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Orthogonal Frequency Division Multiplexing
With Subcarrier Gap Modulation
Ahmad Jaradat∗, Jehad M. Hamamreh†, and H¨
useyin Arslan∗‡
∗Department of Electrical and Electronics Engineering, Istanbul Medipol University, Istanbul, 34810, Turkey
†Department of Electrical and Electronics Engineering, Antalya Bilim University, 07468 Antalya, Turkey
‡Department of Electrical Engineering, University of South Florida, Tampa, FL, 33620, USA
Email: ahmad.jaradat@std.medipol.edu.tr, jehad.hamamreh@antalya.edu.tr, huseyinarslan@medipol.edu.tr
Abstract—A new modulation scheme called orthogonal
frequency division multiplexing with subcarrier gap mod-
ulation (OFDM-SGM) is proposed. The proposed scheme
embeds extra information bits by exploiting the gap
between the active subcarriers in each subblock. The
proposed scheme differs from the OFDM-index modulation
(OFDM-IM), in which information bits are transmitted
using the index of active subcarriers. This OFDM-SGM
technique provides superior spectral and energy efficien-
cies compared to the OFDM-IM, particularly when using
binary phase-shift keying (BPSK)-like low constellation
schemes, that suit the Internet of Things (IoT) appli-
cations that have low complexity. The theoretical error
performance of the proposed scheme is presented, and
the consistency between the theoretically derived error
performance and the simulated one is also provided.
Index Terms—OFDM, index modulation, subcarrier
number modulation, spectral efficiency, subcarrier gap
modulation.
I. INT RODUCT IO N
The forthcoming generations of mobile communi-
cations necessitate the capability to support extreme
requirements such as very high reliability, extremely
low latency, extremely high data rates, enhanced energy
efficiency (EE), and low complexity [1]. Therefore, it
is crucial to specify a proper modulation scheme for a
given application [2]. One interesting solution is pairing
OFDM with a scheme that introduces an extra degree of
freedom to embed additional information bits for each
OFDM symbol. Several existing modulation schemes
for OFDM-based waveforms have been classified, com-
pared, and their future directions presented in [3].
In recent research studies, significant efforts in en-
hancing spectral efficiency (SE) of OFDM can be ob-
served. Index-based schemes, such as OFDM with index
modulation (OFDM-IM) [4], are among the promising
options for OFDM-based waveforms. In these options,
only part of the available subcarriers is classically mod-
ulated along with extra information bits conveyed by
utilizing the index dimension [3].
Inspired by the underlying concept of OFDM-IM, a
novel number-based scheme called OFDM with subcar-
rier number modulation (OFDM-SNM) is proposed in
[5]. The OFDM-SNM implicitly conveys the information
by exploiting the number (not index) of the active
subcarriers along with the complex amplitude and phase
characteristics of a symbol.
With the analogy of the generic design of the OFDM-
IM and OFDM-SNM blocks, we propose a novel gap-
based OFDM scheme named as OFDM-subcarrier gap
modulation (OFDM-SGM). This novel scheme conveys
information implicitly by exploiting the gap, instead of
indices or number, of activated subcarriers as well as the
classical symbols.
The proposed OFDM-SGM transmission scheme im-
proves the system design flexibility by creating ad-
ditional means of conveying information in the gap
dimension. This inherent flexibility can be utilized for
different purposes, such as improving the overall SE of
the communication system while maintaining low detec-
tion complexity. Unlike conventional OFDM, where all
subcarriers are occupied by non-zero elements, exploit-
ing the inactive subcarriers in the proposed OFDM-SGM
scheme could be used to improve the trade-off between
SE and EE.
Our main contributions can be summarized as
shown below:
•A novel OFDM-based transmission scheme named
as OFDM-SGM is proposed in which data bits
are implicitly embedded by a new energy-free ex-
tra information-carrying dimension called the gap
between turned on subcarriers depending on the
incoming bit stream to convey additional informa-
tion besides the conventional quadrature amplitude
modulation (QAM) symbols. Introducing the gap
dimension enables having more active subcarriers
in OFDM-SGM, and thus SE would be improved
compared to OFDM-IM while maintaining low
computational complexity.
