Digital Pre- and Post-Equalizers for
In-Car Data Transmission over
Plasti Opti al Fibers
Von der Fakultät Informatik, Elektrotechnik und Informationstechnik
der Universität Stuttgart zur Erlangung der Würde eines
Doktor-Ingenieurs (Dr.-Ing.) genehmigte Abhandlung
Vorgelegt von
Yixuan Voigt
(geb. Yixuan Wang)
aus Zhuhai, China
Hauptberichter:
Prof. Dr.-Ing. J. Speidel
Mitberichter:
Prof. Dr.-Ing. M. Berroth
Tag der mündlichen Prüfung:
15. Juli 2014
Institut für Nachrichtenübertragung der Universität Stuttgart
2014
This dissertation presents the most important outcome of my research activities con-ducted under the supervision of Professor Joachim Speidel at the institute of telecom-munications, University of Stuttgart, Germany.
Foremost, I would like to express my sincerest appreciation to Professor Speidel, for offering thorough and extremely good feedback on this dissertation and for all his ex-cellent mentorship in the last four years. His influence on me was paramount - not only academically but also personally. He encouraged me to grow as a precision engineering expertise and always supported me to attend international academic conferences which is extremely rewarding in broadening my perspective. He is also a great role model for me in the way that he works hard and behaves gracefully. I owe my deepest gratitude to him.
Special thanks go to my advisory committee members Professor Berroth, Professor Roth-Stielow, Professor ten Brink and Professor Kallfass for their caring and concern about this dissertation.
Special thanks also naturally go to all my colleagues at the institute of telecommunica-tions. Thank you for being not only wonderful colleagues but also my friends. Thank you for being my great support when I am far away from home. Thank you for teaching me German cultures and Swabian dialect, and of course for training me as a "kicker" player. Thank you for the beautiful doctoral hat and the unique poem. Thank you for all the beautiful memories (travel to Berlin, Chinese hotpot, Kara-Okay...). Because of you, I felt for the first time that Stuttgart is my second hometown.
I also want to thank my family and my girls-group. Their prays really helped me going through hard times. To my mom Ning and father Jianfeng, thank you for always telling me that I could achieve anything when I give effort, for actually believing it, for trusting me to do things on my own, for helping me when I couldn’t, and for sending constant love and support across the world for the past 12 years!
Last but least, I would like to give my grateful thank to my husband, Simon. He is really a sweet husband, a big helper and my best friend. The writing of this dissertation would not have been completed without him. I am grateful to Simon not just because he has given up so much time to make my PhD a priority in our lives, but because he considers me as important as himself. Those beautiful flowers and little surprises from him have cheered me up through my ups and downs. I feel so lucky to marry him.
Stuttgart, 2014 Yixuan Voigt
Nomenclature vii List of Symbols xi Abstract xv Kurzfassung xvii 1 Introduction 1 1.1 Motivation. . . 1 1.2 Problem Statement . . . 2
1.3 Previous Works and Suggested Solutions . . . 3
1.4 An Insight into the Considered Equalization Strategies . . . 5
1.5 Outline of the Dissertation . . . 8
2 System Description 11 2.1 The Channel Model . . . 11
2.1.1 Modulation behavior of the standard plastic optical fiber . . . . 12
2.1.2 Optical channel modeling . . . 13
2.1.3 Receiver noise model and the receiver-side SNR estimation . . 15
2.1.3.1 Receiver noise model . . . 16
2.1.3.2 Analytical SNR estimation . . . 20
2.1.3.3 Numerical SNR estimation . . . 21
2.1.4 Experimental setup and results . . . 23
2.1.4.1 Experimental setup . . . 24
2.1.4.2 The measured magnitude responses . . . 24
2.2 Overview of the Transmission System . . . 25
2.2.1 The discrete system model . . . 26
2.2.2 Channel impairments . . . 28
2.2.3 The matched filter bound for band-limited channel . . . 29
3 Equalization of the Optical Channel 33 3.1 General Equalization Techniques . . . 33
3.1.1 Optimal equalization architectures . . . 34
3.1.2 FIR equalizers . . . 41
3.1.2.1 Design of FIR ZF-FFE . . . 41
3.1.2.2 Design of FIR MMSE-FFE . . . 43
3.1.3 Digital adaptive equalizers . . . 44
3.1.3.1 Least mean square (LMS) algorithm . . . 44
3.1.3.2 Normalized least mean square (NLMS) algorithm . . 46
3.1.3.3 Recursive least square (RLS) algorithm . . . 47
3.1.3.4 Comparison of the adaptive algorithms . . . 47
3.2 Design of Transmission Systems with Pre- and/or Post-Equalizers . . . 48
3.2.1 Design of adaptive equalizers . . . 49
3.2.1.1 Structure of the adaptive equalizers . . . 49
3.2.1.2 Configurations of the adaptive equalizers . . . 51
3.2.2 Design of a pre-equalizer . . . 55
3.3 Simulation Results . . . 57
3.3.1 Performances with post-equalization . . . 57
3.3.2 Performances with joint pre- and post-equalization . . . 59
3.3.3 Performances with regard to the MPAM modulation order . . . 62
3.4 Summary . . . 63
4 Tomlinson-Harashima Precoded Systems 65 4.1 Motivation. . . 65
4.2 Principles of the Tomlinson-Harashima Precoding for Channels without Pre-Cursors . . . 66
4.3 Tomlison-Harashima Precoding for Channels with Pre- and Post-Cursors 68 4.4 Design of the Feedforward Equalizer . . . 70
4.4.1 During the start-up . . . 70
4.4.2 After the start-up . . . 74
4.5 Discussion of THP Losses . . . 76
4.6 Evaluation and Simulation Results . . . 80
4.6.1 Comparison of THP-FFE, FFE and DFE at 2 Gbit/s and 3 Gbit/s 81 4.6.2 Effects of a decreased channel bandwidth . . . 83
4.6.3 Performance of the adaptive filter . . . 83
4.6.4 Evaluation of the THP losses . . . 86
4.7 Summary . . . 88
5 Bidirectional Decision Feedback Equalization 91 5.1 Motivation. . . 91
5.2 The System Model . . . 93
5.3 Finite-length BiDFE Algorithms . . . 96
5.3.1 Symbol-wise arbitrated bidirectional arbitrated DFE . . . 97
5.3.2 Block-wise arbitrated trellis-based conflict resolution . . . 97
5.3.3 The novel trellis-based BiDFE algorithm . . . 99
5.5 Simulation Results of the BiDFEs . . . 103
5.5.1 Symbol-spaced BiDFEs . . . 103
5.5.2 Fractionally-spaced BiDFEs . . . 105
5.5.3 Evaluation of the computational complexities in the reconstruc-tion and arbitrareconstruc-tion stage . . . 105
5.6 Summary . . . 107
6 Conclusions 109
6.1 Contribution Summary . . . 109
6.2 Future Directions . . . 111
A Minimum-phase Spectral Factorization 113
AWGN additive white Gaussian noise
BAD bi-directional arbitrated decision feedback equalizer
BER bit error rate
BiDFE bi-directional decision feedback equalizer CBA contradictory block arbitration
DC direct current
DFE decision feedback equalizer DMT discrete multi-tone modulation DSP digital signal processing DVB digital video broadcasting
EDS effective data sequence
EMI electro-magnetic interference
FBF feedback filter
FEC forward error correction FET field-effect transistor
FFE feedforward equalizer
FFF feedforward filter
FFT/IFFT fast Fourier transform/inverse fast Fourier transform
FIR finite impulse response
FS fractionally-spaced
FS-DFE fractionally-spaced decision feedback equalizer FSE fractionally spaced equalizer
FS-FFE fractionally-spaced feedforward equalizer i.i.d. independent and identically distributed IIR infinite impulse response
LC-BiDFE linear combining bidirectional DFE
LD laser diode
LE linear equalizer
LED light emitting diode
LLR log-likelihood ratio
