1
Wideband Beamspace Channel Estimation for
Millimeter-Wave MIMO Systems Relying on Lens
Antenna Arrays
Xinyu Gao, Student Member, IEEE, Linglong Dai, Senior Member, IEEE, Shidong Zhou, Senior Member, IEEE,
Akbar M. Sayeed, Fellow, IEEE, and Lajos Hanzo, Fellow, IEEE
Abstract—Beamspace channel estimation is indispensable for
millimeter-wave (mmWave) MIMO systems relying on lens an-
tenna arrays for achieving substantially increased data rates,
despite using a small number of radio-frequency (RF) chains.
However, most of the existing beamspace channel estimation
schemes have been designed for narrowband systems, while
the rather scarce wideband solutions tend to assume that the
sparse beamspace channel exhibits a common support in the
frequency domain, which has a limited validity owing to the
effect of beam squint caused by the wide bandwidth in practice.
In this paper, we investigate the wideband beamspace channel
estimation problem without the common support assumption.
Specifically, by exploiting the effect of beam squint, we first prove
that each path component of the wideband beamspace channel
exhibits a unique frequency-dependent sparse structure. Inspired
by this structure, we then propose a successive support detec-
tion (SSD)-based beamspace channel estimation scheme, which
successively estimates all the sparse path components following
the classical idea of successive interference cancellation (SIC).
For each path component, its support at different frequencies is
jointly estimated to improve the accuracy by utilizing the proved
sparse structure, and its influence is removed to estimate the
remaining path components. The performance analysis shows
that the proposed SSD-based scheme can accurately estimate the
wideband beamspace channel at a low complexity. Simulation
results verify that the proposed SSD-based scheme enjoys a
reduced pilot overhead, and yet achieves an improved channel
estimation accuracy.
Index Terms—MIMO, millimeter-wave, lens antenna array,
wideband beamspace channel estimation.
I. INTRODUCTION
M
Illimeter-wave (mmWave) multiple-input multiple-
output (MIMO) working at 30-300 GHz has been
X. Gao, L. Dai, and S. Zhou are with the Beijing Nation-
al Research Center for Information Science and Technology (BN-
Rist), Department of Electronic Engineering, Beijing 100084, China
(e-mails: xy-gao14@mails.tsinghua.edu.cn, daill@tsinghua.edu.cn, zhous-
d@tsinghua.edu.cn).
A. Sayeed is with the Department of Electrical and Computer Engi-
neering, University of Wisconsin, Madison, WI 53706, USA (email: ak-
bar@engr.wisc.edu).
L. Hanzo is with the Department of Electronics and Computer Sci-
ence, University of Southampton, Southampton, SO17 1BJ, UK (email:
lh@ecs.soton.ac.uk).
This work was supported by the National Natural Science Founda-
tion of China for Outstanding Young Scholars (Grant No. 61722109),
the National Science and Technology Major Project of China (Grant No.
2018ZX03001004-003), the Royal Academy of Engineering under the UK-
China Industry Academia Partnership Programme Scheme (Grant No. UK-
CIAPP\49), and the US National Science Foundation (Grant Nos. 1629713,
1703389, and 1548996).
recently recognized as a promising technique to substantially
increase the data rates of wireless communications [1], since
it can provide a very wide bandwidth (e.g., 2-5 GHz) [2].
However, in the conventional MIMO architecture working
at sub-6 GHz cellular frequencies, each antenna requires a
dedicated radio-frequency (RF) chain (including the digital-
to-analog/analog-to-digital converter, mixer, and so on) [3],
[4]. Employing this architecture in mmWave MIMO will lead
to unaffordable hardware cost and power consumption due to
the following two reasons [5]: 1) the number of antennas is
usually very large to compensate for the severe path loss (e.g.,
256 antennas may be used at mmWave frequencies instead of 8
antennas at cellular frequencies) [6]; 2) the power consumption
of the RF chain is high due to the increased sampling rate
(e.g., 250 mW/RF chain at mmWave frequencies, compared
to 30 mW/RF chain at cellular frequencies) [7]. To solve this
problem, mmWave MIMO relying on lens antenna array has
been proposed [8]. By employing the lens antenna array (an
electromagnetic lens with power focusing capability and a
matching antenna array with elements located on the focal
surface of the lens [9]), we can focus the signal power
arriving from different directions on different antennas [10],
and transform the mmWave MIMO channel from the spatial
domain to its sparse beamspace representation (i.e., beamspace
channel) [11]. This allows us to select a small number of
power-focused beams for significantly reducing the effective
MIMO dimension and the associated number of RF chains.
