Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 4, No. 1, 2022 

 

131 

Signal	Vector	Design	for	Quadrature	Index	Modulation	
Fuchun Huang 

Guangzhou College of Commerce, Guangzhou 510700, China 

 

Abstract: In this paper, we propose a new technique of quadrature index modulation with signal vector design (QIM-SSD), 
which can directly transmit a three-dimensional (3D) constellation symbol and expand an antenna index (AI) bit. The basic 
principle of QIM-SSD is as follows: two components of a mapped 3D signal symbol are first modulated on the upper and lower 
parts of transmit antennas (TAs) by two specific AI vectors with two blocks of AI bits, respectively, resulting in a space vector 
symbol. The remaining one component is modulated on one TA activated by the specific AI vector with one block of AI bits, 
then forming another space vector symbol. Furthermore, based on the resulted two space vector symbols, an additional bit is 
used to determine the real and imaginary parts of a complex transmit vector. The research of this paper is summarized as follows: 
firstly, the system model of QIM-SSD is introduced in detail. Then, the minimum Euclidean distance and the average bit error 
probability (BEP) are analyzed. With assumption of perfect channel estimation and maximum likelihood (ML) detection, 
simulation results demonstrate that the average BEP performance of QIM-SSD is better than that of the conventional QIM with 
three dimensional constellation and QSM at the same spectral efficiency by MATLAB simulator. 

Keywords: Antenna index, 3D constellation, Euclidean distance, The real and imaginary parts of a complex transmit vector. 
 

1. Introduction 
Index modulation (IM), which exploits multiple index 

domains used to carry additional information, has been 
viewed as a potential key technology to improve the spectral 
efficiency (SE) and enhance the reliability of wireless 
communication. In [1], Quadrature space modulation (QSM) 
extends the spatial domain with the transmit antennas (TAs) 
to the in-phase and quadrature spatial dimensions that 
modulate the real and imaginary parts of the mapped symbol 
(e.g., quadrature amplitude modulation/phase shift 
modulation (QAM/PSK)). Consequently, QSM can achieve 
the additional information bits of twofold base-two logarithm 
of the number of TAs. In order to make use of the idle TAs, 
based on the in-phase and quadrature dimensions, a lot of 
works on generalization of QSM (GQSM) [2-4] have been 
developed to achieve more additional index bits by increasing 
the active AI combinations. However, the additional index 
bits that GQSM achieves are increased at the cost of the 
detection complexity that will be relatively increased with the 
increasing of active TAs. To effectively utilize the idle 
antennas, from the perspective of developing signal 
constellation, quadrature index modulation with three 
dimension (3D) constellation (QIM-TDC) [5] is proposed to 
transmit two components of a 3D symbol by using two active 
TAs in the in-phase dimension and to transmit the remaining 
one by using one active TA in the quadrature dimension, 
respectively. Consequently, compared with QSM, QIM-TDC 
improves not only the squared minimum Euclidean distance 

(MED) but also the additional index bits with the increasing 
of TAs.  

However, it can be observed that QIM-TDC does not make 
full use of the difference between the real part and the 
imaginary part of the transmit vector. Based on this issue, a 
new design, called quadrature index modulation with spatial 
signal design (QIM-SSD), which further exploit the spatial 
domain for additional index information, is proposed to 
design the transmit vector to increase the SE and enhance the 
reliability of communication. Specifically, one spatial vector 
symbol is obtained by modulating the two components of the 
mapped 3D signal constellation point (CP) on two active TAs, 
and another one is obtained by modulating the one component 
of the mapped 3D signal CP on one active TAs. Then, with 
the aid of the one bit, the two spatial vector symbols are 
determined as the real and imaginary parts of the transmit 
vector. Furthermore, the squared MEDs of QIM-SSD are 
provided and compared with that of the QSM and QIM-TDC 
systems. Finally, simulation results demonstrate that QIM-
SSD outperforms the QSM and QIM-TDC systems in terms 
of bit error rate (BER) performance. 

2. The Proposed QIM-SSD 

2.1. System Model 

Considering the QIM-SSD system with the number of tN
TAs, as depicted in Fig.1. In our proposed QIM-SSD system, 
the transmit vector symbol and the input bits sequence is 
denoted by X  and m , respectively. 

