3222 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 5, MAY 2019
QAM signaling, mutual information with and without channel
estimation error, and their detailed mathematical derivations.
II. MIMO S
PATIAL MULTIPLEXING SYSTEM
A. System and Channel Models
We consider a single-user N
t
×N
r
MIMO spatial multiplex-
ing system, where N
t
and N
r
denote the numbers of antennas
at the transmitter and receiver, respectively. We denote a set
of the transmit symbols through N
t
antennas by the column
vector s =(s
1
,s
2
, ···,s
N
t
)
T
∈X
N
t
×1
where X⊂C
denotes a set of the constellation points to be transmitted
from each antenna. The received symbol vector is denoted
by r ∈ C
N
r
×1
, which is expressed as
r = Hs + n, (1)
where n =(n
1
,n
2
, ···,n
N
r
)
T
∈ C
N
r
×1
is an additive white
Gaussian noise (AWGN) vector and H =(h
1
h
2
···h
N
t
) ∈
C
N
r
×N
t
is a complex channel matrix. Each column vector
h
k
=(h
1,k
,h
2,k
, ···,h
N
r
,k
)
T
represents the channel corre-
sponding to the kth transmit antenna with k ∈{1, 2, ···,N
t
}.
In order to analyze the performance in a mathematically
rigorous manner, we further make the following basic assump-
tions throughout this paper: Each element n
i
of the noise
vector n follows an independent and identically distributed
(i.i.d.) circularly symmetric complex Gaussian random vari-
able with zero mean and variance σ
2
n
= N
0
per complex
dimension,i.e.,n
i
∼CN(0,N
0
). Also, the channel is modeled
as uncorrelated Rayleigh fading such that each element of the
channel matrix H is an i.i.d. circularly symmetric complex
Gaussian random variable with zero mean and unit variance,
i.e., h
i,k
∼CN(0, 1). Therefore, E
|h
i,k
|
2
=1for any pair
of i ∈{1, 2, ···,N
r
} and k ∈{1, 2, ···,N
t
},whereE {·}
denotes an expectation operation.
Finally, as is often the case with MIMO spatial multiplexing
systems, we assume that the transmitter does not have the
knowledge of channel state information (CSI) but the receiver
has the perfect knowledge, i.e., H is available at the receiver
side but not at the transmitter side. The assumption of perfect
CSI at the receiver will be relaxed later in Section VI, where
the effect of channel estimation errors is also analyzed.
B. Transmitter Model
Since the transmitter does not have any CSI, we assume
that the transmit energy is distributed equally over the entire
transmit antennas. In other words, each transmit antenna is
assumed to generate a statistically independent symbol without
any precoding and we denote the average symbol energy
per transmit antenna by E
t
.Letthekth transmit symbol be
denoted by s
k
= x
k
+jy
k
where x
k
,y
k
∈ R. Then, in the case
of BPSK, we set y
k
=0and define X = {−
√
E
t
,
√
E
t
}⊂R.
Likewise, for QPSK, both x
k
and y
k
are chosen from X =
{−
E
t
/2,
E
t
/2}. In the case of square-type M-ary QAM
composed of two independent
√
M-pulse amplitude modu-
lation (PAM) constellations, both x
k
and y
k
are chosen from
X = {−(
√
M −1)A, ···, −3A, −A, A, 3A, ···, (
√
M −1)A}
where A =
3E
t
/2(M −1).
Without loss of generality, the reduction of energy due to
path loss and shadowing is normalized such that the average
energy per transmit antenna is expressed in terms of the
average energy of the received symbol E
s
as E
t
= E
s
/N
t
.
(Or, in other words, E
t
is the received symbol energy observed
at each receive antenna normalized by the number of transmit
antennas.) Furthermore, we define the parameter γ
s
= E
s
/N
0
which corresponds to the average SNR per receive antenna.
The SNR per bit, denoted by γ
b
, is commonly defined
as
γ
b
E
s
mN
0
=
γ
s
m
, (2)
where m is the number of bits per symbol (i.e., m =1for
BPSK and m =2for QPSK), and thus
E
t
N
0
=
m
N
t
γ
b
=
γ
s
N
t
. (3)
C. Receiver Model
The primary role of MIMO detector is to output the estimate
ˆs
k
=ˆx
k
+ jˆy
k
corresponding to the transmitted symbol s
k
,
which can be further used by the soft-decision channel decoder
in the case of coded systems.
Given H at the receiver, the general operation of the
MIMO linear detector is to calculate the soft-output vector
ˆs =(ˆs
1
, ˆs
2
, ···, ˆs
N
t
)
T
∈ C
N
t
×1
by linear transformation:
ˆs =
1
√
N
r
W
H
r, (4)
where W =(w
1
w
2
···w
N
t
) ∈ C
N
r
×N
t
is the weight
matrix and X
H
represents the Hermitian transpose of a matrix
X. Note that for analytical convenience and without loss of
generality, we here introduced the scaling factor
1
√
N
r
so as to
normalize the variance of the detector output.
For the ZF detector, the weight matrix is given by the
pseudo-inverse of the channel matrix [33]
W
H
=
H
H
H
−1
H
H
, (5)
which requires the complexity order of O(N
3
t
) due to the
inversion operation of N
t
× N
t
matrix in addition to matrix
multiplication. On the other hand, for the MF detector,
the weight matrix is simply chosen identical to the channel
matrix, i.e.,
W
H
= H
H
, (6)
or equivalently,
w
H
k
= h
H
k
. (7)
Therefore, no matrix inversion is necessary, and thus the
complexity order is dominated by matrix multiplication oper-
ation only, which grows linearly with N
r
per each transmit
antenna. The benefit of MF detector over the other detectors
thus becomes significant especially when the number of the
transmit antennas becomes large, as in the case of massive
MIMO scenarios.