•A tight, closed-form upper bound on the bit error
rate (BER) for OFDM-SGM is derived. We propose
two low-complex detectors for the OFDM-SGM
system based on perfect and imperfect subcarrier
activation pattern estimation. The simulated BER
performance is evaluated for the OFDM-SGM and978-1-7281-4490-0/20/$31.00 c
2020 IEEE
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Secondary
Sub-block
Creator
1
Secondary
Sub-lock
Creator
G
Primary
OFDM
Block
Creator
Conventional
OFDM
Modulator
Gap
Mapper
Conventional
Modulation
Gap
Mapper
Conventional
Modulation
Bits
Splitter
𝑝1
𝑝1
𝑝2
𝑝2
𝑝
𝑝
𝑚bits
Fig. 1. Block diagram of the OFDM-SGM transmitter.
compared with that in its counterparts, such as
OFDM-IM, OFDM-SNM, and traditional OFDM.
The rest of this paper is organized as follows. The
proposed OFDM-SGM technique is illustrated in Section
II. Performance evaluation of the OFDM-SGM in terms
of average bit error probability (ABEP) and computa-
tional complexity is presented in Section III. Section
IV discusses the simulation results. Finally, Section V
provides concluding remarks and future work.1 2
II. SY ST EM M ODEL
The transmitter architecture of the OFDM-SGM is
depicted in Fig. 1. The msource bits are split into
Ggroups using the bits splitter module, each group
includes p=p1+p2bits that form an OFDM-SGM
subblock with a length of L=N/G, where Nis the
FFT size. The gap selecting bits (p1) specify the number
of gaps between active subcarriers in every subblock in
which the subcarrier activation pattern (SAP) is formed
using a proper mapping method.
The gap mapper places the active subcarriers in
each subblock g(g= 1,2, . . . , G) from the set Λ =
{1,2,· · · , L/2−(log2(L/2) −1), L/2, L}. The number
of activated subcarriers in the g-th subblock can be
represented by the gap-dependent variable I(g). This
variable is determined based on the p1incoming bits
that enter the gap mapper. Table I shows the proposed
mapping for L= 4 with p1= log2(L) = 2. As shown in
1Notation: Matrices and column vectors are represented by bold,
capital and lowercase letters, respectively. E(.),(.)T,(.)H,||.||,|.|
represent expectation, transposition, Hermitian transposition, norm,
and absolute value operations, respectively. det(A) represents the
determinant of A.n
k=n!
k!(n−k)! represents the binomial coefficient.
A∼ CN (µ, σ2)is the complex Gaussian distribution of Awith mean
and variance of µand σ2, respectively. ∼ O(.)denotes the complexity
order of a method. Sis the complex signal constellation of size M.
2The simulation codes used to generate the results presented in this
paper can be found at https://www.researcherstore.com/.
Table I, the assignment of zero, one, two, and three gaps
between active subcarriers correspond to p1of [0 0],[0 1],
[1 0], and [1 1], respectively. The SAP in each subblock
can be formulated as cg=c1c2· · · cLT, where
ci∈ {0,1}for i= 1,2,· · · , L. In each subblock,
p2=I(g) log2(M)data symbol bits are available
depending on the SAP. Particularly, these p2bits are
mapped to standard constellations conveyed by the active
subcarriers.
TABLE I
SGM MA PPI NG W IT H p1=2 B IT S AN D L=4
p1cg
[0 0] [1 1 1 1]T
[0 1] [1 0 1 0]T
[1 0] [1 0 0 1]T
[1 1] [1 0 0 0]T
The OFDM-SGM subblock can be represented based
on cgas xg
F=xg
F(1) xg
F(2) ... xg
F(L)T, where
xg
F(i)∈ {0,S} for i= 1,2, ..., L. Then, the OFDM-
SGM block is formed by concatenating Gsubblocks as
xF=xF(1) xF(2) ... xF(N)T. The remaining
steps are performed as in the plain OFDM transmission
process, including IFFT and cyclic prefix (CP) addition.
By applying IFFT to xF, the resultant signal is xt. By
adding NCP CP samples to xt, the output vector can
be written as xCP =xt(N−NC P + 1 : N)xtT.
This CP appended signal would be faded by a wireless
channel with impulse response ht, along with an AWGN
with a variance of No,T in the time-domain.
