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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LWC.2018.2839624, IEEE Wireless
Communications Letters
1
OFDM with Subcarrier Number Modulation
Ahmad M. Jaradat, Jehad M. Hamamreh, and Huseyin Arslan, Fellow, IEEE
Abstract—A new modulation technique, named as Orthog-
onal Frequency Division Multiplexing (OFDM) with Subcar-
rier Number Modulation (SNM), is proposed for efficient data
transmission. In this scheme, the information bits are conveyed
by changing the number of active subcarriers in each OFDM
subblock. The idea of this scheme is inspired from the integration
of OFDM with Pulse Width Modulation (PWM), where the
width of the pulse represents the number of active subcarriers
corresponding to specific information bits. This is different
from OFDM with Index Modulation (OFDM-IM), where the
information bits are sent by the indices of the subcarriers instead
of their number. The scheme is shown to provide better spectral
efficiency than that of OFDM-IM at comparable Bit Error Rate
(BER) performances. Another key merit of the proposed scheme
over OFDM-IM is that the active subcarriers can be located in
any position within the subblock, and thus enabling channel-
dependent optimal subcarrier selection that can further enhance
the system performance.
I. INT ROD UC TI ON
RECENT research studies exhibit the urgent need for de-
signing new advanced waveforms and modulation tech-
niques that are capable of further enhancing spectral efficiency,
energy efficiency, and reliability with minimal complexity
compared to conventional OFDM in order to fit the diverse
needs of future 5G scenarios and services [1], [2].
A novel modulation scheme called spatial modulation that
exploits the spatial domain by selecting the indices of antennas
along with the classical signal constellations (amplitude/phase
modulation) to convey information was proposed in [3]. An
improved spatial modulation scheme, whose spectral efficiency
linearly increases with the transmit antennas’ number rather
than a base-two logarithm factor, was introduced in [4]. The
interesting application of spatial modulation to OFDM system
was proposed under the name OFDM with Subcarrier Index
Modulation (OFDM-SIM) in [5]. In this scheme, OFDM active
subcarriers’ indices vary in each OFDM block to convey infor-
mation bits. A systematic subblock-based transceiver structure,
named as OFDM with Index Modulation (OFDM-IM), that
enables selecting more than one active subcarrier among the
available subcarriers in each subblock was proposed in [6].
In [7], a generalized version of OFDM-IM named as OFDM-
GIM was introduced, where different activation ratios per each
subblock are used to enhance spectral efficiency. Recently, a
comprehensive survey of the recent advances and different
variations of index modulation concept was given in [2].
One can describe and perceive the concept of OFDM-IM [6]
as OFDM combined with Pulse Position Modulation (PPM),
A. J. and J. H. are with the School of Engineering and Natural Sci-
ences, Istanbul Medipol University, Istanbul 34810, Turkey (email: [ajara-
dat,jmhamamreh]@st.medipol.edu.tr). H. A. is with the Department of Elec-
trical Engineering, University of South Florida, Tampa, FL, 33620, USA; and
also with Istanbul Medipol University, Turkey (email: arslan@usf.edu).
where part of the data bits are conveyed by the positions
(indices) of the active subcarriers that can be selected by
a simple look-up table. In this approach, a certain fixed
number of subcarriers are selected in each subblock as active
subcarriers in OFDM-IM to convey data bits.
