798 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 5, MARCH 1, 2010
Novel Power Efficient Optical OFDM Based
on Hartley Transform for Intensity-Modulated
Direct-Detection Systems
Michela Svaluto Moreolo, Raül Muñoz, and Gabriel Junyent
Abstract—We present a novel optical orthogonal frequency
division multiplexing (O-OFDM) scheme, suitable for inten-
sity-modulated direct-detection systems, where the modula-
tion/demodulation processing takes advantage of the fast Hartley
transform algorithm. Due to the properties of the discrete Hartley
transform (DHT), the conventional transmission scheme can be
streamlined. We demonstrate that asymmetrically clipping (AC)
technique can also be applied to DHT-based OFDM; the signal
can be transmitted without the need of a DC bias, resulting in a
power-efficient system, not affected by clipping noise. Hermitian
symmetry is not required for the input signal. Therefore, this
technique supports the double of input symbols compared to
both AC and DC-biased O-OFDM, based on standard Fourier
processing. The analysis in an additive white Gaussian noise
channel shows that the same performance can be achieved by
replacing 4, 16, and 64 QAM (quadrature-amplitude modulation)
AC optical-OFDM with a simpler system based on DHT, using
binary phase-shift keying (BPSK), 4 and 8 PAM (pulse-amplitude
modulation), respectively.
Index Terms—Asymmetrically clipped optical (ACO) OFDM,
discrete Hartley transform (DHT), intensity-modulated direct
detection (IM/DD), optical communication, orthogonal fre-
quency-division multiplexing (OFDM).
I. INTRODUCTION
O
RTHOGONAL frequency-division multiplexing
(OFDM) is a very promising technique, borrowed
from broadband wireless and radio communications, for future
high-speed large-capacity optical networks, and recently it
also represents a novel candidate for optical access networks.
As multicarrier transmission technique, OFDM allows trans-
mission of the signal over several lower-rate subchannels. The
subcarriers are orthogonal to each other, and their spectra are al-
lowed to overlap. This results in a very high spectral efficiency.
Therefore, the use of OFDM in optical networks meets the
twofold requirement of mitigating transmission impairments
Manuscript received July 31, 2009; revised October 09, 2009 and December
07, 2009. First published January 26, 2010; current version published March
05, 2010. This work was supported by the Spanish Ministry of Science and In-
novation under the Project DORADO (TEC2009-07995) and developed within
the Building the Future Optical Network in Europe, a Network of Excellence
funded by the European Commission through the 7th Information and Commu-
nication Technologies-Framework Programme.
M. Svaluto Moreolo and R. Muñoz are with the Centre Tecnològic
de Telecomunicacions de Catalunya, 08860 Barcelona, Spain (e-mail:
michela.svaluto@cttc.es).
G. Junyent is with the Universitat Politècnica de Catalunya, 08034 Barcelona,
Spain and also with the Centre Tecnologic de Telecomunicacions de Catalunya,
08860 Barcelona, Spain.
Digital Object Identifier 10.1109/JLT.2010.2040580
and providing high-data rate transmission. Its resilience to
dispersion impairments can reduce the conventional per-span
compensation and offer an alternative electronic dispersion
compensation method to traditional optical pre- and postcom-
pensation techniques [1], [2]. The high tolerance to chromatic
dispersion and polarization mode dispersion allows extending
the attainable distance before significant distortion to thousands
of kilometers [2]–[9].
The signal processing in the OFDM transmitter/receiver takes
advantage of the efficient algorithm of fast Fourier transform
(FFT) to implement OFDM modulation/demodulation. On one
side, it enables the use of the mature technology and capabilities
of DSP; on the other side, it gives a complex and bipolar signal
that must be transmitted on an optical link.
To solve this critical issue, alternative solutions have been
proposed. Direct detection (DD) and coherent schemes can be
used, trading simplicity against increased sensitivity [5]–[13].
