Performance analysis of optical OFDM transmission systems using PAPR mitigation techniques and alternative transforms.
by
Laia Nadal Reixats Master thesis director
Dr. Michela Svaluto Moreolo Master thesis assistant director
Dr. Gabriel Junyent Giralt
April 2012 A thesis submitted to the Departament of Teoria del Senyal i Comunicacions of the Universitat Politècnica de Catalunya for the degree Master of Science Optical Networking Area Centre Tecnològic de Telecomunicacions de Catalunya (CTTC) Castelldefels, Barcelona
To my family and friends.
Contents
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Introduction 1.1 Optical communications . . . 1.1.1 Optical transmitters . . 1.1.2 Optical fibers . . . . . 1.1.3 Optical amplifiers . . . 1.1.4 Optical receivers . . . 1.2 Background of OFDM systems 1.3 Summary of the thesis . . . .
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Optical-Orthogonal frequency division multiplexing 2.1 Optical-Orthogonal Frequency Division Multiplexing 2.1.1 Orthogonal Frequency Division Multiplexing 2.1.2 Intensity-Modulation . . . . . . . . . . . . 2.1.3 Direct-Detection and Coherent detection . . . Transforms used in O-OFDM 3.1 Fast Fourier Transform . . 3.2 Fast Hartley Transform . . 3.3 Wavelet Transform . . . . 3.3.1 Wavelets functions
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The peak-to-average power ratio problem 4.1 Peak-to-average power ratio definition . . . . . . . 4.2 PAPR reduction techniques . . . . . . . . . . . . . 4.2.1 Selective mapping . . . . . . . . . . . . . 4.2.2 Interleaving . . . . . . . . . . . . . . . . . 4.2.3 Partial transmit sequence . . . . . . . . . . 4.2.4 PAPR reduction techniques using precoding i
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5
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Simulations results 5.1 PAPR evaluation in O-OFDM systems . . . . . . . . . . . . . . . . . . . 5.1.1 Comparison of PAPR reduction techniques applied to FFT and based O-OFDM systems . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Precoding for PAPR reduction in FHT-based O-OFDM . . . . . . 5.1.3 PAPR of OFDM systems based on the DWPT . . . . . . . . . . . 5.2 BER performance of DC biased O-OFDM systems . . . . . . . . . . . . 5.2.1 DC biased O-OFDM system modeled with AWGN channel . . .
. . . FHT . . . . . . . . . . . . . . .
35 . 35 . . . . .
36 39 40 42 42
Conclusions and future work 45 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
ii
List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10
Optical OFDM communication system . . . . . . . . . . . . . . . . . . . . . . 2 Three basic processes of the interaction of light with matter. . . . . . . . . . . . 3 Fabry Perot semiconductor laser structure. . . . . . . . . . . . . . . . . . . . . . 4 MZM external modulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Transfer function for the optical intensity against the drive voltage. . . . . . . . . 6 (a) Refractive index profile for step-index fiber and (b) for graded-index fiber. . . 6 Light confinement in step-index fibers through the total internal reflection. . . . . 7 Light confinement in graded-index fibers through the total internal reflection. . . 8 Optical amplifier principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Possible applications of optical amplifier: (a) In-line amplifiers, (b) pre-amplifiers and (c) Booster or power amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.11 A semiconductor slab used as a photodetector. . . . . . . . . . . . . . . . . . . . 11 1.12 Historical evolution of OFDM [1] . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1 2.2 2.3 2.4 2.5 2.6 2.7
2.8
2.9 3.1 3.2 3.3
OFDM spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time domain signal of one subcarrier for two OFDM frames using CP . . . . . (a) 16-QAM and (b) 16-PSK constellations. . . . . . . . . . . . . . . . . . . . Diagram block of the OFDM transmitter based on the FFT . . . . . . . . . . . Diagram block of the OFDM receiver based on the FFT . . . . . . . . . . . . . Diagram block of a IM/DD system . . . . . . . . . . . . . . . . . . . . . . . . Real-valued OFDM time domain signal with (a) only odd subcarriers modulated and (b) clipped to zero level (ACO-OFDM) and with (c) all subcarriers modulated and (d) clipped to zero level (DCO-OFDM) after adding a bias. . . . . . . Diagram block of a IM/DD system using four different transmission architectures: (a) RF conversion, (b) DMT modulation, (c) FHT tranform and (d) Hilbert transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of a Coherent Optical-OFDM (CO-OFDM) system. . . . . . . .
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14 14 15 15 16 17
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The radix-4 butterfly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Schematic of DMT modulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Synthesis with IDWPT (modulation) and analysis with DWPT (demodulation) for OFDM systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 iii
4.1 4.2
Diagram block of SLM PAPR reduction technique. . . . . . . . . . . . . . . . . 31 Diagram block of PTS PAPR reduction technique. . . . . . . . . . . . . . . . . . 32
5.1
CCDF vs. P AP R0 for (a) FHT-based OFDM signals (using BPSK and N = 64) with SLM, interleaving, PTS, random PTS and without any PAPR reduction technique and for (b) FFT-based OFDM signals (using 4-QAM and N = 64) with and without SLM technique. . . . . . . . . . . . . . . . . . . . . . . . . (a) PAPR as a function of the number of subcarriers at a CCDF of 0.1% and (b) CCDF vs. P AP R0 for different oversampling factors for FHT-based O-OFDM system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CCDF vs. P AP R0 for FHT-based OFDM signals (using 4PAM and 16QAM respectively and N=256) with SLM, interleaving, PTS and random PTS schemes, and without any PAPR reduction technique using 2 and 4 FHT blocks. . . . . . CCDF vs. P AP R0 for FHT-based OFDM signals (using 4PAM and N=256 subcarriers) with SLM and HT or DCT varying the number of transform blocks and without any PAPR reduction technique. . . . . . . . . . . . . . . . . . . . Temporal OFDM signal based on (a) DWPT with Haar wavelet function and (b) FFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PAPR of OFDM symbols based on FFT and DWPT . . . . . . . . . . . . . . . Block diagram of an O-OFDM IM/DD system based on FHT. The signal is limited in amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BER performance at two constant values of Eb /N0 versus clipping level for 8PAM O-OFDM (N = 256) with and without SLM technique in AWGN channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance of DC biased O-OFDM based on FHT with symmetrically clipping in AWGN channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
5.3
5.4
5.5 5.6 5.7 5.8
5.9
iv
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. 40 . 40 . 41 . 42
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List of Tables 3.1
Properties of some wavelet functions. . . . . . . . . . . . . . . . . . . . . . . . 27
5.1
PAPR reduction at a CCDF of 0.1%, of different PAPR reduction techniques, varying the number of IFHT blocks in transmission for N = 256 compared to the unmodified signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 PAPR reduction at a CCDF of 0.1%, of SLM and precoding with HT and DCT PAPR reduction techniques, varying the number of IFHT blocks in transmission for N = 256 compared to the unmodified signal. . . . . . . . . . . . . . . . . . 39
5.2
v
Acknowledgements I would deeply thank my advisors, Michela Svaluto Moreolo and Gabriel Junyent Giralt for their support and help in the development of this thesis and publications on this topic. I would also like to thank all the CTTC staff. I would also give many thanks to my family, friends and boyfriend, who have always been with me. Finally many thanks to the Spanish Ministry of Science and Innovation (MICINN), which has partially supported this work through the project DORADO (TEC2009-07995) and the FPI research scholarship grant BES-2010- 031072.
vii
ACO-OFDM Asymmetrically Clipped Optical-OFDM ADC
Analog to Digital Converter
ASE
Amplified SpontanEous
AWGN
Additive White Gaussian Noise
BER
Bit Error Rate
BPSK
Binary Phase-Shift Keying
CCDF
Complementary Cumulative Density Function
CDF
Cumulative Density Function
CD
Chromatic Dispersion
CO-OFDM Coherent Optical-OFDM CP
Cyclic Prefix
CWT
Continuous Wavelet Transform
DAC
Digital Analog Converter
DCT
Discrete Cosine Transform
DMT
Discrete MultiTone
DSP
Digital Signal Processing
DD
Direct-Detection
DHT
Discrete Hartley Transform
DFT
Discrete Fourier Transform
DWT
Discrete Wavelet Transform
DWPT
Discrete Wavelet Packet Transform
EAM
ElectroAbsorption Modulator
EDFA
Erbium-Doped Fiber Amplifier
FHT
Fast Hartley transform
FFT
Fast Fourier transform
FEC
Forward Error Correction viii
GVD
Group-Velocity Dispersion
HS
Hermitian Symmetry
HT
Hadamard Transform
HDSL
High-Bit-Rate Digital Subscriber Line
ICWT
Inverse Continuous Wavelet Transform
IDFT
Inverse Discrete Fourier Transform
IDWPT Inverse Discrete Wavelet Packet Transform IM
Intensity-Modulation
ICI
InterCarrier Interference
IFHT
Inverse Fast Hartley Transform
IFFT
Inverse Fast Fourier Transform
ISI
InterSymbol Interference
MZM
Mach-Zehnder modulator
MCM
MultiCarrier Modulation
MSM
Metal-Semiconductor-Metal
NA
Numerical Aperture
O-OFDM Optical Orthogonal Frequency Division Multiplexing OFDM
Orthogonal Frequency Division Multiplexing
PAM
Pulse Amplitude-Modulation
PAPR
Peak-to-average power ratio
PS
Parallel to Serial
PMD
Polarization Mode Dispersion
PSK
Phase Shift Keying
PTS
Partial Transmit Sequence
RF
Radio Frequency
RDWPT Real Discrete Wavelet Packet Transform ix
SSB
Single Side Band
SOA
Semiconductor Optical Amplifier
SSMF
Standard Single Mode Fiber
QAM
Quadrature Amplitude Modulation
SLM
SeLective Mapping
SNR
Signal to Noise Ratio
SP
Serial to Paralel
WPT
Wavelet Packet Transform
WT
Wavelet Transform
x
Abstract Orthogonal Frequency Division Multiplexing (OFDM) has recently been introduced in optical communications because of its robustness against channel dispersion and its high spectral efficiency. OFDM is based on the Fast Fourier transform (FFT). In this thesis, we study alternative transforms to create the OFDM symbols: the Fast Hartley transform (FHT) and the Discrete Wavelet Packet Transform (DWPT). Besides the advantages of using Optical Orthogonal Frequency Division Multiplexing (O-OFDM), there are also some disadvantages that must be taken into account. High Peak-to-average power ratio (PAPR) is one of the major drawbacks of OFDMbased systems that can cause intermodulation among the subcarriers due to the nonlinearities of the fiber and devices such as analog-to-digital converters (ADC) and external modulators. Here, PAPR reduction techniques are studied to mitigate the effects of the PAPR. Finally, we demonstrate that applying PAPR reduction techniques to O-OFDM systems using intensity modulation and direct detection (IM/DD) the effects of the clipping noise are mitigated without the need of adding a higher bias to the signal.
