Hierarchical Codebook Design for Beamforming Training in Millimeter-Wave Communication

Abstract

In millimeter-wave communication, large antenna arrays are required to
achieve high power gain by steering towards each other with narrow beams, which
poses the problem to efficiently search the best beam direction in the angle
domain at both Tx and Rx sides. As the exhaustive search is time consuming,
hierarchical search has been widely accepted to reduce the complexity, and its
performance is highly dependent on the codebook design. In this paper, we
propose two basic criteria for the hierarchical codebook design, and devise an
efficient hierarchical codebook by jointly exploiting sub-array and
deactivation (turning-off) antenna processing techniques, where closed-form
expressions are provided to generate the codebook. Performance evaluations are
conducted under different system and channel models. Results show superiority
of the proposed codebook over the existing alternatives.

Full text

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, XXX 20XX 1
Hierarchical Codebook Design for Beamforming
Training in Millimeter-Wave Communication
Zhenyu Xiao, Member, IEEE, Tong He, Pengfei Xia, Senior Member, IEEE, and Xiang-Gen Xia, Fellow, IEEE
Abstract—In millimeter-wave communication, large antenna
arrays are required to achieve high power gain by steering
towards each other with narrow beams, which poses the problem
to efficiently search the best beam direction in the angle domain
at both Tx and Rx sides. As the exhaustive search is time
consuming, hierarchical search has been widely accepted to
reduce the complexity, and its performance is highly dependent
on the codebook design. In this paper, we propose two basic
criteria for the hierarchical codebook design, and devise an effi-
cient hierarchical codebook by jointly exploiting sub-array and
deactivation (turning-off) antenna processing techniques, where
closed-form expressions are provided to generate the codebook.
Performance evaluations are conducted under different system
and channel models. Results show superiority of the proposed
codebook over the existing alternatives.
Index Terms—Millimeter wave communication, mmWave,
beamforming, codebook design, hierarchial search.
I. INT ROD UC TI ON
MILLIMETER-WAVE (mmWave) communication is a
promising technology for next-generation wireless
communication owing to its abundant frequency spectrum
resource, which promises a much higher capacity than the
existing wireless local area networks (WLANs) and the current
cellular mobile communication. In fact, mmWave communi-
cation has received increasing attentions as an important can-
didate technology in both the next-generation WLANs [1]–[7]
and mobile cellular communication [8]–[16]. A fundamental
challenge to mmWave communication is the extremely high
path loss, thanks to the very high carrier frequency on the order
of 30-60 GHz. To bridge this significant link budget gap, joint
Tx/Rx beamforming is usually required to bring large antenna
array gains, which typically requires a large Tx/Rx antenna
array size (e.g., an antenna array size of 36 is used in [4].).
Fortunately, thanks to the small wavelength on the mmWave
frequency, large antenna arrays are possible to be packed into
a small area.
This work was partially supported by the National Natural Science Foun-
dation of China (NSFC) under grant Nos. 61571025, 91338106, 91538204,
and 61231013, National Basic Research Program of China under grant
No.2011CB707000, and Foundation for Innovative Research Groups of the
National Natural Science Foundation of China under grant No. 61221061.
Z. Xiao and T. He are with the School of Electronic and Information
Engineering, Beijing Key Laboratory for Network-based Cooperative Air
Traffic Management, and Beijing Laboratory for General Aviation Technology,
Beihang University, Beijing 100191, P.R. China.
P. Xia is with the School of Electronics and Information Engineering and
the Key Laboratory of Embedded System and Service Computing, Tongji
University, Shanghai, P.R. China.
X.-G. Xia is with the Department of Electrical and Computer Engineering,
University of Delaware, Newark, DE 19716, USA.
Corresponding Author: Dr. Z. Xiao with Email: xiaozy@buaa.edu.cn.
In the mmWave domain, the high power consumption
of mixed signal components, as well as expensive radio-
frequency (RF) chains, make it difficult, if not impossi-
ble, to realize digital baseband beamforming as used in the
conventional multiple-input multiple-output (MIMO) systems.
Instead, analog beamforming is usually preferred, where all
the antennas share a single RF chain and have constant-
amplitude (CA) constraint on their weights [4], [6], [17], [18].
Meanwhile, a hybrid analog/digital precoding structure was
also proposed to realize multi-stream/multi-user transmission
[9], [12], [13], where a small number of RF chains are
tied to a large antenna array. No matter whether the analog
beamforming or the hybrid precoding structure is exploited,
entry-wise estimation of channel state information (CSI) is
time costly due to large-size antenna arrays, and a more
efficient antenna training algorithm is needed.
For the hybrid precoding structure, as the mmWave channel
is generally sparse in the angle domain, different compressed
sensing (CS) based channel estimation methods were pro-
posed to estimate the steering angles of multipath components
(MPCs) [9], [16], [19]–[21], where [19] is for point-to-point
multi-stream transmission, [20] is for multi-user transmission,
while [21], based on a presentation of antenna array with
virtual elements, further enhances the channel estimation over
[19]. For the analog beamforming structure, there in general
exist two different approaches. In [4], [6], [7], [22], an
iterative beamforming training approach is adopted, in which
the beamforming vector on one side (transmitter or receiver)
is alternatively optimized by fixing the beamforming vector
on the other side, and the alternation is repeated iteratively to
improve the beamforming gain one iteration upon another. On
the other hand, in [17], [18], [23], a switched beamforming
approach is adopted, where the beam search space (at the
transmitter and receiver side, respectively) is represented by
a codebook containing multiple codewords, and the best
transmit/receive beams are found by searching through their
respective codebooks. Both approaches have their own merit
and may be useful in different applications.
In this paper, we focus on switched beamforming for one-
stream transmissions. This is motivated by the fact that single-
stream beamforming is actually capacity achieving in the very-
low SNR case [24]. Furthermore, single stream beamforming
can be readily extended to the more complicated hybrid
precoding case [19]. For switched beamforming, an exhaustive
search algorithm may be used, which sequentially tests all
the beam directions in the angle domain and finds the best
pair of transmit/receive beamforming codewords. This is con-
ceptually straightforward. However, the overall search time is

