hz)

Ergodic Rate (bps/Hz)

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8

7

L=1, NRF=1
6

L=3, NRF=1
L=3, NRF=3

8

7

L=1, NRF=1
L=3, NRF=1
6

L=3, NRF=3
L=0.1N t , N RF=3

L=0.1N t , N RF=3
5

4
30

L=0.1N t , N RF=0.1N t

50

L=0.1N t , N RF=0.1N t

5

IT-SU
NIT-SU, full training
NIT-SU, partial training
40

13

60

70

80

90

100

4
30

110

IT-SU
NIT-SU, full training
NIT-SU, partial training
40

50

60

Antenna Number

70

80

90

100

110

Antenna Number

Fig. 6. Ergodic rate comparison of the IT-SU scheme and the NIT-SU scheme with full/partial training for α = 4 (left) and α = 8 (right) with P = 10 dB.
110
100

10 -1

80

Outage proability

Average training length

90

10 0

NIT-MU, full training
IT-MU, exhaustive
IT-MU, max-min
L=1
L=3
L=0.1N t

70
60
50

NIT-MU, full training
NIT-MU, partial training
IT-MU, exhaustive
IT-MU, max-min
L=1
L=3
L=0.1N t

10 -2

40
30
10 -3

20
10
30

40

50

60

70

80

90

100

30

110

40

50

60

70

80

90

100

110

Antenna number

Antenna number

Fig. 7. Average training length of the IT-MU scheme for NRF = U = 3
and ᾱ = 6.

Fig. 8. Outage performance of the IT-MU scheme for NRF = U = 3 and
ᾱ = 6.

other practical issue is the feedback overhead. For both singleuser and multi-user systems with limited feedback capacity,
an important topic is the joint design of interleaved training,
feedback, and transmission scheme. Interleaved training designs and analysis for multi-cell and cooperative systems are
also meaningful future directions.

and

A PPENDIX A
T HE P ROOF OF T HEOREM 1
In calculating the average training length, the probability
that the training length
PNt is i (denoted as Pi ) for i = 1, ..., Nt
is needed. Since
i=1 Pi = 1, it is sufficient to calculate
Pi , i = 1, ...Nt − 1 only.
The training length is 1 when the 1st beam is a non-zero
beam and its effective channel gain is strong enough to avoid
outage. The probability that the 1st beam is non-zero is L/Nt

Pr(kh̄1 k2 > α/Nt ) =

Z

∞

Le−Lxdx = e−αL/Nt .

α/Nt

Thus (14) is obtained for P1 .
The training length is i for i ∈ [2, Nt − 1] when 1) the ith
beam is a non-zero beam, 2) an outage cannot be avoided by
previously trained beams, and 3) an outage can be avoided
with the newly discovered ith beam. To help the presentation,
denote this event as Event X. It can be partitioned into the
sub-events with respect to different j for max{0, L − Nt −
1 + i} ≤ j ≤ min{L − 1, i − 1}, where for the jth event,
there are j non-zero beams within the first i − 1 beams and
L − 1 − j non-zero beams within the beams from i + 1 to Nt .
The probability for the beam distribution of sub-event j thus
(2)
(1)
equals to ξ(i, j) as defined in (12). Notice that bi and bi
are defined as the bounds of j. Further, given the the beam
distribution of sub-event j, the probability of Event X can

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be calculated by considering three cases as follows. To help
the proof, denote the indices of the already trained j non-zero
beams as n1 , · · · , nj and let z , kh̄i k2 .
Case 1 is when j = 0. Event X happens when kh̄i k2 >
2 is when 0 <
α/Nt , whose probability is e−αL/Nt . Case P
j
j ≤ NRF − 1. Event X happens when x , l=1 kh̄nl k2 ≤
α/Nt and z > α/Nt − x. Since x ∼ 1/(2L)X 2 (2j) and
z ∼ 1/(2L)X 2 (2),
Pr[X] = Pr (x ≤ α/Nt , z > α/Nt − x)
!
Z Nα
Z ∞
−Lx
t
−Lz
j j−1 e
=
Le
dz dx
L x
α
(j − 1)!
0
N −x
t

=

j
( Lα
Nt ) e

j!