Different from OFDM-IM, in this work, we propose a new
modulation technique, named as OFDM with Subcarrier Num-
ber Modulation (OFDM-SNM), which basically integrates
OFDM with Pulse Width Modulation (PWM)1. The concept
of OFDM-SNM proposed in this paper is inspired by PWM by
means of representing pulse width by the number of active sub-
carriers. This number is determined depending on the different
combinations of incoming bits. This concept results in creating
a new dimension, i.e., number of active subcarriers, for sending
data in addition to the conventional two dimensional complex
signal constellation plan. These dimensions are jointly utilized
to convey information to the receiver by the number of active
subcarriers (instead of their indices) alongside the information
sent through symbols. The proposed OFDM-SNM results in
better spectral efficiency compared to both OFDM and OFDM-
IM when BPSK is used. Besides, the power efficiency and
reliability of the proposed OFDM-SNM scheme are shown
to be better than that of OFDM, and comparable to that of
OFDM-IM. Similar to OFDM-IM, the proposed OFDM-SNM
has also the potential to reduce Inter-Carrier-Interference (ICI)
and Peak-to-Average Power Ratio (PAPR) due to not activating
all the subcarriers. Different from OFDM-IM, the activated
subcarriers can be placed in any index or position within each
subblock as the information is sent by the subcarriers’ number,
and thus they can be made channel-dependent, resulting in
even much better reliability performance. The exact spectral
efficiency and error performance formulas of the proposed
scheme are derived and shown to be closely matched with
the simulated results.
The rest of the paper is organized as follows. The proposed
OFDM-SNM is illustrated in Section II. Performance analysis
of OFDM-SNM scheme is presented in Section III. Simulation
results are carried out in Section IV. Finally, conclusion and
future works are provided in Section V.2
II. PRO PO SE D OFDM-SNM
The transmitter structure of the proposed OFDM-SNM
system is depicted in Fig. 1. For the transmission of each ith
OFDM block, a variable number of miinformation bits enter
the transmitter of the OFDM-SNM scheme. These bits are split
1Pulse width modulation can also be found in the literature under different
names such as pulse interval modulation or pulse duration modulation.
2Notation: Bold, lowercase and capital letters are used for column vectors
and matrices, respectively. (.)Tand (.)Hrepresent transposition and Hermi-
tian transposition, respectively. det(A) denotes the determinant of A.E(.)
represents the expectation. ~denotes a circular convolution operation.
2162-2337 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LWC.2018.2839624, IEEE Wireless
Communications Letters
2
Secondary
Sub-block
Creator
1
Secondary
Sub-lock
Creator
G
Primary
OFDM
Block
Creator
Conventional
OFDM
Modulator
(IFFT + CP +
DAC)
SNM
Mapper
M- ary
Modulation
SNM
Mapper
M
S
S
M- ary
Modulation
Bits
Splitter
ଵ
ଵ
ଶ
ଶ
݉ bits
Transmitted bits gain compared
to OFDM-BPSK/OFDM-IM (ʹȀͶ)
Information
bits (ଵ) Subcarrier Mapping (ܰൌͶ)
00
01
10
11 2
1
0
-1
Fig. 1. OFDM-SNM transmitter structure.
TABLE I
SNM MA PPE R WI TH p1=2 BITS AND N=4
Information bits Active subcarriers
[0 0] [1 0 0 0]
[0 1] [1 1 0 0]
[1 0] [1 1 1 0]
[1 1] [1 1 1 1]
into Ggroups each containing variable p=p1+p2bits, which
are used to form OFDM subblocks of length N=NF/G,
where NFis the size of the Fast Fourier Transform (FFT).
Unlike OFDM-IM, where constant Kout of Nnumber
of available subcarriers are used to send information, in
our scheme, for each subblock g(g= 1,2, . . . , G), index-
independent variable K∈[1,2,· · · , N ]out of Navailable
subcarriers are activated by a subcarrier number mapper (selec-
tor) according to the corresponding p1= log2(N)bits, while
the remaining N−Ksubcarriers are inactive. Note that Kis
not fixed, but rather changing and taking variable values of the
number of subcarriers according to the incoming information
bits p1as shown in Table I, which presents the case when
N= 4,K∈[1,2,3,4] and p1= log2(4) = 2. For each
subblock g, the subcarrier on-off activation pattern set can be
given as
ig=i1i2· · · iKT, g = 1,2, . . . , G (1)
where ik∈1,0for k= 1,2,· · · , K. This subcarrier on-off
activation pattern procedure can be performed using a look-up
table for smaller Nand Kvalues as shown in Table I. For each
subblock, the remaining p2=K(log2(M)) bits (which change
from one subblock to another based on the number of active
subcarriers) of the p-bit input bit sequence are mapped onto
the M-ary signal constellation in order to determine the data
symbols that are transmitted over the active subcarriers. After
the concatenation of Gsubblocks, the whole OFDM block
can be represented as xF=xF(1) xF(2) ... xF(NF).
