DCT-OFDM with Index Modulation

Marwa Chafii, Justin Coon, Dene Hedges

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Marwa Chafii, Justin Coon, Dene Hedges. DCT-OFDM with Index Modulation.
IEEE Communications Letters, Institute of Electrical and Electronics Engineers, 2017,
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 1

DCT-OFDM with Index Modulation

Marwa Chafii, Member, IEEE, Justin P. Coon, Senior Member, IEEE, and Dene A. Hedges, Member, IEEE

Abstract—In this letter, index modulation (IM) is proposed where N is the number of subcarriers, T is the OFDM
for OFDM systems based on a discrete cosine transform (DCT) symbol period, and ∆F = 1/T is the minimum intercar-
implementation. The new DCT-OFDM-IM scheme is shown to rier
yield spectral efficiency improvements of up to 38% relative to ∫ T spacing required to satisfy ′
the orthogonality constraint
ϕ n (t)ϕn ′ (t)dt = 0 if n ̸= n , with the integral evaluating to
OFDM-IM and up to 55% relative to OFDM while occupying 0
the same bandwidth and maintaining similar robustness against 1 if n = n′ . In DCT-OFDM, the following set of (co)sinusoidal
frequency selective fading channels for low modulation orders. functions is used for data modulation:
√
Index Terms—Discrete cosine transform, orthogonal frequency 2
division multiplexing, index modulation, spectral efficiency. σn (t) = cos(2πnδF t), 0 ≤ t < T, n ∈ [[0, N − 1]], (2)
T
I. I NTRODUCTION where N is the number of subcarriers, and T is the DCT-OFDM
The need for spectrally and energy efficient modulation symbol duration. Orthogonality is satisfied∫ T for a minimum
schemes that will perform well in wideband channels and at intercarrier distance of δF = 1/(2T ): 0 σn (t)σn′ (t)dt = 0
the cell edge in future cellular systems has spurred on devel- if n ̸= n′ , with the integral evaluating to 1 otherwise.
opments in so-called index modulation (IM) schemes [1]–[3]. Let W (Hz) be the available channel bandwidth, and T = T
These techniques, when applied to OFDM waveforms, yield be the symbol duration of both schemes. W can be occupied
the ability to encode information in an index set corresponding by N OFDM subcarriers spaced by ∆F or by N DCT-OFDM
to active subcarriers as well as in the amplitudes and phases subcarriers spaced by δF , so that ∆F = 2δF , and N = 2N .
of those subcarriers. More recently, generalizations of OFDM- For DCT-OFDM, the bandwidth resources can be divided into
IM have been proposed in an effort to yield higher spectral 2N narrow subchannels instead of N compared with OFDM,
efficiencies [4] and better system performance [5]. Furthermore, while maintaining the perfect reconstruction condition.
OFDM-IM variants have been proposed as possible downlink The transmitted (DCT-)OFDM signals are given by
solutions for cell-edge communication in 5G networks [6], [7]. N −1
1 ∑
Instead of using the discrete Fourier transform (DFT), as xOFDM (t) = √ Cn ej2πnt/T (3)
is typical for OFDM, a set of (co)sinusoidal functions has T n=0
been proposed for multicarrier data transmission along with ∑
N −1
a discrete cosine transform (DCT) implementation [8], [9]. xDCT-OFDM (t) = Rn βn cos(πnt/T ), (4)
The resulting system, known as DCT-OFDM, requires only n=0
half the minimum subcarrier spacing needed for OFDM1 , thus √ √
where βn = 1/ T if n = 0 and βn = 2/T otherwise. Cn
doubling the number of subcarriers within the same total
(Rn ) denotes an input symbol from a complex (real) M-ary
bandwidth. In this paper, we show that DCT-OFDM with
(M -ary) constellation mapping, and modulated by subcarrier
IM is a promising technique for enhancing spectral efficiency
index n. The associated
√ alphabet set is denoted by SM (SM ).
compared with OFDM-IM. We describe the proposed scheme
We assume M = M [9] since DCT-OFDM is a real signal.
in Section II. Spectral efficiency is investigated in Section III,
while Section IV contains results on the bit-error rate (BER)
performance of the new method as well as relevant benchmarks. B. DCT-OFDM-IM
Conclusions are drawn in Section V. Let a stream of B input information bits be mapped to a
II. S YSTEM D ESCRIPTION AND B ENCHMARKS DCT-OFDM-IM symbol of duration T . The B bits are split
into G groups, each of P bits (B = P G). Each group of P
A. DCT-OFDM
bits is mapped to a sub-block of length N0 subcarriers. The total
In conventional OFDM, data is modulated using the complex number of subcarriers is N , such that N = GN0 . Each sub-
basis of Fourier exponential functions defined as block contains N0 subcarriers spaced by ∆F /2. For each group
1 g, only K0 subcarriers out of N0 are active. The total number
ϕn (t) = √ ej2πn∆F t , 0 ≤ t < T , n ∈ [[0, N − 1]], (1)
T of active subcarriers is K = GK0 per B-bit transmission. A
diagram of a DCT-OFDM-IM transmitter is shown in Fig. 1.
M. Chafii is with CentraleSupélec/IETR, 35576 Cesson-Sévigné, France (e-
mail: marwa.chafii@supelec.fr). We now briefly describe each transmitter block.
J. P. Coon, and D. A. Hedges are with Department of Engineering Sci- Bit splitter: P bits of each group are divided into two sub-
ence, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK (e-mail: groups⌊with P1 and
(N0 )⌋ 2
P bits each, such that P = P1 + P2 , with
justin.coon@eng.ox.ac.uk, dene.hedges@wadh.ox.ac.uk)
1 We refer to DFT-based OFDM schemes as “OFDM” throughout the paper. P1 = log2 K 0
denoting the number of bits to be mapped
 2