−Lα
Nt

p(x′ , y) =

l=0

.

βj,l [x′ − (NRF − 1)y]

e

′ +(l+1)y
1/L

,

y ≥ 0, x′ ≥ (NRF − 1)y.

Pr[X] = Pr (x′ ≤ α/Nt , y ≤ α/Nt − x′ , z > α/Nt − x′ )


′
Z NαZ min N x′−1 , Nα −xZ
∞
t
t
RF
=
p(x′ , y)Le−Lz dx′ dydz
=

L(Nt − i)(Nt − i) × ... × (Nt − i − L + 2)
Nt × ... × (Nt − L + 1)
PL PL−k (0) k n
L k=0 n=0 Ck,n Nt i
=
,
Nt × ... × (Nt − L + 1)
(0)

where Ck,n is the polynomial coefficient for the term Ntk in .
(m−1)
(m−1)
(m)
for m = 1, · · · , Nt −
Define ∆i , ∆i+1 − ∆i
(0)
L where ∆i = xi . Using the binomial formula, we have
(m)
(m)
∆i = N (∆i )/[Nt×...×(Nt − L + 1)], where


L L−k
n   n−i
X
X (0)
X
n X1 n − i1
(m)
k
Ck,n Nt
N (∆i ) , L
i2
i1 i =1
n=0
i =1
k=0

n−

···

1

Pm−1
j=1

X

im =1

i
j

2

Pm−1 
n − j=1 ij n−Pm
j=1 ij .
i
im
(0)

x
(NRF −2) −

Consequently, for Case 3.2,

0

but straightforward calculations to the following

 O(Nt−2 )
i > Nt + 1 − L
t −i 

,
Pi =
(NL−1
)
−1
−2
1 + O(Nt ) + O(Nt ) i ≤ Nt + 1 − L
 Nt
(L)
 Nt 
t −i
for i = 1, 2, ..., Nt − 1. Since NL−1
/ L has the same
PNt −1
(Nt −
order as or a lower order than O(Nt−1 ), ∀i, and i=1
i)O(Nt−2 ) = O(1), from (13) we have
XNt +1−L
xi + O(1),
(23)
TIT-SU = Nt −
i=1
 Nt 
t −i
where xi , (Nt − i) NL−1
/ L . We rewrite xi as
xi =

Case 3 is when j > NRF − 1, where two sub-cases are
considered. Case 3.1: If NRF = 1, Event X happens when
xl , kh̄nl k2 ≤ α/Nt for all l = 1, ..., j and z > α/Nt . Since
2
xl ’s and z are i.i.d. following 1/(2L)X (2), we have Pr[X] =

j
−αL
−αL
. Case 3.2: If NRF > 1, order the already
e Nt 1 − e Nt
trained j non-zero beams such that kh̄s1 k2 ≥ · · · ≥ kh̄sj k2 ,
where s1 , · · · , sj ∈ {n1 , · · · , nj }. Event X happens when
PNRF −1
kh̄sl k2 ≤ α/Nt , y = kh̄sNRF k2 ≤ α/Nt − x′ ,
x′ , l=1
and z > α/Nt − x′ . Notice that x′ and y are correlated but
both are independent to z. Via utilizing the result of the joint
distributions of partial sums of order statistics [32, Eq. (3.31)],
the joint probability density function (PDF) of x′ and y can
be given as
j−N
XRF

14

α
Nt

0

−x′

(2)
(1)
Pj +Pj ,

(1)

RF −1
], where
where Pj is the integral for x′ ∈ [0, Nαt NN
RF
′
′
(2)
α
x
x
′
min( NRF −1 , Nt − x ) = NRF −1 and Pj is that for x′ ∈
α
x′
α
′
′
RF −1
[ Nαt NN
, Nαt ], where min( NRF
−1 , Nt − x ) = Nt − x . Via
RF
P
n
n
n
m n−m
utilizing (a + b) = m=0 m a b
, the definition of the
lower and upper
incomplete
gamma
functions,
and the indefi

R
nite integral xb−1 Υ(s, x)dx = 1b xb Υ(s, x) + Γ(s + b, x) ,
(2)
(1)
Pj and Pj can be derived as (16) and (17), respectively.
Via the law of total probability and after some simple
reorganizations based on the previous derivations, Pi , i ∈
[2, Nt −1] in (15) can be obtained, which completes the proof.