Now, the remaining steps are performed as done in standard
OFDM modulation. After taking IFFT, the signal can be
represented as xt=I F F T (xF). By appending CP of length
NCP to the transmitted signal, the resulting signal becomes
xCP =xt(NF−NCP + 1 : NF)xt. Now, since we
assume AWGN multi-path channel with channel impulse re-
sponse (ht), the received time-domain signal over the channel
can be written as
yt=xt~ht+nt,(2)
where nt∼ CN (0, No,T )is the AWGN vector. The
operation of the receiver would be the reversal of the
transmitter operations, i.e, removing CP, performing FFT
and SNM demapping and detection as follows. The re-
ceived signal after removing the CP is given by yt=
y0y1... yNF−1. After FFT block, yF=FFT(yt) =
yF(0) yF(1) ... yF(NF−1). Now, to compensate for
the frequency selectivity of the channel, a simple one tap
frequency domain equalizer is used and its output can be
represented as yeq =yF/hF. Then, a simple energy de-
tector is used to extract the active subcarriers pattern using
a properly selected threshold value [8]. It is noted that the
use of a threshold-based detector facilitates low-complexity
receiver for the OFDM-SNM scheme. This detector is much
simpler than ML or LLR based detectors [9]. After that, the
set of active subcarriers is determined and then mapped to
its corresponding bits using SNM demapper, which is the
inversion mapping process used at the transmitter. Now, the
active subcarriers obtained at the receiver for each subblock
is used for constellation symbols detection. Finally, the bits
obtained from both SNM demapper and symbol detection are
combined to form the final estimated subblock bits. By doing
the same procedure for all subblock, the retrieved data stream
is obtained for the whole OFDM block.
III. PER FO RM AN CE A NALYSIS OF OFDM-SNM
To analyze the performance of the proposed OFDM-SNM,
spectral efficiency, pairwise error probability, and power effi-
ciency are derived and investigated as follows.
A. Spectral Efficiency (SE)
The SE (bits/s/Hz) of the proposed OFDM-SNM scheme
can precisely be formulated as
ηOF DM −SN M =PG
g=1 (log2(N) + K(g) log2(M))
NF+NCP
,(3)
where K(g)is the number of active subcarrier in each sub-
block of Nsubcarriers’ length. It can be observed that the
difference between OFDM-SNM and OFDM-IM is in K(g)
parameter. In OFDM-SNM, K(g)varies for each subblock,
which is different from OFDM-IM where K(g)is fixed for
all subblocks. Thus, the proposed OFDM-SNM improves SE
over OFDM-IM as well as conventional OFDM. The amount
of improvement over OFDM-IM depends on K(g)values. For
instance, if N= 4 is selected for both OFDM-SNM and
OFDM-IM (with K= 2), then the SE gain of OFDM-SNM
scheme over OFDM-IM, when BPSK is considered, equals to
ρ= 18/16 = 1.125.
2162-2337 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LWC.2018.2839624, IEEE Wireless
Communications Letters
3
B. Analytical Bit Error Probability
The activated subcarriers in each sub-block are determined
based on the incoming information bits using a mapping pro-
cess that can basically be represented by code. The reason for
this is that there is no information about the exact modulation
used in signal constellation, and the known information is that
the mapping of the bit information into the number of active
subcarriers. Assume there are Ttime slots which represent the
number of bit group, then the transmitted codeword in Ttime
slots X=Xij, where i= 1,2,· · · , T and j= 1,2,· · · , N .