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, XXX 20XX 2
prohibitive, because the number of candidate beam directions
is usually large for mmWave communication. To improve the
search efficiency, a hierarchy of codebooks may be defined [3],
[17], [18], [25]–[27]. For example, a coarse codebook may be
defined with a small number of coarse/low-resolution beams
(or sectors) covering the intended angle range, while a fine
codebook may be defined with a large number of fine/high-
resolution beams covering the same intended angle range, and
that a coarse beam may have the same/similar coverage as that
of multiple fine beams together. A divide-and-conquer search
may then be carried out across the hierarchy of codebooks,
by finding the best sector (or coarse beam) first on the low-
resolution codebook level, and then the best fine beam on the
high-resolution codebook level, with the best high-resolution
beam contained by the best low-resolution beam.
Performances of the switched beamforming schemes, in-
cluding the search time and success rate, are highly dependent
on the hierarchical codebook design. In [17], [18], although
wider beams were proposed to speed up beam search, design
approaches to broaden the beams were not studied. In [25],
codewords with wider beams were generated by summing
two codewords with narrower beams, but the CA constraint
was no longer satisfied. In [26], a sub-array method was
proposed to broaden the beam width by pointing the sub-
array beams in slightly-gapped directions. However, a full
hierarchical codebook was not explicitly designed therein, and
this approach may be not feasible to design very wide or even
omni-directional beams. In [19], the hybrid precoding structure
was adopted to shape wider beams by exploiting the sparse
reconstruction approach, but high-quality wide beams can be
shaped only when the number of RF chains is large enough and
deep sinks within the angle range appear otherwise. In [27], a
binary-tree structured hierarchical codebook was designed by
using antenna deactivation, where wider beams were generated
by turning off part of the antennas. A complete codebook
was designed with closed-form expressions provided therein.
However, the number of active antennas is too small for
very wide or omni-directional beams, which may limit its
application in mmWave communication, where per-antenna
transmission power is limited.
In this paper, we first propose two basic criteria for arbitrary
hierarchical codebook designs, and then devise an efficient
hierarchical codebook by jointly exploiting sub-array and de-
activation (turning-off) antenna processing techniques. Closed-
form expressions are provided to generate the codebook. In
the proposed approach, the beams of the sub-arrays steer
towards widely-gapped directions to broaden beams, which is
essentially different from [26], and the deactivation operates on
the sub-arrays instead of individual antennas like that in [27].
To the best of our knowledge, this is the first to propose these
two criteria and the joint sub-array and deactivation codebook
design. Performance evaluations are conducted under both
line-of-sight (LOS) and non-LOS (NLOS) channels, as well
as with both total and per-antenna transmission power models.
Results show superiority of the proposed codebook over the
existing alternatives, especially when the per-antenna transmit
power is constrained.
The rest of this paper is as follows. In Section II, the
•(0,1)
•(1,1) •(1,2)
t
w
...
r
w
...
RF Chain RF Chain
H
PA LNA
Phase
Shifter
Fig. 1. Illustration of the system.
system and channel models are introduced. In Section III,
the hierarchical codebook design is presented. In Section IV,
performance evaluation is conducted. The conclusion is drawn
lastly in Section V.
Symbol Notations: a,a,A, and Adenote a scalar variable,
a vector, a matrix, and a set, respectively. (·)∗,(·)Tand
(·)Hdenote conjugate, transpose and conjugate transpose,
respectively. E(·)denotes expectation operation. [a]iand [A]ij
denote the i-th entry of aand the i-row and j-column entry
of A, respectively. [a]i:jdenotes a vector with entries being
the i-th to j-th entries of [a].|·|and ∥ ·∥ denote the absolute
value and two-norm, respectively.
II. SY ST EM A ND CH AN NE L MODELS
A. System Model
Without loss of generality, we consider an mmWave com-
munication system with half-wave spaced uniform linear
arrays (ULAs) of NTand NRelements equipped at the
transmitter and receiver, respectively [5], [17], [28], [29], as
shown in Fig. 1, where a single RF chain is tied to the
ULA at both the transmitter and receiver, and thus the analog
beamforming structure is adopted. At the transmitter, each
antenna branch has a phase shifter and power amplifier (AP) to
drive the antenna, while at the receiver, each antenna branch
has a low-noise amplifier (LNA) to amplify the signal and
a phase shifter. It is noted that as the analog beamforming
can be seen as one of the branches of the hybrid precoding
structure, the proposed criteria and codebook design can be
naturally used by the hybrid precoding structure, which will
be shown in Section III-D. Additionally, in our system each
antenna branch can be deactivated or turned off, i.e., there is
a switch in each antenna branch at both sides although not
depicted in the figure. Generally, all the PAs, as well as the
LNAs, have the same scaling factor if activated. Thus, each
element of the antenna weight vectors (AWVs) at the both
sides either has a constant amplitude or is zero.
Letting sdenote the transmitted symbol with unit power,
the received signal is
y=PtotwH
RHwTs+wH
Rn,(1)
where Ptot is the total transmission power of all the active
antennas, wTand wRare the transmit and receive AWVs,
respectively, His the channel matrix, nis the Gaussian noise

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, XXX 20XX 3
vector with power N0, i.e., E(nnH) = N0I. Let W(N)denote
a set of vectors with Nentries as shown in (3), where νis a
normalization factor such that all the vectors have unit power.
We can find that each entry of an arbitrary vector in W(N)
has either an amplitude ν(activated) or is 0 (deactivated).
Consequently, we have wT∈ W(NT), and wR∈ W(NR). It
is noted that this signaling is based on the total transmission
power, and we can further define the total transmission SNR
as γtot =Ptot/N0, and the received SNR with the total
transmission power model as
ηtot =γtot|wH
RHwT|2.(2)
The power gain under this model is
Gtot =ηtot
γtot
=|wH
RHwT|2,(4)
which is also the array gain.
On the other hand, in mmWave communication the scaling
abilities of PAs are generally limited. Thus, a per-antenna
transmission power model is also with significance to char-
acterize the best transmission ability of the transmitter, which
is shown as
y=PperNTactwH
RHwTs+wH
Rn,(5)
where Pper is the per-antenna transmission power, NTact is the
number of active antennas of wT, which varies as different
wT. Also, we have wT∈ W(NT), and wR∈ W(NR).
In addition, the per-antenna transmission SNR is defined as
γper =Pper /N0, and the received SNR with the per-antenna
transmission power model is defined as
ηper =γper NTact|wH
RHwT|2.(6)
The power gain under this model is
Gper =ηper
γper
=NTact|wH
RHwT|2,(7)
which includes both the transmission power gain equal to
the number of active antennas NTact and the array gain
|wH
RHwT|2.
It is worth mentioning that the total and per-antenna trans-
mission power models are suitable for the cases that the scaling
abilities of PA are high enough and limited, respectively.
However, there is no difference for codebook design between
with these two models.
B. Channel Model
Since mmWave channels are expected to have limited
scattering [19], [29]–[32], MPCs are mainly generated by
reflection. That is, mmWave channels have the feature of
directionality. Different MPCs have different physical transmit
steering angles and receive steering angles, i.e., physical
angles of departure (AoDs) and angles of arrival (AoAs).
Consequently, mmWave channels are relevant to the geometry
of antenna arrays. With half-spaced ULAs adopted at the
transmitter and receiver, the channel matrix can be expressed
as [19], [26], [27], [29], [33], [34]
H=NTNR
L

ℓ=1
λℓa(NR,Ωℓ)a(NT, ψℓ)H,(8)
where λℓis the complex coefficient of the ℓ-th path, Lis the
number of MPCs, a(·)is the steering vector function,Ωℓand
ψℓare cos(AoD) and cos(AoA) of the ℓ-th path, respectively.
Let θℓand φℓdenote the physical AoD and AoA of the ℓ-
th path, respectively; then we have Ωℓ= cos(θℓ)and ψℓ=
cos(φℓ). Therefore, Ωℓand ψℓare within the range [−1 1].
For convenience, in the rest of this paper, Ωℓand ψℓare called
AoDs and AoAs, respectively. Similar to [19], [29], λℓcan be
modeled to be complex Gaussian distributed, while θℓand φℓ
can be modeled to be uniformly distributed within [0,2π].a(·)
is a function of the number of antennas and AoD/AoA, and
can be expressed as
a(N, Ω) = 1
√N[ejπ0Ω, ejπ1Ω , ..., ejπ(N−1)Ω]T,(9)
where Nis the number of antennas (Nis NTat the transmitter
and MRat the receiver), Ωis AoD or AoA. It is easy
to find that a(N, Ω) is a periodical function which satisfies
a(N, Ω) = a(N , Ω+2). The channel matrix Halso has power
normalization L