A PPENDIX B
T HE P ROOF FOR L EMMA 1
When Nt ≫ 1, L = O(1), for the special cases of NRF = 1
or L, the Pi values in (14) and (15) can be simplified via long

Since n ≤ L − k and ij ≥ 1, j = 1, ..., m , we have Ck,n = 0
for k > L − m, i.e., n < m. Thus, the highest power of Nt in
(0)
(m)
N (∆i ) is L − m and its scalar coefficient is LCL−m,m m!.
And this term corresponds to i1 = ... = im = 1, which
(L−1)
(0)
is an arithmetic
guarantees CL−m,n 6= 0. Further, ∆i
progression. Then we have
XNt −L+1 Xi−1 (1)
XNt −L+1
XNt −L+1
∆ +...
x1 +
xi =
j1 =1 1
i=1
i=1
i=1
XjL−2 −1 (L−1) XjL−1 −1 (L)
XNt −L+1 Xi−1
∆1 ).
(∆1
+
...
+
jL =1

jL−1 =1

j1 =1

i=1

From the Faulhaber’s formula,


Xn
1 Xp
p+1
kp =
Bj np+1−j ,
k=1
j=0
p+1
j
where Bj is the Bernoulli number, we have
NtX
−L+1 X
i−1
i=1

j1 =1

jm−1 −1

...

X

(m)

∆1

(m)

=

jm =1

∆1
[N m+1 + O(Ntm )]
(m + 1)! t

for m ∈ [1, L] with j0 = i. Since the denominator of x1 is
NtL +O(NtL−1 ) and the numerator of x1 is LNtL +O(NtL−1 ),
we have (Nt − L + 1)x1 = LNt + O(1). Consequently,
NtX
−L+1
i=1

L
X

(m)

∆1
[Ntm+1 + O(Ntm )]
(m
+
1)!
m=1

xi = LNt+O(1)+

= LNt + O(1)
(0)
L
X
LCL−m,m m!NtL−m+O(NtL−m−1 ) m+1
[Nt
+ O(Ntm )]
+
L + O(N L−1 )](m + 1)!
[N
t
t
m=1

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XL

(0)

CL−m,m

+ O(1)
(m + 1)

XL L (−1)m
L
(a)
m
= LNt + LNt
+ O(1) =
Nt + O(1),
m=1 m + 1
L+1

PL
L
L
.
where (a) follows from m=1 m
(−1)m /(m + 1) = − L+1
From this result and (23), (18) can be easily obtained.
= LNt + LNt

m=1

A PPENDIX C
T HE P ROOF OF T HEOREM 2
For a given channel realization with channel path indices
I = {I1 , ..., IL }, with the IT-SU scheme, an outage happens
only when all Nt beams have been trained and the strongest
NRF beams among them cannot avoid an outage. Let SNt
be the set containing the indices of the NRF beams with
the strongest effective channel gains. From the results on the
partial
P sum of order statistics [32, Eq. 3.19], the PDF of
x , l∈SN kh̄l k2 is
t
 NRF NRF −1
L!
x
−Lx L
p(x) =
e
(L − NRF )!NRF !
(NRF − 1)!

N −1
L−N
XRF
NRF RF
(L − NRF )!
NRF +l−1
+L
(−1)
(L − NRF − l)!l!
l
l=1
 lxL

−
× e NRF − A(l, x) , x ≥ 0,
(24)
( P
where
NRF −2
m=0
A(l, x) ,
Thus

m

, NRF ≥ 2
− NlxL
RF
.
0
otherwise

1
m!

out(IT-SU) = Pr(x ≤ α/Nt ),

which leads to (19) by using (24).
A PPENDIX D
T
HE P ROOF OF L EMMA 2


− Lα
Nt and N
Since Υ 1,Lα
RF = 1, from (19), we
Nt = 1 − e
have
Lα
XL−1
(L − 1)! e(−1−l) Nt − 1
l
(−1)
out(IT-SU) = L
l=0
(L − 1 − l)!l!
−1 − l


XL−1
Lα
L!
e−(1+l) Nt − 1
(−1)l+1
=
l=0
(L − (l + 1))!(l + 1)!
XL
Lα
L!
(a)
e−t Nt − 1
(−1)L−t
= (−1)L
t=0
(L − t)!t!
XL
Lα
L!
(b)
+ 1 = (1 − e− Nt )L ,
(−1)t
−
t=0
(L − t)!t!

where (a) and (b) follow
Pnfromthe variable substitutions t =
l + 1 and (x + y)n = l=0 nl xn−l y l , respectively.