LMS least mean square
MF matched filter
MFB matched filter bound
ML maximum likelihood
MLSE maximum likelihood sequence estimation
MMSE minimum mean squared error
MOST media oriented system transport MPAM M-ary pulse amplitude modulation
MSE mean squared error
NLMS normalized least mean square
OFDM orthogonal frequency division multiplexing
PD photo-diode
PIN PD positive-intrinsic-negative photo-diode POF plastic optical fiber
PSD power spectral density
RLS recursive least square
PAM pulse amplitude modulation
PMMA poly-methylmetacrylate
QAM quadrature amplitude modulation
RE recursive equalizer
RS Reed-Solomon
SER symbol error rate
SINR signal-to-interference-plus-noise ratio SI-POF step-index plastic optical fiber
SNR signal-to-noise ratio
SS symbol-spaced
SS-DFE symbol-spaced decision feedback equalizer
SSE symbol spaced equalizer
SS-FFE symbol-spaced feedforward equalizer
THP Tomlinson-Harashima precoding
TBCR trellis-based conflict resolution TB-BiDFE trellis-based bi-directional DFE VSLMS variable step size LMS
WLAN wireless local area network
a[k] M-ary pulse amplitude modulated symbol sequence . . . .27
at[k] the training sequence . . . .73
A linear fiber loss . . . .13
Ah a real-valued constant . . . .39
b[n] bit sequence after the Reed-Solomon encoder . . . .27
b′[m] information bit sequence . . . . 27
B receiver bandwidth . . . .13
B3dB 3 dB modulation bandwidth . . . .13
B(z) the product of H(z) and W (z) . . . .68
Ca input capacitance of an amplifier . . . .16
Cd capacitance of a photo-diode . . . .16
CT total capacitance of a photo-diode and an amplifier . . . .17
C a selection matrix . . . .42
d the minimum distance between two constellation points . . . 28
e[k] the error between the equalized sample and a reference sample . . . .43
Eh the energy of the channel impulse response . . . .30
FB(z) the discrete-time feedback filter in the DFE . . . .38
FF(z) the discrete-time feedforward filter in the DFE . . . .38
Fn amplifier noise figure . . . .18
G(z) a causal, monic and minimum-phase discrete-time filter . . . .39
G0 the amplifier gain . . . .16
h[k] the discrete-time channel impulse response . . . .26
ha(t) the channel impulse response . . . .14
he[k] the estimated discrete-time channel impulse response . . . .73
H(z) z-transform of h[k] . . . .34
He(z) the estimated channel transfer function . . . .69
Hpof(f ) the POF modulation transfer function . . . .12
HR(f ) the receive filter . . . .34
h the discrete-time channel impulse response vector . . . .28
he the estimated discrete-time channel impulse response vector . . . 96
H a matrix containing the channel information . . . .42
ia the amplifier noise current . . . .16
in the total noise current . . . .20
ip the signal current generated by a photo-diode . . . .16
it the thermal noise current . . . .17
is the shot noise current . . . .16
I[k] the inter-symbol-interference . . . .28
Idark the dark current generated by a photo-diode . . . .18
Ip the average signal current generated by a photo-diode . . . .18
i the desired discrete-time channel impulse response . . . .71
I an identity matrix . . . .73
J the mean-squared-error . . . .43
Jo the minimum MSE . . . .45
kB the Boltzmann’s constant . . . .18
K the oversampling factor . . . .49
Lc length of the discrete-time channel impulse response . . . .26
Lcontr the mean length of the conflict events . . . .102
Le length of the estimated discrete-time channel impulse response . . . .96
Lpof the fiber length . . . .13
M the modulation order of MPAM . . . .27
N the length of the data sequence . . . .73
N0 the single-sided PSD of the AWGN . . . .30
Ncontr the mean number of the conflict events . . . .102
Nm an arbitrary integer number . . . .67
Ny the length of the received sample sequence . . . .94
n[k] AWGN sample . . . .27
n′[k] the colored noise sample . . . .34
nf the length of a FIR feedforward filter . . . .42
nb the length of a FIR feedback filter . . . .50
p the pre-cursors ISI vector . . . .71
Po the root mean square optical power . . . .20
Ps the symbol error rate . . . .30
PS,M AP Mthe MPAM symbol error rate . . . .80
Pb the bit error rate . . . .30
q the elementary charge . . . .18
Q the Q-function . . . .80
r[k] the reference sample . . . .43
Ra the amplifier input resistance . . . .16
Rb the detector bias resistor . . . .16
RT the parallel resistance of Rband Ra . . . .17
R0 the responsivity of a PIN photo-diode . . . .21
Ry the correlation matrix of y[k] . . . .43
s[k] it equals either a[k] or x[k] . . . .79
tc the central time . . . .14
t0 the sampling delay . . . .26
T the sampling duration . . . .26
T0 the absolute temperature . . . .18
Ts the symbol duration . . . .26
v[k] the effective data sequence . . . .67
w the coefficient vector of W (z) . . . .42
w[k] the coefficient vector of a time-variant FIR filter . . . .45
wo the optimum coefficient vector . . . .43
W half of the window length . . . .68
W (z) a linear post feedforward equalizer . . . .68
WD(z) a discrete-time filter in FF(z) . . . .38
x[k] the transmit signal . . . .27
x[k2 k1] indicate a sequence (x[k1], x[k1+ 1], · · · , x[k2]) . . . .99
ˆ xF[k] the decided symbol of the forward DFE in a BiDFE . . . .95
ˆ xR[k] the decided symbol of the reverse DFE in a BiDFE . . . .95
y the received sample vector . . . .43
y[k2 k1] indicate a sequence (y[k1], y[k1+ 1], · · · , y[k2]) . . . 99
˜ y the time-reversed received sample vector . . . .43
y′[k] the received sample after the received filter . . . .34
ˆ yF the estimated received sample by the forward DFE in a BiDFE . . . .96
ˆ yR the estimated received sample by the reverse DFE in a BiDFE . . . .96
ˆ yF the estimated received sample by the forward DFE in a BiDFE . . . .96
ξ2 the Euclidean distance . . . .99
ξa the mean power of the MPAM data sequence . . . .28
ξx the mean power of the x[k] . . . .29
γSIN R the signal-to-interference-plus-noise ratio . . . .29
γmodulo the THP modulo loss . . . .80
γprecoding the THP precoding loss . . . .79
λ the forgetting factor . . . .47
λmax the largest eigenvalue of the correlation matrix Ry . . . .46
µ the step size of LMS . . . .44
˜ µ the step size of NLMS . . . .46
µa the mean value of a[k] . . . .78
µs the mean value of s[k] . . . .79
µx the mean value of x[k] . . . .78
Φnn PSD of the AWGN . . . .30
Φn′n′ PSD of the colored noise . . . .34
σ standard deviation of the POF impulse response . . . .13
σn standard deviation of the AWGN . . . .27
σ′ n standard deviation of the colored noise . . . .36
σ2 a variance of a[k] . . . .77
ˆ σ2 n the estimated variance of the AWGN . . . .73
σ2 s variance of s[k] . . . .79
σ2 x variance of x[k] . . . .77
Lately, a hot topic in the automobile industry is the development of the in-vehicle infotainment communication network based on the media oriented system transport (MOST) standard, where a cost-effective optical physical layer composed of light emit-ting diodes (LED), plastic optical fibers (POF) and positive-intrinsic-negative photo-diodes (PIN PD) is used by the in-car network. The latest MOST150 standard has specified a transmission speed of 150 Mbit/s, while the next MOST generation is tar-geted at multi-Gbit/s. Obviously, the very limited bandwidth of the current physical layer will weigh on the future MOST generations. However, it is important to evaluate the potential of the current physical layer, for the reason that the car-manufacturers may continue using the low-cost and easily operable POFs and LEDs.