At the receiver, the decoder may decode another codeword
ˆ
X=ˆ
Xij. Our system model has just those two codewords (X
and ˆ
X), so here the analytical BER considers only pairwise
error probability (P(X−→ ˆ
X)). It should be noted that the
receiver might cause errors on the two consecutive detection
processes, i.e., the number of active subcarriers as well as
the M-ary symbols. In the considered model, we assume that
the frequency selective channel is fixed within one block and
follows Rayleigh distribution. The input-output relationship in
the frequency domain can be written in the following form
y=Xh +nz.(4)
The transmitted sequence Xcould be detected correctly or
erroneously as ˆ
X. An optimal detector for the proposed system
can be represented mathematically as
P(X=X|y)≥P(X=ˆ
X|y).(5)
We assume that nz∼ CN (0, No,F ), then the detector is simply
ML detector defined as
|y−X|<|y−ˆ
X|.(6)
The distance from yto Xis less than that to ˆ
X, so we can say
that Xis the nearest neighbor to y. Since we have only two
probabilities then the threshold value is assumed to be at the
middle point between Xand ˆ
Xas X+ˆ
X
2. So, the probability
of choosing Xcan be computed as
P(y<X+ˆ
X
2|X=X) = P(z>||X−ˆ
X||
2)
=Q(||X−ˆ
X||
2pNo/2),
(7)
where Q(.)is the Q-function [10] and No=No,F .
Accordingly, the error probability depends only on the
distance between the sequences: Xand ˆ
X; with the effect
of the channel (h), the Q function becomes [11]
P(y<X+ˆ
X
2|X=X) = Q(s||(X−ˆ
X)h||2
2No
)
=Q(shH(X−ˆ
X)H(X−ˆ
X)h
2No
).
(8)
Equation (8) can be rewritten as
P(y<X+ˆ
X
2|X=X) = Q(shHAh
2No
) = Q(rδ
2No
),(9)
TABLE II
SIM ULATI ON PAR AM ETE RS
Modulation type BPSK (M= 2)
IFFT/FFT size (NF) 64
CP Guard Interval (samples) 8
Number of subblocks in each OFDM symbol (G) 16
Number of available subcarriers in each subblock (N) 4
Number of bits mapped to each subblock (p1) 2
Multipath channel delay samples locations [0 3 5 6 8]
Multipath channel tap power profile (dBm) [0 -8 -17 -21 -25]
where δ=hHAh =hH(X−ˆ
X)H(X−ˆ
X)h=||(X−ˆ
X)h||2,
and the Amatrix equals to (X−ˆ
X)H(X−ˆ
X). The channel
frequency response hFis assumed to be zero-mean Circularly
Symmetric Complex Gaussian (CSCG) random variable with
unity variance hF∼ C N (0,1).hFcan be completely de-
scribed by its mean and covariance matrix K=E[hFhFH],
where Kis a Hermiation matrix with a dimension of N < NF.
So, a submatrix KNwith size N×Nis sufficient to represent
the channel frequency response in each subblock. If we define
a complex orthogonal matrix Qwhere QHQ=I,has a
subset of hFcan be represented as orthonormal basis h=Qu,
and Das a diagonal matrix with rank r1< N, where
D=E[uuH] = E[hhH], and KNcan be given as
KN=E[hhH] = QDQH.(10)
The PDF of the orthonormal basis ufollows the PDF of h
and can be expressed as
f(u) = exp(−uHD−1u)
πr1det(D).(11)
The Conditional Pairwise Error Probability (CPEP) is the same
as equation (9) and the PDF of the fading channel is repre-
sented in equation (11). Then, by taking the expectation for
CPEP with respect to the channel, and using the approximation
of Q function found in [10], the Unconditional Pairwise Error
Probability (UPEP) can be obtained as
P(X−→ ˆ
X) = 1/12
det(IN+q1KNA)+1/4
det(IN+q2KNA),(12)
where q1= 1/(4No,F )and q2= 1/(3No,F ). The overall bit
error probability is of great interest rather than individual PEP.
The Average Bit Error Probability (ABEP) can be calculated
as follows [6]:
Pb(E)≈1
p nxX
XX
ˆ
X
P(X−→ ˆ
X)e(X,ˆ
X),(13)
where pis the number of information bits per subblock
transmission, nxrepresents the number of realizations of X,
and e(X,ˆ
X)is the number of information bit errors committed
by choosing ˆ
Xinstead of X. It is shown analytically that the
BER of OFDM-SNM is similar to that of OFDM-IM [6]. This
analytical result will be verified by simulation as well.
