3388 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 5, MAY 2017
weight r, then down-converts the obtained N
s
analog signals
into baseband data using its RF chains. Specifically, we have
y = W
H
r =
√
PW
H
HFs + W
H
n (4)
where W =[w
1
, w
2
, ···, w
N
s
]∈C
N
r
×N
s
represents the
receiver beamforming matrix.
Here it is worthy to point out that the above transmitter
and receiver beamforming matrices: F and W,relyonanalog
processing. However, in the analog domain, signals may be
only controlled by a phase shifter network [19], [24], [42],
which will greatly restricts the feasible set of beamforming
matrices. To overcome such hardware limitation, hybrid analog
and digital beamforming is suggested [19], [24], which divides
beamforming among the analog and digital domains. In other
words, we have that F = F
RF
F
BB
and W = W
RF
W
BB
,
where {F
RF
∈ C
N
t
×N
RF
, W
RF
∈ C
N
r
×N
RF
} represent the
analog beamforming matrices with the entries bearing equal
gain, {F
RF
, F
BB
}∈C
N
RF
×N
s
represent the baseband beam-
forming matrices. When incorporating F
BB
and W
BB
,the
system architecture in Fig.1 is also applicable for hybrid
beamforming.
With the signal model in (4), the achievable spectral effi-
ciency is given by [6], [9], [19]
R = log
2
I
N
s
+
P
N
s
σ
2
W
H
W
−1
W
H
HFF
H
H
H
W
(5)
It is well known that the optimal beamforming matrix pair
(F
o
, W
o
) maximizing R is determined by the SVD of the
channel matrix H [6]. Assume that the SVD of H bear the
form below
H = UV
H
= λ
1
u
1
v
H
1
+ λ
2
u
2
v
H
2
+···+λ
N
u
N
v
H
N
(6)
where U =[u
1
, ···, u
N
r
] is an N
r
× N
r
unitary matrix,
is an N
r
× N
t
diagonal matrix with non-negative singular
values {λ
i
}
N
i=1
on the diagonal, N = min(N
t
, N
r
),andV =
[v
1
, ···, v
N
t
] is an N
t
×N
t
unitary matrix. The vectors u
i
and
v
i
(i = 1, ···, N) are referred to as the left and right singular
vectors, respectively, corresponding to the singular value λ
i
.
Without loss of generality, hereafter we assume that N = N
t
and λ
1
>λ
2
> ··· >λ
N
. When considering N
s
≤ N data
streams, the optimal transmitter and receiver beamforming
matrices are F
o
=[v
1
, ···, v
N
s
] and W
o
=[u
1
, ···, u
N
s
],
respectively. The conventional way of obtaining F
o
and W
o
is to first estimate the channel matrix H and then conduct
the SVD of H. However, the involved channel estimation is
formidably challenging for MIMO MMW systems with large
arrays but a limited number of RF chains. Motivated by this,
in next sections we will develop two efficient beamforming
training schemes to realize the SVD beamforming, which
circumvent the burdensome channel estimation.
III. P
OWER ITERATION BASED BEAMFORMING TRAINING
A. Power Iteration (PWI) Principle
Note that u
H
i
u
j
= δ
i, j
, v
H
i
v
j
= δ
i, j
, i, j = 1, ···, N,
where δ
i, j
denotes the Kronecker delta. We can easily obtain
from (6) that
H
t
= H
H
H = λ
2
1
v
1
v
H
1
+···+λ
2
N
v
N
v
H
N
(7)
which in fact is the eigenvalue decomposition (EVD) of H
t
.
Specifically, we have H
t
v
i
= λ
2
i
v
i
.Thismeansthat{λ
2
i
}
N
i=1
and {v
i
}
N
i=1
, respectively, are the eigenvalues and eigenvectors
of H
t
.
Let us first consider single-stream (SS) beamforming, i.e.
N
s
= 1. In this case, the optimal transmitter and receiver
beamforming matrices reduce to v
1
and u
1
, respectively. Given
an initial vector v
0
∈ C
N×1
, power iteration (PWI) [29] tries
to obtain v
1
by exploiting that
˜v = lim
n→∞
H
n
t
v
0
1
= lim
n→∞
a
1
λ
2n
1
v
1
+ a
2
λ
2n
2
v
2
+···+a
N
λ
2n
N
v
N
2
= lim
n→∞
a
1
λ
2n
1
v
1
+
a
2
λ
2n
2
a
1
λ
2n
1
v
2
+···+
a
m
λ
2n
N
a
1
λ
2n
1
v
N
3
= a
1
λ
2n
1
v
1
(8)
where in the 1
st
step we assume that v
0
=
N
i=1
a
i
v
i
and
use (7), in the 3
rd
step we use the fact λ
1
>λ
2
> ···>λ
N
.
Obviously, ˜v/˜v=v
1
.SinceHv
1
= λ
1
u
1
, we can easily
obtain u
1
= Hv
1
/Hv
1
.
For multi-stream (MS) beamforming, we need to obtain
F
o
=[v
1
, ···, v
N
s
], which can be achieved by using N
s
-stage
PWI. During the first stage, the same process as in (8) is used
to obtain v
1
. During the I-th (2 ≤ I ≤ N
s
) stage, we try
to obtain v
I
with the knowledge of {v
i
}
I−1
i=1
. Specifically, we
choose an initial vector v
0
∈ C
N×1
, and project it onto the
null space of {v
i
}
I−1
i=1
, thereby obtaining
¯v
(I)
0
= (I
m
− P
(I)
)v
0
= v
0
−
I−1
i=1
(v
H
i
v
0
)v
i
=
N
i=I
b
i
v
i
(9)
where P
(I)
=[v
1
, ···, v
I−1
][v
1
, ···, v
I−1
]
H
is the orthogo-
nal projection matrix associated with {v
i
}
I−1
i=1
, and we assume
that v
0
=
N
i=1
b
i
v
i
. Similar to (8), we have that ¯v =
lim
n→∞
H
n
t
¯v
(I)
0
= b
I
λ
2n
I
v
I
and thus ¯v/¯v=v
I
. For the receiver
beamforming matrix W
o
=[u
1
, ···, u
N
s
], the vector u
i
can
be obtained as u
i
= Hv
i
/Hv
i
(i = 1, ···, N
s
).
B. PWI Based Beamforming Training
Let us focus on the SS beamforming for the moment. In the
beginning, the transmitter adopts an N
t
× 1 beam vector
f
0
=[1, 0, ···, 0]
T
. Note that f
0
is omnidirectional, since
at present we do not know the position of the receiver with
respect to the transmitter. If we use directional beam vector, the
receiver may encounter extremely low SNR if it does not lie in
the beam direction. For f
0
, we only activate the first antenna
element while keeping the other antenna elements inactive.
Considering that per-antenna transmit power is limited, we can
repeatedly transmit f
0
with n
p
times, and average the received
signals at the receiver, which can improve the received SNR by
10 log
10
n
p
dB. Simulations (to be given in Section V) show
that this doing can lead to satisfactory performance.
With F = f
0
,theN
r
× 1 signal in (3) changes to
r =
√
PHf
0
+ n
r
=
P
L
L
l=1
{α
l
a
H
t
(φ
l
)f}a
r
(θ
l
) + n
r
(10)