 



 

132 

xm

3D Signal
Constellation 　

tN

2Spatial

Mapper

ne

1V

ym

zm

2B

4B

1B bitsm
1

3B
iA

lB

2V

5B

AI Vector
Selector A

AI Vector
Selector B

AI Vector
Selector C

Bits
Splitter

 
Figure 1. System Model of QIM-SSD 

 
As depicted in Fig. 1, the input information bits m  are 

divided into five blocks: 1B  , 2B  , 3B  , 4B  , 5B  . For the 

first block of 1B , containing 2log ( )M  bits, it is mapped 

into a M-ary 3D CP symbol ( , , )x y zs m m m  , 

 1, , M L  from the 3D signal constellation [5] denoted 

by  1, , Ms s  L  , e.g., 16-ary 3DCII: 

    1, 1, 2 , 1, 2, 1        . Through three antenna 

index (AI) vector selectors: A, B, C�three blocks of 2B , 3B , 

4B information bits are mapped into three AI vectors from the 

AI vector sets: υ , κ , e , respectively. Specifically, in order 

to consider that three components ( , ,x y zm m m  ) of the 

mapped ( , , )x y zs m m m   are modulated by the three AI 

vectors simultaneously in a QIM-SSD symbol period without 

interference, the two components ( ,x ym m ) are modulated on 

two different active TAs, and the component zm   is 

modulated on one active TA. The design rules of two spatial 

vectors symbol ( 1V  , 2V  ) for the in-phase or quadrature 

dimension  are follow :  
The design of the spatial vector symbol 

 +x i y lm m1V A B . 

The two block of 2B  , containing 2 tlog ( /2)N   bits, is 

used to select one AI vector iA  ,  t1, , / 2i N    from 

the AI vector set  
t1 /2, , Nυ A A  ; the block of 3B  

containing 2 tlog ( /2)N  bits, is used to select one AI vector 

lB  �  t1, , / 2l N    from the AI vector set 

 
t1 /2, , Nκ B B  . Then, the two AI vectors iA   and 

lB   respectively modulate the two components ,x ym m  of 

the mapped 3D symbol ( , , )x y zs m m m  , resulting in two 

spatial vector symbols: x im A  and y lm B . To avoid mutual 

interference, x im A   and y lm B   are transmitted on the 

upper and lower parts of TAs. Thus, a spatial vector symbol 

 
T

x i y lm m   1V A B  is obtained as by combining x im A  

with y lm B  , where iA   with t 2 1N    dimensions 

represents a vector with a non-zero element equaling to 1 in 
the i -th row and is used to make the i -th TA be in active 

state, lB   with t 2 1N    dimensions represents a vector 

with a non-zero element equaling to 1 in the  -th row and is 

used to make the t 2N   -th TA be in active state. 

The design of the spatial vector symbol 2 z nm V e  

For the last block of 4B , containing 2 tlog ( )N  bits, it is 

used to selected a AI vector ne  with t 1N  dimensions from 

the AI vector set  
t1, , Ne e e , where 

 t1, ,n N  . Then, the selected AI vector ne  is used to 

modulate the component zm  of the the mapped 3D symbol 

( , , )x y zs m m m  , resulting in a spatial vector symbol 

2 z nm V e , where ne  represents a vector with a non-zero 

element equaling to 1 the n -th row and is used to make the 
n -th TA be in active state.  

Here, to further explain the three AI vector sets, the 

following examples are provided, assumed that t 4N  ,  

 

1 0

0 1

 
  
 

υ , 
1 0

0 1

 
  
 

κ , 

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

 
 
 
 
 
 

e  (1) 

 
In order to further transmit more additional information bits, 

we use one index bit to determine the real part and the 
imaginary part of the complex transmit vector X . Specifically, 

when the block of 5 =0B  , the vector symbol 1V   is 

considered as the real part of the complex vector X , while the 

vector symbol 2V  is considered as the imaginary part of the 

complex vector X  . Otherwise, the vector symbol 1V   is 

considered as the imaginary part of the complex vector X , 

the vector symbol 2V  is considered as the real part of the 



 