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Yongming Huang received the B.S. and M.S. degrees from Nanjing University, Nanjing, China, in
2000 and 2003, respectively, and the Ph.D. degree
in electrical engineering from Southeast University,
Nanjing, China, in 2007. Since 2007, he has been
a Faculty Member with the School of Information Science and Engineering, Southeast University,
where he is currently a Full Professor. From 2008
to 2009, he visited the Signal Processing Laboratory,
School of Electrical Engineering, Royal Institute of
Technology, Stockholm, Sweden. He has authored
over 200 peer-reviewed papers, hold over 50 invention patents, and submitted
over 10 technical contributions to the IEEE standards. His current research
interests include MIMO wireless communications, cooperative wireless communications, and millimeter wave wireless communications. He has served
as an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING, IEEE WIRELESS COMMUNICATIONS LETTERS, EURASIP
Journal on Advances in Signal Processing, and EURASIP Journal on Wireless
Communications and Networking.

Cheng Zhang received the B.Eng. degree from
Sichuan University, Chengdu, China in June 2009,
and the M.Sc. degree from Institute No.206 of China
Arms Industry Group Corporation, Xian, China in
May 2012. He worked as a Radar signal processing
engineer at Institute No.206 of China Arms Industry
Group Corporation, Xian, China from June 2012 to
Aug. 2013. Since Mar. 2014, he has been working towards Ph.D. degree at Southeast University,
Nanjing, China. From Nov. 2016 to Nov. 2017, he
was a visiting student with University of Alberta,
Edmonton, Canada. His current research interests include space-time signal
processing and application of learning algorithm in channel estimation and
transmission design for millimeter-wave massive MIMO communication systems.

Yindi Jing received the B.Eng. and M.Eng. degrees
from the University of Science and Technology of
China, in 1996 and 1999, respectively. She received
the M.Sc. degree and the Ph.D. in electrical engineering from California Institute of Technology,
Pasadena, CA, in 2000 and 2004, respectively. From
Oct. 2004 to Aug. 2005, she was a postdoctoral
scholar at the Department of Electrical Engineering
of California Institute of Technology. Since Feb.
2006 to Jun. 2008, she was a postdoctoral scholar
at the Department of Electrical Engineering and
Computer Science of the University of California, Irvine. In 2008, she joined
the Electrical and Computer Engineering Department of the University of
Alberta, where she is currently an associate professor. She was an Associate
Editor for the IEEE Transactions on Wireless Communications 2011-2016
and currently serves as a Senior Area Editor for IEEE Signal Processing
Letters (since Oct. 2017) and a member of the IEEE Signal Processing Society
Signal Processing for Communications and Networking (SPCOM) Technical
Committee. Her research interests are in massive MIMO systems, cooperative
relay networks, training and channel estimation, robust detection, and fault
detection in power systems.

Luxi Yang received the M.S. and Ph.D. degree
in electrical engineering from the Southeast University, Nanjing, China, in 1990 and 1993, respectively. Since 1993, he has been with the School
of Information Science and Engineering, Southeast
University, where he is currently a full professor of
information systems and communications, and the
Director of Digital Signal Processing Division. His
current research interests include signal processing
for wireless communications, MIMO communications, cooperative relaying systems, and statistical
signal processing. He has authored or co-authored two published books and
more than 150 journal papers, and holds 30 patents. Prof. Yang received the
first- and second-class prizes of Science and Technology Progress Awards of
the State Education Ministry of China in 1998, 2002 and 2014. He is currently
a member of Signal Processing Committee of Chinese Institute of Electronics.