ℓ=1
E(|λℓ|2) = 1.(10)
C. The Problem
From a system level, joint Tx/Rx beamforming is required
to maximize the received SNR, i.e.,
Maximize ηtot =γtot|wH
RHwT|2or
ηper =γper NTact|wH
RHwT|2,
Subject to wR∈ WR,wT∈ WT.
(11)
Clearly, if His known at the transmitter and receiver,
and there is no CA constraint, the optimal wTand wR
can be easily solved by singular value decomposition (SVD).
However, in mmWave communication it is too time costly
for entry-wise estimation of the channel matrix, which has a
large scale due to large antenna arrays, and there exists the CA
constraint. Thus, the SVD approach is basically not feasible
for mmWave communication.
Fortunately, according to (8) the mmWave channel has
uncertainty mainly on AoDs/AoAs at both sides. In such a
case, the one-stream beamforming problem in (11) can be
simplified to find the AoD/AoA of an arbitrary strong MPC
(or better the strongest MPC)1, and set wTand wRas the
Tx/Rx steering vectors pointing to these AoD/AoA.
To this end, a straightforward way is to evenly sample the
angle domain [−1,1] with a small interval, e.g., 1/N for
an Nantenna-array, and sequentially test all these sampled
angles with corresponding steering vectors at both sides. This
is the exhaustive search method. Clearly the codebooks for
1Basically, under LOS channel, where there is a LOS component signif-
icantly stronger than the other MPCs, the best MPC (the LOS component)
needs to be found; while under NLOS channel, where all the MPCs have
similar strengths, an arbitrary strong MPC can be feasible.

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, XXX 20XX 4
W(N) = {ν[β1ejθ1, β2ejθ2, ..., βNejθN]Tβi∈ {0,1}, θi∈[0,2π), i = 1,2, ..., N }(3)
exhaustive search are composed by only steering vectors.
Although exhaustive search is feasible and can always find
the best MPC, it has a time complexity O(N2)[27], [35],
which is too high for large arrays. Thus, hierarchical search is
widely used to reduce the search time. In fact, the search time
is highly dependent on the design of the Tx/Rx codebooks,
which are subsets of WTand WR, respectively. Hence, we
focus on the codebook design in this paper.
III. HIERARCHICAL CODEBOOK DESIGN
In this section, we design a hierarchical codebook composed
by codewords (or AWVs) with different beam widths, which
helps the search efficiency in finding the steering vectors of
a strong or the strongest MPC at both sides. It is noted that,
based on the specific structure of the mmWave channel model,
the codebook design is to establish a relationship between
the codewords in the angle domain to speed up the beam
search. Thus, the codebook design is in fact irrelevant to
the instantaneous channel response. When beamforming is
required in practice before data transmission, a beam search
process needs to be launched based on the designed codebook
to find the suitable beamforming weights (steering vectors) for
a given channel. For different channels, the codebook is the
same, but the searched optimal steering vectors are different
depending on the channel responses.
Although several hierarchial search schemes have been
proposed for beam search in both literatures [25]–[27] and
some standards, like IEEE 802.15.3c and IEEE 802.11ad [3],
[17], [18], to the best of our acknowledge, there are no
criteria proposed to judge whether a codebook is suitable or
not, and there are few complete hierarchical codebooks with
closed-form expressions provided for mmWave communica-
tion. Therefore, in this section, we first propose two basic
criteria to design a hierarchical codebook, and then design
one jointly using sub-array and deactivation techniques based
on the proposed criteria.
A. Two Criteria
Before introducing the two criteria, we first introduce two
definitions here. Let A(w,Ω) denote the beam gain of walong
angle Ω, which is defined as
A(w,Ω) = √Na(N, Ω)Hw=
N

n=1
[w]ne−jπ(n−1)Ω,(12)
where Nis the number of elements of w.
Let CV(w)denote the beam coverage in the angle domain
of AWV w, which can be mathematically expressed as
CV(w) = Ω|A(w,Ω)|> ρ max
ω|A(w, ω)|,(13)
where ρis a factor within (0,1) to determine the beam
coverage of w. It is easy to find that the coverage is smaller
as ρbecomes greater. When ρ= 1/√2, the beam coverage
is the 3dB coverage, and the beam width is the well-known
3dB beam width. Different codebook design methods may
have different values of ρ, and codewords with different beam
widths in the same codebook may also have different values
of ρ.
Hierarchical search is simply layered search, i.e., the AWVs
within the codebook are layered according to their beam width.
AWVs with a lower layer have larger beam width. Letting
w(k, n)denote the n-th codeword (or AWV) in the k-th layer,
the two criteria are presented as follows.
Criterion 1: The union of the beam coverage of all the
codewords within each layer should cover the whole angle
domain, i.e.,
Nk

n=1 CV(w(k, n)) = [−1,1], k = 0,1, ..., K, (14)
where Nkis the number of codewords in the k-th layer, K
it the maximal index of the layer (there are K+ 1 layers in
total).
Criterion 2: The beam coverage of an arbitrary codeword
within a layer should be covered by the union of those of
several codewords in the next layer, i.e.,
CV(w(k, n)) ⊆
m∈Ik,n CV(w(k+ 1, m)), k = 0,1, ..., K −1,
(15)
where Ik,n is the index set with indices of the codewords in
the (k+1)-th layer for the n-th codeword in the k-th layer. For
convenience, we call w(k, n)a parent codeword, and {w(k+
1, m)|m∈ Ik,n}the child codewords of w(k, n).
It is clear that Criterion 1 guarantees the full coverage, i.e.,
there is no miss of any angle during the beam search, while
Criterion 2 establishes a tree-fashion relationship between the
codewords, which enables hierarchical search. If each parent
codeword has Mchild codewords, all the codewords in the
codebook constitute an M-way tree with respect to their beam
coverage in the angle domain. In such a case, hierarchical
search can be easily realized by using the tree search algorithm
in both the receiver and transmitter as following [19], [27].
The Hierarchical Search: Initially, we fix the transmitter
to be in an omni-directional mode, and run an M-way tree
search for logM(NR)stages to find the best receive codeword.
And then we fix the receiver to be in a directional mode
with the found best receive codeword, and then run an M-
way tree search for logM(NT)stages to find the best transmit
codeword. In each stage, we have Mcandidate codewords,
which are the Mchild codewords of a parent codeword found
in the last stage. We need to test all the Mcodewords one by
one to find the best one, and treat it as a new parent codeword
for the next-stage search. Therefore, the search time (number
of tests) for Tx/Rx joint training is
T=MlogMNT+MlogMNR.(16)

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, XXX 20XX 5
•(0,1)
•(1,1) •(1,2)
•(2,1) •(2,2) •(2,3) •(2,4)
•(log ! , 1) •(log ! , !)
…
Angle Domain -1 +1
The 0-th Layer
The 1-st Layer
The 2-nd Layer
The Last Layer
Fig. 2. Beam coverage of a binary-tree structured codebook.
In the next subsection we will design a codebook with
M= 2, for the reason that when M= 2 the codebook
tree is a typical binary tree, and the number of antennas
is powers of two, which is generally used in antenna array
design. Nevertheless, extending the proposed method to other
values of Mis straightforward.
B. The Deactivation Approach
As a basis of the joint sub-array and deactivation approach,
we first introduce the deactivation (DEACT) approach in this
subsection to design a binary-tree codebook, which has the
beam coverage shown in Fig. 2, where there are log2(N) + 1
layers with indices from k= 0 to k= log2(N), and the
number of codewords in the k-th layer Nk= 2k. Here N
denotes the number of antennas of an arbitrary array. Thus,
N=NTat the transmitter and N=NRat the receiver.
Besides, we have
CV(w(k, n)) = CV(w(k+ 1,2n−1)) ∪ CV (w(k+ 1,2n)),
k= 0,1, ..., (log2(N)−1), n = 1,2,3, ..., 2k.
(17)
In our method, we define
CV(a(N , Ω)) = Ω−1
N,Ω + 1
N,(18)
which means that the steering vectors have a beam width 2/N
centering at the steering angle [24]. In other words, within
the beam coverage of a(N, Ω), it has the maximal beam gain
along the angle Ω, while the minimal beam gain along the
angles Ω±1/N. Thus, we can compute the value of ρfor our
codebook as
ρ=
a(N, Ω−1/N)Ha(N, Ω)
a(N, Ω)Ha(N , Ω) 
or 
a(N, Ω + 1/N)Ha(N, Ω)
a(N, Ω)Ha(N , Ω) 
=1
N
N