133 

complex vector X . Consequently, with the determining of the 

block 5B , the complex vector X  is obtained as 

              

 
 

 

1 2 5

2 1 5

5

5

,       0

, ,      1

,      0
   =

( ) ,   1

x i y l z n

z n x i y l

j B

j B

m m jm B

m j m m B









    
  

   


     

V V
X

V V

A B e

e A B

      (2) 

 

where, av1 E  , avE  denotes the average energy of 

each complex transmit vector X . 
To further explain the working principle of forming the 

complex transmit vector X , examples for which are given in 

following Table I, assumed that t =4N , 

1 ( , , )x y zB s m m m  , and three AI vector sets are 

obtained according to (1). 
 

2B  3B  4B  5B  X  

0 0 0 0 0 [mx+jmz,0,my,0]T

0 1 0 1 0 [mx, jmz,0,my]T

1 0 1 0 0 [0,mx, my+jmz ,0]T

1 1 1 1 0 [0,mx, 0, my+jmz]T

0 0 0 0 1 [jmx+mz,0,jmy,0]T

0 1 0 1 1 [jmx, mz,0,jmy]T

1 0 1 0 1 [0,jmx, jmy+mz ,0]T

1 1 1 1 1 [0,jmx,0 , jmy+mz]T

2.2. Receiver and Detection 
At the receiver, after transmitting the complex transmit 

vector X   over the MIMO channel matrix H   with an 
additive white Gaussian noise (AWGN) vector N  , the 
receive signal vector can be expressed as  

 
   Y = HX + N               (3) 

 

where r 1NC Y  , r tN NC H   is a Rayleigh-fading 
channel matrix, whose each entry is assumed to be an 
independent and identically distributed complex Gaussian 

random variable with  0,1CN  . r 1NC N   is the 

Gaussian white noise vector with zero-mean and 2  

covariance, i.e., 
r

2(0, )NCN  I . 

Under the conditions of the ideal channel estimation and 
maximum likelihood (ML) detection, for better detection of 
the original information bits, the algorithm of the ML decoder 
is given by  

 

5

2

5
, , , ,

ˆ ˆˆ ˆ ˆ[ , , , , ] arg  min
i l n s B

i l n B s  Y - HX      (4) 

where ŝ  denotes the estimate of the 3D mapped symbol

s  . î  , l̂  , n̂  denote the estimated index number of the AI 

vectors iA  , lB  , ne  . 5B̂   denotes the estimate of the block 

5B . 

3. Performance Analysis 

3.1. Bit Error Probability 
In this paper, the channel state information is assumed to 

be perfect, and the estimated transmit vector of the 

normalized transmit vector X  is expressed as X̂ .  
Based on the union bound technique [6], the average PEP 

can be derived by  

ˆ

1 ˆ ˆ( ) ( )
2

APEP PEP e   m
X X

X X X X
m

 (5) 

where ˆ( )e X X  is the total number of erroneous bits. 

Based on the moment generating function [6], the average 
pairwise error probability (PEP) conditioned on H  can be 
calculated by     

 
 

   

r

r r

r

2 2

2
20

0 0

12
12

20
0,min2

0

ˆ ˆ
1ˆ( )

2 4 sin

1 sin 1 1

2 2
sin

4

N

N kN
N k
k

k

PEP E Q E d
N N

d C
d

N






 

  





 



                
          

 
 

                
 





H H

X

H X X H X X
X X

   (6) 

 
 

where ( )Q    denotes the Gaussian Q function�

 2 2
,min 0 ,min/ 4d N d  X X �

2
2

,min ''

ˆd   
 X X X . 

Thus, the average PEP will be derived by substituting the 
E.q. (6) into the E.q. (5). Moreover, it can be observed from 
the E.q. (6), when the number of receive antennas is fixed, the 
larger the squared MED is, the smaller it is, and the better the 
asymptotic system performance is. 