III. S PECTRAL E FFICIENCY A NALYSIS

The key benefit that DCT-OFDM-IM offers relative to the
aforementioned benchmarks is a higher spectral efficiency,
which is defined as the rate (in bits per second) of communica-
tion in a given bandwidth. To explore this benefit, we assume
here that DCT-OFDM, OFDM-IM, OFDM-IQ-IM are allocated
the same channel bandwidth W and have the same symbol
period T . Moreover, we assume that the number of groups
considered in DCT-OFDM-IM is equal to that considered in
OFDM-IM and OFDM-IQ-IM, i.e., G = G, which gives us
Fig. 1: DCT-OFDM-IM transmitter. N = 2N and N0 = 2N0 . Since the grouping is related to
the detector complexity, this assumption is justified by the fact
that for each group in OFDM-IQ-IM, the complexity is related
to an index set of active tones and P2 = K0 log2 (M ) denoting
to the detection of 2N0 real symbols (N0 for in-phase symbols
the number of bits mapped to M -ary constellation symbols.
and N0 for quadrature symbols), which is similar to the
Index selector: By applying a selection procedure complexity needed for one group in DCT-OFDM-IM of length
such as a look-up table or combinatorial mapping2 , the 2N0 . Moreover, OFDM-IM employs a complex constellation,
P1 bits for each group g are mapped into an index set and thus the detector complexity in this case is comparable to
Ag = {ag,1 , ag,2 , . . . , ag,K0 } of K0 active subcarriers. that of DCT-OFDM-IM and OFDM-IQ-IM under the constant
Constellation mapper: A real M -ary constellation is used group assumption noted above.
for DCT-OFDM-IM (e.g., M -ary PAM).
Generation of IDCT input vector: Based on the index A. Spectral Efficiency for Finite Numbers of Subcarriers
selector and mapper outputs, a vector V = [V0 , V1 , . . . , VN ] of
K0 (K0IQ resp.) tones are selected out of N0 for OFDM-IM
length N is created as follows:
 (OFDM-IQ-IM resp.), and log2 (M) data bits are assigned to
R , if n ∈ ∪ A
 G each selected tone. Knowing that the bandwidth W = ∆F N ,
n g we can calculate the spectral efficiency for OFDM-IM and
Vn = g=1 (5)