DD systems are simpler and can be implemented by using com-
mercial components [6]; no laser is required as local oscillator
at the receiver. The simplicity of this cost-effective solution is at
expenses of the spectral efficiency, and its effectiveness depends
on the system linearity [6], [10]. Coherent detection allows to
directly implement the standard RF OFDM scheme, but it is
more costly and sensitive to phase noise and frequency offset;
the RF-to-optical up-conversion and the optical-to-RF down-
conversion require narrow linewidth lasers. In this paper, DD
is considered. In particular, we provide an alternative OFDM
processing for intensity-modulated (IM) optical systems. An
OFDM signal to be transmitted on an IM system must be con-
verted into real and positive. To generate real OFDM symbols,
the input signal mapped into a complex constellation is forced to
have Hermitian symmetry. Adding a DC bias to real signals is an
inefficient solution in terms of optical power, to obtain unipolar
signals. Usually, the bias value is at least two times the stan-
dard deviation of the signal. In presence of high negative picks,
the noise due to the clipping at zero level affects the transmis-
sion. In 2006, Armstrong and Lowery proposed asymmetrically
clipping (AC) as a power-efficient technique to transmit OFDM
signals on optical systems, without clipping noise [14]. Only
the odd subcarriers are modulated by a signal with Hermitian
symmetry.
An alternative optical OFDM scheme can be based on a real
trigonometric transform to directly deal with real signal. In this
paper, we propose a novel optical OFDM scheme based on the
discrete Hartley transform (DHT) for power efficient transmis-
sion in IM/DD systems.
0733-8724/$26.00 © 2010 IEEE
SVALUTO MOREOLO et al.: NOVEL POWER EFFICIENT OPTICAL OFDM 799
Fig. 1. Block diagram of an IM/DD system, using optical OFDM based on Hartley transform. The input sequence is mapped into a real constellation and OFDM
modulated by the
N-order IFHT. The demodulation uses the direct FHT.
Hartley transform is particularly attractive for the processing
of real signals. The direct and inverse transforms are identical,
and the Hartley transform of a real signal is real. Fourier
transform always implies a complex processing and the phase
carries fundamental information, while Hartley transform is a
real trigonometric transform. Furthermore, the real and imag-
inary parts of the discrete Fourier transform (DFT) coincide
with the even and the negative odd parts of the DHT, respec-
tively: the transform kernels only differ for the imaginary unit
[15]. DFT is used to perform the OFDM modulation, because
it can be seen as a bank of modulators, whose narrowband
channels have mutually orthogonal subcarriers. Similarly,
the mirror-symmetric sub-bands of DHT ensure subcarriers
orthogonality, and the spectral behavior enable to carry the
data symbols for the parallel processing [16]. Therefore, fast
Hartley transform (FHT) can replace FFT algorithm to furnish
an alternative OFDM scheme. If the input symbols are real,
the inverse FHT (IFHT) gives real OFDM signals. When the
OFDM signals are real valued, the multicarrier transmission
technique is considered a special case of OFDM. No in-phase
and quadrature modulation onto an RF carrier is required and
in the literature it is referred as discrete multi-tone modulation
(DMT) [17], [18]. Because of the DHT real processing, a
simpler transmission system can be achieved, as demonstrated
for high-speed wireless communications [19]. We adapt the
DHT-based solution proposed in [19] to optical systems. To
the best of our knowledge, no demonstration for IM/DD op-
tical OFDM systems based on Hartley transform exists in the
literature.
The paper is organized as follows: in Section II, we describe
the optical OFDM system based on DHT. In Section III, we
demonstrate that the AC technique can be applied to this alter-
native modulation scheme, without Hermitian symmetry con-
strain for the input signal. In Section IV, we present the perfor-
mance of DHT-based AC optical OFDM (ACO-OFDM) in an
additive white Gaussian noise (AWGN) channel. We compare it
with DC-biased DHT-based OFDM and analyze different real
constellations. We also compare our results to the performance
of DC-biased and ACO-OFDM systems based on Fourier trans-
form. Finally, in Section V conclusions are drawn.
