1

On the Achievable Spectral Efficiency of Spatial
Modulation Aided Downlink Non-Orthogonal
Multiple Access
arXiv:1705.06466v1 [] 18 May 2017

Xuesi Wang, Student Member, IEEE, Jintao Wang, Senior Member, IEEE,
Longzhuang He, Student Member, IEEE, Zihan Tang, Student Member, IEEE, and Jian Song, Fellow, IEEE

Abstract—In this paper, a novel spatial modulation aided nonorthogonal multiple access (SM-NOMA) system is proposed.
We use mutual information (MI) to characterize the achievable
spectral efficiency (SE) of the proposed SM-NOMA system. Due
to the finite-alphabet space-domain inputs employed by SM,
the expression of the corresponding MI lacks a closed-form
formulation. Hence, a lower bound is proposed to quantify the MI
of the SM-NOMA system. Furthermore, its asymptotic property
is also theoretically investigated in both low and high signal-tonoise ratio (SNR) regions. The SE performance and its analysis
of our proposed SM-NOMA system are confirmed by simulation
results.
Index Terms—Non-orthogonal multiple access (NOMA); spatial modulation (SM); spectral efficiency (SE); mutual information (MI); lower bound.

I. I NTRODUCTION
Non-orthogonal multiple access (NOMA) constitutes a
promising technique in the fifth-generation (5G) mobile networks [1]. Different from the traditional orthogonal multiple
access (OMA), different users are designed to access the
same time, frequency and code domain resources, but different
power levels in NOMA. Compared to OMA, NOMA is more
flexible and is capable of offering a higher sum rate and lower
outage probability [1][2].
Moreover, the combination of NOMA and multiple-input
multiple-output (MIMO) has recently attracted substantial research interest [3]. However, in conventional MIMO systems,
the simultaneous use of multiple transmit antennas (TAs)
requires a large amount of radio frequency (RF) chains, which
significantly increases the corresponding power dissipation and
the implementation complexity. To circumvent this problem,
spatial modulation (SM) technique has recently been applied
to MIMO systems [4]-[7]. In conventional SM, only one
TA is activated for each symbol’s transmission, hence, only
one single RF chain is needed. Since the active antenna is
uniform-randomly selected from the transmit antennas in an
SM transmitter, the information is thus carried both by the
active antenna indices and by the transmitted amplitude-phase
modulation (APM) symbols.
For SM-MIMO systems, lots of literature has considered
the spectral efficiency (SE) of SM systems through mutual
X. Wang, J. Wang, L. He, Z. Tang and J. Song are with the Department of
Electronic Engineering, Tsinghua University, Beijing, 100084, China (e-mail:
wxs.tsinghua@gmail.com).
This work was supported by the National Natural Science Foundation of
China (Grant No. 61471221) and the Tsinghua Fudaoyuan Research Fund.

information (MI) [8][9]. In [9], the authors explored the
channel capacity of SM associated with a large array of
antennas and maximized the mutual information by optimizing
the distribution of the channel input. In addition, the current
research on multiple access methods based on SM mainly
focuses on the uplink transmission [10] and the receive SM
(R-SM) schemes over the MIMO broadcast channels [11]. To
the best of our knowledge, NOMA is a novel multiple access
method for downlink multi-user SM systems.
In this paper, we propose a novel SM aided NOMA (SMNOMA) system and use MI to characterize its achievable
SE. As the exact expression of the corresponding MI lacks
a closed-form formulation, computational-massive methods,
such as numerical integration or Monte Carlo method, are
usually required. In this context, a closed-form lower bound is
proposed in this paper to provide an accurate approximation
to the exact MI of the proposed system. Meanwhile, the
asymptotic values in both high and low signal-to-noise ratio
(SNR) regions of the MI are also derived to characterize the
tightness of the proposed closed-form MI’ lower bound with
respect to its true value. Finally, the SE performance and its
analysis of our proposed SM-NOMA system are validated by
simulation results.
The rest of this paper is organized as follows. Section II
describes the proposed SM-NOMA system model. Section
III analyzes its achievable SE through MI. In Section IV,
the simulation results are provided to validate the proposed
expressions and the system’s performance. Finally, Section V
concludes this paper.