Consequently, the high power consumption and hardware cost
of mmWave MIMO systems can be mitigated [12]–[14].
To select the power-focused beams, a high-dimensional
beamspace channel is required at the base station (BS). How-
ever, this is not a trivial task in mmWave MIMO systems
relying on lens antenna arrays, since the number of RF chains
is much smaller than the number of antennas so that we cannot
directly observe the complete channel in the baseband [15].
To circumvent this problem, some beamspace channel estima-
tion schemes have been proposed in [16]–[20]. For example,
in [16], a training-based scheme is proposed. It first scans all
the beams and only retains a few strong beams. Then, the
least squares (LS) algorithm is employed for estimating the
reduced-dimensional beamspace channel. In [17], a modified
version of [16] is proposed, where the overhead of beam
training is reduced by simultaneously scanning several beams
with the help of power splitters at the BS. In [18], a support
detection based scheme is proposed for further reducing the
2
Lens
Antenna
Array
N
Adaptive
Selection
Network
Baseband
Signal
Processing
User 2
User K
RF
Chain
Remove
CP
M-point
FFT
K
RF
N N<
RF
Chain
Remove
CP
M-point
FFT
RF
Chain
Add
CP
M-point
IFFT
User 1
RF
Chain
Add
CP
M-point
IFFT
RF
Chain
Add
CP
M-point
IFFT
Fig. 1. Architecture of wideband mmWave MIMO-OFDM system relying on lens antenna array.
pilot overhead. It exploits the sparsity of the beamspace
channel to directly estimate the channel support (i.e., the index
set of nonzero elements in a sparse vector). However, all of
these schemes have been designed for narrowband systems,
while realistic mmWave MIMO systems are more likely to be
of wideband nature for achieving high data rates. For wideband
systems, there are only a few recent contributions. In [19],
a simultaneous orthogonal matching pursuit (SOMP)-based
scheme is proposed. It first regards the wideband beamspace
channel estimation problem as a multiple measurement vector
(MMV) problem associated with a common support (i.e.,
the channel support at different frequencies is assumed to
be the same), and then solves it by the SOMP algorithm.
In [20], an orthogonal matching pursuit (OMP)-based scheme
is proposed. It first estimates the support of the wideband
beamspace channel at some frequencies independently by the
OMP algorithm. Then, it combines them into the common
support at all frequencies. Unfortunately, the common support
assumption in [19], [20] has limited validity in the practical
wideband mmWave MIMO systems. As discussed in [21], the
combination of a wide bandwidth and a large number of an-
tennas will make the channel spreading factor defined in [21]
larger than one, and the effect of “beam squint” becomes more
obvious, where “beam squint” is used to imply that the indices
of the power-focused beams are frequency-dependent [22]. As
a result, the support of wideband beamspace channels also
tends to be frequency-dependent, and the existing wideband
solutions [19], [20] relying on the common support assumption
will suffer from an obvious performance loss in practice.
In this paper, inspired by the classical successive inter-
ference cancellation (SIC) conceived for multi-user signal
detection [23], we propose a successive support detection
(SSD)-based wideband beamspace channel estimation scheme
without the common support assumption. Specifically, the
contributions of this paper can be summarized as follows:
1) By exploiting the effect of beam squint, we first prove
that each path component of the wideband beamspace chan-
nel exhibits a unique frequency-dependent sparse structure.
Specifically, for each sparse path component, we demonstrate
that: i) its frequency-dependent support is uniquely determined
by its spatial direction at the carrier frequency; ii) this spatial
direction can be estimated by tentatively generating several
beamspace windows (BWins) to capture the path power.