0, OFDM-IQ-IM to be
otherwise. (⌊ ( )⌋ )
G N0
IDCT modulation: consists in applying an IDCT of length ξOFDM-IM = log2 + K0 log2 (M) (7)
N K
N at the transmitter side (a DCT is performed for the demod- ( ⌊ ( 0 )⌋ )
G N0
ulation at the receiver side). The transmitted DCT-OFDM-IM ξOFDM-IQ-IM = 2 log2 + K0IQ log2 (M) . (8)
N K0IQ
signal is therefore expressed as
Similarly, one can easily calculate the spectral efficiency of
∑
N −1
DCT-OFDM-IM to be
xDCT-OFDM-IM (t) = Vn βn cos(πnt/T ). (6) (⌊ ( )⌋ )
G N0 K0
n=0 ξDCT-OFDM-IM = log2 + log2 (M) . (9)
N K0 2
C. Other OFDM-IM Variants B. Maximum Spectral Efficiency for Many Subcarriers
Conventional OFDM-IM employs a nearly identical principle In order to provide insight into the behaviour of the spectral
to that of DCT-OFDM-IM, but where the complex field is used efficiency, we explore the asymptotic regime (i.e., large number
for modulation (i.e., the constellation mapper supports M-ary of subcarriers) to characterize the number of active subcarriers
QAM). We define the following notation for OFDM-IM: B is that maximizes the spectral efficiency. To make progress with
the number of input information bits, G is the number of groups, the analysis, we ignore the floor function ⌊·⌋ in (7)-(9). Let
each of length N0 subcarriers and having K0 active subcarriers. K0 = αN0 , K0 = βN0 , and K0IQ = γN0 , where α, β,
K = GK0 is the total number of active subcarriers. and γ are parameters in the interval [0, 1]. By using Stirling’s
OFDM-IQ-IM is a variant of OFDM-IM that was proposed approximation to simplify the binomial coefficients, it is easy
in [4]. Its principle consists of equally splitting the input bits to find asymptotic expressions for the spectral efficiencies of
dedicated for IM into two parts: one for in-phase components the three IM schemes. Taking derivatives of these expressions
and one for quadrature components. The M-ary constellation with respect to α, β, and γ, and letting N → ∞, one can show
(M-ary QAM) symbol is then constructed by combining the that the spectral efficiency maximizing values are given by
output of these two index modulations. Let G be the number √
M M
of groups considered in OFDM-IQ-IM, each of length N0 α= and β = γ = √ . (10)
subcarriers and having K0IQ active subcarriers. M+1 M+1
These maximizing proportions of active subcarriers are surpris-
2 See [3] for more information on these mapping techniques. ingly accurate for finite numbers of subcarriers (cf. Fig. 2).
 3

3 100
OFDM, 1.77 bits/s/Hz
DCT, 1.77 bits/s/Hz
2.5 10-1 OFDM-IM, 1.77 bits/s/Hz
OFDM-IQ-IM, 2.22 bits/s/Hz
Spectral Efficiency

DCT-OFDM-IM, 2.44 bits/s/Hz

2
10-2

BER
1.5
10-3
10-2
1
10-4
DCT−OFDM−IM 20 22 24
0.5 OFDM−IM
OFDM−IQ−IM
0 5 10 15 20 25 30 35 40 45
0 Eb/N0 in dB
0 5 10 15 20 25 30 35
Number of Active Subcarriers Fig. 3: BER performance comparison (M = 2, M = 4).
Fig. 2: Spectral efficiency vs. number of active subcarriers.
100
OFDM, 3.55 bits/s/Hz
DCT, 3.55 bits/s/Hz
C. Spectral Efficiency vs. Number of Active Subcarriers 10-1
OFDM-IM, 3.44 bits/s/Hz
OFDM-IQ-IM, 3.55 bits/s/Hz
Fig. 2 shows the spectral efficiency of DCT-OFDM-IM for DCT-OFDM-IM, 3.77 bits/s/Hz
various numbers of active tones K0 ∈ [[1, 32]], and that of 10-2
OFDM-IM and OFDM-IQ-IM for K0 , K0IQ ∈ [[1, 16]]. The
comparison is performed for M = 4, N = 16, and G = 1. An BER
10-3
initial observation of the three curves illustrates the potential
for spectral efficiency improvements offered by DCT-OFDM- 10-4 10-2
IM. It is also observed from Fig. 2 that the maximum spectral 18 20 22 24 26
efficiency of OFDM-IQ-IM is similar to that of DCT-OFDM-
IM in this example. However, it should be noted that this is an 0 5 10 15 20 25 30 35 40 45
Eb/N0 in dB
artefact that results when we consider a single group. Dividing
the input information bits into G > 1 groups will result in Fig. 4: BER performance comparison (M = 4, M = 16).
reduced peak spectral efficiency for the three IM schemes
discussed here; however, this approach must be employed to
reduce implementation complexity, which can be prohibitively The gain is less significant for higher order constellations.
high for large N due to the combinatorial nature of IM Interestingly, although OFDM-IQ-IM exhibits good spectral
techniques. As we increase G, we will see below that the peak efficiency gains relative to most other benchmarks, the proposed
spectral efficiency of DCT-OFDM-IM is consistently higher DCT-OFDM-IM scheme has the highest spectral efficiency of
than that of the OFDM-IQ-IM scheme. all schemes, even for the case where the DFT-based methods
exploit 16-QAM. These observations strongly motivate further
study of DCT-OFDM-IM variants.
D. Spectral Efficiency vs. Number of Groups
To compare the spectral efficiencies of the IM schemes IV. BER PERFORMANCE
against non-IM benchmarks, we let α, β, and γ satisfy (10) The BER performance of multicarrier schemes with IM
and set N = 128. We define the spectral efficiency increase of depends on the detection method employed. Several schemes
technique x relative to y as for index set detection have been reported in the literature [3].
ξx − ξy Here, for a fair comparison of the studied schemes, a frequency
ρx/y = . (11) domain equalizer using zero forcing (ZF) inversion of the chan-
ξy
nel is performed for the received signal (additional FFT/IFFT
In Tab. I, a spectral efficiency comparison between different operations are required for DCT-OFDM-IM scheme), and a
modulation schemes is presented3 . We observe that the spectral minimum distance detection is used to recover the index set
efficiency of the different IM schemes is markedly increased and the transmitted symbols.
relative to (DCT-)OFDM for low-modulation orders (more than Here, the BER performance of DCT-OFDM-IM is shown
55% for (M = 2, G = 1) when using DCT-OFDM-IM). in a frequency selective fading channel and compared with
3 Here, we ignore the guard interval. When considering the same length of
OFDM-IM and OFDM-IQ-IM using the following simulation
the guard interval for all schemes, the absolute spectral efficiency for each will parameters: N = 128, G = 8, ∆F = 15 kHz and α, β, γ satisfy
decrease, but the relative spectral efficiency will remain the same. (10). The channel used in this comparison is the extended
 4