II. S YSTEM M ODEL
In this paper, we consider a base station (BS) equipped with
M TAs. Since the NOMA system is interference-limited, we
adopt the hybrid multiple access scheme [2], i.e., users in one
cell are divided into several groups where NOMA is implemented within each group and the inter-group interference can
be eliminated by adopting OMA among different groups. We
assume that each group contains K users, while each user is
equipped with a single antenna.
In our proposed SM-NOMA system, the signal vector
transmitted by the BS can be considered as the superposition
of the SM-signal vectors intended for the multiple users.

2

! Nk+1
!
NK
K
X
X
X

1
(nt )2 2
2
2
·
···
CN 0, σv + σs
br,t αt , ΩK,K ∼ CN 0, σv2
Nt
nk+1 =1
nK =1
t=k+1
t=k+1
! N
!
N
K
K
k
K
X
X
X (n )2
Y 1
t
2
2
2
·
···
CN 0, σv + σs
br,t αt .
Nt
n =1
n =1
K
Y

Ωr,k ∼

Yr,k ∼

t=k

t=k

K

k

Mathematically, the transmit signal vector can be expressed
as follows:
K
X
(n )
x=
αk sk wk k ,
(2)
k=1

where
sk denotes the k-th user’s transmitted APM symbol,

E |sk |2 = 1, and αk denotes the k-th user’s transmission
(n )
power. Besides, wk k ∈ CM×1 represents the space-domain
input signal vector of the k-th user, which is uniform-randomly
(1)
(2)
(N )
selected from Γk , {wk , wk , · · · , wk k }. The nonzero
(nk )
elements subscripts of wk
represent the active antennas
(n)
indices and we have normalized the energy, i.e., kwk k22 = 1
(n = 1, 2, · · · , Nk ). In addition, Nk represents the total
(n )
amount of wk k belongs to the k-th user.
The design of Γk is closely related to the specific spacedomain alphabet design adopted by user k. In this paper, the
conventional SM regime is adopted by the k-th user, i.e., we
have Γk = {e1 , e2 , . . . , eM }, where en represents the nth column of an (M × M )-dimensional identity matrix. In
this case, each user randomly activates one of the M TAs.
Moreover, the number of required RF chains can be represents
as NRF = kxk0 . In our adopted conventional SM regime,
obviously, the number of RF chains used is equal to the
number of users currently served, i.e., NRF = K.
Upon assuming a flat-fading MIMO channel, the k-th user’s
received symbol can be represented as
X
(n )
(n )
(3)
yk = hTk wk k αk sk +
hTk wt t αt st +vk ,
t6=k

|

{z

}

intra-group interference

in which hTk ∈ C1×M denotes the narrowband channel
vector associated with the k-th users, vk denotes the independent identically distributed (i.i.d) additive white Gaussian
noise (AWGN), and its corresponding random variables V ∼
CN (0, σv2 ).
The intra-group interference can be partly eliminated by
employing SIC [3]. Without loss of generality, we assume the
decoding order as (1, · · · , K). A user can thus successfully
decode the messages intended for those users having a smaller
decoding order than himself, while the messages intended for
the remaining users are simply handled as interference [3].
To realize this assumption, the message sk intended for the
k-th user must be decoded by the r-th (k ≤ r ≤ K) user
correctly as
(nk )

yr,k = hTr wk

αk sk +

K
X

(nt )

hTr wt

αt st + vr .