2) Inspired by the idea of SIC, we propose to decompose the
wideband beamspace channel estimation problem into a series
of sub-problems, each of which only considers a single path
component. For each path component, its support observed at
different frequencies is estimated jointly to improve the accu-
racy by utilizing the proved sparse structure, and then its influ-
ence is removed to estimate the remaining path components.
The performance analysis shows that the proposed scheme can
accurately estimate the wideband beamspace channel at a low
complexity.
3) We provide extensive simulation results to verify the ad-
vantages of the proposed SSD-based scheme. We demonstrate
that our scheme achieves a satisfactory channel estimation
accuracy at a lower pilot overhead than the existing schemes.
We also show that our wideband scheme performs well in
narrowband systems.
The rest of the paper is organized as follows. In Section
II, the system model of wideband mmWave MIMO-OFDM
relying on lens antenna array is introduced, and the problem
of wideband beamspace channel estimation is formulated
when single-antenna users are considered. In Section III, the
proposed SSD-based scheme is specified, together with its
performance analysis. In Section IV, the proposed SSD-based
scheme is extended to the scenario with multiple-antenna
users. In Section V, our simulation results are provided to
verify the advantages of the proposed SSD-based scheme.
Finally, our conclusions are drawn in Section VI.
Notation: Lower-case and upper-case boldface letters a and
A denote a vector and a matrix, respectively; A
T
, A
H
, A
−1
,
and A
†
denote the transpose, conjugate transpose, inverse, and
pseudo inverse of matrix A, respectively; ∥A∥
2
and ∥A∥
F
denote the spectral norm and Frobenius norm of matrix A,
respectively; ∥a∥
2
denotes the l
2
-norm of vector a; |a| denotes
the amplitude of scalar a; |S| denotes the cardinality of set S;
A (S, :) and A (:, S) denote the sub-matrices of A consisting
of the rows and columns indexed by S, respectively; a (S)
denotes the sub-vector of a indexed by S. Finally, I
N
is the
identity matrix of size N × N.
II. SYSTEM MODEL
As shown in Fig. 1, we consider an uplink time division
duplexing (TDD) wideband mmWave MIMO-OFDM system
with M sub-carriers. The BS employs an N -element lens
3
antenna array and N
RF
RF chains to simultaneously serve
K users. In this section, we assume that each user employs
single antenna, while in Section IV, multiple-antenna users
will be considered. Next, we will first introduce the wideband
beamspace channel. Then, the wideband beamspace channel
estimation problem will be formulated.
A. Wideband beamspace channel
We commence with the wideband mmWave MIMO channel
in the conventional spatial domain. To characterize the disper-
sive mmWave MIMO channel [24], we adopt the widely used
Saleh-Valenzuela multipath channel model presented in the
frequency domain. The N × 1 spatial channel h
m
of a certain
user at sub-carrier m (m = 1, 2, ··· , M ) can be presented
as [4], [21], [25]
h
m
=
N
L
L
l=1
β
l
e
−j2πτ
l
f
m
a (φ
l,m
), (1)
where L is the number of resolvable paths, β
l
and τ
l
are the
complex gain and the time delay of the l-th path, respectively,
φ
l,m
is the spatial direction at sub-carrier m defined as
φ
l,m
=
f
m
c
d sin θ
l
, (2)
where f
m
= f
c
+
f
s
M
m −1 −
M−1
2
is the frequency of
sub-carrier m with f
c
and f
s
representing the carrier fre-
quency and the bandwidth (sampling rate), respectively, c
is the speed of light, θ
l
is the physical direction, and d
is the antenna spacing, which is usually designed accord-
ing to the carrier frequency as d = c/2f
c
[4]. Note that
in narrowband mmWave systems with f
s
≪ f
c
, we have
f
m
≈ f
c
, and φ
l,m
≈
1
2
sin θ
l
is frequency-independent. How-
ever, in wideband mmWave systems, f
m
= f
c
, and φ
l,m
is
frequency-dependent. Finally, a (φ
l,m
) is the array response
vector of φ
l,m
. For the typical N-element uniform linear
array (ULA), we have a (φ
l,m
) =
1
√
N
e
−j2πφ
l,m
p
a
, where
p
a
=
−
N−1
2
, −
N+1
2
, ··· ,
N−1
2
T
[4].