n=1
ejπ(n−1)/N .
(19)
Although the value of ρdepends on N, we have ρ≈0.64
given that Nis large, e.g., N≥8. Even when Nis small, ρ
is still close to 0.64, e.g., ρ= 0.65 when N= 4.
Notice that the Ncodewords in the last layer cover an angle
range [−1,1] in total, which means that all these codewords
must have the narrowest beam width 2/N with different
0.5
1
1.5
2
30
210
60
240
90
270
120
300
150
330
180 0
w(2,1)
w(2,2)
w(1,1)
w(1,2)
w(0,1)
Fig. 3. Beam patterns of w(2,1),w(2,2),w(1,1),w(1,2) and w(0,1)
for the DEACT approach, where N= 128.
steering angles. In other words, the codewords in the last layer
should be the steering vectors with angles evenly sampled
within [−1,1]. Consequently, we have CV(w(log2(N), n)) =
[−1 + 2n−2
N,−1 + 2n
N],n= 1,2, ..., N . With the beam
coverage of the last-layer codewords, we can further obtain
that of the codewords in the other layers in turn as an order
of descending layer indices, i.e., obtain CV(w(log2(N)−
1, n)),CV(w(log2(N)−2, n)), ..., CV(w(0, n)) in turn.
Finally, the beam coverage of all the codewords can be
uniformly written as
CV(w(k, n)) = [−1 + 2n−2
2k,−1 + 2n
2k],
k= 0,1,2, ..., log2N, n = 1,2,3, ..., 2k.
(20)
Comparing (20) with (18), it is clear that when
w(k, n) = [a(2k,−1 + 2n−1
2k)T,0T
(N−2k)×1]T,(21)
(20) is satisfied. This is just the deactivation approach that was
proposed in [27], where the number of active antennas is 2k
in the k-th layer, and the other antennas are all turned off. Fig.
3 shows an example of beam pattern of the DEACT approach
for the case of N= 128. From this figure we find that the
beam coverage of w(0,1) is just the union of those of w(1,1)
and w(1,2), while the beam coverage of w(1,1) is just the
union of those of w(2,1) and w(2,2).
C. The Joint Sub-Array and Deactivation Approach
It is noted that for the deactivation approach, when kis
small, the number of active antennas is small, or even 1 when
k= 0. This greatly limits the maximal total transmission
power of an mmWave device. In general, we hope the number
of active antennas is as large as possible, such that higher
power can be transmitted, because in mmWave communication
per-antenna transmission power is usually limited. To achieve
this target, we consider jointly using the sub-array and deac-
tivation approach here. As the key of this approach is BeaM
Widening via Single-RF Subarray, we term it BMW-SS.

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, XXX 20XX 6
We also want to design a codebook with the beam coverage
shown in Fig. 2. According to (18), in the k-th layer, each
codeword has a beam width of 2/2k. For the codewords of
the last layer, we can also adopt the steering vectors according
to (21). Compared with the codewords in the last layer, those
in the lower layers have wider beams, and according to (20),
codewords in the same layer have the same beam widths but
different steering angles. Thus, there are two basic tasks in the
codebook design, namely to rotate the beam along required
directions and to broaden the beam by required factors. We
first introduce beam rotation.
1) Beam Rotation: Beam rotation can be realized according
to the following theorem.
Theorem 1. CV(w◦√Na(N, ψ )) = CV(w) + ψ, where ◦
represents entry-wise product (a.k.a. Hadamard product), N
is the number of elements of w,ψis an arbitrary angle. A+ψ
is a new angle set with elements being those of the angle set
Aadded by ψ.
The proof is referred to Appendix A, and this theorem can
be used not only for the BMW-SS approach, but also for other
codebook design methods.
Theorem 1 implies that given an arbitrary codeword w,
we can rotate its beam coverage CV(w)by ψwith w◦
√Na(N, ψ). This theorem helps to design all the other
codewords in the same layer once one codeword in this layer
is found. To explain this, we need to emphasize that all the
codewords in the same layer have the same beam widths but
different steering angles according to (20), which means that
the beam coverage of all the codewords can be assumed to
have the same shape but different offsets in the angle domain.
Thus, we can obtain one codeword based on another in the
same layer as long as we know the angle gap between them
according to Theorem 1. In particular, suppose we find the first
codeword in the k-th layer w(k, 1). According to (20), we do
know that the angle gap between the n-th codeword in the
k-th layer, i.e., w(k, n), and w(k, 1) is 2n−2
2k,n= 2,3, ..., 2k.
Then we can obtain the all the other codewords in this layer
based on w(k, 1) according to Theorem 1 (see Corollary 1
below).
Corollary 1. Given the first codeword in the k-th layer
w(k, 1), all the other codewords in the k-th layer can be
found through rotating w(k, 1) by 2n−2
2k,n= 2,3, ..., 2k,
respectively, i.e., w(k, n) = w(k, 1) ◦√Na(N, 2n−2
2k).
Proof: To prove this corollary, we need to prove that,
according to (20), when w(k, n) = w(k, 1) ◦√Na(N, 2n−2
2k),
w(k, n)∈ W(N)and CV(w(k, n)) = [−1 + 2n−2
2k,−1 + 2n
2k].
Since
[w(k, n)]i= [w(k, 1) ◦√Na(N, 2n−2
2k)]i
=[w(k, 1)]iejπ(n−1) 2n−2
2k,
(22)
we have |[w(k, n)]i|=|[w(k, 1)]i|. As w(k, 1) ∈ W(N),
w(k, n)∈ W(N).
In addition, w(k, 1) has a beam coverage [−1,−1 + 2
2k].
According to Theorem 1,
CV(w(k, n))
=CV(w(k, 1) ◦√Na(N, 2n−2
2k))
=CV(w(k, 1)) + 2n−2
2k
=[−1,−1 + 2
2k] + 2n−2
2k
=[−1 + 2n−2
2k,−1 + 2n
2k].
(23)
2) Beam Broadening: The remaining task is to broaden the
beam for each layer. Given an N-element array, generally we
would expect a beam width of 2/N . Nevertheless, this beam
width is in fact achieved by concentrating the transmission
power at a specific angle Ω0, i.e., by selecting AWV to
maximize |A(w,Ω0)|. Intuitively, if we design the AWV to
disperse the transmission power along different widely-spaced
angles, the beam width can be broadened. More specifically,
if a large antenna array is divided into multiple sub-arrays,
and these sub-arrays point at sufficiently-spaced directions, a
wider beam can be shaped.
To illustrate this, let us separate the N-antenna array into M
sub-arrays with NSantennas in each sub-array, which means
N=MNS. In addition, letting fm= [w](m−1)NS+1:mNS, we
have [fm]n= [w](m−1)NS+n,m= 1,2, ..., M.fmcan be seen
as the sub-AWV of the m-th sub-array. Therefore, the beam
gain of wwrites
A(w, ω) =
N

n=1
[w]ne−jπ(n−1)ω
=
M

m=1
NS

n=1
[w](m−1)NS+ne−jπ((m−1)NS+n−1)ω
=
M

m=1
NS

n=1
e−jπ(m−1)NSω[fm]ne−jπ(n−1)ω
=
M

m=1
e−jπ(m−1)NSωA(fm, ω),
(24)
which means that the beam gain of wcan be seen as the
union of those of fm. According to (18), by assigning fm=
ejθma(NS,−1 + 2m−1
NS), where ejθmcan be seen as a scalar
coefficient with unit norm for the m-th sub-array, the m-th
sub-array has beam coverage CV(fm)=[−1 + 2m−2
NS,−1 +
2m
NS],m= 1,2, ..., M . Hence, whas the beam coverage
CV(w) =
M

m=1 CV(fm) = [−1,−1+ 2M
NS
] = [−1,−1+ 2M2
N],
(25)
i.e., the beam width has been broadened by M2by using the
sub-array technique, where a broadening factor Mcomes from
the number of sub-arrays, while another factor Mresults from
the reduction factor of the sub-array size.
However, in the above process, the mutual effects between
different sub-arrays are not taken into account. In the case of