II. O
PTICAL
OFDM SYSTEM
BASED ON
DHT
The block diagram of the DHT-based optical OFDM system
is depicted in Fig. 1. The IFHT and FHT are used in place of the
inverse FFT (IFFT) and FFT, to perform the OFDM modulation
and demodulation, respectively. According to the definition of
DHT [15], the OFDM symbol is given by
(1)
where
indicates the symbol sequence and represents
the number of symbols processed in parallel. The DHT kernel
is real, and it can also be indicated as [15]
(2)
If a real constellation [e.g., binary phase-shift keying (BPSK),
M-PAM (pulse-amplitude modulation)] is used for the subcar-
riers modulation, the OFDM symbol
is real. If the input
symbols are complex, as the 16 symbols 4 QAM (quadrature-
amplitude modulation) of Fig. 2,
is complex. Fig. 2 shows
the data blocks at the input and output of a 16-order IFHT, com-
pared to a 16-order IFFT. Both the discrete transforms have a
nonzero imaginary part. The IFFT has been evaluated so that the
forward and inverse transform has the same normalizing factor
(3)
In order to obtain real-valued IFFT, the input vector must have
Hermitian symmetry. Fig. 3(a) shows the real-valued OFDM
symbol at the output of a 32-order IFFT. The information se-
quence is mapped into 4 QAM symbols; the second-half of the
input vector is given by their complex conjugate values [indi-
cated in Fig. 3(a) as QAM
]. The dc and Nyquist frequencies
that are
and , respectively, are set to zero.
800 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 5, MARCH 1, 2010
Fig. 2. From the top, input sequence of 16 symbols 4 QAM; outputs of its
discrete 16-order IFFT and 16-order IFHT.
For a given OFDM signal bandwidth, when the number of
subcarriers is doubled the carrier spacing decreases accord-
ingly. On the other hand, for a given bit rate, by maintaining the
same carrier spacing, if only half of the available subcarriers
are used to carry data, the required bandwidth to transmit the
same data signal increases accordingly. For a given constel-
lation size, the data rate that can be supported by the double
of subcarriers is greater. In DMT systems, only half of the
IFFT points are used to process the information data symbols
(independent complex values); the second-half is required to
process the complex conjugate vector, due to the Hermitian
symmetry constrain [18]. When DHT is used, Hermitian sym-
metry is not required: if the input vector is real, the IFHT is
real-valued and the number of subcarriers carrying information
symbols (independent real-valued values) coincides with the
DHT points, as shown in Fig. 3(b). So that to transmit the same
data signal, a lower constellation size (BPSK) is required. In the
remainder of the paper, we compare O-OFDM system based
on IFHT and real-valued IFFT of order
able to transmit the
same data signal.
As shown in Fig. 1, by using the DHT, the input data stream is
serial-to-parallel converted and mapped into a real constellation
to generate a real OFDM signal. After parallel-to-serial conver-
sion, the real OFDM signal can be processed by one single dig-
ital-to-analog converter (DAC). The analog signal is real, but it
is still bipolar and must be converted into unipolar to be IM. In
the following section, we demonstrate that both AC and DC-bi-
ased solutions, adopted in standard optical OFDM, are possible
for DHT-based optical OFDM. Adding a DC-bias requires more
power and residual clipping noise can affect the transmission.
By adopting AC technique the average optical power is reduced
without affecting the transmission.
At the receiver, after DD, the signal is converted to digital by
one single analog-to-digital converter (ADC) and the sequence
is recovered by DHT processing and demodulating the signal.
As in the standard implementation of OFDM, a cyclic prefix
(CP) can be used to mitigate intersymbol interference and in-
tercarrier interference [19]. The choice of prefix length should
take into account that this overhead reduces the supported data
rate, to cope with the delay introduced by the channel. Gener-
ally, the CP is a small fraction of the OFDM symbol, but to be
effective should be longer than the delay spread. Mandyam in
[20] demonstrates that sinusoidal transforms with a symmetric
extension may outperform DFT, when delay spread due to the
channel is longer than CP duration. Merched extends the study
to the Hartley domain [21]. However, due to the double-side
band optical spectrum, ACO-OFDM does not enable to com-
pensate for significant fiber chromatic dispersion [22]. In fact,
the intensity modulation generates an OFDM signal on both
sides of the optical carrier frequency. DD is more robust to dis-
persion impairments when combined with optical single-side
band modulation than when direct IM is applied [5].