The objective of this dissertation is to increase the data-rate for the next MOST gen-eration from 150 Mbit/s to 2 ∼ 3 Gbit/s, based upon the current MOST150 optical physical layer. The main emphasis lies in investigating electronic signal processing techniques to detect the multi-level pulse-amplitude modulated (MPAM) signal trans-mitted through the noisy dispersive POF-based optical channel. To be specific, four different transmission schemes are studied respectively: the post-equalization scheme using either linear or decision-feedback equalizer, the joint pre- and post-equalization scheme, the non-linear Tomlinson-Harashima precoding (THP) scheme, and the bidi-rectional decision feedback equalization (BiDFE) scheme. In the BiDFE scheme, a novel trellis-based BiDFE (TB-BiDFE) equalizer is proposed. Their performances are investigated by means of theoretical analysis and computer simulations. As will be shown, with the help of electronic equalizers and error-correcting code, the final bit-rate is able to reach 3 Gbit/s over a 10 m standard step-index POF, despite the use of a low-cost LED transmitter.
In den letzten Jahren verstärkte sich der Trend im Automobilbereich, optische Netz-werke für Information, Kommunikation und Unterhaltung (Infotainment) im Fahrzeug einzusetzen. Eine erste weltweite Standardisierung erfolgte mit dem Media Oriented System Transport und einer Bitrate von 25 Mbit/s (MOST25). Kürzlich wurde das System MOST150 spezifiziert, das eine höhere Bitrate von 150 Mbit/s erlaubt. Beide Systeme setzen auf der Bit-Transportschicht (Physical Layer) kostengünstige optische Komponenten, wie lichtemittierende Dioden (LED), optische Plastikfasern (POF) und preisgünstige Photodioden ein. Durch die stürmischen Entwicklungen des Internets, des digitalen Fernsehens mit hoher Auflösung, der Navigation und dem vielfältigen Einsatz von Videokameras in Fahrzeugen wird der künftige Bitratenbedarf sehr stark ansteigen. In der vorliegenden Dissertation werden daher Übertragungsverfahren untersucht, die eine um etwa Faktor 20 höhere Bitrate erlauben sollen, d.h. ca. 3 Gbit/s zur Verfü-gung stellen. Da die Systeme weiterhin kosteneffizient bleiben müssen, wird als harte Randbedingung formuliert, dass derselbe Physical Layer wie bei MOST150 Anwen-dung finden kann. Besonders die begrenzte Bandbreite der optischen VerbinAnwen-dung stellt für hohe Bitraten einen kritischen Erfolgsfaktor dar. Da spezielle optische Komponen-ten in den nächsKomponen-ten Jahren immer noch deutlich teurer sein werden als die Elektronik, wird in dieser Arbeit der Schwerpunkt auf elektronische Verfahren der Codierung, Mo-dulation und Signalverarbeitung im Sender und Empfänger gelegt. Aus Gründen des Aufwands kommen statt komplexer Modulationsverfahren nur M-stufige Puls-Amp-lituden-Modulation (MPAM) in die engere Wahl. Darüber hinaus bilden sich vier ver-schiedene Verfahren heraus, die näher untersucht werden: Erstens, die digitale Vorent-zerrung im Sender, bei der entweder ein linearer oder ein entscheidungsrückgekoppel-ter Entzerrer verwendet wird; zweitens, eine Kombination aus einem Vorentzerrer beim Sender und einem Entzerrer beim Empfänger; drittens, das nichtlineare
Vorentzerrungs-und Codierverfahren nach Tomlinson-Harashima Vorentzerrungs-und viertens ein bidirektionaler ent-scheidungsrückgekoppelter Entzerrer (BiDFE) im Empfänger. Dabei wird auch ein neuartiges Trellis-basiertes BiDFE-Verfahren vorgeschlagen. Die Leistungsfähigkeit aller Varianten wird, wenn möglich analytisch, in jedem Fall auch durch Rechnersimu-lation ermittelt. Kanalmodelle werden durch Messungen im Labor bestätigt.
In der vorliegenden Arbeit wird gezeigt, dass durch effiziente Modulations-, Codier-und Entzerrungsverfahren aufwandsgünstige elektronische Schaltungen möglich wer-den, um auf einem bestehenden Physical Layer einer Standard-Stufenindex-Plastikfaser von etwa 10 m Länge Bitraten bis ca. 3 Gbit/s zu erzielen.
Introdu tion
1.1 Motivation
Since 30 years when the first vehicle networks were built to exchange control in-formation between car devices, car-makers are dedicated to improve the transmission speed of these networks. One of the well-known networks is the controller area net-work (CAN), which operates at a transmission rate of up to 1 Mbit/s in order to transmit control signals [1]. Over the last few years, the rapidly developed automobile info-tainment applications greatly increased the number of automobile devices involved, for example, many cars are equipped with the high-grade sound system, navigation system, telephone systems, DVD changers and the voice operation [2]. These devices are in great demand of interacting with each other and exchanging not only control but also audio and video signals. However, CAN is no longer adequate to support the required fast communication among them because of its narrow bandwidth. Then, the media oriented systems transport (MOST) infotainment backbone was designed to meet this specific need. MOST backbone is based on a cost-effective optical physical layer that consists of red light emitting diodes (LED), standard poly-methylmetacrylate step-index plastic optical fibers (PMMA SI-POF) and PIN photo-diodes (PD). The optical physical layer connects various infotainment automobile devices in a single-fiber unidirectional or in a two-fiber bidirectional optical ring.
One of the most obvious benefits of adopting an optical physical layer for the in-vehicle communications network is its non-sensitivity to the electro-magnetic interfer-ence (EMI) generated by many noisy electronic devices in the automobile environment. In contrast to the commonly used glass fiber, plastic optical fiber (POF) possesses many unique advantages and is especially suited for short-range transmission: apart from
having lower-cost and lighter-weight, the large core diameter of POF enables an easier installation and more robust coupling of light sources into the fiber. Moreover, POF is more mechanically flexible as well as simpler to fabricate [3]. These characteristics make POF a quite appropriate transmission media for the MOST networks. In addition to POF, MOST utilizes the LED transmitter, which is also better suited for automo-bile applications than the laser diode, thanks to their features like less expensive, less sensitive to temperature variations, simpler to modulate and more reliable [4].