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, XXX 20XX 7
fm=ejθma(NS,−1 + 2m−1
NS), we have
A(w, ω)fm=ejθma(NS,−1 + 2m−1
NS
)
=NS
M

m=1
e−jπ(m−1)NSωejθm×
a(NS, ω)Ha(NS,−1 + 2m−1
NS
).
(26)
As the steering vector has the properties that a(NS,−1 +
2m−1
NS)Ha(NS,−1 + 2n−1
NS) = 0 when m̸=n, the beam gain
of fmalong the angle −1 + 2m−1
NSis affected little by fn. It
is clear that |A(w,−1 + 2m−1
NS)|=√NSfor m= 1,2, ..., M ,
which means that the beam gains along angles ω= 1 + 2m−1
NS
are significant.
Additionally, to reduce the beam fluctuation, it is required
that the intersection points in the angle domain of these
coverage regions, i.e., ω=−1 + 2n
NS,n= 1,2, ..., M −1, also
have high beam gain, and this can be realized by adjusting
coefficients ejθm. Concretely, we have (28), where in (a) we
have used the fact that a(NS, ω1)Ha(NS, ω2)is small and can
be neglected when |ω1−ω2|>2/NS, in (b) we have exploited
the condition that NSis even in this paper. To maximize
|A(w, ω)|2, we face the problem
maximize
∆θ|f(NS,∆θ)|2,(27)
which has a solution that ∆θ= (2k−NS−1
NS)π, where k∈
Z. Thus, we may choose θm=−jm NS−1
NSπ, which satisfies
∆θ=π, to reduce the fluctuation of the beam.
In summary, by using the sub-array method and setting
fm=e−jm NS−1
NSπa(NS,−1 + 2m−1
NS),m= 1,2, ..., M , we
obtain a codeword wwith a beam width 2M
NS=2M2
N. If we
jointly using the sub-array and deactivation method, we may
obtain codewords with beam widths 2NA
NS=2MNA
Nby setting
as
fm=


e−jm NS−1
NSπa(NS,−1 + 2m−1
NS
), m = 1,2, ..., NA,
0NS×1, m =NA+ 1, NA+ 2, ..., M.
(29)
where NAis the number of active sub-arrays.
3) Codebook Generation: Recall that we need to design
w(k, n)with beam widths 2/2kin the k-th layer.
When k= log2(N), we have w(log2(N), n) = a(N, −1 +
2n−1
N),n= 1,2, ..., N .
When k= log2(N)−ℓ, where ℓ= 1,2, ..., log2(N), we
obey the following procedures to compute w(k, n):
•Separate w(k, 1) into M= 2⌊(ℓ+1)/2⌋sub-arrays with
fm= [w(k, 1)](m−1)NS+1:mNS, where ⌊·⌋ is the flooring
integer operation, m= 1,2, ..., M ;
•Set fmas (29), where NA=M/2if ℓis odd, and NA=
Mif ℓis even;
•According to Corollary 1, we have w(k, n) = w(k, 1) ◦
√Na(N, 2(n−1)
N), where n= 2,3, ..., 2k;
•Normalize w(k, n).
Fig. 4 shows an example of the beam pattern of the BMW-
SS approach in the case of N= 128. From this figure we find
0.5
1
1.5
2
2.5
30
210
60
240
90
270
120
300
150
330
180 0
w(2,1)
w(2,2)
w(1,1)
w(1,2)
w(0,1)
Fig. 4. Beam patterns of w(2,1),w(2,2),w(1,1),w(1,2) and w(0,1)
for the BMW-SS approach, where N= 128.
that the beam coverage of w(0,1) is just the union of those of
w(1,1) and w(1,2), while the beam coverage of w(1,1) is
just the union of those of w(2,1) and w(2,2). Comparing the
beam pattern of DEACT shown in Fig. 3 with that in Fig. 4, it
can be observed that although there are small-scale fluctuations
for BMW-SS, the beams of BMW-SS appear flatter than those
of DEACT within the covered angle.
On the other hand, for BMW-SS all the codewords either
have all the antennas activated, or have half of them activated,
which shows a significant advantage over DEACT in terms
of the maximal total transmission power, especially for the
low-layer codewords. Fig. 5 shows the comparison of beam
patterns of BMW-SS, DEACT, and the approach in [19]
(termed as Sparse) with the per-antenna transmission power
model, where all the weights of active antennas have a unit
amplitude. From this figure, we find that BMW-SS offers
much higher beam gains than DEACT due to exploiting much
greater number of active antennas. In addition, for the Sparse
codebook, when the number of RF chains is small, there
are deep sinks within the beam coverage of the wide-beam
codewords, and the sink is more severe when the number of
RF chains is smaller, which are in accordance with the results
in [19] (Fig. 5 therein). Clearly, if the AoD or AoA of an
MPC is along the sink angle, it cannot be detected with the
codeword, which results in miss detection of the MPC. By
contrast, BMW-SS does not have such deep sinks.
It is noted that the corresponding hierarchical search of the
designed codebook will eventually converge to a codeword
of the last layer, i.e., a steering vector, at both ends. We
can find that the angle resolution of the last layer is 2/N.
Thus, the designed codebook is just coarse codebook, while
the corresponding search method is coarse search, like those in
[27]. If a higher angle resolution is required, a fine codebook
composed by steering vectors with a smaller sampling gap
than 2/N is necessary. Details are referred to [27].

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, XXX 20XX 8
A(w, ω)fm=ejθma(NS,−1 + 2m−1
NS
), ω =−1 + 2n
NS
=NS
M

m=1
e−jπ(m−1)NSωejθma(NS, ω)Ha(NS,−1 + 2m−1
NS
)
(a)
≈NSe−jπ(n−1)NSωejθna(NS,−1 + 2n
NS
)Ha(NS,−1 + 2n−1
NS
)+
NSe−jπnNSωejθn+1 a(NS,−1 + 2n
NS
)Ha(NS,−1 + 2n+ 1
NS
)
=1
√NS
e−jπ(n−1)NSωejθn×NS