Chapter 1 Introduction OFDM has emerged as a leading modulation technique [2] and [3] in the optical domain. It is also used in wireless and wireline applications and in almost every major communication standards. The use of OFDM in optical communications mitigates transmission impairments and, at the same time, provides high-data rate transmission across dispersive optical media. The progress in Digital Signal Processing (DSP) technology can make processing at optical data rates feasible. O-OFDM introduces spectral efficiency and tolerance to impairments such as chromatic dispersion (CD) and polarization mode dispersion (PMD) to the system. It belongs to a broader class of MultiCarrier Modulation (MCM) in which data information is carried over many lower rate subcarriers. The subcarriers are orthogonal to each other, and their spectra can overlap. This results in a very high spectral efficiency. The insertion of a Cyclic Prefix (CP) makes OFDM an effective solution to InterSymbol Interference (ISI) and InterCarrier Interference (ICI), caused by a dispersive channel, that can degrade the performance of the system. This CP consists of an identical copy of the first samples of the frame that are added at the end of it, implying an increase of the signal bandwidth. The signal processing in the OFDM transmitter/receiver is based on the FFT to implement the OFDM modulation/demodulation. So the symbols can be generated in a very computationally efficient way. Other transforms are also suitable to generate the OFDM symbols. This is the case of the Fast Hartley Transform (FHT) that has recently been introduced in Intensity-Modulation (IM)/Direct-Detection (DD) systems [4] because a simpler transmission system can be achieved with real processing. The use of Discrete Wavelet Packet Transform (DWPT) reduces the Peak-to-average power ratio (PAPR), one of the major drawbacks that presents OFDM systems. DWPT doesn’t need the use of the CP because it mitigates ISI effects and it is more robust respect to ICI, due to very high spectral containment properties of wavelet filters. IM/DD is a cost effective solution for O-OFDM systems implementation, where real positive signals are required. Whereas the OFDM symbols are bipolar and complex. To generate real OFDM signals Hermitian Symmetry (HS) must be forced when using the FFT. As the FHT is a real transform, a real signal is achieved at the output of the transform by previously mapping the signal into a real constellation such as Binary Phase-Shift Keying (BPSK) or M-Pulse 1
Amplitude-Modulation (PAM) modulation format. On the other hand DC biased O-OFDM is used to achieve positive OFDM signals [5]. It consists of clipping and adding a bias to the signal. Clipping the signal causes clipping noise that degrades the performance of the system. Indeed, the use of a high bias reduces the clipping noise despite it increases the electrical power. Alternatively, Asymmetrically Clipped Optical-OFDM (ACO-OFDM) can be also used to have a positive signal. ACO-OFDM doesn’t introduce clipping noise but it is less spectral efficient than DC biased O-OFDM. In fact, the bandwidth of an ACO-OFDM signal is the double of a DC biased O-OFDM one as half of the subcarriers are set to zero. In addition to clipping noise, the PAPR is a drawback of the OFDM that must be reduced in order to avoid signal distortion. It exists a big effort in developing techniques [6] that can be applied to the OFDM system to mitigate this effect. In general all the PAPR reduction techniques, have a trade-off between performance and other parameters such as Bit Error Rate (BER) reduction, power increase, distortion or extra bandwidth requirements. In this thesis we propose to apply PAPR reduction techniques to mitigate the effects of the PAPR and the clipping noise in DC biased O-OFDM systems. This chapter gives an introduction on optical communications. Specially, optical transmitters, optical fibers, optical amplifiers and optical receivers will be briefly described. Then we go through the OFDM background and finally the outline of the thesis is provided.
1.1
Optical communications
The goal of optical signal transmission is to achieve a predetermined BER between any two nodes in an optical network. This BER must be at least 10−9 (Forward Error Correction (FEC) can be assumed). The increasingly Internet traffic growth requires the deployment of optical transmission systems supporting high data rates. Unfortunately at high data rates some nonlinear effects due to the fiber appear in the communication. Using advanced modulation formats mitigates these effects transmitting the signal to much lower symbol rates. An optical transmission system consists of a transmitter, a receiver and the fiber. We can also introduce amplifiers, equalizers, filters and other components in the link depending on the system requirements [3]. The role of the transmitter is to modulate the signal and generate the optical version to launch it to the fiber. Figure 1.1 shows an example of an optical OFDM system using IM/DD or coherent detection. The OFDM symbols are created and then transmitted through the fiber. Finally, the
Data
OFDM transmitter
IM or coherent transmitter
DD or coherent detection
channel ASE filter
Transmitter
Figure 1.1: Optical OFDM communication system
2
OFDM receiver
Receiver
Data
receiver demodulates the OFDM symbols after detecting them with one or more photodetectors (PD) depending on the type of detection that is used.
1.1.1
Optical transmitters
OFDM symbols are created in transmission by modulating the mapped information in orthogonal frequencies using the FFT. In this work, we also propose alternative transforms such as the FHT [4] and the DWPT [7]. In order to convert the signal into the optical domain, a laser and one or more modulators are needed. We can directly modulate the laser or use an external modulator such as a Mach-Zehnder modulator (MZM) or a ElectroAbsorption Modulator (EAM). The laser is an optical source based on the concept of population inversion. Laser means Light Amplification by Stimulated Emission of Radiation. The light generation process occurs in certain semiconductors materials due to the recombination of electrons and holes in p-n conjunctions, which are the heart of a semiconductor optical source. It is formed by bringing a p-type and an n-type semiconductor into contact. A semiconductor is made n-type or p-type by doping it with impurities whose atoms have an excess valence electron or one less electron compared to the semiconductor atoms. There are three basic processes in semiconductor materials, as it is illustrated in figure 1.2, by which light interacts with matter: absorption, spontaneous emission, and stimulated emission. In the case of spontaneous emission, photons are emitted in random directions with no phase relationship among them. Stimulated emission, by contrast, is initiated by an existing photon. One important feature of stimulated emission is that the emitted photon matches the original photon in energy and in other characteristics, such as the direction of propagation. Finally absorption is also initiated by a photon which energy is taking up by matter. In figure 1.2, hν represents the photon energy, (h) is Planck’s constant and ν is the optical freSpontaneous emission
Absorption E1 (Excited state) E0 (Ground state)
E1
Stimulated emission E1
hv
hv
hv
hv hv
E0
E0
Figure 1.2: Three basic processes of the interaction of light with matter.
quency which is proportional to the energy difference between the energy levels (E1 − E0 ). If N0 and N1 are the atomic densities in the ground and the excited states, respectively, and ρem is the spectral density of the electromagnetic energy, the rates of spontaneous emission, stimulated emission and absorption can be written as [8] Rspon = AN1 Rstim = BN1 ρem Ra bs = CN0 ρem 3
(1.1)
Where A,B,C are constants. In normal conditions, the number of electrons in the ground state (with energy E0 ) N0 is greater than in the excited state (with energy E1 ) N1 . In thermal equilibrium their ratio follows the Boltzman’s statistics [8] N1 −hν −E1 − E0 = exp = exp , (1.2) N0 kB T kB T where kB is the Boltzman constant and T is the absolute temperature. Following [8] we can write the ratio between Rstim and Rspon as −1 hν Rstim = exp −1 1. Rspon kB T
(1.3)
According to (1.3) for radiation in the visible or near-infrared region (hν ∼ 1eV ), spontaneous emission always dominates over stimulated emission in thermal equilibrium, at room temperature (kB T ≈ 25mV ). Therefore Rstim can only exceed Rabs when N1 > N0 . This condition is referred to as population inversion, and it is never achieved in thermal equilibrium state. For Injection current n1
p-type
n2 Active region n-type
n3
Gain medium z=L
z=0 Mirrors
Figure 1.3: Fabry Perot semiconductor laser structure.
this reason it is necessary a pump source to force this effect. The population inversion is a prerequisite for laser operation, and in an atomic system, it is achieved by using three- and fourlevel pumping scheme, such that an external energy source raises the atomic population from the ground state to an excited state lying above the energy state, E1 in figure 1.2. When the population inversion effect takes place, an input signal (x) propagating inside the active layer would then amplify as exp(gx), where g is the gain coefficient. The optical gain alone is not enough for laser operation. Optical feedback which converts an amplifier into an oscillator is also required. In most lasers the feedback is provided by placing the gain medium inside a Fabry 4
RF signal
Bias
V1(t) Ein(t)
Eout(t)
V2(t)
Figure 1.4: MZM external modulator.
Perot cavity formed by using two mirrors. Therefore, there are three basic components to sustain stimulated emission: the pump source, the active medium or cavity and the feedback mirrors. Depending on the active medium, which can be liquid, gaseous or solid we have different lasers useful for different applications. One typical laser is the semiconductor laser, which emit light through stimulated emission. Figure 1.3 shows the scheme of a Fabry Perot semiconductor laser. On the other hand in transmission it is necessary an external modulator to convert the electrical signal to the optical field, mixing the Radio Frequency (RF) signal with the output of the laser. MZM is commonly used and it is based in the electro-optic effect. This effect is related to the change of refractive index (n) (in certain materials) with respect to the voltage (V ) applied across the electrodes. The MZM is a planar waveguide structure deposited on the substrate, with two pairs of electrodes. One is for the DC bias voltage and the other one is for the RF signal. V1 (t) and V2 (t) denotes the electrical drive signal on the upper and lower electrodes, respectively as it is shown in figure 1.4. The output electrical field Eout (t) can be related with the input electrical field Ein (t), that comes from the laser source, by π π 1 exp j V1 (t) + exp j V2 (t) Ein (1.4) Eout (t) = 2 Vπ Vπ where Vπ is the switching voltage. The bias voltage (Vbias ) is included in both V1 and V2 . Vbias and Vπ are related between each other. It exists two different detection structures in optical communications, explained in chapter 2: direct detection and coherent detection. The transfer function of the MZM for the direct detection is the optical intensity against the drive voltage, whereas in the case of coherent detection it is the I or Q component of the optical field against the drive voltage. It is seen in figure 1.5 that the optimal MZM bias for optical intensity modulation is the quadrature point and for optical field modulation it is the null point. The choice of Vbias and Vπ must ensure that we work in the linear part of the transfer function. On the other hand another commonly used external modulator is the EAM. The EAM is a semiconductor-based planar waveguide composed of multiple p-type and n-type layers. The MZM modulation speed is comparable to the one of the EAM. The MZM presents a higher extinction ratio (the ratio of average powers corresponding to symbol 1 and symbol 0). 5
Optical intensity
Quadrature point
V 0
-Vπ
Vπ
Optical field
Figure 1.5: Transfer function for the optical intensity against the drive voltage.
1.1.2
Optical fibers
Optical fibers are used to transport optical signals from source to destination. They present low loss and extremely large bandwidth allowing the transmission of high-speed signals over long distances before the regeneration becomes necessary. Nowadays every 80 km of fiber approximately, an optical amplifier is introduced. The simplest form of an optical fiber consists of a cylindrical core of silica glass surrounded by a cladding whose refractive index is lower than that of the core. The core has a refractive index n1 and the cladding’s refractive index is n2 . The majority of the power is concentred in the core and there is a difference in both refractive n()
n()
n1 n1 n2
r1
n2 r2
r1
r2 Radial index
Radial index
(a)
(b)
Figure 1.6: (a) Refractive index profile for step-index fiber and (b) for graded-index fiber.
indexes (n1 > n2 ), which is achieved by a mix of dopants added to the fiber core. The fibers that present an abrupt change of the refraction index at the core-cladding interface are called step-index fibers. Whereas the ones that the refractive index decreases gradually inside the core, 6
are known as graded-index fiber. Figure 1.6 depicts both refractive index profiles. The light confinement by the total internal reflection in a step-index fiber is shown in figure 1.7. Consider the geometry of a step-index fiber, where a ray making an angle θi with the fiber axis is incident at the core center. Because of refraction at the fiber-air interface, the ray bends toward the normal. The angle θr of the refracted ray is given by equation (1.5)[8]. n0 sin(θi ) = n1 sin(θr )
(1.5)
However the ray will be totally reflected from the core-cladding interface if equation (1.6) is satisfied [3]. q (1.6) n0 sin(θi ) < n21 − n22 where n2 , n1 and n0 are the refractive indices of the fiber cladding, core and air, respectively. Unguided ray Guided ray
θr
core
n1>n2
θi n0=1
cladding
n2
Figure 1.7: Light confinement in step-index fibers through the total internal reflection.