CP removal, FFT operation, gap demapping, and de-
tection constitute the operations that performed at the
OFDM-SGM receiver. The frequency-domain received
signal vector of dimension N×1after the CP removal
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and after applying FFT can be written as
yF=XFhF+nF,(1)
where XFis a diagonal matrix with dimension N×N,
and its elements on the main diagonal are represented by
xF(1),xF(2), . . . , xF(N).hFis the channel vector in
the frequency domain with a size of N×1, and it follows
the distribution of CN (0,IN), where INrepresents the
identity matrix with size N×N. The frequency-domain
channel vector is related with its time representation
as hF=WNh0
t, where WNrepresents the discrete
Fourier transform (DFT) matrix of dimension N×N
with WNHWN=NIN, and h0
t= [ht,0, ..., 0]T
denotes the zero-padded version of htwith length N.
nF∼ C N (0, No,F )is the frequency-domain AWGN
vector [4].
Thereafter, a one-tap frequency domain equalizer is
applied to compensate for the frequency selectivity of
the multi-path channel. A maximum likelihood (ML)
detection would detect the SAP. For detection of the data
code p1, a similar Table I is used at the receiver.
Subsequently, the gap demapper module is used to get
the SAP, and then the gap bits could be estimated in each
subblock. Then, M-ary signal constellation detection
is done depending on the received SAP in each sub-
block. At the final stage, the detected bits from the gap
demapping and classical symbols detection jointly form
the latest estimated subblock bits. The recovered data
sequence is obtained for the whole block by performing
similar steps to all subblocks.
To reduce the detection complexity of the employed
optimal ML, we adopt two low-complexity detectors,
namely, perfect SAP estimation (PSAPE), and imperfect
SAP estimation (ISAPE). The PSAPE detector neglects
the errors caused by the wrong detection of subcarriers
in the received SAP, whereas, the detection in ISAPE
is error-prone. In the both detectors, the demodulation
stage of the classical constellation symbols would be
unsuccessful to extract p2bits correctly when an erro-
neous SAP detection happens. This occurs because of the
demodulation of some incorrectly detected subcarriers.
So, the transmitted p1and p2bits are erroneous when
the detected SAP is wrong. On the other hand, if SAP is
correctly detected, then p1is correct but it is not certain
that p2is correctly estimated.
The PSAPE detector is taken into account for the
conducted simulations due to its low complexity com-
pared to that of the ISAPE. It is worthy to note that
the performance comparison between PSAPE and ISAPE
detectors is presented as well. A log-likelihood ratio
(LLR) detector is also implemented for the proposed
scheme to reduce its computational complexity. Assum-
ing BPSK is adopted in the proposed scheme, the LLR
values can be represented as [4]
λ(α) = max(a, b) + ln(1 + exp(−|b−a|)) + |yF(α)|2
No,F
,
(2)
where a=−|yF(α)−hF(α)|2
No,F and b=−|yF(α)+hF(α)|2
No,F .
This LLR detector decides on specific active subcarri-
ers that have maximum LLR values. Then, these selected
active subcarriers would be mapped to p1bits, and the
symbols carried over these active subcarriers would be
demodulated to get the p2bits.
III. PER FO RMANC E EVALUATION OF TH E PRO POSED
OFDM-SGM
In this section, the assessment of the proposed scheme
is done based on some key metrics like ABEP and the
complexity associated with the detectors.
A. Average Bit Error Probability
The ABEP should be calculated based on the eval-
uation of pairwise error probability (PEP) since the
transmission bits are carried over the gap dimension as
well as the conventional symbols [5]. The conditional
PEP (CPEP) could be expressed by the Q-function [6]
P(cg−→ ˆcˆg|ht) = Q sPt
No,F
||ht∆||2!,(3)
where Ptis the total transmitted power, ∆ = cg
pI(g)−
ˆcˆg
pI(ˆg).I(g)and I(ˆg)represent the number of activated
subcarriers in the g-th and ˆg-th OFDM-SGM subblock,
respectively, cgand ˆcˆgare transmitted and detected
sequences. The formula of CPEP in (3) can be approxi-
mated using the Q-function as [7]
Q(x)≈
2
X
j=1
ρjexp(−ηjx2),(4)
where ρ1= 1/12 and ρ2= 1/4,η1= 1/2and η2= 2/3.