i=1
e−jπ(i−1)/NS+ejπNSωej(θn+1 −θn)
NS

i=1
ejπ(i−1)/NS
(b)
=1
√NS
e−jπ(n−1)NSωejθn×NS

i=1
e−jπ(i−1)/NS+ej∆θ
NS

i=1
ejπ(i−1)/NS
∆
=1
√NS
e−jπ(n−1)NSωejθnf(NS,∆θ),
(28)
1
2
3
4
5
30
210
60
240
90
270
120
300
150
330
180 0
w(3,3) for Sparse, 2 RF Chains
w(3,3) for Sparse, 4 RF Chains
w(3,1) for BMW−SS
w(3,2) for BMW−SS
w(3,1) for DEACT
w(3,2) for DEACT
Fig. 5. Comparison of the beam patterns of BMW-SS, DEACT, and the
approach in [19] (termed as Sparse) with the per-antenna transmission power
model, where N= 32.Ld= 1 for the Sparse approach.
D. Generalization
1) For the Hybrid Precoding Structure: In this paper we
adopt an analog beamforming structure, and both the proposed
two criteria and the BMW-SS approach are based on the
analog beamforming structure. However, they are naturally
feasible for the hybrid precoding structure, because the analog
beamforming structure can be seen as one of the branches of
the hybrid precoding structure [19].
To realize multi-stream transmission with the hybrid pre-
coding structure, the AoDs and AoAs of multiple MPCs need
to be searched. The search process with BMW-SS based on
the beamforming structure in this paper can be adopted to
search the AoD and AoA of each single MPC. In fact, similar
extension from one-stream transmission to multi-stream trans-
mission has been studied for the Sparse codebook in [19].
2) For Other Types of Antenna Arrays: In this paper we
adopt a ULA model. There are other types of antenna arrays in
practice, e.g., uniform planar array (UPA) and uniform circular
array (UCA). The proposed criteria and the BMW-SS approach
can be easily extended to the UPA model. In particular, for
a typical 2-dimensional grid UPA with m×nelements, its
steering vector can be expressed as the Kronecker product of
those of two ULAs with m×1and n×1elements, respectively
[26]. The search process as well as the codebook design could
be extended to the UPA model and will be studied later.
On the other hand, for a UCA model, the two criteria
are also feasible, but that the beam coverage in Criterion 1
should be extended to a 2-dimensional angle range, including
both the azimuth and elevation angle ranges. However, the
proposed BMW-SS approach can hardly be extended to the
UCA model, because the relation between the elements of
a steering vector changes. It would be indeed interesting to
design a new codebook according to the proposed criteria with
a UCA mode.
3) For Arbitrary Number of Antenna Elements: In this
paper we require that the number of elements of a ULA (N) is
Mpfor some positive integer p, which is because the BMW-
SS approach needs to divide the array or a sub-array into M
smaller sub-arrays. For a ULA with an arbitrary number of
elements, the sub-array technology is infeasible if Nis not
Mto an integer power. Hence, the proposed approach may
not be extended to with arbitrary number of antenna elements.
There are two possible choices in practice. One is to select a
ULA with Nbeing Mto an integer power when designing
the system, which is reasonable because the beamforming
method should be considered in system planning. The other
one is to exploit BMW-SS for beamforming with M⌊logMN⌋
antennas while deactivating the other ones, where ⌊·⌋ is the
floor operation. Afterwards, further beam refinement can be
launched with all the antennas activated.
IV. PERFORMANCE EVALUATION
In this section, we evaluate the performance of the designed
hierarchical codebook by the BMW-SS approach, and compare
it with the alternatives. We consider two different system
models on the transmission power, namely total transmission

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, XXX 20XX 9
power and per-antenna transmission power, which correspond
to the signal models in (1) and (5), respectively. The total
transmission power signal model reflects the performance for
the case that the transmission power on each antenna branch
can be high enough, while the per-antenna transmission power
signal model reflects the limit performance for the case that
the transmission power on each antenna branch is limited2.
The per-antenna transmission power model makes more sense
in mmWave communication, where the output power of a
single power amplifier is generally limited [2], [4]. The
activation/deactivation operations of a codebook are irrelevant
to the power models. In particular, no matter which power
model is adopted, the codewords of BMW-SS either have all or
half of the antennas activated, those of DEACT have varying
numbers of the antennas activated and the number may be
quite small, while those of Sparse always have all the antennas
activated.
Besides, in the simulations, both LOS and NLOS channel
models are considered based on (8). For LOS channel, the
first MPC has a constant coefficient and random AoD and
AoA, while the other NLOS MPCs have complex Gaussian-
distributed coefficients and random AoDs and AoAs [26],
[29]. The LOS MPC is generally much stronger than the
NLOS MPCs. For NLOS channel, all the MPCs have complex
Gaussian-distributed coefficients with the same variance and
random AoDs and AoAs [19], [26], [29]. Both the LOS
and NLOS channels are sparse in the angle domain, because
the number of MPCs is much smaller than the numbers of
the Tx/Rx antennas [19], [26], [29]. For all the codebooks,
the hierarchical search method introduced in Section III-A
is used. The performances of received SNR and success rate
are all averaged on the instantaneous results of 104random
realizations of the LOS/NLOS channel.
A. Total Transmission Power Model
In this subsection, the total transmission power signal model
shown in (1) is used. With this model, the deactivation of
antennas will not affect the total transmission power, i.e., the
total transmission power is the same for the involved schemes.
Fig. 6 shows the received power during each search step
with the BMW-SS and DEACT codebooks under both LOS
and NLOS channels, where NT=NR= 64,L= 3,Ptot = 1
W, and N0= 10−4W, i.e., the SNR for beam training
is sufficiently high, which means the length of the training
sequence is sufficiently long. The upper bound is achieved
by the exhaustive search method. For the LOS channel, the
LOS component has 15dB higher power than that of an
NLOS MPC. From this figure we can find that the received-
power performance of these two codebooks is similar to each
other. Under both channels, at the beginning, i.e., in the first
two steps, DEACT behaves slightly better than BMW-SS;
while in the following steps, BMW-SS slightly outperforms
DEACT, until both methods achieve the same performance
after the search process, because they have the same last-
layer codewords. Meanwhile, both approaches reach the upper
2The limit performance here refers to the performance in the case that all
the active antennas transmit with maximal power.
0 2 4 6 8 10 12
0
5
10
15
20
25
30
35
Steps of the Hierarchical Search
Received Power (dBW)
LOS, DEACT
LOS, BMW−SS
LOS, Upper Bound
NLOS, DEACT
NLOS, BMW−SS
NLOS, Upper Bound
Fig. 6. Received power during each search step with the BMW-SS and
DEACT codebooks under both LOS and NLOS channels, where NT=NR=
64,L= 3,Ptot = 1 W, and N0= 10−4W. Step 1 to Step 6 is for Rx
training, while Step 7 to Step 12 is for Tx training.
bound under LOS channel, while achieve a performance close
to the upper bound under NLOS channel. This is because
under LOS channel, the LOS component is the optimal MPC,
and it is acquired by all BMW-SS, DEACT and the exhaustive
search. However, under NLOS channel, BMW-SS and DEACT
acquire an arbitrary MPC of the LNLOS MPCs, which may
not be the optimal one acquired by the exhaustive search.
Fig. 7 shows the success rate of hierarchical search with the
BMW-SS, DEACT and Sparse (proposed in [19]) codebooks
under LOS channel, where NT=NR= 64,L= 3.ηis
the power difference in dB between the LOS component and
an NLOS MPC. From this figure, it is observed that both the
transmission SNR γtot and ηaffect the success rate. For all
the codebooks, the success rate improves as γtot increases.
However, due to the mutual effect of MPCs (i.e., spatial
fading), the success rate improves little when γtot is already
high enough. Basically when ηis bigger, the mutual effect is
less, and the success rate is higher. For the Sparse codebook,
the performance also depends on the number of RF chains.
When the number of RF chains is small, e.g., only 2, the
deep sinks within the covered angle (See Fig. 5) will sharply
reduce the success rate, as shown in Fig. 7. Furthermore, we
can find that the success rate with the BMW-SS codebook is
higher than that with the DEACT codebook. This is because
the beams of the BMW-SS codebook are flatter than those of
the DEACT codebook; thus they are more robust to the spatial
fading. Also, the success rate with the BMW-SS codebook is
higher than that with the Sparse codebook when the number
of RF chains is not large.
Fig. 8 shows the success rate of hierarchical search with
the BMW-SS, DEACT and Sparse codebooks under NLOS
channel, where NT=NR= 64. From this figure, the same
performance variation with respect to the transmission SNR
γtot can be observed as that in Fig. 7, and Sparse with 2 RF
chains also has the poorest performance. In addition, BMW-SS
basically outperforms DEACT and Sparse with 4 RF chains,