(4)

(nt )
t=k+1 br,t αt st

+ vr

t=k+1
(n )

(nk )

Let br,kk = hTr wk

and ωr,k =

PK

(1)

denotes the intra-group interference as well as the AWGN,
(n )
we get yr,k = br,kk αk sk + ωr,k .
According to (4), the SE of the r-th user decoding the k-th
message can be characterized via MI between the k-th message
received by the r-th user, i.e., yr,k , and the transmitted APMdomain message sk comes with the space-domain message
(n )
wk k [8][9], which is represented by Ir,k and can be given
as follows:
Ir,k = I (Yr,k ; X, Br,k ) ,
(5)
(n )

in which yr,k , sk , and br,kk are realizations of the random
variables Yr,k , X, and Br,k , respectively. Furthermore, the
achievable SE performance of the proposed system
PK is characterized by the sum MI of all the users, i.e., k=1 Ik,k .
III. M UTUAL I NFORMATION A NALYSIS

Because I (Yr,k ; X, Br,k ) = h (Yr,k ) − h (Yr,k |X, Br,k ), (5)
can be simplified as
I (Yr,k ; X, Br,k ) = h (Yr,k ) − h (Ωr,k ) ,

(6)

where the random variable Ωr,k corresponds to the random realization ωr,k . Therefore, the MI analysis can be decomposed
into the entropy calculation of the receive signal yr,k as well
as the interference ωr,k .
In SM, for each symbol’s transmission to the k-th user, the
(n )
space-domain information wk k randomly selects one of the
Nk antenna selection vectors in Γk , according
to auniform

(n )
probability distribution [8]. Hence, P Br,k = br,kk = N1k .
Besides, we assume a complex-valued Gaussian input, i.e.,
X ∼ CN (0, σs2 ).
Meanwhile, when the value of Br,t is determined, Ωr,k
and Yr,k can be seen as the superposition of several complex
Gaussian
random variables, e.g., (Ω
 r,k |Br,k+1 , · · · , Br,K ) ∼

PK
(n )2
CN 0, σv2 + σs2 t=k+1 br,tt α2t . Therefore, Ωr,k and Yr,k
are subject to Gaussian mixture distribution (GMD) [12], as
given in (1).
Without loss of generality, the probability density function
(PDF) of a scalar complex GMD A can be represented as [12]
fA (a) =

L
X
l=1

PL

βl fl (a) =

L
X
l=1

βl

1 −
e
πσl2

1
e
πσl2

−

|a−µl |2
σ2
l

|a−µl |2
σ2
l

,

(7)

where l=1 βl = 1, and fl (a) =
.
The
entropy
of
A
can
be
represented
as h(A) =
R
− fA (a) log2 [fA (a)] da. Due to the logarithm of a sum
of exponential funcations, the entropy of a GMD random
variable has no closed-form formulation [12]. It thus relies
on numerical integration or Monte Carlo method. In order to

3

avoid the huge computational complexity required by Monte
Carlo method, we use Lemma 1 in [13] to provide the lower
and upper bounds of h(A):
Lemma 1: The lower and upper bounds of h(A) are given
by:
!
L
L
X
X
βl log2
hLB (A) = −
βt zlt
hUB (A) =

t=1

l=1
L
X

βl log2

l=1



πeσl2
βl



(8)

,

R
where zlt = fl (a)ft (a)da.
Proof: Because the function log2 (·) is concave, according
to Jensen’s inequality, we have a lower bound as
R
PL
h(A)= − l=1 βl fl (a) log2 [fA (a)] da
R

PL
≥ − l=1 βl log2 fl (a)fA (a)da
(9)


PL
PL
= − l=1 βl log2
t=1 βt zlt .

Similarly, we have an upper bound as
R
P
h(A)= − L
fl (a) log2 [fA (a)] da
l=1 βl
R
PL
≤ − l=1 βl fl (a) log2 [βl fl (a)] da
 2
PL
πeσ
= l=1 βl log2 βl l .