Following equation (1.6) we can define the Numerical Aperture (NA) as q √ N A = n21 − n22 ≈ n1 2∆, ∆ 1.
(1.7)
2 ∆ is the normalized index difference ∆ = n1n−n and it should be made as large as possible in 1 order to couple maximum light into the fiber. This phenomenon is caused when different rays travel along paths of different lengths. As a result, these rays disperse in time at the output end of the fiber. On the other hand the ray trajectories of the light inside the graded-index fibers, which are depicted in figure 1.8, are obtained solving an equation of harmonic oscillator
1 dn d2 ρ = 2 dz n dρ
(1.8)
where ρ is the radial distance of the ray from the axis. The use of graded-index fibers reduces qualitatively the intermodal or multipath dispersion (where the light distributed among several modes). This is due to the fact that with a suitable choice of the refractive-index profile the rays can arrive at the same time to destination. In the medium where the refractive index is low the rays travel faster. Furthermore an optical fiber can be single mode or multimode. Single mode fibers only supports the principal mode of the fiber. The fiber is designed such that all higher order modes are cut off at the operating wavelength. The main advantage of single-mode fibers is 7
Unguided ray Guided ray core
n1>n2 n0=1
cladding
n2
Figure 1.8: Light confinement in graded-index fibers through the total internal reflection.
that intermodal dispersion is absent. This is due to the energy of the injected pulse is transported by a single mode. However, pulse broadening does not disappear altogether [9]. Different spectral components of the pulse travel at slightly different group velocities, a phenomenon referred to as Group-Velocity Dispersion (GVD), intramodal dispersion, or fiber dispersion. Intramodal dispersion has two contributions, material dispersion (DM ) and waveguide dispersion (DW ), see equation (1.9). The first contribution, material dispersion, occurs because the refractive index of silica, the material used for fiber fabrication, changes with the optical frequency ω. 2πc d D=− 2 λ dω
1 vg
= DM + DW ,
(1.9)
where vg is the group velocity defined in [8] at the frequency ω and at λ wavelength and c is the light speed. DM and DW are defined by 1 dn2g 2πc dn2g = 2 λ dω c dλ 2 2π∆ n2g V d2 (V b) dn2g d(V b) =− 2 + . λ n2 ω dV 2 dω dV
DM = − DW
(1.10)
Here n2g is the group velocity of the cladding material, V is the cutoff condition and b is a normalized propagation constant b, see [8]. Moreover some other impairments that appear in the fibers are [9]: chromatic dispersion, where the light is distributed over a range of wavelengths and Polarization Mode Dispersion (PMD), which is a consequence of the light being distributed over different polarizations. Another limitation that appears in the fibers is the losses, which reduce the signal power reaching the receiver, and also the nonlinear effects. As optical receivers need a certain minimum amount of power for recovering the signal accurately, the transmission distance is inherently limited by fiber losses. Some limiting factors that can be present in the fiber in long-haul communications are: the attenuation, material absorption, Rayleigh scattering and others [8]. Nonlinearities can also be present in the fiber due to the waveguide geometry that confines light to a small cross section over long fiber lengths. Some nonlinearities can be stimulated light scattering, nonlinear phase modulation and four wave mixing explained in [9]. 8
1.1.3
Optical amplifiers
The purpose of an optical amplifier is to restore the signal power level without any optical-toelectrical conversion. Fiber attenuation is often a factor that limits the length of an optical fiber link. Hence, for very long fiber spans, it is necessary to periodically amplify the light signal. The optical amplifier is simply a higher-power replica of what came in the input. The main drawback of optical amplifiers is that they do not regenerate a signal that has been degraded by dispersion. However the use of dispersion compensation techniques reduce this effect. An example of one dispersion compensation technique can be to introduce at the end of the fiber a second piece of fiber with a dispersion coefficient of sign opposite that the first. The general form of an optical Pump power Optical fiber
Optical fiber Pin
Optical amplifier medium (G)
Pout
Figure 1.9: Optical amplifier principle.
amplifier is depicted in figure 1.9 and most of them uses the same mechanism that the lasers but without the feedback part. The amplification factor is defined as the ratio between the amplifier output (Pout ) and the amplifier input (Pin ), G=
Pout . Pin
(1.11)
All amplifiers degrade the Signal to Noise Ratio (SNR) of the amplified signal because of spontaneous emission that adds noise to the signal during its amplification. The amplifier noise figure (Fn ) is defined as the ratio of SNR at the input (SN R)in to SNR at the output (SN R)out Fn =
(SN R)in (SN R)out
(1.12)
The optical amplifier has three main applications: booster, in-line amplifiers and pre-amplifier, that are depicted in figure 1.10. The most important application for long-haul systems consists of using amplifiers as in-line amplifiers, figure 1.10 (a), which replace electronic regenerators. Many optical amplifiers can be cascaded in the form of a periodic chain as long as the system performance is not limited by the cumulative effects of fiber dispersion, fiber nonlinearity, and amplifier noise. Another way to use the optical amplifiers is as a booster or power amplifier, figure 1.10 (c), placing it just after the transmitter in order to increase the transmitted power. Moreover, optical transmitters can be also placed before the receiver to increase, in this case, the received power, figure 1.10 (b). Such amplifiers are called optical preamplifiers and are commonly used to improve the receiver sensitivity. There are different types of optical amplifiers [8]: 9
Rx
... Tx
(a)
In-line amplifiers
Tx
(b)
Tx
Rx
(c)
Pre-amplifier
Rx
Booster
Figure 1.10: Possible applications of optical amplifier: (a) In-line amplifiers, (b) pre-amplifiers and (c) Booster or power amplifier.
Erbium-Doped Fiber Amplifier (EDFA), which has the greatest impact on fiber optic communications, Semiconductor Optical Amplifier (SOA) and Raman amplifier. In the case of the EDFA the operating wavelength and the gain bandwidth are determined by the dopants. Rare-earth elements, such as erbium, and others can be used to amplify the signal. They also have become very attractive because the amplification peak is near of the wavelength region 1.55µm (3rd window). On the other hand SOA amplifiers experience a relatively large feedback because of reflections. Finally, Raman amplifiers use stimulated Raman scattering, which occurs in silica fibers when an intense pump beam propagates through it [3].
1.1.4
Optical receivers
The main component of the optical receiver is the photodetector [8] that converts light into electricity through the photoelectric effect by optical absorption. The purpose of the receiver is to recover the signal as clean as possible. The photodetector should have high sensitivity, fast response, low noise, low cost, and high reliability. In IM/DD systems the use of one photodetector in reception is enough to recover the signal. Whereas in coherent detection it is needed two photodetectors and one additional laser is needed to recover phase and quadrature information. A basic concept of a photodetector is the responsivity (R), which is defined as R=
Ip Pin
(1.13)
where Pin is the incident optical power and Ip the photocurrent. Detectors with a large responsivity are preferred since they require less optical power. There are different commonly used photodetectors: p-n photodiodes, p-i-n photodiodes, avalanche photodiodes and MetalSemiconductor-Metal (MSM) photodetectors. The first group, p-n photodiodes, is based on a reverse biased p-n junction, which is known as depletion region. When using p-n photodiodes, optical power decreases exponentially as the incident light is absorbed inside the depletion region. Its responsivity is high because of a high quantum efficiency. In order to increase the 10
r to on du c m ic Se hm
O ic n co ct ta
Figure 1.11: A semiconductor slab used as a photodetector.
depletion region width, p-i-n photodiodes can be used. This photodiodes has an undoped or slightly doped semiconductor material between the p-n junction. Because of its nature, the middle i-layer offers a high resistance, and most of the voltage drop occurs across it. Therefore, changing the middle layer thickness, the width W of the depletion region can be controlled. In contrast avalanche diodes, can achieve large values of responsivity reducing the required optical power.
1.2
Background of OFDM systems
The first proposal to use OFDM for transmission appeared in 1966 and was introduced by Robert W. Chang [10]. This document presents the way to implement OFDM and also explains its main concept. OFDM symbols are created by filtering the signal and then multiplying the outputs by different frequencies. Thus, the subcarriers are orthogonaly created, they overlap between each other and they are band-limited. Therefore, the spectra is produced without causing ICI and ISI. This method supposed a huge revolution in the communication world and in 1969 appeared the Discrete Fourier Transform (DFT) as a way to generate the orthogonal subcarriers [11]. This paper highlight the use of the DFT because it can be entirely implemented by digital circuitry and it can be computed with fast algorithms (FFT). Later on, in the 80’s appeared the concept of CP whose purpose was to resolve the channel dispersion-induced ICI and ISI. In 1995 Telatar and Foschini studied OFDM, [12] and [13], for multiple antenna systems opening a new investigation area. OFDM also began to be considered in wireless communications after the publication of Cimini of Bell labs in 1985 [14]. While in 1987 Lassalle and Alard proposed OFDM for digital broadcasting for mobile receivers [15]. This article explains the benefits of using OFDM to overcome the adverse effects of severe multypath propagation. Cioffi and others at Stanford demonstrated in 1990 the potential to apply OFDM in wireline communications [16], designing a Discrete MultiTone (DMT) transceiver for High-Bit-Rate Digital Subscriber Line (HDSL). So with the advancement of powerful silicon DSP technology, OFDM triumphed 11
in a broad range of applications such as the RF domain from digital audio/video broadcasting (DAB/DVB) to wireless local area networks (LANs). OFDM has been recently applied to optical communications. In 2001 appeared the first paper based on OFDM for optical wireless [17] and in 2005 it was used in optical fiber communication systems [18], [19]. Figure 5.2 depicts OFDM
Figure 1.12: Historical evolution of OFDM [1]
historical evolution towards O-OFDM [1]. Theoretical basis are shown above the line whereas papers on application of OFDM to particular fields are presented under the line.
1.3
Summary of the thesis
In chapter 2 of this thesis, O-OFDM is introduced. In chapter 3, different alternative transforms to the FFT to create the OFDM symbols, are explained. In chapter 4, PAPR is defined and different distortionless techniques to mitigate its effects are described. In chapter 5, simulation results are presented. PAPR reduction techniques are evaluated and compared using the Complementary Cumulative Density Function (CCDF). Furthermore, BER performance of applying such techniques to DC biased O-OFDM systems is evaluated. Finally in chapter 6, conclusions are drawn.
12
Chapter 2 Optical-Orthogonal frequency division multiplexing In this chapter the OFDM is introduced. Moreover, two different architectures suitable for optical communications are presented: IM/DD and coherent detection.