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, XXX 20XX 10
0 10 20 30 40 50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
Success Rate
DEACT, η = 10dB
BMW−SS, η = 10dB
Sparse, 2 RF Chains, η = 10dB
Sparse, 4 RF Chains, η = 10dB
DEACT, η = 20dB
BMW−SS, η = 20dB
Sparse, 2 RF Chains, η = 20dB
Sparse, 4 RF Chains, η = 20dB
Fig. 7. Success rate of hierarchical search with the BMW-SS, DEACT and
Sparse codebooks under LOS channel, where NT=NR= 64,L= 3.ηis
the power difference in dB between the LOS component and an NLOS MPC.
0 10 20 30 40 50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
Success Rate
DEACT, L=1
BMW−SS, L=1
Sparse, 2 RF Chains, L=1
Sparse, 4 RF Chains, L=1
DEACT, L=4
BMW−SS, L=4
Sparse, 2 RF Chains, L=4
Sparse, 4 RF Chains, L=4
Fig. 8. Success rate of hierarchical search with the BMW-SS, DEACT and
Sparse codebooks under NLOS channel, where NT=NR= 64.
and the superiority depends on L. When L= 1, i.e., there is
only one MPC, both BMW-SS and DEACT achieve a 100%
success rate when γtot is high enough, because there is no
spatial fading. In contrast, Sparse cannot achieve a 100%
success rate even when L= 1, due to the deep sinks within
the covered angle. When L > 1, all these schemes can hardly
achieve a 100% success rate, due to the mutual effect of
multiple MPCs.
It is noteworthy that the superiority of BMW-SS versus
DEACT in Fig. 6 is different from that in Figs. 7 and 8. Fig.
6 just shows the variation of the received power during the
search process with a high SNR, while Figs. 7 and 8 show
the search results over a wide SNR range. Fig. 6 actually
corresponds to the search process for a set of points in Figs.
7 and 8 with SNR equal of 40dB, and the received-power
superiority of BMW-SS over DEACT in Fig. 6 will become
larger if smaller SNR values are adopted.
B. Per-Antenna Transmission Power Model
In this subsection, the per-antenna transmission power
signal model shown in (5) is used to compare the limit
performances of BMW-SS and DEACT with the same per-
antenna transmission power. With this model, the deactivation
of antennas will significantly affect the total transmission
power. In particular, the total transmission power is lower if
the number of active antennas is smaller.
Fig. 9 shows the received power during each search step
with the BMW-SS and DEACT codebooks under both LOS
and NLOS channels, where NT=NR= 64,L= 3,Pper = 1
W, and N0= 10−4W. The upper bound is achieved by the
exhaustive search method. For the LOS channel, the LOS
component has 15dB higher power than that of an NLOS
MPC. Comparing this figure with Fig. 6, we find a significant
difference that with the per-antenna transmission power model
BMW-SS has a distinct superiority over DEACT during the
search process, especially at the beginning of the search
process. The superiority is about 15 dB at the beginning, and it
becomes less as the search goes on, until vanishes at the end of
beam search, i.e., the two methods achieve the same received
SNR after the search process. The superiority of BMW-SS
results from the fact that the number of the active antennas
for the codewords with wide beams is significantly greater
than that for DEACT, and thus BMW-SS has a much higher
total transmission power than DEACT when the per-antenna
transmission power is the same.
Moreover, the increasing speed of received power is the
same from Step 1 to Step 6 for both of the two schemes,
but from Step 7 to Step 12, the increasing speed for BMW-
SS varies, and that for DEACT becomes greater than that
from Step 1 to Step 6. This is because with per-antenna trans-
mission power, there are two power gains during the search
process according to (4), namely the array gain provided by
narrowing the Tx/Rx beams and the total transmission power
gain provided by increasing the number of active transmit
antennas. For DEACT, there is only Rx array gain from Step
1 to Step 6, where Rx training is performed, while there are
both Tx array gain and total transmission power gain from
Step 7 to Step 12, where Tx training is performed; thus, the
increasing speed of received power is greater from Step 7
to Step 12. For BMW-SS, there is also only Rx array gain
from Step 1 to Step 6 for Rx training; thus the received
power consistently increases with the same speed as DEACT.
But from Step 7 to Step 12 for Tx training, although the
Tx beam consistently becomes narrower, which means that
Tx array gain is consistently improved, the number of active
antennas alternatively changes between NTand NT/2, which
means that the total transmission power may become larger
or smaller. Hence, when both the Tx array gain and total
transmission power increase, the received power improves
with a speed the same as DEACT, while when the Tx array
gain increases but the total transmission power decreases, the
received SNR does not improve and may even decrease.
It is noted that the superiority of BMW-SS over DEACT at
the beginning of the search process is with big significance

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, XXX 20XX 11
0 2 4 6 8 10 12
0
5
10
15
20
25
30
35
40
45
50
55
Steps of the Hierarchical Search
Received Power (dBW)
LOS, DEACT
LOS, BMW−SS
LOS, Upper Bound
NLOS, DEACT
NLOS, BMW−SS
NLOS, Upper Bound
Fig. 9. Received SNR during each search step with the BMW-SS and DEACT
codebooks under both LOS and NLOS channels, where NT=NR= 64,
L= 3,Pper = 1 W, and N0= 10−4W. Step 1 to Step 6 is for Rx training,
while Step 7 to Step 12 is for Tx training.
for mmWave communication, where per-antenna transmission
power is generally limited. This superiority guarantees that
with the BMW-SS codebook, the success rate of beam search
will be upgraded with the same transmission distance, or the
transmission distance will be extended with the same success
rate of beam search.
Figs. 10 and 11 show the success rates of hierarchical search
with the BMW-SS and DEACT codebooks under LOS and
NLOS channels, respectively. The same simulation conditions
are adopted as those in Figs. 7 and 8, respectively, and the
same results can be obtained from Figs. 10 and 11 as those
from Figs. 7 and 8, respectively, except that the superiority of
BMW-SS over DEACT becomes more significant in Figs. 10
and 11, which benefits from not only the fact that the beams
of the BMW-SS codebook are flatter than those of the DEACT
codebook, but also that the number of the active antennas of
the BMW-SS codewords is basically much greater than that of
DEACT, which offers much higher total transmission power.
Also, Figs. 10 and 11 reveal that even with low per-antenna
transmission power, the success rate of BMW-SS can be close
to 100%, which is evidently better than those of DEACT and
Sparse.
V. CONCLUSIONS
In this paper hierarchical codebook design has been studied
for mmWave communication. Firstly, two basic criteria have
been proposed for the codebook design. Next, a complete
binary-tree structured hierarchical codebook has been designed
by jointly using sub-array and deactivation techniques, i.e.,
the BMW-SS approach. Performance evaluation has been
conducted with both a total transmission power and a per-
antenna transmission power system models. Results show
that the BMW-SS codebook offers two advantages over the
deactivation codebook, namely flatter beams and much larger
number of active antennas. Both of these two advantages
−20 −10 0 10 20 30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
Success Rate
DEACT, η = 10dB
BMW−SS, η = 10dB
Sparse, 2 RF Chains, η = 10dB
Sparse, 4 RF Chains, η = 10dB
DEACT, η = 20dB
BMW−SS, η = 20dB
Sparse, 2 RF Chains, η = 20dB
Sparse, 4 RF Chains, η = 20dB
Fig. 10. Success rate of hierarchical search with the BMW-SS and DEACT
codebooks under LOS channel, where NT=NR= 64,L= 3.ηis the
power difference in dB between the LOS component and an NLOS MPC.
−20 −10 0 10 20 30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
Success Rate
DEACT, L=1
BMW−SS, L=1
Sparse, 2 RF Chains, L=1
Sparse, 4 RF Chains, L=1
DEACT, L=4
BMW−SS, L=4
Sparse, 2 RF Chains, L=4
Sparse, 4 RF Chains, L=4
Fig. 11. Success rate of hierarchical search with the BMW-SS and DEACT
codebooks under NLOS channel, where NT=NR= 64.
basically provide performance superiorities in terms of re-
ceived power during the search process and the success rate of
beam search under both transmission power models, and the
performance superiority is especially significant with the per-
antenna transmission power system model. In addition, the
BMW-SS codebook also outperforms the Sparse codebook,
since there are no deep sinks within the beam coverage for
BMW-SS.
APPENDIX A
PROO F OF TH EO RE M 1
Given an arbitrary N-element vector wand two arbitrary
angles ψand Ω, we want to prove that CV(w◦√Na(N, ψ )) =
CV(w) + ψ, where A+ψis a new angle set with elements
being those of the angle set Aadded by ψ. Note that w◦
√Na(N, ψ)is actually a new vector achieved based on w