The very first MOST generation was the MOST25 infotainment backbone, which was equipped in BMW 7-series in the year 2001 and is able to transfer data 25 times faster than the CAN. Later on, many other car manufactures such as Mercedes, Audi, and Volvo followed [2]. Since then, POF-based in-vehicle communication networks have been well accepted and have become popular. The first upgrade of MOST25 was the MOST50, where the transmission speed was accelerated to 50 Mbit/s. It was then upgraded to the state-of-the-art MOST150 operating at 150 Mbit/s. As a matter of fact, despite MOST150 is 3 times faster than MOST50, it is still incompetent for supporting those bandwidth-demanding infotainment applications, such as parallel transmission of HD-videos, multiple side and front view cameras, camera-based driver assistance systems or high-speed WLAN. With the purpose of serving these applications, a 2 to 3 Gbit/s transmission capability is of great interest for the next MOST generation, which means, a 10 to 20 times improvement in the current transmission capability is needed. Equally important as increasing the data-rate, the migration from MOST150 to its future generation should be smooth for cost reasons [2]. Upgrading the current optical phys-ical layer to the Gbit/s range by glass fibers and lasers is more expensive, compared to the use of sophisticated electrical signal processing techniques at the transceivers on the basis of the already existing MOST150 optical networks with POFs and LEDs. On ac-count of this consideration, this dissertation centers on designing and investigating var-ious electrical signal processing techniques, so as to meet the need for a cost-effective high-speed physical layer in the automobile communications network. With these tech-niques, it is expected that the speed limit on the MOST150 can be successfully raised to 3 Gbit/s, despite the use of bandwidth-limited LED and POF.
1.2 Problem Statement
Although MOST150 optical physical channel has the potential for high-speed data transmission over short distances, we are faced with a few problems.
From a communication point of view, the first problem to consider is the channel band-width that is a measure of the information-carrying capacity of the given channel. In the MOST150 optical physical layer, the LED transmitter might decrease the signal band-width and distort the signal through non-linearity. Similarly, the transmission media POF is bandwidth-limited, and so is the PD receiver. Consider the cascade of LED, POF and PD as an overall channel, its bandwidth is below 140 MHz for a 10∼15 m long POF. If such a channel is used for 2∼3 Gbit/s data transmission, it is not surpris-ing that the current detected optical pulse will be heavily corrupted by previous and post received optical pulses arising from the inter-symbol-interference (ISI) distortion, and become indistinguishable at a conventional receiver. Hence, the first task we are confronted with is to overcome the ISI term.
Another problem that influences the transmission quality is the channel noise composed of thermal noise and shot noise. Because noise may disturb the received signal ampli-tude and increase the bit error rate (BER), it should be handled simultaneously with ISI.
In brief, the intended transmission at multi-Gbit/s over the MOST150 optical physi-cal layer will be corrupted by strong ISI and noise, whose negative impacts must be mitigated to ensure a reliable transmission.
1.3 Previous Works and Suggested Solutions
There are various previous researches dedicated on improving the data-rate for short-range optical communications systems by the use of electronic signal processing tech-niques. Previous works on this topic mainly offer three solutions: one solution is to use a linear prefilter/peaking at the transmitter for enlarging the transmission band-width, another solution is to use optimized bandwidth-utilization techniques such as the base-band orthogonal frequency division multiplexing (OFDM) technique, and the third solution is to use post-equalization techniques together with the multi-level pulse amplitude modulation signaling. For example, a 1.25 Gbit/s transmission utilizing pre-filtered four-level pulse amplitude modulation (4PAM) signal and fractionally-spaced (FS) post-equalization is presented in [5]. Nevertheless, the prefilter scheme becomes impractical for transmissions at multi-Gbit/s, because the amplitude range of the pre-distorted input signal increases along with the data-rate and consequently, a large direct current (DC) offset is required in the LED transmitter, which leads to a large power consumption and a decrease in the receiver’s dynamic range. In [6], 1 Gbit/s transmis-sion was demonstrated using the discrete multi-tone modulation (DMT) or the so-called
base-band OFDM. However, it is well known that DMT technique requires high linear transceivers over a wide input optical power range, which is very difficult for the circuit design especially at a high rate of speed. Besides, the quadrature amplitude modulation (QAM) modulator/demodulator and fast Fourier transform/inverse fast Fourier trans-form (FFT/IFFT) components needed by DMT noticeably raise the system cost and complexity. Furthermore, most systems employing the third solution operated at either lower speed or, if around 3 Gbit/s, costly laser diodes must be used as fast transmit-ters. Up to now, there is no report to our knowledge of utilizing a simple optical set-up based on LED and PIN for high data-transfer rates in the multi-Gbit/s range, since lit-tle effort was done previously to design an advanced equalizer whose performance is good enough to combat the strong ISI generated. Therefore, this dissertation intends to achieve this goal.
Despite of that, before looking for solutions in the electronic domain, solutions in the optical domain are looked into. To cope with ISI, there are approaches that use lenses or similar principles to restrict the modes excited in a fiber. Because the number of modes involved in a transmission is reduced, the modal dispersion becomes less prob-lematic and the bandwidth of the fiber can be increased. This approach, however, might have a large power loss because all higher-order modes will be rejected. Moreover, even if the bandwidth of POF is successfully extended, the bandwidth of LED will still restrict the speed of the overall optical link. Another drawback of the optical-domain implementation is that the installation and maintenance of the optical devices might be expensive and inconvenient. By contrast, the use of conventional optical transceivers together with electronic signal processing techniques can be quite advantageous. In one aspect, it is more flexible: if later on there is a change in the fiber length, the same optical transceivers can be kept without redesigning the system layout but by adjusting the corresponding electronic signal processing part. In another aspect, it is more sta-ble: after the transmission link is established, most slowly time-varying factors in the channel such as the operating temperature or aging degradation of the optical devices can be captured and compensated by a digital adaptive equalizer. For the reasons men-tioned above, the optical-domain approaches is finally dropped, and this dissertation concentrates on developing solutions in the electrical domain.
For simplicity and cost-reasons, the LED transmitter is modulated by intensity modula-tion, where the information is described by the intensity of a carrier light. Correspond-ingly, the PD receiver performs a non-coherent direct detection. In addition, with the intention to reduce the equalization complexity, the bandwidth efficiency is enhanced by means of modulation techniques. Among which, the classical multi-level pulse am-plitude modulation scheme (MPAM) is chosen, for it is probably the most simple and
adequate method that can be combined with the LED intensity modulation to increase the spectrum efficiency. For example, by use of 8PAM, the bandwidth efficiency is improved three-fold, so the remaining equalization part can be less complex. A short introduction about the equalizer design will be discussed in the upcoming section.
1.4 An Insight into the Considered Equalization
Strategies
Among all kinds of equalizers, maximum-likelihood sequence estimation (MLSE) [7] is the most powerful joint equalization and detection technique in the field of combating ISI. It minimizes probability of errors by applying the trellis-based Viterbi algorithm [8]. However, the complexity of the Viterbi algorithm grows exponentially with the channel order and the order of the modulation scheme. For the considered transmission, the MLSE equalizer is nearly impractical to be implemented because of its high complexity. By contrast, the linear feedforward equalizer (FFE) is well known for its simplicity but with sub-optimal performance. Thanks to the linear transversal filter structure, its complexity in proportion to the filter length grows only linearly with the channel order. For a channel introducing weak to moderate ISI, its performance is often sufficient. However, the linear equalizer enhances the noise in the process of suppressing ISI. So eventually, as the channel distortion becomes severe, the performance of a linear equalizer can be limited by the noise enhancement in an obvious manner. Another widely used sub-optimal equalizer is the decision-feedback equalizer (DFE) [9, 10]. It improves the performance of a linear equalizer by employing a non-linear structure, where a feedback filter (FBF) and a decision device are used in addition to a feedforward filter (FFF). Assume correct decisions, the previous ISI can then be subtracted from the current symbol by feeding back the previously decided symbols through the FBF. The FFF suppresses the contribution of the precursor ISI, although it enhances the noise at the same time, the noise magnification is not that severe as in the case of a linear equalizer. Besides, the noise can be eliminated at the decision device at the cost of decision errors.