2.1
Optical-Orthogonal Frequency Division Multiplexing
OFDM has recently been applied to optical communications. One major reason is the growing demand for increased data rates across dispersive optical media [1] and [3]. O-OFDM has been proposed to cope with upgrades to the next transmission speed in highly reconfigurable networks. It is also a good option for optical long-haul communications specially for the reason that it can be designed to be extremely tolerant to chromatic dispersion. O-OFDM late appearance in optics was mainly due to the lack of a mature DSP technology at optical speed and also due to the signal restrictions in IM/DD systems. Typical OFDM symbols created with the FFT are bipolar and complex. Whereas IM/DD systems requires positive and real signals before the modulator. Therefore, a lot of effort was devoted to adapt OFDM signal to these requirements. The first part of this section introduces the OFDM concept. Then IM/DD systems and different way to achieve a positive and real-valued OFDM symbol are explained. Finally in the last section coherent detection is briefly described.
2.1.1
Orthogonal Frequency Division Multiplexing
OFDM is a special class of multicarrier modulation (MCM) that consists of transmitting a signal over several lower-rate orthogonal subchannels [1] and [3]. In figure 2.1, the different subcarriers that form one OFDM symbol are depicted. The spectrum of an individual subcarrier has 13
a |sin(x)/x|2 form, so each OFDM subcarrier has significant sidelobes over a frequency range which includes many other subcarriers. This effect is the cause of one of the major disadvantages of OFDM: the sensitivity to frequency offset and phase noise. As already mentioned OFDM is
f1
...
fN
Figure 2.1: OFDM spectrum
very robust against channel dispersion because the symbol is divided into narrow subbands. The CP consist in adding at the beginning of each frame an identical copy of the end of the OFDM symbol, as it can be seen in figure 2.2. If the receiver FFT window is aligned with the start of the main symbol period of the first arriving signal and the delay spread, introduced in the system by the channel, is smaller than the CP, then no ICI or ISI occurs. In order to create
Figure 2.2: Time domain signal of one subcarrier for two OFDM frames using CP
the orthogonal subcarriers we use the Inverse Fast Fourier Transform (IFFT)/ FFT for the modulation/demodulation of the signal. Usually, the signal is mapped in a Quadrature Amplitude Modulation (QAM) modulation format before doing the IFFT. In this case, symbols have different energy levels. Also Phase Shift Keying (PSK) format can be used, where symbols are distributed in a circle of unitary energy. Using QAM, as the symbols have more distance between each other, the receiver can recover the signal with less errors. Nevertheless more power is needed to transmit the symbols. Both modulation formats (for 16 symbols) are depicted in figure 2.3. The block diagram of the transmitter is shown in figure 2.4 and it is composed by a Serial to Paralel (SP) converter, a mapper, an IFFT of N points, a block to insert the CP and 14
16-QAM
16-PSK
2
4 3
1
2 1
0
0 -1
-1
-2 -3 -4 -4
-2
0
2
4
(a)
-2 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
(b)
Figure 2.3: (a) 16-QAM and (b) 16-PSK constellations.
a Parallel to Serial (PS) converter. With the mapping, a certain number of bits is represented by each symbol. The IFFT block generates an OFDM symbol with N orthogonal subcarriers. The channel dispersion can destroy the orthogonality between subcarriers so the CP is added to combat the dispersion of the channel and avoid ISI and ICI. Moreover equalization is necessary to mitigate the effects of the channel such as dispersion that can lead to ISI, enhancing the performance of the system. Finally the resulting signal is serialized, digital-to-analog converted and it is sent through the channel.
Filter Filter
Im{X(k)}
DAC
Re{X(k)}
DAC
. . .
Parallel to Serial
. . .
CP insertion
. . .
X (k )
N‐IFFT
. . .
Mapper
input data
Serial to Parallel
x(n )
FFT‐based OFDM transmitter Figure 2.4: Diagram block of the OFDM transmitter based on the FFT
In the receiver, figure 2.5, the inverse process takes place. Firstly, after the detection of the signal, the data is serial to parallel converted in order to remove the CP in the following step. Then the FFT is implemented and finally the resulting signal is demapped, serialized and analogto-digital converted in order to recover the original bit stream. The robustness against channel dispersion and its ease of phase and channel estimation in a time-varying environment make OFDM a suitable advanced modulation format for optical communications systems. However OFDM also presents some drawbacks that must be taken into account in order to mitigate their 15
. . .
Parallel to Serial
. . .
Demapper
. . .
Equalization
. . .
N-FFT
. . .
CP removal
ADC ADC
Serial to Parallel
Filter
Q
Filter
I
output data
FFT-based OFDM receiver
Figure 2.5: Diagram block of the OFDM receiver based on the FFT
effects. High PAPR is one of the major problems that arise when using such modulation. Chapter 4 tackles this PAPR problem and proposes some techniques to reduce it. Furthermore, OFDM is very sensitive to frequency and phase noise which lead to ICI. Both effects appear due to the long OFDM symbols length. Hopefully, all these impairments can be mitigated and OFDM becomes very attractive to be used in optical communications, RF communications and others.
2.1.2
Intensity-Modulation
In IM systems the information is carried on the optical intensity. Therefore the transmitted signal must be unipolar. The OFDM signals are complex and bipolar so they must be modified in order to work with real positive symbols. Two possible solutions to make the signal positive are: DC biased O-OFDM and ACO-OFDM.
DC biased systems DC biased O-OFDM consists of adding a bias to the signal and then clip as it is shown in figure 2.6. It exists different alternatives to do the clipping: asymmetrically clipping and symmetrically clipping. Usually the bias value is at least twice the standard deviation of the signal [1]. The problem of this technique is that it causes clipping noise that degrades the performance of the system. Using a higher bias reduces the noise but at the same time, more bit electrical energy normalized to the noise power spectral density (Eb /N0 ) is needed to achieve an acceptable BER. Hence, there is a trade off between power efficiency and noise in the selection of the bias. Asymmetrically clipping consists of adding a bias to the analog signal and then clipped it to zero level. The clipped signal Xc (t) can be written as: X(t) + bias ≤ 0 0, Xc (t) = (2.1) X(t) + bias, X(t) + bias > 0 16
FHT and FFT-based OFDM Tx
IM
Optical channel
DD
FHT and FFT-based OFDM Rx
bias
Figure 2.6: Diagram block of a IM/DD system
Where X(t) is the analog OFDM signal. On the other hand, symmetrically clipping consists of limiting the amplitude of the signal and then add a bias (B). The symmetrically clipped OFDM signal can be represented by: |X(t)| ≤ B X(t), Xc (t) = (2.2) B · sign(X(t)), |X(t)| > B Where B is the maximum allowed signal amplitude before clipping and bias. The clipping level in dB is defined as: B2 A = 10 · log10 . (2.3) E[|X(t)|2 ] DC biased O-OFDM performance depends on the choice of the bias, which is related with the constellation size and the clipping level. For high constellation sizes a high clipping level is required in order to achieve an acceptable BER performance. According to [20], when the signal is mapped into a 4QAM format a 7dB clipping level is enough. Whereas when the signal is mapped into a 64QAM format we need to increase the clipping level to 9dB. Therefore, the clipping level must be adjusted depending on the constellation size in order to guarantee a target BER. Power efficient optical OFDM ACO-OFDM is another alternative that can be used in IM/DD systems in order to obtain positive signals. It consists of modulating only the odd subcarriers and set to zero the even ones. Then we can clip the signal to zero level without adding noise and without loosing information [19]. Thus only the odd frequencies have the property that: x(n, k) = −x(n + N/2, k).
(2.4)
Where N is the number of subcarriers and x is the input vector of the transform block. With this scheme the optical power is substantially reduced, but in contrast it only carries useful information on half of the available signal bandwidth. In figure 2.7 (a) it is shown an OFDM frame with only the odd subcarriers modulated (ACO-OFDM). Therefore, as the signal has odd symmetry, we can clip the signal without loosing data as it can be seen in 2.7 (b). In 2.7 (c) it is represented one OFDM frame with all the subcarriers modulated while in 2.7 (d) it is shown this signal clipped to the zero level after adding a bias of twice the standard deviation of the 17
3
3
2
2
1
1 Amplitude
Amplitude
original signal (DC biased O-OFDM). With DC biased O-OFDM, we are able to transmit more information with the same bandwidth, implying higher spectral efficiency, but at the same time clipping noise affect the transmission. Both, DCO-OFDM and ACO-OFDM are used in IM/DD
0
0
-1
-1
-2
-2
-3
-3 5
10
15 Time
20
25
30
5
10
(a)
15 Time
20
15 Time
20
25
30
(b)
6
3
5 2
4 3 Amplitude
Amplitude
1
0
2 1 0
-1
-1 -2
-2 -3
5
10
15 Time
20
25
-3
30
5
10
(c)
25
30
(d)
Figure 2.7: Real-valued OFDM time domain signal with (a) only odd subcarriers modulated and (b) clipped to zero level (ACO-OFDM) and with (c) all subcarriers modulated and (d) clipped to zero level (DCO-OFDM) after adding a bias.
systems. In the case of ACO-OFDM, the clipping noise must fall in the even subcarriers while the useful information needed to recover the signal in the odd ones. Hence, the received constellation points have half of the power than in transmission. Moreover, with ACO-OFDM we are able to achieve an optimum performance with the same design. Whereas with DCO-OFDM, we have already seen that the bias must be constantly adjusted depending on the constellation size. Therefore, ACO-OFDM can be easily used in adaptive systems to transmit the OFDM symbols previously mapped with mixed formats [20].
2.1.3
Direct-Detection and Coherent detection
Direct detection systems have lower complexity and cost, whereas coherent systems achieve better performance in receiver sensitivity, spectral efficiency, and robustness against polarization dispersion. During the last two decades, big efforts have been devoted to investigate IM/DD systems. However, coherent detection communications have recently also progressed. Developments in OFDM signal processing has driven the evolution of optical communications. 18
Direct-Detection DDO-OFDM can be used in a broader range of applications and it is very attractive due to its low cost. DD systems, as it is shown in figure 2.8, uses a laser at the transmitter to create the optical signal by directly or externally modulating the electrical signal. A typically used external modulator is the MZM. Then in the receiver, the signal is recovered using a photodiode that can be modeled with a square law characteristic. As only real data can be modulated using a single (a) Re
Electrical OFDM transmitter
fi
90º
Im
(b)
...
...
0 1 QAM N/2 0 QAM* N
Data in
DMT transmitter
Real data
(c) ...
1
N/2 ...
MPAM
N
OFDM transmitter based on the FHT
Real data
Data out
Bias Electrical OFDM signal
fc-fi fc fc+fi
OFDM receiver
fc fc+fi
MZM channel
SSB filter
ASE filter Photodetector
0
...
1
N/2 ...