(10)

(n )

Considering the distribution of br,kk and sk , we can simplify
Lemma 1 as Proposition 1 by setting βl = L1 and µl = 0:
Proposition 1: The lower and upper bounds of h(A) can be
simplified as
!
L
L
X
1
1X
log2
hLB (A) = log2 (πL) −
L
σ 2 + σt2
t=1 l
l=1
(11)
L

1X
log2 σl2 .
hUB (A) = log2 (πeL) +
L
l=1

According to (6) and based on (11), by substituting Ωr,k
and Yr,k into ILB (Yr,k ; X, Br,k ) = hLB (Yr,k ) − hUB (Ωr,k ),
we propose the lower bound of MI. Considering the user
pairing strategies in NOMA1 [2][3], we derive the mathematical expression of ILB when K = 2, as given in (14),
(n m )2
σ2
where i ∈ {1, 2}, ρ = σs2 denotes the SNR and br,kt t =
(n )2

(m )2

v

br,kt + br,kt . It is worthnoting that h(Ω2,2 ) = log2 (πeσv2 )
because Ω2,2 ∼ CN 0, σv2 [12].
Moreover, to better characterize the tightness of the proposed closed-form MI’ lower bound with respect to its true
value, by reducing ρ to zero and increasing ρ without limit in
(14), we arrive at Proposition 2 and Proposition 3.
Proposition 2: In low SNR region, the asymptotic values
of MI are all 0 while their lower bounds approach:
ILB (Yi,1 ; X, Bi,1 ) ≈ 1 − log2 (eN2 )
ILB (Y2,2 ; X, B2,2 ) ≈ 1 − log2 (e).

(12)

1 The impact of user pairing on the performance of SM-NOMA systems is
beyond the scope of this paper, which will be handled with more consideration
in our future work.

Proposition 3: In high SNR region, the asymptotic values
of I (Y2,2 ; X, B2,2 ) and ILB (Y2,2 ; X, B2,2 ) do not exist while
others approach:
α21
)
α22
(13)
α2
ILB (Yi,1 ; X, Bi,1 ) ≈ log2 (1 + 12 ) + 1 − log2 (eN2 ).
α2
I (Yi,1 ; X, Bi,1 ) ≈ log2 (1 +

Proof: Assuming σs2 ≫ σv2 , we can omit σv2 in (1).
Besides, based on our adopted conventional
SM regime,

we can obtain Ωi,1  ∼ CN 0, σs2 b2 α22 , and Yi,1 ∼
CN 0, σs2 b2 (α21 + α22 ) according to [14], where b2 =
PN
(n1 )2
(n )2
1 PN1
≈ N12 n22=1 bi,22 . Therefore,
n1 =1 bi,1
N1

I (Yi,1 ; X, Bi,1 ) = h (Yi,1 ) − h (Ωi,1 )



= log2 πeσs2 b2 (α21 + α22 ) − log2 πeσs2 b2 α22
α2
= log2 (1 + 21 ).
α2
Similarly, based on (14), we can thus obtain the approximation
of ILB (Yi,1 ; X, Bi,1 ).
Furthermore, it is observed that a constant shift exists
between the MI and its lower bound. Particularly, the constant
shift can be written as Ci,1 = 1 − log2 (eN2 ) and C2,2 =
1 − log2 (e). More importantly, a constant shift imposes no
impact on the optimization of the MI’s lower bound.
IV. S IMULATION R ESULTS
In this section, we provide serveral simulation results to
confirm our proposed lower bound and asymptotic analysis.
Besides, in order to clarify the benefits of the proposed SMNOMA system, we provide the MISO-NOMA scheme and
the time division multiple access method based on SM (SMTDMA) as the counterparts.
In simulations for the SM system, we set M = 4 BS
antennas, K = 2 users by user pairing strategies, and each
user uniform-randomly selects one of the M transmit antennas.
Therefore, the number of required RF chains is NRF = 2.
Moreover, we assume that the receiver side has no channel
state information (CSI) feedback, so the zero-mean complex
Gaussian distributed channels are considered without precoding. In order to obtain the exact value of the targeted MI of
the SM system, we adopt the Monte Carlo method to calculate
the entropy h(A), i.e., h(A) = −Ea∼fA (a) {log2 [fA (a)]}.
Besides, in MISO-NOMA scheme, considering the fairness
of comparison, we set the number of TAs as M ′ = 2 without
′
precoding, which also needs NRF
= 2 RF chains.
Fig.1 shows I (Y1,1 ; X, B1,1 ) and I (Y2,2 ; X, B2,2 ) of different systems as well as the lower bound2 in SM-NOMA system.
Obviously, our proposed SM-NOMA system outperforms the
conventional MISO-NOMA system. Compared to the SMTDMA system, although the MI performance of the first
user is worse at high SNR regions, the proposed SM-NOMA
system still has a larger sum MI, as shown in Fig.2 (a).
2 According to (14), I (Y
1,1 ; X, B1,1 ) and I (Y2,1 ; X, B2,1 ) has almost
the same property and lower bound, so we only illustrate the former.