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, XXX 20XX 12
and the steering vector a(N, ψ). Let us first see the beam
gain of this new vector.
A(w◦√Na(N, ψ),Ω)
(a)
=√Na(N, Ω)H(w◦√Na(N , ψ))
(b)
=
N

n=1
[w]nejπ(n−1)ψe−jπ(n−1)Ω
=
N

n=1
[w]ne−jπ(n−1)(Ω−ψ)
(c)
=A(w,Ω−ψ),
(30)
where (a) and (c) are according to the definition of the beam
gain in (12), while (b) is obtained according to definition of
the entry-wise product.
Thus, we further have
CV(w◦√Na(N , ψ))
(a)
={Ω| |A(w◦√Na(N, ψ),Ω)|>
ρmax
ω|A(w◦√Na(N, ψ), ω)|}
(b)
={Ω| |A(w,Ω−ψ)|> ρ max
ω|A(w, ω −ψ)|}
(c)
={Ω| |A(w,Ω−ψ)|> ρ max
ω|A(w, ω)|}
(d)
={Ω0+ψ| |A(w,Ω0)|> ρ max
ω|A(w, ω)|}
(e)
={Ω0| |A(w,Ω0)|> ρ max
ω|A(w, ω)|} +ψ
=CV(w) + ψ,
(31)
where (a) is according to the definition of beam coverage in
(13), (b) is according to (30), (c) is based on the fact that the
maxima of |A(w,Ω−ψ)|does not depend on the angle offset
ψ, (d) is obtained by letting Ω = Ω0+ψ, and (e) is obtained
according to the definition of an angle set plus a single angle
in Theorem 1.
ACKNOWLEDGMENTS
The authors would like to thank the editor and the anony-
mous reviewers for their many useful and detailed com-
ments that have helped to improve the presentation of this
manuscript. The authors would also like to thank the authors
of [19] to share their source code online, and particularly thank
Dr. Ahmed Alkhateeb for his kind help to explain how to use
the source code.
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Zhenyu Xiao received the Ph.D. degree in the
department of Electronic Engineering from Tsinghua
University, Beijing, China, in 2011, and B.E. degree
in the department of Electronics and Information
Engineering from Huazhong University of Science
and Technology, Wuhan, China, in 2006. From 2011
to 2013, he served as a post doctor in the E.E.
department of Tsinghua University, Beijing, China.
Since 2013, he has been a lecturer in Beihang
University, Beijing, China.
Dr. Xiao has published over 50 papers, and served
as reviewers for IEEE Transactions on Signal Processing, IEEE Transactions
on Wireless Communications, IEEE Transactions on Vehicular Technology,
IEEE Communications Letters, etc. He has been TPC members of IEEE
GLOBECOM’12, IEEE WCSP’12, IEEE ICC’15, etc. His research interests
are communication signal processing and practical system implementation for
wideband communication systems, which cover synchronization, multipath
signal processing, diversity, multiple antenna technology, etc. Currently his is
dedicated in millimeter-wave communications and UAV networks.
Tong He received the B.E. degree in the department
of Electronic and Information Engineering from
Beihang University in 2012. Now he is pursuing
his Ph.D. degree in the department of Electronic
and Information Engineering of Beihang University.
His research interest is millimeter-wave communi-
cations.
Pengfei Xia Pengfei Xia received his Ph.D. from
Department of Electrical and Computer Engineering,
University of Minnesota, Twin Cities, MN in 2005.
Currently, he is a Full Chair Professor, College
of Electronics and Information, Tongji University,
Shanghai. His research interests lie at the inter-
section of wireless communications, wireless net-
works, signal and data processing, including LTE
cellular systems, IEEE 802.11 WLANs, millimeter
wave communications, transceiver beamforming, un-
licensed band communications. He has published
over 50 IEEE journal and conference papers, and have more than 70 US
patents and/or patent applications. He has received the IEEE Signal Processing
Society Best Paper Award 2011, and co-edited the first book on 60 GHz
millimeter wave communication systems. Currently an IEEE Senior Member,
he serves as the Editor for IEEE Transactions on Signal Processing, IEEE
Communications Magazines, and Technical Committee Member for the IEEE
Signal Processing Society.
Xiang-Gen Xia (M’97, S’00, F’09) received his
B.S. degree in mathematics from Nanjing Normal
University, Nanjing, China, and his M.S. degree in
mathematics from Nankai University, Tianjin, China,
and his Ph.D. degree in electrical engineering from
the University of Southern California, Los Angeles,
in 1983, 1986, and 1992, respectively.
He was a Senior/Research Staff Member at
Hughes Research Laboratories, Malibu, California,
during 1995-1996. In September 1996, he joined the
Department of Electrical and Computer Engineering,
University of Delaware, Newark, Delaware, where he is the Charles Black
Evans Professor. His current research interests include space-time coding,
MIMO and OFDM systems, digital signal processing, and SAR and ISAR
imaging. Dr. Xia is the author of the book Modulated Coding for Intersymbol
Interference Channels (New York, Marcel Dekker, 2000).
Dr. Xia received the National Science Foundation (NSF) Faculty Early
Career Development (CAREER) Program Award in 1997, the Office of Naval
Research (ONR) Young Investigator Award in 1998, and the Outstanding
Overseas Young Investigator Award from the National Nature Science Foun-
dation of China in 2001. He also received the Outstanding Junior Faculty
Award of the Engineering School of the University of Delaware in 2001.
He is currently serving and has served as an Associate Editor for numerous
international journals including IEEE Transactions on Signal Processing, IEEE
Transactions on Wireless Communications, IEEE Transactions on Mobile
Computing, and IEEE Transactions on Vehicular Technology. Dr. Xia is
Technical Program Chair of the Signal Processing Symp., Globecom 2007 in
Washington D.C. and the General Co-Chair of ICASSP 2005 in Philadelphia.