Both FFE and DFE can apply the zero-forcing (ZF) or minimum-mean squared error (MMSE) [11,12] criterion to calculate their coefficients based on the channel informa-tion. ZF aims at completely removing the ISI regardless of possible noise enhancement, while MMSE criterion provides a better trade-off between the noise enhancement and the ISI elimination. A pre-condition of performing these criteria is to know the chan-nel information. Yet in reality, the chanchan-nel characteristics are usually unknown, and in
many cases, are time-variant. In such situations, the equalizer coefficients shall be able to converge automatically to a ZF or a MMSE solution, irrespective of what their ini-tial values are. Meanwhile, the equalizers shall be adapted such that any time variation in the channel will be compensated. Therefore, the application of adaptive equalizers is also an emphasis of this dissertation. Among the most popular adaptive algorithms, the least mean square (LMS) algorithm and its variations are considered due to their simplicity. Compared to LMS, the recursive least squares (RLS) algorithm has a faster convergence rate and more fidelity, yet at the expense of an increased computational load. In typical applications, the adaptive equalizer starts with a training mode to gather information about the channel, and later on switches to the decision-directed mode to follow the moderate variations during the transmission.
The deduction of the DFE design typically presupposes correct past decisions. How-ever, once a decision error is made, this error will propagate in the FBF, which increases the error probabilities of the subsequent decisions and may eventually cause error bursts. This well-known error propagation effect is more serious when the tap weights and/or the number of the feedback taps become larger [13]. In order to overcome the prob-lem, one of the commonly used precoding techniques at the transmitter [14,15,16,17], the Tomlinson [18]-Harashima [19] precoding (THP), can be applied. It requires chan-nel information at the transmitter, so an up-link is a requisite for sending the estimated channel information backwards from the receiver. In this dissertation, THP is combined with a feedforward equalizer at the receiver to construct, depending on the FFE type, the zero-forcing THP (ZF-THP) [20,19], the MMSE-THP [21] or the adaptive THP-FFE. DFE has another drawback except for the error propagation, that is, the sub-optimal performance in comparison to the matched filter bound (MFB) [22]. The MFB is de-fined as the signal-to-noise ratio (SNR) at the receiver output under the assumption that a matched filter receiver is used and only one symbol is transmitted. Through sending a single symbol, the transmission is not affected by ISI despite of a band-limited channel, so the use of the matched filter can provide the maximum output SNR and the minimum symbol error rate (SER). Under this circumstance, the MFB provides a lower bound on SER [23,24] and a pseudo-bound on BER. It is thus a good figure of merit to evaluate the BER performance of an equalizer.
To cope with the limitation of combating severe ISI and the error propagation problem in a DFE, we are motivated to search for an alternative suboptimal nonlinear equalizer with low computational complexity. The new type of equalizer is the bi-directional DFE (BiDFE) [25,26,27], which improves the performance of a single DFE without paying
Equalization
Linear DFE MLSE/MAP
Error Propagation Gap from MFB
Decision Device Optimization Filter Length Optimization Bidirectional DFE Non-causal DFE Computational complexity Low Low High
Performance Poor Moderate Good
Impairments
Figure 1.1: The equalizationproblem
the price for an up-link as in the case of THP. This approach brings in the receive di-versity by use of two post DFEs in parallel at the receiver: it uses a reverse-mode DFE for equalizing the time-reversed received signal and simultaneously, a forward-mode DFE for equalizing the received signal as in the conventional way. BiDFE is able to mitigate the error propagation effect because in the two DFEs, decisions are made in opposite directions and thus the decision errors will propagate in opposite directions, consequently the most erroneous locations in the decisions of both DFEs are different. Moreover, even without decision errors, the performance of a BiDFE is superior to a single DFE because the noises at the outputs of both DFEs (before the decision device) exhibit a low correlation with each other. That means, a diversity combining of the decisions from both DFEs can be performed for improving the reliability of the results. As proposed by [28], a weighted linear combination of the soft decisions from both DFEs resulted in a smaller value of noise enhancement in comparison to either of the two constituent DFEs. In addition, it has been shown that when both DFEs are allowed to have infinite length, the linear combined BiDFE (LC-BiDFE) will be capable of at-taining the matched filter bound under the ideal feedback assumption. However, in the presence of error propagation and if the filter length is constrained, the LC-BiDFE then will perform worse than another variety of BiDFE - the bidirectional arbitrated DFE (BAD) [29]. BAD combines the decisions of both DFEs using a reconstruction based arbitration technique with a higher computational cost. To reduce the complexity of BAD, the trellis-based conflict resolution (TBCR) [30] algorithm is investigated there-after. Motivated by TBCR, a novel BiDFE technique - the trellis-based bi-directional DFE (TB-BiDFE) - is proposed in this dissertation, which is able to provide superior performance than both BAD and TBCR and at the same time being less complex than BAD.
Figure 1.1 summarizes the equalization problems. This dissertation is concerned pri-marily with the techniques marked with solid boxes. Techniques marked with dashed boxes are not taken into consideration, because they are inconvenient to implement in the scope of automotive environment.
1.5 Outline of the Dissertation
Chapter 2 begins with modeling the modulation transfer function of the plastic opti-cal fiber. The overall optiopti-cal channel model including the LED transmitter and the PD receiver is subsequently presented with respect to its electrical transfer function and 3 dB modulation bandwidth. Based on the channel model, a proper assessment of the signal-to-noise ratio, which is later on necessarily required for evaluating the BER per-formances, is carried out by use of a more comprehensive electronic equivalent receiver model. The end-to-end transmission system is then described, and the transmission im-pairments that must be handled by equalization are indicated. After that, the matched filter bound is deduced to provide a lower bound on the BER.
To manage the transmission impairments, Chapter 3 first outlines the conventional equalization strategies and a class of adaptive algorithms on the basis of the mean squared error (MSE) criterion. The adaptive algorithms are essential for compensating the variations in channel characteristics, or for performing equalization without know-ing the precise channel information. To be specific, the least mean square (LMS), the normalized LMS (NLMS), the variable step size LMS (VSLMS) and the recursive least squares (RLS) adaptive algorithms are considered. Their characteristics including the speed of convergence and computational complexity are analyzed as well.
In the second part of Chapter3, performances of two transmission strategies, which are the pre- and post-equalization strategy and the post-equalization strategy, are examined through computer simulations. The equalizers are developed based on either symbol-spaced or fractionally-symbol-spaced structure and their coefficients are computed via LMS and RLS adaptive algorithms. The resulting BER curves and the computational complexities are compared with each other, with the aim to find out a fair trade-off between the performance and the complexity. Optimization in some key parameters of the adaptive equalizers is also demonstrated.
The performance of the two transmission strategies in Chapter3is limited either by the increased transmit power or by the error propagation problem. To surpass the limited
performance, an alternative transmission strategy employing the nonlinear THP is stud-ied in Chapter 4. In this chapter, THP at the transmitter is used together with a linear feedforward equalizer at the receiver, whose taps are either optimized by the MMSE criterion or automatically adjusted by the normalized LMS algorithm. For analysis purpose, the THP losses in comparison to an ideal DFE without error propagation are particularly elucidated.