A. Computational Complexity
We compare the FHT algorithms with optimized algorithms
to evaluate the FFT of real-valued sequences. When FFT
exploits the Hermitian symmetry property of the transform,
FHT algorithms require about the same number of multipli-
cations but more additions [23]–[25]. Nevertheless, additional
resources must be used for calculating the complex conjugate
vector to deal with real-valued FFT. In the case of radix-2
algorithm, reported in [25], for both the decimation-in-time
and decimation-in-frequency, the number of multiplications
required by the DHT is
and the number
of additions is
, with the trans-
form order. The FHT-based algorithm has the same number
of multiplications and
more additions than the cor-
responding FFT algorithm optimized for a real input vector.
Similarly, the number of additions required by FHT slightly
exceeds the ones required by FFT of a real-valued sequence
for radix-4, split radix, prime factor, and Winograd transform
algorithms, as demonstrated in 1985 by Sorensen et al. [25].
In 1986, Duhamel and Vetterli proposed the fastest algorithm
implementing the DHT. Its improved version requires only
two more additions than the FFT algorithms for real-valued
signal with minimum arithmetic complexity [26], [27]. This
improvement increases the computational speed in DSP de-
vices. The minimum number of multiplications for both FFT
and FHT is
, while the number
of additions is
and
, respectively. The number
of total operations required for 32-order transforms is indicated
in Fig. 3, and it is also shown where additional computational
resources are needed.
Moreover, although the FFT and IFFT are very similar, the
same FHT routine can be applied to calculate the forward and
inverse transforms, because DHT is self-inverse. Therefore, the
same DSP device can be used for modulation and demodulation.
To change from the forward to the IFFT transform, an additional
control is required to reverse the sign of the imaginary unit in
the transform kernel.
SVALUTO MOREOLO et al.: NOVEL POWER EFFICIENT OPTICAL OFDM 801
Fig. 3. (a) Real-valued discrete outputs of a 32-order IFFT. The input sequence has Hermitian symmetry, and it is given by the half-length vector of 4 QAM
symbols and the corresponding complex conjugate vector. The dc and Nyquist frequencies are set to zero. (b) Real-valued discrete DHT-based OFDM symbol,
evaluated as the 32-order IFHT of a BPSK vector of length 32. The number of multiplications (P) and additions (A) reported in the figure have been evaluated
according to the FFT and FHT algorithms with the minimum arithmetic complexity [27].
III. ASYMMETRICALLY CLIPPED DHT-BASED O-OFDM
In this section, we demonstrate that AC technique can also be
applied to OFDM signals generated by using the Hartley trans-
form. With a suitable choice of the subcarriers to be modulated,
the DHT-based OFDM signal can be clipped at zero level and
correctly recovered without clipping noise.
In fact, as for DFT, also for DHT it is easy to demonstrate that
for
odd
(4)
Therefore, if only the odd subcarriers are nonzero, the DHT-
OFDM symbol is redundant, as shown in Fig. 4(a), and can be
clipped without losing information. The nonredundant OFDM
symbol is shown in Fig. 4(b); it is obtained by clipping the IDHT
evaluated for a BPSK sequence of length
. The sequence
can be recovered by simply performing the Hartley trans-
form of the IDHT-OFDM symbol
(5)
By making explicit the symbol elements in the summation, it
can be rewritten in the following form
(6)
Fig. 4. (a) BPSK signal OFDM modulated with a 64-order DHT, in digital
h
(
k
)
and analog
h
(
t
)
form; (b) corresponding clipped DHT-OFDM symbol.
Stated (4) and according to the assumption that only the odd
subcarriers are nonzero
(7)
where the summation in (6) has been split in two, depending on
the sign of
. For clipped signal
(8)
802 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 5, MARCH 1, 2010
Fig. 5. Analog signal representing 50 OFDM symbols mapped into BPSK con-
stellation and modulated with 64-order DHT (a) DC-biased and (b) clipped.
only one of the two terms of (6) is nonzero, depending on the
sign of
; therefore, the recovered symbol sequence will be
(9)
This means that, as in ACO-OFDM based on FFT [14], the
symbol sequence can be recovered from the odd subcarriers,
and the constellation points have the half of the original values.