(d)
0 QAM
Electrical OFDM transmitter
Re{analytic signal}
N
Optical IQ modulator
Im{analytic signal}
Figure 2.8: Diagram block of a IM/DD system using four different transmission architectures: (a) RF conversion, (b) DMT modulation, (c) FHT tranform and (d) Hilbert transform.
modulator, it is necessary to manipulate the OFDM symbol. There are different ways to proceed. A first option, figure 2.8(a), is based on a RF modulation of the OFDM base band signal. Once the OFDM symbol is created, the real and imaginary part are converted to the analog domain separately. The next step is to modulate the signal to RF , multiplying the real part by a cosine and the imaginary part by a sine and then adding both signals obtaining a double side real signal. An oscillator to the frequency fi is required in order to implement the modulation. Finally the OFDM signal is modulated. The second, figure 2.8 (b) uses the HS property in order to have a real valued signal at the output of the IFFT. This implementation is called DMT [21] and it is explained in chapter 3. But basically consists of filling with data the firsts subcarriers (from the second input to the (N/2 − 1) input), where N is the total number of subcariers. The Nyquist frequencies are set to zero and the other transform inputs are filled with conjugated and reflected 19
data in order to implement the HS. Hence, only half of the subcarriers support data symbols. In order to generate a real-valued signal, an alternative transform, the FHT, has been recently proposed in [4]. This real trigonometric transform, figure 2.8 (c), supports the double of the input symbols of a standard real-valued FFT using a simpler implementation. It has the same DSP in transmission and reception and works with real algebra. The last scheme, figure 2.8 (d), uses the properties of the Hilbert transform. Modulating the real and imaginary parts of an analytic signal, an optical signal with no-negative frequency components is obtained. In order to have an analytic signal at the output of the IFFT, half of the elements of the input vector must be set to zero. Using one of these already presented architectures we are able to have real-valued OFDM systems and therefore we can work with IM/DD systems. Once we have real OFDM symbols they are modulated in order to transmit the optical signal to the fiber. To optimize the system performance it has been observed that both the carrier and the OFDM signal must have the same power. The MZM creates a double side band spectrum with respect to the optical carrier. So in order to ensure that the OFDM subcarriers are represented only once by the optical frequencies and avoid chromatic dispersion fading, Single Side Band (SSB) modulation can be adopted. In short reach applications double side band can be used. Finally, in reception, the data is detected with a photodetector. Then it is electrically amplified, analogically converted and demaped with the OFDM demodulator in order to recover the original bit stream. Coherent detection CO-OFDM was proposed to combat fiber chromatic dispersion [22]. It shows better performance than IM/DD based OFDM in terms of bandwidth efficiency, robustness against polarization dispersion and receiver sensitivity, but it requires higher complexity in the transceiver design than DD systems [3]. CO-OFDM block diagram is depicted in Fig.2.9. The OFDM is modulated in I
90º Hybrid MZ M
-
MZM
Data
+
OFDM transmitter
channel MZ M
MZM
Q
OFDM Data receiver
-
90º
+
Local laser
Figure 2.9: Block diagram of a CO-OFDM system.
phase and quadrature separately with a laser, so two modulators, for example MZM, are needed in transmission. Then the modulated signal is transmitted to the channel. In reception, one laser, one 90o hybrid and 4 photodetectors are needed to recover the signal, 2 for the quadrature and 2 for the phase of the received signal. Finally, both components are filtered and demodulated in the OFDM receiver. 20
Chapter 3 Transforms used in O-OFDM In this chapter different transforms to create the OFDM symbols are introduced. The signal processing in the OFDM transmitter/receiver is usually based on the FFT to implement the OFDM modulation/demodulation. Here, we present two alternative transforms to create the OFDM symbols: the FHT and the DWPT. Moreover we will see the application of both in optical systems.
3.1
Fast Fourier Transform
OFDM systems were previously implemented using oscillators and filters before the Inverse Discrete Fourier Transform (IDFT) was proposed to create the different orthogonal subcarriers [3]. Later on, it was seen that the N points, IDFT had increasing complexity when the number of subcarriers (N ) was large. Therefore in 1965 the FFT was introduced in order to reduce the computational cost, and the necessary number of operations needed to implement the OFDM symbols. The fundamental principle of the FFT algorithm is based on the decomposition of the DFT of a sequence of length N to smaller DFTs. An N points IDFT requires (N 2 ) complex multiplications, which are phase rotations. It also requires (N 2 − 1) complex additions (without taking into account the evaluation of the base functions: sines and cosines). The hardware required to implement an addition has lower complexity than the one used for a multiplication. The IFFT drastically reduces the amount of calculations. Hence it is used to implement the OFDM symbols. The FFT can be seen as a bank of modulators, whose narrowband channels have mutually orthogonal subcarriers equally separated by T1s . Where T s is the symbol period. Using the radix-2 algorithm [23], an N point IFFT requires only (N/2)log2 (N ) complex multiplications. Thus, using the IFFT algorithm we achieve to reduce hardware complexity. If we use a radix-4 algorithm the number of multiplication is reduced even further. This radix-4 algorithm is depicted in figure 3.1. Using this scheme the transform is split into a number of trivial fourpoint transform (1, −1, j, −j). Thus, we don’t need a full multiplier. We only need to add and subtract and switch of real and imaginary parts. Therefore using the radix-4 algorithm we only 21
x0
x1
x2
y0=x0+x1+x2+x3
+j -1-j -1
y1=x0+jx1-x2-jx3
y2=x0-x1+x2-x3 -1
x3
-j -1 +j
y3=x0-jx1-x2+jx3
Figure 3.1: The radix-4 butterfly.
require ((3/8)(N log2 N − 2)) phase rotations or complex multiplications and (N log2 N ) complex additions. The spectrum of an individual OFDM subcarrier has a |sin(x)/x|2 form and it overlaps with the spectra of other subcarriers. Moreover, if the channel is linear the orthogonality between subcarriers is preserved and they can be recovered in reception without interference and without using analog filtering techniques. The inverse fast Fourier transform (IFFT) is defined as: N −1 1 X x(n)exp(2jπkn/N ) k = 0, 1, ..., N − 1, X(k) = √ N n=0
(3.1)
where x(n) indicates the symbol sequence, previously mapped with a m-QAM or M-PSK format, and N represents the number of subcarriers. The FFT is defined by: N −1 1 X X(k)exp(−2jπkn/N ) n = 0, 1, ..., N − 1. x(n) = √ N k=0
(3.2)
These forms of the IFFT and FFT transform pair, of equations (3.1) and (3.2) respectively, have the advantage that the discrete signals at the input and the output of the transform for each symbol have the same total energy and average power, simplifying the analysis of many OFDM functions. To model the probability distribution of X it is need N ≥ 64 to assume Gaussianity. Moreover, the FFT has been used to implement O-OFDM in optical communications, allowing the transmission to high data rates. We have already seen that a possible architecture to work in optics, is IM-DD, where the information is carried in the optical intensity. Therefore a real positive version of the original signal is needed and the FFT must be adapted. The most common solution is to implement DMT, which is based on the HS property [21]. Discrete Multitone Modulation DMT consists on allocating the information in the input of the FFT such a way that real data is obtained at the output. The IFFT is a complex transform 22
0 1
0 Real data X(1) X(2)
Complex conjugate
N/2-1 N/2 N/2+1
X(N/2-1) N-IFFT
Real data
0 X*(N/2-1)
X*(2) N-1
Real data
X*(1)
Figure 3.2: Schematic of DMT modulation.
which output symbols have real and imaginary components. The use of DMT allows to work with real-valued signals using the IFFT. The principle of DMT is depicted in figure 3.2. The mapped symbols are divided into N subcarriers. The first half, from input 1 to input N/2 − 1 of the IFFT, carry useful data. Whereas the first and the N/2 inputs, the Nyquist frequencies, are set to zero. Then following the HS property, defined in (3.3), the second half of the available inputs carry the flipped complex conjugate version of the first half. x(N − n) = x∗ (n)
(3.3)
Therefore only half of the inputs of the IFFT carry information, but at the output we obtain a realvalued signal. DMT allows us to work with real valued O-OFDM signals, which is a prerequisite of cost effective IM/DD systems.
3.2
Fast Hartley Transform
The FHT is particulary attractive because it has the same DSP in transmission and reception and works with real algebra [4]. Moreover, it is a real trigonometric transform. Hence, if the input data of the transform is mapped in a real constellation such as M-PAM or BPSK, the output will be also real. Furthermore, the Discrete Hartley Transform (DHT) kernel only differs to the DFT in the imaginary unit. An OFDM symbol obtained with the Inverse Fast Hartley Transform (IFHT) can be written as: N −1 1 X h(n)cas(2πkn/N ) k = 0, 1, ..., N − 1. H(k) = √ N n=0
23
(3.4)
In equation (3.4) we define, cas(2πkn/N ) = cos(2πkn/N ) + sin(2πkn/N ).
(3.5)
h(n) represents the real data modulated with a BPSK modulation or M-PAM. The FHT has the same routine in transmission and reception. Therefore, the reconstructed signal at the output of the FHT is also defined by equation 3.4. Using the FHT, we don’t need to force HS. Thus, it supports the double of the input symbols of a standard real-valued FFT using a simpler implementation. Hence, the use of the FHT becomes very attractive in IM/DD systems, where it is necessary to work with real and positive signals. In order to compare the performance of both the FFT and the FHT we must transmit the same data signal. Therefore, if we modulate the signal with a M -PAM format using the FHT, we must map the information in a M 2 -QAM format to be able to compare the results. In terms of computational cost, when we work with IM/DD systems and the FFT exploits the HS property of the transform, the FHT algorithm require about the same number of multiplications but more additions. On the other hand, additional resources must be used for calculating the complex conjugate vector to deal with real-valued FFT. Bracewell presented in [24] a radix-2 decimation-in-time FHT algorithm. In 1986, Duhamel and Vetterli proposed the fastest algorithm implementing the DHT. Later on an improved version which required only two more additions than the FFT algorithms for real-valued signal appeared [25]. This last improvement increased the computationally speed. Here in this thesis, we will study the application of these transforms in IM/DD optical systems. The FHT has been investigated in the literature. For example, simulations in Additive White Gaussian Noise (AWGN) channel have been developed in [4] and in [26] for high modulation formats demonstrating similar performance with the FFT based O-OFDM system. Moreover, performance simulations of FHT based O-OFDM signals, have been done for both IM/DD [27] and coherent systems [28].
3.3
Wavelet Transform
The Wavelet transform has been proposed to be used as an alternative of the FFT for O-OFDM systems. It consists of decomposing a signal of interest into a set of basis waveforms, called wavelets. The wavelet transform is a multi-resolution analysis mechanism where an input signal is decomposed into different frequency components, and then each component is studied with resolutions matched to its scales. This process is similar to the Fourier transform but with the difference that the basis functions used are not sines and cosines. Sinusoids have infinitely lengths, whereas wavelets have finite duration. Therefore wavelets have both frequency and time localization. Thanks to this, wavelets can combat better ISI without being necessary to use the CP. Moreover it gives good time resolution and poor frequency resolution at high frequencies and a good frequency resolution and poor time resolution at low frequencies. In contrast, they are very sensitive to the PMD. Wavelet transform is classified in Continuous Wavelet Transform (CWT) and Discrete Wavelet Transform (DWT) [29]. On the one hand CWT of a continuous signal y(t) is defined as the sum of all time of the signal multiplied by scaled, shifted versions of the wavelet 24
ϕ(t): 1 Y (s, τ ) = √ s
Z
∞
y(t)ϕ −∞
∗
t−τ s
dt.
(3.6)
In equation (3.6), s is the scale factor and τ is the translation factor. y(t) is the time domain signal to be modulated, ϕ(t) is the wavelet scaling function and Y (s, τ ) is the resulting signal after the wavelet transformation.The DWT analyzes the signal at different frequency bands with different resolutions by decomposing the signal into an approximation and detail. DWT employs two sets of functions, known as scaling (ϕ[n]) and wavelet functions (ψ[n]), which are associated with low pass and high pass filters. So according to [30] we define: Y [k] =
∞ X
y[n]ϕn [k] k = 0, 1, ..., N − 1.