In Chapter5, a bi-directional DFE used in the electronic part of the receiver is proposed as the fourth type of transmission strategies. The objective of introducing BiDFE is to reach the target BER at a lower SNR comparing to the strategies suggested by the previ-ous two chapters, under the constraint of low to moderate complexity. First, the BiDFE structure and two well-known BiDFE algorithms (BAD and TBCR) are presented. At the next step, one of the key contributions of this dissertation, the novel TB-BiDFE algorithm is presented in detail and explained by examples. In the last section, the simulation results as well as a comparison of the BiDFE complexities are given. In the end, this dissertation is concluded with Chapter6.
System Des ription
To begin with, it is useful to develop a numerical channel model for the POF based MOST150 optical physical layer. Based on the channel model, an overview of the digi-tal optical transmission system and its performance specifications, such as bit error rate (BER), power consumption and complexity, are subsequently provided. This chapter also contains the study of the noises on the channel, and provides the essential informa-tion of the average receiving SNR. Furthermore, a brief descripinforma-tion of an experimental set-up for measuring the modulation transfer function of the optical channel is included. The closing section of the chapter points out the major transmission impairments and the necessity of implementing equalization. For that, a performance upper bound for equalization, called the matched filter bound, is deduced. As it is well beyond the scope of this chapter to treat the optical communication in depth, we concentrate just on what is needed for understanding this dissertation.
2.1 The Channel Model
Despite the use of an optical transmission link, the information sinks are automo-bile electronic devices working with electronic signals. Consequently, the electronic-optical-electronic conversion must be performed. For this reason, we aim to model the end-to-end optical link to its electrical equivalent channel. Then, we estimate the av-erage receiving SNR at the channel output, which is an important parameter that will be used by computer simulations and results validation in later chapters. Finally, the feasibility of the numerical channel model is proven by experimental measurements.
2.1.1 Modulation behavior of the standard plasti opti al
ber
The plastic optical fiber is a type of multi-mode waveguide, since there are usually dif-ferent optical paths propagating along the fiber after an optical pulse is launched by the transmitter. For a step-index type of POF, light on each path arrives at the receiver with a different time delay, leading to an overlap-add of multiple copies of the transmit-ted optical pulse which results in a broadened received optical pulse. This well-known phenomenon is called the modal dispersion, which depends solely on the POF char-acteristics and almost determines the modulation bandwidth (or electrical bandwidth) of the POF. It should be noted that the bandwidth of POF in this dissertation always refers to the 3 dB bandwidth of its modulation transfer function (or the equivalent low-pass transfer function) rather than the optical frequency transfer function. The modula-tion bandwidth of POF can be hardly increased once its length, material, structure and launching condition were specified.
Generally speaking, there exist various types of dispersions in a fiber which limit the electrical bandwidth. They can be categorized into two types: the propagation path dependent modal dispersion and the wavelength dependent chromatic dispersion. Ta-ble2.1 summarizes all kinds of them in terms of the fiber type [31]. Because the con-sidered POF is multi-mode and short in length, material and chromatic dispersion can be ignored for their minor effects on a short fiber. Consequently, the modal dispersion imposes the most significant constraint on the modulation bandwidth for the considered POF type.
Table 2.1: Dispersions inFibers
modaldispersion hromati dispersion
polarization
mode disp.
pro ledisp. material disp. waveguide
disp.
pro le disp.
(singlemode) (multimode) (multi-and
singlemode)
(singlemode) (multimode)
By taking into account the measured values of modal dispersion, MOST150 oPHY automotive physical layer sub-specification [32] has specified that the base-band mod-ulation transfer function Hpof(f ) of a standard step-index POF behaves like a Gaussian
low-pass filter:
Hpof(f ) = Ae−2(πσf )
2
e−j2πf τ Lpof, 0 ≤ f ≤ B
where A is the linear fiber loss, σ = 0.132/B3dB is the standard deviation, B3dB =
1009 · (Lpof
m ) −0.8747
MHz is the 3 dB modulation bandwidth, τ = 4.97 · 10−9 s/m is the
transmission delay per meter, Lpof is the fiber length in meters and B is the receiver
bandwidth.
It should be pointed out that except for [32], some literature have reported different modulation bandwidths for POFs with the same length. Actually, there are many fac-tors affecting the calculation of the theoretical modulation bandwidth, such as the fiber’s numerical aperture, amount of mode coupling, launch condition, temperature and so forth. All these factors influence the amount of modal dispersion and thus the mod-ulation bandwidth. For example, an under-filled launch condition tends to produce a higher bandwidth than an overfilled condition, because only a portion of the modes are excited in the fiber which results in a smaller total modal dispersion. It has been often reported that this limited launching effectively increases the POF’s bandwidth. Because [32] cites the result from measurements carried out according to the MOST150 stan-dard launching condition, we also use (2.1) as the theoretical POF modulation transfer function throughout the dissertation.
2.1.2 Opti al hannel modeling
Figure2.1depicts the end-to-end transmission system including the digital signal pro-cessing (DSP) component by a set of block diagrams. The optical transmission link, which is marked by the dashed box, comprises a red light emitting diode (LED) ana-log transmitter, a common polymethyl methacrylate (PMMA) step-index optical fiber (POF) with 1 mm core diameter, and a PIN photo diode (PD) receiver applying the direct detection. The LED transmitter modulates the intensity of light pulses propor-tional to the electrical signal at the DAC output. The light pulses propagating along the POF get distorted by fiber dispersions and attenuation, and are finally captured and converted by the PD receiver to a proportional electric current, which is then converted to voltage by a subsequent front-end amplifier. If necessary, the voltage signal will be further amplified by a main amplifier. Note that the optical receiver bandwidth must be less than one half of the ADC sampling rate in order to meet the Nyquist sampling criterion. Finally, we define the electrical equivalent representation of the optical link from the DAC output up to the ADC input as the electrical equivalent channel, which is abbreviated as channel in the following.
The channel can be seen as linear if both LED and PD work in their linear response regime. Based on this assumption, the channel frequency response Ha(f ), which is a
Information Symbols
Electrical domain
DAC ModulatorLED
x[k] x(t) Transmitter Offline DSP Electrical domain ADC y[k] y(t) Demodulator PD Receiver POF Channel Optical link
Figure 2.1: Opti al ommuni ation systemmodel
cascade of the LED transmitter, POF and the PD receiver, can be decomposed into a product of three transfer functions: one for the LED transmitter, one for the POF, and one for the PD receiver. Among the three, the inherent bandwidth of LED can be larger than 150 MHz depending on the type and the biasing condition, PD provides normally the greatest bandwidth, yet a POF of length 10 m has a modulation bandwidth of about 135 MHz. Because the transfer functions of LED and PD are normally flat up to the 3 dB bandwidth, and the modulation bandwidth of POF is smaller than the ones of LED and PD, the channel transfer function Ha(f ) can be approximated by the POF transfer
function in (2.1) up to the 3 dB bandwidth:
Ha(f ) = Hpof(f ), 0 ≤ f ≤ B3dB, (2.2)
However, we apply the Gaussian low-pass approximation for the transfer function up to the receiver bandwidth in the dissertation, as this pessimistic approximation provides us more conservative simulation results. Finally, the electrical equivalent channel transfer function is given by:
Ha(f ) = Ae−2(πσf )
2
e−j2πf τ Lpof,
0 ≤ f ≤ B. (2.3)
Correspondingly, the electrical equivalent channel impulse response is: ha(t) = A √ 2πσe −(t−τ Lpof ) 2 2σ2 . (2.4)
ha(t) is a Gaussian impulse with area A (equals to the linear fiber loss) and variance σ2,
0 0.02 0.04 0.06 0.08 0.1 0 0.01 0.02 0.03 0.04 Time [us]
Normalized channel impulse response (10m)
ha (t ) 0 50 100 150 200 −8 −6 −4 −2 0 Frequency [MHz]
Normalized modulation transfer function (10m)
3dB bandwidth 135MHz H a (f )/ H a (0 )
Figure 2.2: The normalized hannel impulse response ha(t) and the hannel transfer fun tionHa(f )(assuming zero phaseshift)basedon a10 mPOF
POF used by MOST automotive applications is 10 to 15 meters long [32], we consider here a 10 m POF whose 3 dB modulation bandwidth is about 135 MHz calculated by using (2.3). Figure2.2shows the normalized theoretical channel impulse response and its transfer function (assuming zero phase shift) for a 10 m POF .