The spatial channel h
m
can be transformed to its beamspace
representation by employing the lens antenna array, as shown
in Fig. 1. Essentially, this lens antenna array plays the role of
an N × N-element spatial discrete fourier transform (DFT)
matrix U
a
1
, which contains the array response vectors of N
orthogonal directions (beams) covering the entire space as [8]
U
a
= [a ( ¯φ
1
) , a ( ¯φ
2
) , ··· , a ( ¯φ
N
)], (3)
where ¯φ
n
=
1
N
n −
N+1
2
for n = 1, 2, ··· , N are the spatial
directions pre-defined by the lens antenna array. Accordingly,
1
The reason why the lens antenna array realizes the spatial DFT can be
found in [10, Lemma 1]. Explicitly, it is shown that the power-focusing
capability of the lens relies on the spatial phase shifters on the lens’ aperture,
which usually cannot be adjusted according to different frequencies. As a
result, the response of the lens antenna array cannot be frequency-dependent
as in (1). However, we would like to surmise that it may be possible but rather
challenging to conceive a frequency-dependent lens antenna array capable of
compensating for the effect of beam squint. In this case, the proposed SSD-
based scheme can be further simplified to its narrowband version as we have
proposed in [18], since the beamspace channel at different sub-carriers will
have the common support.
the wideband beamspace channel
˜
h
m
at sub-carrier m can be
presented by
˜
h
m
= U
H
a
h
m
=
N
L
L
l=1
β
l
e
−j2πτ
l
f
m
˜c
l,m
, (4)
where ˜c
l,m
denotes the l-th path component at sub-carrier m
in the beamspace, and ˜c
l,m
is determined by φ
l,m
as
˜c
l,m
= U
H
a
a (φ
l,m
) (5)
= [Ξ (φ
l,m
−¯φ
1
) , Ξ (φ
l,m
− ¯φ
2
) , ··· , Ξ (φ
l,m
−¯φ
N
)]
T
,
where Ξ (x) =
sin N πx
sin πx
is the Dirichlet sinc function [12].
Based on the power-focusing capability of Ξ (x) [12], [18],
we know that most of the power of ˜c
l,m
is focused on only
a small number of elements. Additionally, due to the limited
scattering in mmWave systems, L is also small [24], [26].
Therefore,
˜
h
m
should be a sparse vector [27]. However, since
φ
l,m
in (5) is frequency-dependent in wideband mmWave
systems (i.e., f
m
= f
c
), the beam power distribution of the l-
th path component should be different at different sub-carriers,
i.e., ˜c
l,m
1
= ˜c
l,m
2
for m
1
= m
2
. This effect is termed as
beam squint [21], which is a key difference between wideband
and narrowband systems. For example, when we consider a
narrowband system with θ
l
= −π/4, N = 32, and f
c
= 28
GHz, the beam power distribution of the l-th path component
is shown by the black line in Fig. 2, which is fixed. By contrast,
when we extend this system to a wideband one with M = 128
and f
s
= 4 GHz, the beam power distributions of ˜c
l,1
and ˜c
l,M
are shown by the blue line and red line in Fig. 2, respectively.
We observe that ˜c
l,1
only has a single strong beam ˜c
l,1
(6),
while ˜c
l,M
has 2 strong beams, namely ˜c
l,M
(4) and ˜c
l,M
(5),
which are different. Fig. 3 shows the effect of beam squint
from another perspective, where the parameters are the same
as in Fig. 2, and the curve indexed by n represents the power
variation of the nth element (beam) of ˜c
l,m
over frequency. We
observe from Fig. 3 that in contrast to the narrowband systems
where the power of each beam is frequency-independent [18],
the power of each beam in wideband systems varies signifi-
cantly over frequency. Due to beam squint and the fact that
the beamspace channel is the summation of several resolvable
path components, we can conclude that the support of the
beamspace channel should be frequency-dependent, which is
different from the common support assumption considered in
the existing beamspace channel estimation schemes
2
.