4



(n )2
N1 X
N2
N1
N2
X
X
X
1 + ρα22 bi,22
1


−
·
ILB (Yi,1 ; X, Bi,1 ) = log2
log 
N1 N2 n =1 n =1 2 m =1 m =1 2 + ρ α2 b(n1 m1 )2 + α2 b(n2 m2 )2
1
2
1
2
2 i,2
1 i,1
#
" N
 
N
2
2
X
N2
1 X
1
.
ILB (Y2,2 ; X, B2,2 ) = log2
−
·
log2
2 (n2 m2 )2
e
N2 n =1
m2 =1 2 + ρα2 b2,2
2


N1
e



14

5

3

SM-NOMA
MISO-NOMA
SM-TDMA
Lower bound
Lower bound with shift

6
5
4

2
3

1

2

0
-1

1

-2

0

SM-NOMA
MISO-NOMA
SM-TDMA

8

Mutual Information (bits/s/Hz)

SM-NOMA
MISO-NOMA
SM-TDMA
Lower bound
Lower bound with shift
Asymptotic value

4

Mutual Information (bits/s/Hz)

(14)

SM-NOMA: User1
SM-NOMA: User2
Lower bound
Lower bound with shift
Asymptotic value

12
10

6

8
6

4
4
2
2
0

-10

-5

0

5

10

15

20

-10

-5

(a) SNR (dB)

0

5

10

15

20

(b) SNR (dB)

-2

0
-10

-5

0

5

10

(a) SNR (dB)

15

20

0

2

4

6

8

10

(b) The transmit power ratio α21/α22

Fig. 1. MI of different systems and its lower bound of the SM-NOMA system,
given α21 = 4 and α22 = 1: (a) I (Y1,1 ; X, B1,1 ); (b) I (Y2,2 ; X, B2,2 ).

Fig. 2. (a) Sum MI of different systems, given α21 + α22 = 5; (b)
I (Y2,2 ; X, B2,2 ) and I (Y1,1 ; X, B1,1 ) at SNR = 30 dB.

In addition, our proposed lower bounds are confirmed with
the aforementioned constant shift C1,1 and C2,2 . Because of
h(A)’s relatively loose upper bound, ILB (Y1,1 ; X, B1,1 ) has
much weaker bound tightness than ILB (Y2,2 ; X, B2,2 ) and
C1,1 > C2,2 . Meanwhile, because of the SIC decoding order,
I (Y2,2 ; X, B2,2 ) > I (Y1,1 ; X, B1,1 ), which is shown in Fig.2
(b). As SNR increases, I (Y1,1 ; X, B1,1 ) changes from powerlimited to interference-limited and approaches a fixed value in
high SNR region, which is analyzed in (13). The only way to
enhance I (Y1,1 ; X, B1,1 ) in high SNR region is to increase
α2
the transmit power ratio α12 , as shown in Fig.2 (b).

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2

V. C ONCLUSION
In this paper, we propose and analyze a novel SM-NOMA
system from the point of view of its MI. The SE performance
of our proposed SM-NOMA system is confirmed by simulation results. In our future work, we will analyze the bit
error ratio (BER) of the proposed system and focus on the
optimization of transmitting power allocation and extend our
proposed SM-NOMA system to the generalized SM (GSM)
scenarioes.
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