(3.7)
n=−∞
DWT is applied to create filter banks and using quadrature mirror filters. Quadrature mirror filters are commonly used in signal processing, and they satisfy, h1 [L − 1 − k] = (−1)k h0 [k].
(3.8)
In equation (3.8) h0 and h1 are the impulse responses of the low pass and high pass filters respectively. The complete scheme of transmission is decomposed in two parts: the analysis and synthesis. In the analysis we create the approximation of the original signal filtering with a low pass filter (h0 ) and then downsampling by a factor of 2. Consequently the details are created with a high pass version of the filter (h1 ) and with the same downsampling factor. Therefore the analysis basis are: ϕ2n [k] = H0 [2n − k] ϕ2n+1 [k] = H1 [2n − k]
(3.9)
In the synthesis the opposite scheme of the analysis is followed. Firstly the approximation is upsampled with a factor of 2 and then filtered with a low pass filter. Whereas the details are also upsampled by the same factor and the filtered with a high pass filter. Consequently the synthesis basis are defined as: ϕ˜2n [k] = G0 [k − 2n] ϕ˜2n+1 [k] = G1 [k − 2n]
(3.10)
Iterating both schemes we can decompose the original signal into multiple levels and the reconstructed it. DWT, the approximation is decomposed so the spectra of the signal at the output is narrower at every decomposition. As our purpose is to implement a OFDM system, we need equally separated subcarriers. This effect is achieved by using DWPT with some modifications. DWPT consist of decomposing the approximation as well as the details as it is explained in [31] and [32]. Here in order to construct a OFDM system,in the transmitter the synthesis is performed and in the receiver the analysis is done as it is depicted in figure 3.3. OFDM systems based on 25
2
H0
G1
2
2
H1
G0
2
2
H0
G1
2
G0
2
2
H1
2
H0
2
2
G1
H0 H1
2
H0
2
H1
2
2
H0 H1
G1
2
G0
2
H1 2
2
G0
2
G1
2
G0
2
G1
2
G0
2
2
2
H0
G1
2
H1
G0
2
Figure 3.3: Synthesis with IDWPT (modulation) and analysis with DWPT (demodulation) for OFDM systems.
DWPT combat ISI effects better than the ones based on the FFT or the FHT. Moreover they are more robust respect to ICI, due to very high spectral containment properties of wavelet filters. Wavelet Packet Transform (WPT) have recently been used in optical communications. In [7], WPT-OFDM is used instead of the conventional FFT in a CO-OFDM system. They compare the performance of both transforms, concluding that WPT systems are very sensitive to the PMD but they have better performance than FFT for short haul transmission. Using WPT a Chromatic Dispersion (CD) of 3.380ps/nm at 112 Gb/s can be mitigated without adding a CP.
3.3.1
Wavelets functions
It exists a large set of wavelet functions. Haar is the simplest wavelet system that exists. Its scaling and wavelet functions are defined respectively by if 0 ≤ t < 12 1 1 if 0 < t < 1 1 −1 ≤t= δ(n − l) P αn =< f, ϕn > f (t) = n αn ϕn (t).
(3.12) (3.13)
By the same way biorthogonal bases can be written as < ϕn , ψl >= δ(n − l) 26
(3.14)
αn =< f, ϕn > βn =< f, ψn > f (t) =
P
n αn ψn (t) =
P
n
βn ϕn (t).
(3.15)
In Table 3.1 some wavelets properties are summarized. Db. M is the Daubechies wavelet of order M. Whereas Bior. Mr Md is the biorthogonal wavelet with orders Mr for reconstruction and Md for decomposition. WAVELET
Orthogonality Biorthogonality Linear Phase Vanishing moments
Haar
Yes
Yes
Yes
1
Db M
Yes
Yes
No
M
Bior. Mr Md
No
Yes
Yes
Mr
Symlet M
Yes
Yes
No
M
Coiflet M
Yes
Yes
No
2M-1
Table 3.1: Properties of some wavelet functions.
27
28
Chapter 4 The peak-to-average power ratio problem OFDM modulation allows to transmit information with high data rate through an optical fiber link. However, OFDM can present high signal peaks that can distort the signal. This effect is called PAPR and it is presented and studied in this chapter. Moreover, some PAPR reduction techniques to mitigate its effects are proposed and analyzed for IM/DD systems based on both the FFT and the FHT.
4.1
Peak-to-average power ratio definition
High peak-to-average power ratio is one of the major drawbacks of OFDM signals. The PAPR is defined as the ratio between the maximum peak power and the average power of the transmitted OFDM signal: max |F (k)|2 0≤k≤N L−1 . (4.1) P AP R = E[|F (k)|2 ] In equation (4.1), N is the number of subcarriers, L is the oversampling factor, F is the modulated signal and E[.] denotes the expectation. The PAPR is calculated from the digital signal, meaning that the true maximum value of the O-OFDM signal may not be included in the sampled points. Therefore, we need to introduce an oversampling factor in order to provide sufficiently accurate results in the measure of it [6]. A fourfold oversampling factor (L = 4) is enough to consider the missing peaks. The theoretical limit of PAPR can be derived from equation (4.1) considering k = 0. It can be written in dB as: P AP R = 10log10 N.
(4.2)
Therefore this theoretical limit only depends on the number of subcarriers, and it is fortunately rarely obtained. On the other hand, one of the more common techniques used to measure the 29
PAPR is the Complementary Cumulative Density Function (CCDF). The CCDF is defined as: CCDF = P (P AP R > P AP R0 ) = 1 − P (P AP R ≤ P AP R0 ) = 1 − CDF,
(4.3)
and gives the probability that the PAPR exceeds a threshold (P AP R0 ). The CDF is the cumulative density function. With the CCDF we are able to evaluate the performances of the different PAPR reduction techniques presented in the following section. Expression (4.3) is valid only for a large number of subcarriers (N ≥ 64). For low number of subcarriers, it is not accurate since we can not assume a Gaussian distribution.
4.2
PAPR reduction techniques
A lot of effort has been developed to mitigate the effects of the PAPR. Therefore a number of approaches have been proposed in the literature to deal with this problem [6]. They can introduce distortion to the signal, reduce the available bandwidth or suppose a power increase. So depending on the application, one PAPR reduction technique is more suitable than another [6]. As our purpose is to apply PAPR reduction techniques in IM/DD systems, we must transmit a real signal. Therefore, when the FFT is used the HS must be preserved after applying PAPR reduction techniques. In contrast if the FHT is used, HS must not to be preserved and PAPR reduction techniques are easier to implement. In this section different distortionless PAPR reduction techniques are analyzed. Firstly, we introduce SeLective Mapping (SLM) PAPR reduction technique for both the FFT and the FHT. Afterwards we describe the interleaving technique, Partial Transmit Sequence (PTS) and precoding techniques based on Hadamard Transform (HT) and Discrete Cosine Transform (DCT) only for the FHT as it has reduced complexity.
4.2.1
Selective mapping
SLM is a PAPR reduction technique suitable for a wide range of applications. This method consists of generating OFDM frames representing the same information by multiplying the mapped data by a vector P(u) with u = 1, 2, ..., U [33]. Then IFFT or IFHT is applied to the different signal representations in order to select the one with minimum PAPR. Therefore, U is the number of IFHT or IFFT blocks. In figure 4.1, X(u) and H(u) indicate the u-th OFDM sequence F(u) , modulated by using the IFFT and IFHT, respectively. Vector P(u) (with u = 1, 2, ..., U ), has N elements belonging to the set {±1} as we are working with IM/DD systems. All the elements of P(1) are always set to 1, in order to consider also the original OFDM frame. So when FHT is used we continue having a real signal after multiplying by the vector P(u) . In the case of the real-valued FFT the vector P(u) must preserve HS so we have less freedom in the construction of this vector. The PAPR reduction depends on the number of vectors (U ): increasing this set increases the peak power reduction, but also the number of required IFFT or IFHT blocks and 30
P(1)
P(2)
X(1)/H(1)
IFFT
P(3) Input data
M-QAM S/P
IFHT
IFFT
IFHT
IFFT m-PAM
. . .
. . .
P(U) IFFT
IFHT
Minimum PAPR selection
X(U)/H(U) IFHT
Figure 4.1: Diagram block of SLM PAPR reduction technique.
thus the hardware resources for the system implementation. SLM needs log2 (U ) bits of side information for the correct frame reception [33]. Working with 4 transform blocks, only 2 bits are required to transmit side information to the receiver.
4.2.2
Interleaving
Interleaving technique consists of permuting or reordering the original data to create different sequences that carries the same information; then the one with minimum PAPR is selected, as in the SLM technique [34]. Using the FHT the technique can be easily applied. Whereas with the FFT, HS constraint must be preserved. The original input data vector with N components h = [h0 h1 ... hn ... hN −1 ] becomes h0 = [h00 h01 ... h0n ... h0N −1 ] where the indexes of the vector elements are related by the one-to-one mapping (hn ) → (h0n ). In this case U also denotes the number of transformation blocks and also the number of interleavers. In the particular case of U = 4 the required side information is log2 U = log2 4 = 2bits.
4.2.3
Partial transmit sequence
The main idea of PTS PAPR reduction technique is to divide the original frame in different subvectors h(v) , v = 1, 2, ..., V [6]. V represents the number of FHT blocks. These subvectors are created taking into account that all the subcarriers positions, which are represented in other subvectors, are set to zero. The total number of zeros at the input of each IFHT block is N/V (V − 1). A possible choice for the vector partitioning is based on adjacent selection [35], whereas another implementation is randomly selecting the subvectors [36]. For example, in the case of V = 2 and N = 8, the resulting vectors h(v) obtained after doing adjacent partitions of s = [s1 s2 s3 s4 s5 s6 s7 s8 ] can be: h(1) = [s1 s2 s3 s4 0 0 0 0] and h(2) = [0 0 0 0 s5 s6 s7 s8 ]. Whereas doing random partitions of s, one possible distribution of h(v) vectors can be: h(1) = [s5 0 s2 s4 0 0 0 s1 ] and h(2) = [0 s7 0 0 s6 s3 s8 0]. Once the subvector partition is done, the IFHT is performed and then the output is multiplied by the components of a weighted vector p(u) . Finally the signal is recombined and then the R(p(u) ) with minimum PAPR is selected for 31
transmission. Due to the linearity of the IFHT we can write: (u)
R(p
V X (v) ) = IF HT { p(u) v ·h } v=1
=
V X
p(u) v
(v)
· IF HT {h } =
V X
(v) p(u) v ·H
v=1
v=1
u = 1, ..., U.
(4.4)
(u)
The elements of p(u) are real values in the set {±1} with p0 always set to 1. All the elements of p(1) are set to 1, in order to consider the original vector. Therefore, the total number of optimization vectors is 2(V −1) , and it coincides with the total number of signal representations U . The required side information is (V − 1)log2 (W ) bits, where W is the possible different values p OPTIMIZATION
p(u)1
h(1) IFHT
Input data S/P
BPSK or MPAM
x/h
H(1) p(u)1 (u) 2
p
IFHT
Subvector Partitioning
. .
(v) .
h
p(u)3
IFHT . . .
R(p)
p(u)v
IFHT
H(v) p(u)v
Figure 4.2: Diagram block of PTS PAPR reduction technique.
of the components of the vector p(u) ; for example in this case as p(u) components are in the set {±1}, W = 2. In the particular case of having 4 IFHT blocks (V = 4) and W = 2, the required side information is 3 bits. Compared with SLM and interleaving with 4 IFHT blocks, PTS needs only one additional bit to carry side information.