B. Problem formulation
In TDD systems, the users are required to transmit pilot
sequences to the BS for uplink channel estimation, and the
channel is assumed to remain unchanged during this peri-
od [29], [30]. In this paper, we adopt the widely used or-
thogonal pilot transmission strategy, and therefore the channel
estimation invoked for each user is independent [31]. Let us
consider a specific user without loss of generality, and define
s
m,q
as its transmitted pilot at sub-carrier m and instant q (each
2
It is worth pointing out that beam squint also exists in wideband mmWave
MIMO systems using the conventional phased arrays [28]. The proposed
channel estimation scheme in this paper can be also used in such systems.
4
1 32
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.4
Fig. 2. Beam power distributions of the l-th path component in a narrowband
system and in a wideband system.
1 20 40 60 80 100 120 128
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
|˜c
l,m
(n)|
2
Sub−carrier m
n = 5
n = 4
n = 6
Fig. 3. Beam power variation over frequency.
user transmits one pilot per instant) before the M-point IFFT
and cyclic prefix (CP) adding [20]. Then, as shown in Fig. 1,
the N
RF
× 1 received pilot vector y
m,q
at the BS after receiver
combining (realized by the adaptive selection network [18]),
CP removal, and M -point FFT can be presented as [20]
y
m,q
= W
q
˜
h
m
s
m,q
+ W
q
n
m,q
, m = 1, 2, ··· , M, (6)
where W
q
of size N
RF
× N is the receiver combining matrix
(fixed at different sub-carriers due to the analog hardware
limitation [20]) and n
m,q
∼ CN
0, σ
2
I
N
of size N × 1 is
the noise vector with σ
2
representing the noise power. After
Q instants of pilot transmission, we can obtain the overall
measurement vector ¯y
m
=
y
T
m,1
, y
T
m,2
, ··· , y
T
m,Q
T
as
¯y
m
=
¯
W
˜
h
m
+ n
eff
m
, m = 1, 2, ··· , M, (7)
where we assume s
m,q
=1 for q =1, 2, ··· , Q without loss
of generality [32], and define n
eff
m
as the effective noise
vector. Furthermore, we define
¯
W =
W
T
1
, W
T
2
, ··· , W
T
Q
T
of size QN
RF
× N as the overall combining matrix, which
is designed according to the hardware realization of the
adaptive selection network. For example, if the adaptive se-
lection network is realized by low-cost 1-bit phase shifters
as in [18]
3
, the elements of
¯
W can be randomly selected
from the set
1
√
QN
RF
{−1, +1} with equal probability. Here
the normalization factor
1
√
QN
RF
is used for guaranteeing
that
¯
W has unit-norm columns [34]. The reason we adopt
a randomly selected matrix is that it has been shown to
have a low mutual-column coherence, and therefore can be
expected to achieve a high recovery accuracy according to
well-established compressive sensing theory [35]. Finally, it
should be noted that hardware impairments are indeed imposed
on the adaptive selection network, leading to an element-wise
gain/phase offset in W
q
, which cannot be fully captured in
the estimated channel. This is a common problem inherent in
most of the popular channel estimation schemes conceived for
hybrid analog and digital architectures [4], since the analog
modules (e.g., phase shifter network) are usually involved
in the channel estimation procedure. Fortunately, since the
gain/phase offsets are usually not serious in practice, the
channel estimation accuracy degradation caused by hardware
impairments will not be significant.
According to (7), we now can recover
˜
h
m
given ¯y
m
and
¯
W. Since
˜
h
m
is sparse, this problem can be solved relying
on compressive sensing (CS) algorithms with a significant-
ly reduced number of instants for pilot transmission (i.e.,
Q ≪ (N/N
RF
)) [27], [36]. However, most of the existing
schemes using CS algorithms have been designed for nar-
rowband systems [16]–[18], while mmWave MIMO systems
are more likely to be of wideband nature for achieving
high data rates. For wideband systems, only the SOMP-
based scheme [19] and the OMP-based scheme [20] have
been proposed, but they assume that
˜
h
1
,
˜
h
2
, ··· ,
˜
h
M
share
a common support, which is not strictly valid in practice due
to the effect of beam squint, as shown in Fig. 2 and Fig. 3 [21].