Using (4), the CPEP can be formulated as [8]
P(cg−→ ˆcˆg|ht) =
2
X
j=1
ρj
L
Y
l=1
exp(−ηjPt
No,F
U(l)|∆(l)|2),
(5)
where U(l) = |ht(l)|2,|∆(l)|2=|cg(l)
pI(g)−ˆcˆg(l)
pI(ˆg)|2.
The unconditional PEP (UPEP) can be calculated by
averaging CPEP over htas [9]
P(cg−→ ˆcˆg) = X
ˆc ˆg6=cg
EhtP(cg−→ ˆcˆg)|ht).(6)
Furthermore, a more approximative Q-function ex-
pression in [10] could be used for a tighter upper bound
of BER as
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Q(x)≈0.168e−0.876x2+0.144e−0.525x2+0.002e−0.603x2
(7)
By using the spectral theorem [11] and (7), (6) can be
simplified to [4]
P(cg−→ ˆcˆg)≈1
1/0.168 det IL+0.876
2No,F KLA+
1
1/0.144 det IL+0.525
2No,F KLA+
1
1/0.002 det IL+0.603
2No,F KLA,
(8)
where ILrepresents L×Lidentity matrix, KL=
EhthH
t,A = (cg−ˆcˆg)H(cg−ˆcˆg).
Considering all UPEP events and assuming informa-
tion bits are equiprobable, the upper bound for ABEP in
the OFDM-SGM could be attained [4]
Pb(E) = 1
pgnx
G
X
g=1 X
ˆc ˆg6=cg
P(cg−→ ˆcˆg)e(cg,ˆcˆg),(9)
where pgis the length of the vector that constitutes data
bits corresponding to an OFDM-SGM subblock, nxis
the number of legitimate realizations of the transmitted
sequence, and e(cg,ˆcˆg)represents the number of bits
in a difference between cgand ˆcˆg. The aforementioned
theoretical analysis for the error performance of OFDM-
SGM is verified with its corresponding simulation one,
as shown in section IV.
B. Complexity analysis
To compare the PSAPE and ISAPE, and LLR detector
in terms of their computational complexity with the
optimal ML, the average number of metric calculations
per subcarrier is considered as a key performance metric.
The complexities associated with the considered OFDM-
based modulation options are presented in Table II. As
observed from Table II, the computational complexity
of the ML detector is highly critical to G,L, and
M; however, the complexity of the PSAPE and ISAPE
detectors is only determined by G, which is much less
than the optimal ML detector. Furthermore, Table II
presents the comparison between different detectors for
OFDM-SGM with its counterparts. It can be seen that
the OFDM-SGM with an LLR detector offers compa-
rable complexity performance compared to OFDM-IM
and classical OFDM. Therefore, the LLR detector is a
preferable choice for practical OFDM-SGM systems.
IV. SIM ULATI ON RE SU LTS
In this section, throughput, spectral and energy ef-
ficiency trade-off, and BER of the proposed OFDM-
SGM scheme are compared with other existing schemes
including OFDM-IM, OFDM-SNM, and plain OFDM.
In our simulations, Nand Lare set to 64 and 4,
respectively, the number of subblocks G=N/L = 16,
and SGM bits are p1= log2(L) = 2 bits in each OFDM-
SGM subblock. We assume the BSPK modulation of
active subcarriers to utilize the full capability of the
OFDM-SGM scheme. The simulated wireless channel is
Rayleigh distributed and frequency-selective, as adopted
in [3]. Similar signal-to-noise ratio (SNR) (or Eb/No,T )
is considered to provide a fair comparison between
the featured schemes, where Ebis the bit energy. The
CP length (NCP ) is set to be longer than the number
of channel taps to avoid intersymbol interference [12].
Furthermore, CSI is assumed not to be available at the
transmitter.
The throughput performance of the OFDM-SGM com-
pared to its counterparts under BPSK is demonstrated
in Fig. 2. As presented in Fig. 2, the SE of OFDM-
SGM under BPSK is higher than that of its competitive
schemes because of having a more average number of ac-
tive subcarriers in OFDM-SGM as compared to OFDM-
IM. The degree of freedom offered by introducing the
gap between active subcarriers as the new transmission
medium enables an improved throughput performance of
OFDM-SGM compared to the plain OFDM.