4.2.4
PAPR reduction techniques using precoding
Here we present different precoding techniques based on two different transforms: the Hadamard Transform (HT) and the Discrete Cosine Transform (DCT). PAPR reduction using Hadamard transform Another alternative PAPR reduction technique is precoding using Hadamard transform. HT is based in the Hadamard square matrix (HN ) of dimensions N × N , which elements are +1 or -1 [37]. Therefore, it is easy to compute. The rows of this matrix are mutually orthogonal, so it is used to lower the correlation relationship of the mapped sequences at the input of the 32
IFHT. So that, the use of this transform reduces the occurrence of the high peaks, compared with the original OFDM frame, adding low computational complexity. One advantage of this PAPR reduction technique is that doesn’t need side information as the Hadamard matrix is already known. The Hadamard matrix of 1, 2 and N orders are: 1 1 1 H1 = (1); H2 = √ (4.5) 2 1 −1 1 HN/2 HN/2 HN = p . (4.6) N/2 HN/2 −HN/2 At the receiver the transmitted signal is recovered by applying the inverse of the corresponding Hadamard matrix (H−1 N ). HT can be used jointly with other PAPR reduction techniques such as SLM to obtain higher reduction of the PAPR. But of course, this is at expense of the computational cost of using two transforms at the same time. Moreover a direct implementation of this scheme with the real-valued FFT is not possible due to the HS constraint. Whereas using the IFHT, the HT can be easily applied. PAPR reduction using discrete cosine transform Another precoding PAPR reduction technique is based on the use of the DCT [38]. The DCT is a real transform that consist of multiplying the data by a cosine. The resulting signal after applying the DCT to the mapped signal is: C
(u)
(k) = w(k)
N X
c
(u)
(n)cos
n=1
w(k) =
√1 N
(2n + 1)kπ 2N
, k = 1, ..., N
(4.7)
k=1
q 2
N
(4.8) 2≤k≤N
Where c(u) (n) is the component n of the vector c(u) with u = 1, ..., U . In this case, using the real-valued FFT we have the same problem than using the HT. We can’t directly apply the DCT without destroying the HS. In contrast, with the FHT, this scheme can be applied without any restriction.
33
34
Chapter 5 Simulations results In this section we evaluate the performance of different distortionless PAPR reduction techniques for IM/DD systems based on the FFT and the FHT using the CCDF. We also introduce the use of wavelets and we study the PAPR of different wavelet functions. Additionally, we evaluate the BER performance of DC biased O-OFDM. Finally, we show the BER performance of distortionless PAPR reduction techniques in O-OFDM systems affected by AWGN noise.
5.1
PAPR evaluation in O-OFDM systems
Here we evaluate distortionless PAPR reduction techniques using the CCDF. We firstly analyze the O-OFDM system depicted in figure 5.7 based on both the FFT and the FHT. For comparison purpose, when the FFT is used, the signal is mapped into M 2 -QAM format, whereas with the FHT scheme a M -PAM format is required, according to [4]. Firstly, we evaluate transforms of N = 64 subcarriers without oversampling and with 4-QAM format when the FFT is used, whereas with the FHT scheme we use a simple BPSK format [39]. Later on, we increase the number of subcarriers to N = 256 with an oversampling factor of 4 (L = 4). With these last parameters, we will evaluate in section (5.2) the performance of distortionless PAPR reduction techniques using the FHT. Finally, we study the PAPR of different wavelet functions for a OFDM system.
35
5.1.1
Comparison of PAPR reduction techniques applied to FFT and FHT based O-OFDM systems
0
10
-1
10
FHT-PTSRand-V=4 FHT-PTS FHT-SLM FHT-PTSRand-V=3 FHT-interleaver FHT
-2
10
-3
10
Pr(PAPR>PAPRo)
Pr(PAPR>PAPRo)
10
3
4
5
6
7
8
9
10
11
12
13
PAPRo(dB)
(a)
10
10
10
0
-1
-2
FFT-SLM FFT
-3
4 (b)
6
8 PAPRo(dB)
10
12
Figure 5.1: CCDF vs. P AP R0 for (a) FHT-based OFDM signals (using BPSK and N = 64) with SLM, interleaving, PTS, random PTS and without any PAPR reduction technique and for (b) FFT-based OFDM signals (using 4-QAM and N = 64) with and without SLM technique.
Since the analyzed optical systems are IM/DD, they require real-valued OFDM signals. Therefore, for implementing SLM and PTS techniques, P(u) and p(u) vectors must be real. The HS required by FFT-based O-OFDM is not preserved by arbitrarily permuting or reordering the input vector (interleaving technique) or by partitioning it into subvectors (PTS technique). Therefore, in order to be applied to FFT-based O-OFDM, these techniques must be modified to force the HS in the interleaving operation or in the subvector partitions, resulting in a restricted choice of the possible combinations. For this reason, we analyze interleaving and PTS only for FHT-based O-OFDM, while SLM technique is applied to both FFT- and FHT-modulated signals, by using P (u) vectors able to preserve the HS. Figure 5.1 (a) shows the CCDF of FHTbased O-OFDM signals for the proposed PAPR reduction techniques. For V = 4 (IFHT blocks), W = 2, and U = 8 at a CCDF of 10−3 , the PTS technique gives a PAPR reduction of 3.1 dB compared to the system without peak-power reduction. With random PTS and the same parameters, we achieve a reduction of 3.7 dB. PTS with random subvector partitions provides the best performance. If the IFHT blocks are reduced to 3 (V = 3), the same PAPR reduction as SLM with U = 4 (4 IFHT blocks) is obtained (2.8 dB). Interleaving provides a lower reduction of the PAPR: by using 4 interleavers (U = 4), the PAPR is reduced of 2.7 dB. SLM technique applied to O-OFDM based on FFT, with the same parameters used for FHT-modulation (U = 4 and same P (u) sequences with values in the set {±1}), gives the same PAPR reduction, as shown in figure 5.1 (b). Increasing the number of subcarriers also the PAPR increases. This evolution is depicted in figure 5.2 (a). On the other hand the oversampling factor (L) is necessary in order 36
0
14.5
10
Pr(PAPR>PAPRo)
PAPR(dB)
14
13.5
13
-1
10
-2
10
L=1 L=4 L=12 L=20
12.5
12 0
-3
500
1000 1500 Number of subcarriers
2000
10
2500
6
(a)
8
10 PAPRo(dB)
12
14
(b)
Figure 5.2: (a) PAPR as a function of the number of subcarriers at a CCDF of 0.1% and (b) CCDF vs. P AP R0 for different oversampling factors for FHT-based O-OFDM system.
to measure more accurately the PAPR [6]. In figure 5.2 (b) it can be observed that at a BER of 10−3 the PAPR increases of 0.5 dB when we use use an oversampling factor of 4 instead of 1. If the signal is oversampled a factor 12 or 20, the CCDF of the PAPR is about the same as the one with L = 4. In fact as demonstrated in [21] with a fourfold oversampling (L = 4) is sufficient to avoid missing peaks of the signal and provide correct measurements of the PAPR. For this reason PAPR technique
1 IFHT
2 IFHT
3 IFHT
4 IFHT
Random PTS
-
1.5 dB
2.4 dB
3.1 dB
PTS
-
0.7 dB
-
2.6 dB
SLM
-
1.5 dB
-
2.4 dB
Interleaver
-
1.5 dB
-
2.4 dB
Table 5.1: PAPR reduction at a CCDF of 0.1%, of different PAPR reduction techniques, varying the number of IFHT blocks in transmission for N = 256 compared to the unmodified signal.
we evaluate the performance of PAPR reduction techniques increasing the number of subcarriers to N = 256 and with a fourfold oversampling factor (L = 4) for 4-PAM O-OFDM system. After analyzing the PAPR reduction using either the FFT or the FHT, we focus our study in the FHT as HS constraint is not required and therefore, PAPR reduction techniques can be applied easily. It can be observed from figure 5.3 that PTS technique with 3 blocks achieves the same PAPR reduction, 2.4dB, than SLM with 4 transform blocks compared to the unmodified signal PAPR (13.3dB). By using 2 IFHT blocks with SLM, interleaving or random PTS technique, the 37
10
Pr(PAPR>PAPRo)
10
10
10
0
-1
-2
FHT-PTSRand-V=4 FHT-PTSadj-V=4 FHT-PTSRand-V=3 FHT-SLM-U=4 FHT-interlever-U=4 FHT-SLM-U=2 FHT-interlever-U=2 FHT-PTSRand-V=2 FHT-PTSadj-V=2 FHT
-3
4
5
6
7
8
9 10 PAPRo(dB)
11
12
13
14
Figure 5.3: CCDF vs. P AP R0 for FHT-based OFDM signals (using 4PAM and 16QAM respectively and N=256) with SLM, interleaving, PTS and random PTS schemes, and without any PAPR reduction technique using 2 and 4 FHT blocks.
probability that the PAPR exceeds 11.8dB is less than 0.1%, resulting in a PAPR reduction of 1.5dB. Using PTS with 2 IFHT the reduction is 0.7dB. All these results are summarized in table 5.1, where it can be seen the PAPR reduction obtained when applying the proposed techniques at a probability of 0.1%, varying the number of IFHT blocks in transmission. It is worth noting that at the receiver side just one FHT block is required.
38
5.1.2
Precoding for PAPR reduction in FHT-based O-OFDM
Here, we propose to use SLM technique, using 2 and 4 IFHT blocks, in combination with precoding using the HT and DCT. There is a trade off between complexity and performance of the system. Figure 5.4 shows the performance in terms of CCDF. Using SLM scheme and HT or DCT, high PAPR reduction is achieved. In order to do a fair comparison we should consider the HT or the DCT as one more transform blocks. So, firstly, we evaluate the case of using 2 blocks for the IFHT and 2 for the HT or DCT in transmission. With DCT precoding technique we achieve a PAPR reduction of 4.4dB. Whereas with the HT the probability that the PAPR is larger than 11.2dB is less than 0.1% (2.1dB reduction) compared to the unmodified signal PAPR (13.3dB). Using HT reduces also the PAPR without adding much complexity (compared to DCT) to the system. As explained in section (4.2.4) the Hadamard matrix is very easy to compute and its elements are 1, −1. At the increase of the number of IFHT blocks, also the number of precoding blocks increases. PAPR technique
4 blocks
8 blocks
SLM-HT
2.1 dB
2.9 dB
SLM-DCT
4.4 dB
4.9 dB
Table 5.2: PAPR reduction at a CCDF of 0.1%, of SLM and precoding with HT and DCT PAPR reduction techniques, varying the number of IFHT blocks in transmission for N = 256 compared to the unmodified signal.
At a CCDF of 0.1% and using 8 blocks in transmission (4 IFHT and 4 DCT), we reduce the PAPR of 4.9dB whereas using 4 IFHT and 4 HT this reduction is 2.9 dB. It is important to note that when precoding techniques with the DCT and the HT are used, one additional transform block is required at the receiver. However, no side information is needed. These results are summarized in Fig. 5.2.