III. WIDEBAND BEAMSPACE CHANNEL ESTIMATION
In this section, we first explicitly demonstrate that the wide-
band beamspace channel exhibits a sparse structure. Then, we
propose an efficient SSD-based scheme. Finally, the associated
performance analysis is provided to quantify the advantages
of our scheme.
A. Sparse structure of wideband beamspace channel
As shown in Fig. 2 and Fig. 3, the common support
assumption is not strictly valid in practice due to the effect
of beam squint. Fortunately, the wideband beamspace channel
still exhibits a unique frequency-dependent sparse structure.
This will be proved by the following lemmas, which constitute
the basics of the proposed SSD-based scheme.
Lemma 1. Consider the l-th path component of the wideband
beamspace channel. The frequency-dependent support T
l,m
of ˜c
l,m
for m = 1, 2, ··· , M is uniquely determined by the
3
It is worth pointing out that during data transmission, an adaptive selection
network relying on 1-bit phase shifters can also be configured to realize
conventional beam selection [12]. To achieve this, we can turn off some phase
shifters to realize “unselect” [33] and set some phase shifters to shift the phase
0
◦
to realize “select”.
5
spatial direction φ
l,c
of the l-th path at the carrier frequency
f
c
, which is defined as φ
l,c
= (f
c
/c) d sin θ
l
= (1/2) sin θ
l
.
Proof: Based on the analysis in [18], the index of the
strongest element n
⋆
l,m
of ˜c
l,m
is determined by φ
l,m
as
n
⋆
l,m
= arg min
n
|φ
l,m
− ¯φ
n
|, (8)
where ¯φ
n
is defined in (3). Then, the support of ˜c
l,m
can be
obtained by
T
l,m
= Θ
N
n
⋆
l,m
− Ω, ···n
⋆
l,m
+ Ω
, (9)
where Θ
N
(x) = mod
N
(x − 1) + 1 is the mod function guar-
anteeing that all elements in T
l,m
belong to {1, 2, ··· , N}, and
Ω determines how much power can be preserved by assuming
that ˜c
l,m
is a sparse vector with support T
l,m
. For example,
when N = 256 and Ω = 4, at least 96% of the power can be
preserved [18]. The reasonable nature of (9) can be explained
as follows. In practice, φ
l,m
is arbitrary, which is usually
different from the pre-defined beam directions ¯φ
1
, ¯φ
2
, ··· ¯φ
N
.
In this case, the power of ˜c
l,m
will be distributed across several
beams. According to the properties of Ξ (x) in ˜c
l,m
, Ξ (x) is
larger when x is closer to 0, and we know that the indices of
these power-focused beams should be adjacent. The detailed
proof can be found in [18, Lemma 2].
On the other hand, based on (2) and the definition of φ
l,c
,
φ
l,m
can be rewritten following [21] as
φ
l,m
=
1 +
f
s
Mf
c
m − 1 −
M − 1
2
φ
l,c
, (10)
which is only determined by φ
l,c
(M, f
c
, f
s
are given system
parameters). As a result, once φ
l,c
is known, the support T
l,m
of ˜c
l,m
for m = 1, 2, ··· , M can be obtained based on (8)
and (9).
Lemma 1 implies that φ
l,c
is a crucial parameter for deter-
mining T
l,m
for m = 1, 2, ··· , M . In the following Lemma
2, we will provide some insights about how to estimate φ
l,c
.