2.1.3 Re eiver noise model and the re eiver-side SNR
esti-mation
Throughout this dissertation, the commonly used BER in digital communications is uti-lized as a figure of merit to evaluate the performance of our digital optical transmission system. BER however, closely relies on the SNR at the receiver which is determined by the noise quantity if a steady signal power is received. Mostly, noise in an optical transmission link comes from the receiver and is a mixture of thermal and shot noise. In this section, noises as well as SNR at the receiver side are investigated.
2.1.3.1 Re eiver noise model
During the process of converting an optical signal into an electrical current, PD gener-ates not only the signal current but also additional noise currents. To predict the amount of noise currents for sake of SNR estimation, a receiver noise model is first established, whose block diagram is shown in Figure2.3.
photo-diode ip G0 front-end amplifier noise filter y(t)
Figure 2.3: Opti al re eiver blo k diagram
Without loss of generality, the optical receiver compromises a photo-detector, a front-end amplifier and a noise filter. The functionality of the preamplifier is providing a low-noise interface for receiving the weak detected photo-current. Ideally, it should be a current-in/voltage-out amplifier with high bandwidth. Unfortunately, most solid-state field-effect transistor (FET) amplifiers are of the in/out or voltage-in/current-out variety [33]. To overcome this problem, a bias resistor Rb can be
con-nected in series to the photo-diode, for converting the photo-current into a voltage signal which can then be amplified by a voltage amplifier. This set-up is known as the high-impedance or low-high-impedance front-end, depending on the value of Rb. Here, we focus
on analyzing the SNR performance for a low-impedance front-end receiver as plotted in Figure2.4, for its wide bandwidth is suitable for high-speed transmission and its worse noise performance delivers a conservative SNR estimation. The noise filter after the amplifier is used to reject the noise power outside the receiver bandwidth.
Then, we draw the equivalent circuit of the low-impedance front-end receiver including various associated noise sources in Figure2.5. Here, the photo-diode is represented by a current source ip, a shunt capacitance Cd and a shot noise source is. Rb is the bias
resistor. And the amplifier has an input resistance Ra, an input capacitance Caas well
as a gain of G0. The noise current generated by the amplifier at its input is indicated
Rb current-to voltage converter (resistor) G0 Amplifier noise filter Vout VB ip
Figure 2.4: Low-impedan e front-endre eiver
po(t) ip′+ in′ 1 Cd Rb Ra Ca G0 ip(t) is(t) it(t) ia(t) Noise Filter y(t) 2
Photo diode Amplifier Noise freeamplifier
Figure 2.5: Smallsignalmodel ofthere eiver
circuit has a total capacitance of
CT = Cd+ Ca, (2.5)
and a total load resistance of
RT =
RbRa
Rb+ Ra
, (2.6)
whose associated thermal noise source is it. Because this front-end has a low-pass
characteristic, its 3 dB bandwidth is determined by the time constant CT · RT, namely:
B ≤ 2πR1
TCT
Obviously, RT should be kept small to ensure a large bandwidth B. This is also why
the front-end is a low-impedance type: it allows the thermal noise to dominate and sacrifices the noise performance to trade for high receiver bandwidth.
The various noise sources associated with the front-end receiver can be described by their equivalent noise currents, which are summarized as follows:
• Shot noise current is(t) from dark current and quantum noise:
the dark current Idark is referred to as the current generated by a photo-diode
even in the absence of an incident light. And the quantum noise comes from the fact that the light flows as discrete photons rather than a continuous fluid [34]. The sum of these two noises gives the total shot noise is in a photo-diode. The
mean-square value of the shot noise can be expressed as: i2
s = 2q(Ip+ Idark)B, (2.8)
where q = 1.602 × 10−19C is the elementary charge, B in Hz is the bandwidth
of the receiver,
Ip =
q i2
p (2.9)
is the root mean square value of ip(t), and ¯ denotes the mean operation. The
power spectral density (PSD) of is(t) is approximately constant (white).
• Thermal noise current it(t) from the detector bias resistor Rb and the amplifier
input resistance Ra. Its mean square value is:
i2 t =
4kBT0B
RT
, (2.10)
where kB = 1.38054 × 10−23J/K is the Boltzmann’s constant, T0 is the absolute
temperature in Kelvin, and RT is the parallel resistance of Rb and Ra given by
(2.6) in Ohm. The PSD of it(t) is constant (white).
• Thermal noise current ia(t) from the front-end amplifier. In general, this noise is
related to the type of the amplifier and its PSD is approximately constant (white) within the receiver bandwidth. For the ease of calculation, here we adopt the amplifier noise figure Fn, which is a parameter that can be used to calculate the
thermal noise without considering the internal structure of the amplifier. The noise figure Fn is referred to as the ratio of the input SNR to the output SNR
G0 (N3) P1 N1 P2 N2
Figure 2.6: Anampli er model
To better explain how to calculate the noise generated by an amplifier by using the noise figure Fn, an amplifier with gain G0 = P2/P1 is plotted on Figure2.6,
where Pν and Nν (ν = 1, 2) represent the signal and the noise mean power at
node 1 or 2, respectively. Now we define the input SNR of the amplifier as P1/N1,
and the output SNR of the amplifier as P2/N2. We also define the noise power
generated by the amplifier at its input as N3. So we could write
N2 = G0· (N1+ N3).
Because Fnis defined as:
Fn= P1/N1 P2/N2 = 1 G0 · N2 N1 = G0· (N1+ N3) G0· N1 = 1 + N3 N1 , Fnequals: Fn = 1 + N3 N1 . (2.11)
In the amplifier circuit in Figure2.5, we consider the noise power at the amplifier input N1 = i2t and N3 = i2a. Then using (2.11) and (2.10), we get the mean square
value of ia: i2 a = (Fn− 1) · i2t = 4kBT0B(Fn− 1) RT . (2.12)
2.1.3.2 Analyti al SNR estimation
The signal-to-noise ratio should be estimated at node 2 in Figure 2.5. However, we could estimate the SNR value at node 1 instead. This approach is feasible because the block with G0in Figure2.5is noise free, and the SNR value at node 2 is the same as at
node 1.