Table I: Spectral efficiency comparison for N = 128.

Modulation Order M = 2, M = 4 M = 4, M = 16
number of groups G 1 2 4 8 1 2 4 8
ξDCT-OFDM-IM 3.1 3.06 2.93 2.75 4.57 4.53 4.43 4.25
ξOFDM-IM 2.26 2.21 2.12 2 4.04 4 4 3.87
ξOFDM-IQ-IM 3.06 2.93 2.75 2.5 4.53 4.43 4.25 4
ξOFDM = ξDCT-OFDM 2 4
ρDCT-OFDM-IM/OFDM-IM 37.24% 38.02% 38.23% 37.5% 13.13% 13.28% 10.94% 9.68%
ρDCT-OFDM-IM/OFDM-IQ-IM 1.53% 4.25% 6.81 % 10% 1.03% 2.11% 4.41 % 6.25%
ρDCT-OFDM-IM /(DCT-OFDM, OFDM) 55.46 % 53.12 % 46.87 % 37.50% 14.45 % 13.28% 10.93 % 6.25%
ρOFDM-IM /(DCT-OFDM, OFDM) 13.28 % 10.93 % 6.25 % 0% 1.17 % 0% 0% -3.13%
ρOFDM-IQ-IM /(DCT-OFDM, OFDM) 53.12 % 46.87 % 37.5 % 25 % 13.28 % 10.94 % 6.25 % 0%

IM and OFDM-IQ-IM are not particularly useful when high-

100
order constellations are applied since they do not achieve any
significant gain, in terms of spectral efficiency or performance,
10-1 relative to OFDM. Moreover, the DCT-OFDM-IM detection
10-2 scheme is linear, not optimal. Also, no forward error correction
10-2 16 18 20 22 is employed, and optimal index sets are not chosen. Hence,
DCT-OFDM-IM, K=7, 2-PAM
BER

OFDM-IQ-IM, K=5, 4-QAM there is a lot of scope for improvement. These observations
DCT-OFDM-IM, K=4, 2-PAM
10-3 OFDM-IM, K=4, 4-QAM
again motivate further study on DCT-OFDM-IM, particularly
DCT-OFDM-IM, K=10, 4-PAM with respect to enhanced receiver design.
OFDM-IQ-IM, K=6, 16-QAM
DCT-OFDM-IM, K=9, 4-PAM
10-4
OFDM-IM, K=7, 16-QAM
OFDM-IM, K=6, 4-QAM V. C ONCLUSION
OFDM, 4-QAM
In this paper, we proposed DCT-OFDM with IM, and showed
0 5 10 15 20 25 30 35 40
Eb/N0 in dB that it can achieve significant spectral efficiency gains relative to
relevant benchmarks. In fading wireless channels, the new tech-
Fig. 5: BER performance comparison for the same spectral nique yields comparable BER performance to these benchmarks
efficiency. when minimum distance detection is employed along with ZF
channel equalization. It is hoped that the potential of DCT-
OFDM-IM, which was highlighted herein, will motivate further
typical urban (ETU) model of the LTE standard [10]. A research on DCT-based IM techniques, particularly with respect
cyclic prefix is added to the transmitted signals of the studied to receiver design and operation with high-order constellations,
schemes. A comparison of the BERs for the three schemes with a view to enhancing next generation systems.
is shown in Fig. 3 for (M = 2, M = 4) and in Fig. 4 for
(M = 4, M = 16). For low modulation order (Fig. 3), all R EFERENCES
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