Lemma 2. Let us define C
n
= [˜c
l,1
, ˜c
l,2
, ··· , ˜c
l,M
], where we
assume φ
l,c
= ¯φ
n
. Then, the power of the s-th row C
n
(s, :)
of C
n
can be calculated as
∥C
n
(s, :)∥
2
2
=
M
α
n
α
n
2
−
α
n
2
Ξ
2
n − s
N
+ ∆φ
d∆φ, (11)
where α
n
= f
s
¯φ
n
/f
c
. Moreover, if we define a beamspace
window (BWin) Υ
n
=Θ
N
{n−∆
n
, ··· , n+∆
n
} centered
around n, the ratio γ between the power of the sub-matrix
C
n
(Υ
n
, :) and the power of C
n
can be presented as
γ =
∥C
n
(Υ
n
, :)∥
2
F
∥C
n
∥
2
F
=
1
α
n
∆
n
i=−∆
n
α
n
2
−
α
n
2
Ξ
2
i
N
+ ∆φ
d∆φ.
(12)
Proof: Based on (5), the power of the s-th row C
n
(s, :) of
C
n
can be calculated as
∥C
n
(s, :)∥
2
2
=
M
m=1
Ξ
2
(φ
l,m
− ¯φ
s
). (13)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0. 2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0. 2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0. 2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BWin
(b)
(c)
( )
5 , 5
l c
j j
=
C
3
¡
(
)
2
3
2
1,:C
( )
2
3
2
2,:C
( )
2
3
2
3,:C
(a)
( )
3 , 3
l c
j j
=
C
n
BWin
3
¡
( )
2
0
X
Not include
( )
2
0
X
Include
( )
2
0
X
Include
Fig. 4. Illustration of the power distribution: a) C
3
; b) C
5
; c) C
3
(1, :),
C
3
(2, :), and C
3
(3, :).
Defining ∆φ
m
=
f
s
φ
l,c
Mf
c
m − 1 −
M−1
2
, we can rewrite (13)
based on (10) as
∥C
n
(s, :)∥
2
2
=
M
m=1
Ξ
2
(φ
l,c
+ ∆φ
m
− ¯φ
s
)
(a)
=
M
m=1
Ξ
2
n − s
N
+ ∆φ
m
, (14)
where (a) is valid since φ
l,c
= ¯φ
n
. Note that M is usually a
large number (e.g., M = 512). Therefore, ∆φ
m
is small and
the summation in (14) can be well-approximated by its integral
form as
∥C
n
(s, :)∥
2
2
=
M
α
n
α
n
2
−
α
n
2
Ξ
2
n − s
N
+ ∆φ
d∆φ, (15)
where the integral interval is determined by ∆φ
1
and ∆φ
M
with
M−1
M
≈ 1. Furthermore, based on (15), the power of
C
n
(Υ
n
, :) can be written as
∥C
n
(Υ
n
, :)∥
2
F
=
M
α
n
i∈Υ
n
α
n
2
−
α
n
2
Ξ
2
n−i
N
+∆φ
d∆φ (16)
(a)
=
M
α
n
∆
n
i=−∆
n
α
n
2
−
α
n
2
Ξ
2
i
N
+∆φ
d∆φ,
where (a) is due to the fact that Υ
n
is centered around n. On
the other hand, since
N
s=1
Ξ
2
(φ
l,m
−¯φ
s
)=1, the total power
of ∥C
n
∥
2
F
is M. Then, the conclusions can be derived.
According to Lemma 2 , we observe that if φ
l,c
= ¯φ
n
,
the most power of C
n
can be captured by a carefully de-
signed BWin Υ
n
centered around n. For example, given
N = 256, f
c
= 28 GHz, f
s
= 4 GHz, φ
l,c
= ¯φ
1
, we can
capture γ ≈ 92% of the power of C
1
by using the BWin
Υ
1
= Θ
256
{1 − 8, ··· , 1 + 8} (∆
1
= 8). On the other hand,
if φ
l,c
= ¯φ
1
, e.g., φ
l,c
= ¯φ
10
, using Υ
1
to capture the power
of C
10
will lead to serious power leakage, where we only
have γ ≈ 47%. This observation is further illustrated in Fig.
4 (a) and (b). Therefore, we can conclude that the BWin Υ
n
centered around n can be considered as a feature specialized
for φ
l,c
= ¯φ
n
, which can be exploited for estimating φ
l,c
.