3.2. Squared Minimum Euclidean Distance 
Based on the above mentioned design and analysis, the 

squared MED 2
,mindX   between the transmit vectors for the 

QIM-SSD may be calculated by 
 

 2
2

,min '
'

ˆmind   
 X X X             (7) 

Then, according to the E.q.(7), the squared MEDs 2
,mindX  

are listed in Table I and compared with that of both QSM and 
QIM-TDC at different SEs, where pTXmb denotes 
transmitting m bits in each time slot with pTAs and Signal C. 
is Abbreviation of Signal Constellation. 

 

 
 
 
 



 

134 

Table 2. Comparison of the squared MED at different SEs and TAs. 

 
4TX8b 4TX9b 8TX11b 8TX12b 

Signal C. 2
,mindX

 Signal C. 2
,mindX

 Signal C. 2
,mindX

 Signal C. 2
,mindX

 

QSM 16QAM 2/10 32QAM 2/20 32QAM 2/20 64QAM 2/42 
QIM-TDC 16-3DCII 2/6 32-3DCII 2/8 16-3DCII 2/6 32-3DCII 2/8 
QIM-SSD 8-3DCI 2/3 16-3DCII 2/6 8-3DCII 2/3 16-3DCII 2/6 

 
It can be observed from Table II that the squared MEDs 

2
,mindX

of the QIM-SSD have the larger value than the QSM 

and QIM-TDC systems at the same SE. Furthermore, at high 
signal noise ratio (SNR), due to that the asymptotic 
performance is mainly determined by the worst-case PEP, 
which corresponds to the squared MED between the transmit 
vectors. According to the analysis of Table II, in high SNR 
region, the BER performance of the presented QIM-SSD 
system is more reliable than both QSM and QIM-TDC in 
wireless communication network. 

4. Results and Discussion 
In this section, the BER vs SNR performance for the QIM-

SSD by implementing the ML detector at the receiver is 
numerically expounded and compared with that of the QSM 
and QIM-TDC at the same spectral efficiency and TAs. Based 
on the analysis of Table II, simulation results using Monte 
Carlo for QIM-SSD are provided to verify the advantage of 

QIM-SSD with t r[ , ] [4, 4]N N  . In our simulation results, 

the channel state information is assumed to be perfect at the 
receiver. 

In Fig. 2, the BER vs SNR curves of QIM-SSD are depicted 
and compared with that of QSM and QIM-TDC at different 
SEs (4TX8b and 4TX9b b/s/Hz) and TAs. At the scenario of 
4TX8b, since the QIM-SSD achieves one more extra index 
bit than that the QIM-TDC achieve, the QIM-SSD only 
transmit the 8-ary 3DC signal CP symbol, but the QIM-TDC 
and QSM need to transmit the 16-ary 3DCII and the 16QAM 
signal CP symbols, respectively. Thus, according to the 
analysis of Table II, the squared MED ( e.g., 2/3) of QIM-SSD 
is bigger than that ( e.g., 2/6, 2/10) of both QIM-TDC and 
QSM. So that the QIM-SSD achieves better BER 
performance than the QIM-TDC and QSM. It can be observed 
from Fig. 2 that 1.8 dB SNR gain over QIM-TDC, 5 dB SNR 
gain over QSM are achieved for QIM-SSD at the BER value 
of 10-4. Similarly, the QIM-SSD with 16-ary 3DCII has bigger 
squared MED than the QIM-TDC with 32-ary 3DCII and the 
QSM with 32QSM, so as to improve the BER performance. 

 

 
Figure 1. BER performance comparison of QIM-SSD with the QIM-TDC and QSM at 4TX8b and 4TX9b. 

 

5. Conclusion 
Based on the advantages and characteristics of 3D 

constellation mapping, a new design, called QIM-SSD, is 
proposed to enhance the spectral efficiency and the reliability 
of the wireless communication system. Furthermore, the 
squared MED comparisons between QIM-SSD and the QIM-
TDC and QSM systems are provided. Under the conditions of 
ideal channel estimation and maximum likelihood detection, 
simulation results demonstrate that the proposed QIM-SSD 
achieves the better BER performance. In a word, the 
exploiting of the spatial domain and the increasing of the 
squared MED are good means to further improve the BER 
performance of mobile communication. 

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