39
Pr(PAPR>PAPR0)
10
10
10
10
0
-1
-2
-3
FHT-SLM-DCT-8blocks FHT-SLM-DCT-4blocks FHT-SLM-HT-8blocks FHT-SLM-HT-4blocks FHT
4
6
8 10 PAPR0(dB)
12
14
Figure 5.4: CCDF vs. P AP R0 for FHT-based OFDM signals (using 4PAM and N=256 subcarriers) with SLM and HT or DCT varying the number of transform blocks and without any PAPR reduction technique.
PAPR of OFDM systems based on the DWPT 4
4
3
3
2
2
1
1 Amplitude
Amplitude
5.1.3
0
0
-1
-1
-2
-2
-3
-3
-4
200
400
600
800 1000 1200 Time samples
1400
1600
1800
-4
2000
(a)
200
400
600
800 1000 1200 Time samples
1400
1600
1800
2000
(b)
Figure 5.5: Temporal OFDM signal based on (a) DWPT with Haar wavelet function and (b) FFT.
In the literature it has been demonstrated that OFDM frames created with DWPT presents lower PAPR than the ones that uses the FFT [32]. DWPT allows a wide choice of wavelets varying some parameters as the number of vanishing moments, biorthogonality properties and linearity of the phase. Moreover O-OFDM based on WPT has already been introduced in optical communications as a promising transformation block [7]. It has been implemented in coherent optical systems. Here in this thesis we do a first step, evaluating the PAPR of a WPT based 40
OFDM signal for further study of this transform in the optical field. In figure 5.5 is depicted an OFDM signal with 2000 samples, using the Inverse Discrete Wavelet Packet Transform (IDWPT) and the IFFT respectively. It can be seen that the OFDM signal based on the FFT presents higher and more peaks, after setting a threshold in 1.5 amplitude value. Therefore wavelets can be 0
10
-1
Pr(PAPR>PAPRo)
10
DWPT-Haar (Daubechies1) DWPT-Bior1.3
-2
10
DWPT-Simlet2 DWPT-Daubechies2 DWPT-Bior5.5 DWPT-Coiflet 2 FFT -3
10
3
4
5
6
7
8
9
10
PAPRo(dB)
Figure 5.6: PAPR of OFDM symbols based on FFT and DWPT
used to reduce the PAPR. We evaluate the PAPR performance of different wavelet functions for bipolar signals. We simulate 1048576 bits mapped into 4-QAM modulation for a N = 16 points IFFT. Figure 5.6 depicts the CCDF of the PAPR of an OFDM signal based on FFT and DWPT using Haar (Daubechies 1), biorthogonal 1.3, biorthogonal 5.5, Coiflet 2, Symlet 2, and Daubechies 2 with an oversampling factor of 4 (L=4). We can see that Daubechies 2 and Symlet 2 give the best PAPR performance of the studied wavelets. In fact the probability that the PAPR exceeds 8.8 dB is very low ( P (P AP R > 8.8) = 10−3 ). Using a IDWPT transformation with Daubechies or Symlet wavelet of order two, the PAPR is reduced of 0.8dB compared to the OFDM signal obtained by an IFFT modulation. However, using a Haar wavelet a PAPR reduction of 0.2 dB for a given probability of P (P AP R > P AP R0 ) = 10−3 is obtained. Biorthogonal 1.3 and Biorthogonal 5.5 wavelets reduce the PAPR of 0.4 dB and 0.5 dB respectively, when comparing with the signal constructed with the IFFT (9.6 dB). The worst value of the PAPR, 9.1 dB, is given with the Coeiflet 2 wavelet when it is compared with Symlet 2 and Daubechies 2. These results are related with the number of vanishing moments (Nv ). Having a Coiflet of order 2 means that 3 moments are canceled. Because both parameters are related with the formula Nv = 2M − 1; Where M is the order of the wavelet. So the choice of the wavelet function becomes crucial in order to reduce the PAPR of the transmitted signal. If we use a biorthogonal and linear phase wavelet, then PAPR is reduced compared to the FFT. Whereas when biorthogonality and orthogonality are preserved and hence there is no linear phase, the number of vanishing moments affects to the values of the PAPR. Having two vanishing moments gives the best PAPR performance. 41
5.2
BER performance of DC biased O-OFDM systems
In this section, we evaluate the performance in terms of BER of the DC biased O-OFDM system based on the FHT depicted in figure 5.7 in AWGN channel. We show how the BER is affected at the varying of the clipping level. Moreover, we present the performance in terms of BER for different modulation formats (BPSK, 4PAM and 8PAM) applying and without applying PAPR reduction techniques and using a 7.3dB clipping level. The goal of this section is to present PAPR reduction techniques as a possible solution to be used in DC biased O-OFDM systems to reduce clipping noise and PAPR.
. . .
. . .
P/S
. . .
Demapper
Direct Detection
N-FHT
Optical channel
S/P
Intensity Modulation
ADC
. . .
DAC
. . .
P/S
. . .
N-FHT
input data
mQAM / MPAM
DC biased O-OFDM system modeled with AWGN channel
S/P
5.2.1
output data
12 10 Amplitude
BIAS DHT-based OFDM transmitter
8 6
DHT-based OFDM receiver
4 2 0 50
100
150 Time
200
250
Figure 5.7: Block diagram of an O-OFDM IM/DD system based on FHT. The signal is limited in amplitude.
Here, we analyze the BER performance of applying SLM, SLM with the DCT and random PTS techniques to the system of Fig. 5.7. These techniques are the best in terms of PAPR -1
10
-2
Bit Error Rate
10
-3
10
-4
10
w/o tech. Eb/N0=25dB SLM Eb/N0=25dB w/o tech. Eb/N0=28dB SLM Eb/N0=28dB 5
6
7
8
9
10
11
12
13
14
15
Clipping level (dB)
Figure 5.8: BER performance at two constant values of Eb /N0 versus clipping level for 8PAM O-OFDM (N = 256) with and without SLM technique in AWGN channel.
42
reduction using between 2 and 4 IFHT blocks as shown in Fig.5.3. The channel is modeled as an AWGN channel. Figure 5.8 shows simulation results of the BER versus the clipping level with 10
0 8PAM 8PAM-SLM-U=2 8PAM-PTS-U=4 8PAM-SLM-DCT-4blocks 4PAM
Bit Error Rate
10
-1
4PAM-SLM-U=2 4PAM-PTS-U=4 4PAM-SLM-DCT-4blocks BPSK BPSK-SLM-U=2
10
10
10
BPSK-PTS-U=4 BPSK-SLM-DCT-4blocks
-2
-3
-4
0
5
10
15 20 EbN0 (dB)
25
30
35
Figure 5.9: Performance of DC biased O-OFDM based on FHT with symmetrically clipping in AWGN channel.
and without SLM PAPR reduction technique for 8PAM format and two different Eb /N0 values. When 8PAM format is used a 7dB clipping level is not enough to ensure a target BER of 10−3 . Therefore, a higher clipping level is needed. According to [4] and [21], 9dB is the optimum clipping level when 8PAM format is used. In order to enhance the power efficiency of the system, PAPR reduction techniques can be applied, as it is depicted in Fig. 5.8. Two values of Eb /N0 are considered: 25dB and 28dB which ensure a target BER of 10−3 and 10−4 , respectively for a 9dB clipping level [26]. Fixing an Eb /N0 = 25dB and using SLM a target BER of 10−3 can be achieved using 8dB clipping level, whereas without applying techniques it is not possible to ensure this BER. The optimum clipping level, using and without using SLM, is 9dB. Increasing the Eb /N0 = 28dB, with 7.3dB and 8.3dB clipping level we are able to guarantee a target BER of 10−3 and 10−4 respectively when SLM is used. On the other hand, when no PAPR reduction techniques are applied, the required clipping levels to ensure the same values of BER (10−3 and 10−4 ) are higher, 7.8dB and 9.2dB respectively. The optimum clipping levels, in this case, is 9.5dB when SLM is used and 10dB when no techniques are applied. Therefore, the use of PAPR reduction techniques reduce the clipping noise, without the need of a higher clipping level. Moreover, the power efficiency of the system is increased. In the following simulations a clipping level of 7.3dB has been considered in order to show the performance in terms BER of applying different PAPR reduction techniques to the system of Fig.5.7. This clipping level has been selected from Fig.5.8, as is the lowest value that ensures a target BER of 10−3 for an Eb /N0 of 28dB and 2 blocks SLM. Figure 5.9 shows the BER improvement after applying SLM with the DCT using 2 IFHT blocks and 2 additional block for the DCT, random PTS with 4 IFHT and SLM with 2 IFHT blocks for different modulation formats (BPSK, 4PAM, 8PAM) and a clipping level of 7.3dB. At a target BER of 10−3 the required bit electrical energy normalized 43
to the noise power spectral density (Eb /N0 ) for BPSK is 15dB, approximately the same for the three techniques. For 4-PAM when SLM with DCT is used, the (Eb /N0 ) required to ensure a target BER of 10−3 is reduced of 1.2dB compared to the BER curve obtained without applying PAPR reduction techniques (Eb /N0 = 19.8dB) . When using random PTS with 4 IFHT blocks this reduction is 1dB. Moreover, if we reduce the number of blocks to 2 IFHT, using SLM, this reduction becomes 0.5dB. Finally, for 8PAM, with SLM-DCT and random-PTS, the floor induced by clipping noise doesn’t occur up to 10−4 BER with only 7.3dB clipping level. It is seen that the techniques perform better for higher modulation orders.
44
Chapter 6 Conclusions and future work 6.1
Conclusions
In this thesis, we have presented the main drawbacks of DC bised O-OFDM systems; the clipping noise and the PAPR. We have proposed solutions to tackle this problem applying distortionless PAPR reductions techniques. Firstly, the PAPR of OFDM signals based on the FFT and on alternative transforms such as FHT and DWPT has been analyzed. It has been seen that the FFT and the FHT based OFDM signals have similar behavior in terms of PAPR. Highlighting that the FHT allows an easier implementation of the techniques because it doesn’t require the HS constraint. We have demonstrated that random PTS is a promising technique that achieves high PAPR reduction. Moreover, we have introduced SLM and precoding using either the HT or the DCT, obtaining high PAPR reduction. SLM with precoding DCT using 2 IFHT achieves good results in terms of PAPR, but at the same time the receiver needs 2 transform blocks ( 1 for the DCT and 1 for the FHT) instead of only 1. Hence, there is a trade off between performance and complexity when choosing the most suitable PAPR reduction technique. Finally, we have presented the improvements in terms of BER performance applying PAPR reduction techniques to DCO-OFDM systems based on the FHT. We have analysed the proposed system in AWGN channel, varying the clipping level. It has been demonstrated that the proposed techniques reduce the clipping noise of the system, achieving a target BER with a lower clipping level and improving the BER and power efficiency of the system.
6.2
Future work
In order to mitigate the PAPR in OFDM systems, DWPT can be exploited for IM/DD systems. Thanks to the wavelet properties low PAPR can be achieved. Hence, applying PAPR reduction techniques will further improve the performance of optical OFDM systems. 45
On the other hand, the FHT offers a wide range of possibilities to implement an IM/DD system. Thanks to its properties, PAPR reduction techniques can be easily applied. For this reason, our work is focused on O-OFDM systems based on the FHT. Our idea is to continue studying the PAPR reduction for all actual implementation in optical OFDM schemes, reducing the required resources and improving the transmission performance. Moreover we are also interested in adaptively modulation techniques that enable to transmit at different bit rates filling the subcarriers with different modulation formats.
46
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