0 5 10 15 20 25 30
Eb/No,T(dB)
0.65
0.7
0.75
0.8
0.85
0.9
0.95
Throughput (bps/Hz)
Proposed OFDM-SGM
OFDM-IM
Classical OFDM
Fig. 2. Throughput of the proposed OFDM-SGM scheme and its
counterparts under BPSK.
As commonly done in the literature, we investigate
the trade-off between the SE and EE for the considered
modulation schemes. This trade-off is determined by the
number of active subcarriers in the OFDM block [13].
In particular, more saved energy and less SE result from
having less number of active subcarriers. Fig. 3 exhibits
the SE and EE ratios of the proposed scheme, OFDM-
IM, OFDM-SNM, and the reference OFDM scheme
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TABLE II
COM PLE XI TY C OMPA RI SON B ET WE EN DI FFE RE NT DE TE CTO RS F OR T HE PR OPO SE D SC HEM E AN D IT S CO UNT ER PARTS
Modulation Scheme Detector type Complexity order
Proposed OFDM-SGM
Optimal ML ∼ O(G M L/2)
PSAPE ∼ O(G)
ISAPE ∼ O(G)
LLR ∼ O(M)
OFDM-SNM ML ∼ O(L, I (g), M)
OFDM-IM Near optimal LLR ∼ O(M)
Conventional OFDM ML ∼ O(M)
under BPSK. It is obvious from Fig. 3 that the OFDM-
SGM offers higher SE and EE ratios as opposed to the
OFDM-IM with low activation ratio (AR), and the plain
OFDM.
s1 s2 s3 s4
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
SE ratio
1
1.5
2
2.5
EE ratio
SEr
EEr
Fig. 3. The SE ratio (SEr) and EE ratio (EEr) of the proposed
scheme and its counterparts. The symbols s1, s2, s3, s4 correspond
to the proposed scheme, OFDM-IM with AR = 0.25, OFDM-IM
with AR = 0.5, and conventional OFDM, respectively.
Fig. 4 shows that OFDM-SGM has improved BER
performance over classical OFDM at high SNR values.
Moreover, the proposed scheme with an LLR detector
has comparable BER performance to OFDM-SNM and
OFDM-IM. The reason for that is the gap between active
subcarriers in OFDM-SGM increases the influence of
active gap bits on the BER more.
Moreover, the BER performance of the proposed
OFDM-SGM when using the ISAPE detector has worse
BER performance compared to that of the PSAPE detec-
tor. The reason behind this degraded BER performance
of the ISAPE detector is that more error sources result
from employing ISAPE as compared to the PSAPE
detector. Fig. 4 also shows that with the growth of SNR,
the obtained theoretical results become closer to the
simulative ones.
V. CO NC LUSIO N
This paper introduces a novel energy-efficient multi-
carrier modulation scheme termed as OFDM-subcarrier
0 5 10 15 20 25 30
Eb/No,T(dB)
10-4
10-3
10-2
10-1
100
101
BER
Proposed OFDM-SGM, PSAPE
Proposed OFDM-SGM, ISAPE
Proposed OFDM-SGM, LLR
Proposed OFDM-SGM, Theo.
OFDM-IM
OFDM-SNM
Classical OFDM
Fig. 4. The BER performance of the OFDM-SGM, OFDM-IM,
OFDM-SNM, and plain OFDM under frequency-selective Rayleigh
channel with BPSK.
gap modulation (OFDM-SGM) that transmits extra in-
formation by the gap between the activated subcarri-
ers beside the classical modulation symbols. The pro-
posed OFDM-SGM has low-complex transceivers. Also,
OFDM-SGM performs better than its counterparts in
terms of throughput under low modulation order. Fur-
thermore, the upper bound on the BER of the OFDM-
SGM agrees with the analyzed one. As future work, the
proposed scheme will be investigated with different mod-
ulation orders and subblock sizes. Another potential re-
search direction is combining the proposed scheme with
the other existing OFDM-based modulation schemes to
provide a further enhancement in different performance
metrics, especially SE [14]–[16].
ACK NOWLEDGM EN T
The author Jehad M. Hamamreh is supported in part
by the Scientific and Technological Research Council of
Turkey (TUBITAK) under Grant 119E392.
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