C. Power Efficiency
Since not all subcarriers are occupied where only active sub-
carriers carry modulated data, OFDM-SNM approach achieves
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Communications Letters
4
0 2 4 6 8 10 12 14 16 18 20
Eb/No,T(dB)
10-3
10-2
10-1
100
BER
Plain OFDM
OFDM-IM, N=4, K=2, without Interleaving
OFDM-SNM, N=4, without Interleaving
OFDM-GIM, N=4, K=[1,2,3], with Interleaving
OFDM-SNM, N=4, with Interleaving
OFDM-IM, N=4, K=2, with Interleaving
Fig. 2. BER of OFDM-SNM compared to conventional schemes.
better power efficiency compared to conventional OFDM. In
conventional OFDM, the transmitted power (Ptx) is distributed
equally among all subcarriers so that the average power per
subcarrier equals to (Ptx/NF). The distribution of observing
the number of active subcarriers in each subblock is assumed
to follow binomial distribution as
P(Nac =K) = N
KpK
r(1 −pr)N−K,(14)
where Nac represents the number of active subcarriers which
takes values of K= 1,2,· · · , N. Also, pris the probability of
the event (Nac =K). By assuming that the total transmitted
power allocated to OFDM-SNM block is Ptx, and the power
is equally distributed to all subblocks, the power consumed by
OFDM-SNM scheme can be written as
Pc=Ptx
G N
G
X
g=1
K(g)P(Nac =K(g)),(15)
where K(g)and P(Nac =K(g)) represent the number of
active subcarriers and their corresponding probability in the
subblock g, respectively. Thus, the average power allocated
per subcarrier equals to Pc/NF.
IV. SIM UL ATION RESULTS
In this section, BER and throughput performances of
OFDM-SNM system are simulated and compared with both
standard OFDM and OFDM-IM. The simulation parameters
used are shown in Table II. It is assumed that the multi-
path channel is Rayleigh distributed. Fig. 2 shows BER vs.
Eb/No,T of the proposed OFDM-SNM system compared to
its competitive systems such as conventional OFDM, OFDM-
IM [6] (with K= 2 and N= 4) and OFDM-GIM [7] (with
N= 4 and K= [1,2,3] ). As seen from Fig. 2, both OFDM-
SNM and OFDM-IM have similar BER performance which
is better than that of classical OFDM for high SNR values
(SNR >10 dB). Also, it is observed that the BER can be
improved further when interleaving is adopted [12]. Fig. 3
demonstrates that the throughput performance of the proposed
OFDM-SNM (in both cases when CP is included and excluded
from the throughput calculation) is better than that of OFDM-
IM by a factor of 1.125 and than that of OFDM-GIM by a
factor of 1.1 at equivalent BER performances.
0 5 10 15 20 25 30
Eb/No,T(dB)
0.7
0.8
0.9
1
1.1
Throughput (bps/Hz)
OFDM-SNM, N=4 - excluding CP
OFDM-IM, N=4, K=2 - excluding CP
OFDM-GIM, N=4, K=[1,2,3] - excluding CP
OFDM-SNM, N=4 - including CP
OFDM-IM, N=4, K=2 - including CP
X: 25
Y: 1.125
Fig. 3. Throughput of OFDM-SNM compared to conventional schemes.
V. CO NC LU SI ON
This paper introduces a novel multi-carrier modulation
scheme called OFDM-SNM that sends information not only
by symbols but also by the number (instead of indices) of
active subcarriers in each subblock. OFDM-SNM improves the
spectral efficiency compared to that of OFDM-IM by 12.5%.
Besides, different from OFDM-IM, since the active subcarriers
in OFDM-SNM send information by their number instead
of indices, their mapping can be configured to be floating
or contiguous based on the channel quality or the ICI level
between subblocks, resulting in even a better performance.
Validating these extra merits alongside investigating OFDM-
SNM with different modulation orders and subblock sizes will
be key subjects of future research studies.
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