All the clipping noise falls into the even subcarriers that can be
easily discarded.
As clipping is a memoryless nonlinearity, the attenuation of
the input sequence and the presence of additive noise directly
follow the Bussgang’s theorem. It can be applied because
can be assumed to have a Gaussian distribution for .
Therefore, the same analysis of clipping reported in [14] can be
considered valid also for the DHT-OFDM signal.
A. Simulation Results
To prove the correctness of the theoretical approach, we sim-
ulate the transmission of 50 OFDM symbols in a back-to-back
system based on DHT processing. The input sequence is
mapped into BPSK constellation points and modulated by
a
DHT. Only the odd subcarriers are modulated;
therefore, in an ACO-OFDM system based on N-order DHT,
only
data symbols can be transmitted. However, as shown
in Fig. 5, the optical power, which is proportional to the elec-
trical OFDM signal, is substantially reduced, compared to the
DC-biased signal, for which all the subcarriers can be mod-
ulated. The bias value should be at least two times the signal
standard deviation. The choice of a fixed bias value trades
power efficiency and additional noise. In fact, in DC-biased
optical systems the reduction of clipping noise is at expenses
of power consumption.
At the receiver side, the DHT of the clipped signal is per-
formed to recover the BPSK sequence. Fig. 6 shows the received
constellation: the symbols, with half of the original value, can
be recovered from the odd subcarriers, and the even subcarriers
represent the noisy component of the signal.
B. Comparison Between DHT-Based and DFT-Based
ACO-OFDM Transmission Systems
An OFDM system based on
-order DHT can transmit
independent real-valued symbols. In DFT-based OFDM, the
Fig. 6. Received constellation in a DHT-based OFDM system, using AC. The
recovered BPSK signal with the half of the original values is carried by the odd
subcarriers and the clipping noise falls into the even subcarriers.
number of independent symbols that can be transmitted with
a
-order DFT is reduced to , due to the Hermitian sym-
metry constrain on the input sequence.
Consider transmitting a certain bit rate over
subcar-
riers. ACO-OFDM requires half the carriers to be unused. At
each parallel processing, 32 bits are mapped into 4 QAM con-
stellation and transmitted over
odd subcarriers. The
remaining 16 are used to carry the complex conjugate vector.
By using Hartley transform, all the
carriers carry
information symbols. The input bits are mapped into a simpler
BPSK constellation to transmit the same bit rate.
Fig. 7 shows the transmission of the same bit sequence by
using either an IFHT or an IFFT of order
and the re-
sulting real-valued OFDM signals. The 32 even-indexed subcar-
riers are set to zero in order to avoid signal distortion due to the
clipping noise. According to [27],
multiplications and
additions are required for the FHT. That is only two
more additions compared to the real-valued FFT algorithm with
the minimum arithmetic complexity.
IV. P
ERFORMANCE ANALYSIS
We analyze the performance of the proposed IM/DD optical
OFDM communication systems based on DHT, in an AWGN
channel. We add a Gaussian noise source in the electrical do-
main, after DD, to the system represented in Fig. 1. In order
to compare the performance of the proposed system, with AC
and DC-biased O-OFDM systems based on FFT presented by
other authors [28], [29], we consider the same assumptions. The
comparison that we propose is between DMT systems based on
-order DHT, and DFT able to transmit the same data signal
per OFDM symbol, i.e., the same information bit sequence per
parallel processing. We assume that the impulse response of the
optical channel is unitary, and we do not consider the CP. For
the analysis of DC-biased solution, we evaluate the performance
![Fig. 7. 16 symbols 4 QAM and 32 symbols BPSK, representing the same input sequence of 32 bits. The corresponding real-valued discrete OFDM symbols using (a) a 64-order IFFT and (b) a 64-order IFHT have been evaluated modulating only the odd subcarriers. The even subcarriers are set to zero. The second-half of IFFT odd subcarriers are modulated by the complex conjugate values of the 16 symbols 4 QAM. The number of multiplications (P) and additions (A) reported in the figure have been evaluated according to [27].](/figures/fig-7-16-symbols-4-qam-and-32-symbols-bpsk-representing-the-20w23yne.webp)