At node 1, we define ip′ as the signal current generated by ip, and in′ as the total noise
current generated by is, itand ia. We also define the total noise current as:
in = is+ it+ ia. (2.13)
Because the noise currents are statistically independent, we obtain: i2
n= i2s+ i2t + i2a. (2.14)
Using (2.8), (2.10) and (2.12), we get i2 n= 2q(Ip+ Idark)B + 4kBT0B RT + 4kBT0B(Fn− 1)) RT = 2q(Ip+ Idark)B + 4kBT0FnB RT . (2.15)
Assume that the PSD of the signal ip(t) is Ip2/2B for |f| ≤ B, and zero elsewhere.
Since the transfer functions between each current source and the current at node 1, respectively, are the same, and the PSD of the signal and noise currents are constant (white) for |f| ≤ B, and the signal and the noise currents are statistically independent with each other, the SNR at node 1 in the frequency range |f| ≤ B results in:
γSN R = i2 p′ i2 n′ = i 2 p i2 n , (2.16) Inserting (2.9) in (2.16) yields: γSN R = I2 p i2 n . (2.17)
The signal current ip(t) generated by a photo-diode varies linearly in proportion to the
received optical power po(t):
with R0the responsivity of a PIN photo-diode. With the root mean square optical power Po = q p2 o(t), (2.19) we get from (2.18): Ip = R0· Po, (2.20)
Inserting (2.20) and (2.15) into (2.17) yields the electrical SNR for the low-impedance front-end receiver with an amplifier at node 1:
γSN R =
(R0Po)2
2q(R0Po+ Idark)B + 4kBRT0TFnB
(with an amplifier) (2.21)
2.1.3.3 Numeri al SNR estimation
To evaluate SNR in (2.21), the receiver bandwidth B should be decided at the first step. As discussed in Section2.1.3.1, a suitable receiver bandwidth should be larger than the bandwidth of LED and POF in cascade, implying that B should be at least 135 MHz, if a 10 m POF is considered. As B is decided by RT according to (2.7), the maximum
load resistor RT that produces a sufficient receiving bandwidth Bmin = 135 MHz is
then:
max{RT} =
1 2πBminCT
. (2.22)
Table 2.2: Transmitter andre eiverbasi spe i ation
LED
(IF: forward urrent)
ondition typi al unit
peakwavelength IF = 10mA 650 nm
output power IF = 20mA 1.8 dBm
rise and fall time 4.8 ns
bandwidth IF = 20mA 110 MHz
PD
(VR: reverse voltage,VF: forward voltage)
ondition typi al unit
dark urrent VR = 10V, T =105
◦
C 300 nm
total apa itan e VF = 1.6V 2.5 pF
where CT = Cd+ Cais the total capacitance of the photo-diode and the amplifier, and
Cd = 2.5 pF is the capacitance of the photo-diode given by Table 2.2. Table2.2 also
lists other relevant parameters of LED and PD for the SNR estimation.
Without loss of generality, we assume an amplifier capacity Ca= 2.5 pF and an
ampli-fier noise figure Fn = 5 dB ≈ 3, which are some typical values of a start-of-the-art FET
amplifier [35]. For Bmin = 135 MHz and a total capacitance CT = 5 pF, the maximum
load resistance RT according to (2.22) is 235 Ω. Normally, it is important to select the
total load resistance RT as big as possible in order to keep the associated thermal noise
at a low level. However, for the sake of conservative SNR estimation, a relatively small value of RT = 150 Ω is chosen, such that the thermal noise power can be relatively
high. Again, according to (2.7), RT = 150 Ω allows a maximum receiver bandwidth of
212 MHz that is higher than the signal bandwidth 135 MHz. Nevertheless, the receiver bandwidth B should to set to 135 MHz. The reason is that the excessive noise power outside the signal bandwidth can be rejected by a subsequent low-pass noise filter after the amplifier. Thus from this point of view, only the signal bandwidth is relevant for the SNR estimation. Consequently, we calculate γSN R in (2.21) with B = 135 MHz and
RT = 150 Ω.
After determining the receiver bandwidth, the next step is to find out the average re-ceived signal power. According to (2.20), its minimum and maximum value can be calculated via R0 = 0.35 A/W, and a minimum received optical power
Pmin = −22 dBm = 6.3 × 10−6W
or a maximum received optical power
Pmax = −2 dBm = 6.3 × 10−4W
defined by [32], respectively. As a result, the minimum average signal current has a value of:
min{Ip} = R0· Pmin
= 0.35 × 6.3 × 10−6A = 2.2 × 10−6A, (2.23)
while the maximum average signal current is: max{Ip} = R0· Pmax
The following step is to calculate the noise power. Since the mean-squared noise current contributed by load resistance and amplifier is irrelevant to the received signal power, they can be calculated explicitly. Applying (2.10), (2.12), RT = 150 Ω, B = 135 MHz,
Fn = 3 and T0 = 293 K, we get: i2 t + i2a = 4kBT0BFn RT = 5.52 × 10 −23· 293 · 135 × 106· 3 150 A 2 = 4.37 × 10−14A2. (2.25)
In the last step, by inserting (2.23), (2.24) and (2.25) into (2.21), the minimum and maximum SNR value can be finally evaluated numerically. With Idark = 300 nA, the
minimum and maximum SNR at the receiver yields:
min{γSN R} = (min{I p})2 2q(min{Ip} + Idark)B + 4kBRT0FnB T ≈ (2.2 × 10 −6)2 2 · 1.6 × 10−19· 2.5 · 135 + 4.37 × 10−14 ≈ 20.4 dB, (2.26) and max{γSN R} = (max{I p})2 2q(max{Ip} + Idark)B + 4kBTR0FnB T ≈ (2.2 × 10 −4)2 2 · 1.6 × 10−17· 2.2 · 135 + 4.37 × 10−14 ≈ 59.5 dB, (2.27)
respectively. From the calculations above, it is easy to notice that the thermal (AWGN) noise is the most conspicuous noise source and largely dominates over the shot noise in the receiver. Because of this effect, an AWGN noise model can be used to describe the noise generated by the optical receiver.
2.1.4 Experimental setup and results
This section presents the channel magnitude response measured by experimental setup, with the purpose to verify the previous statement that POF is the limiting factor for the overall optical channel.
2.1.4.1 Experimental setup
The MOST150 standard headers and 1 mm step-index POFs with different lengths are used in the measurements. These standard headers are embedded with coupling of the headers to the POF, as well as optical devices (red LED and PIN photo-diode) whose pa-rameters of communications interest are given by Table2.2. The considered POF length is 1, 3 and 15 meters respectively, where 15 m is the maximum length allowed between two communication nodes according to the MOST150 physical layer specification. The LED is driven by a very simple analog circuit where the information signal directly modulates the light intensity. The transmitter is not equipped with any special func-tionality such as analog equalizers/peaking circuit or linearizing circuit. The receive circuit has a low-impedance front-end structure. The transceivers as well as their elec-tronic driving and receiving circuits are built on to a single-sided copper clad board with the copper side used as a ground plane. The ground pins of the different components are directly soldered to the ground plane, while the other pins are air-wired above the ground plane. The input and output of this board are connected to a network analyzer to study the base-band transfer function of the optical link. During the measurement, the network analyzer produces a small swept RF signal imposed on the transmitter and sweeps the output frequency of the receiver repeatedly over the range of our interest, i.e., from 10 kHz to 300 MHz. The RF modulation power is varied from -20 dBm (at small bias) to 0 dBm, and at the same time, the biasing current is adjusted to ensure that the modulation is in the small signal regime and there is no observable change in the measured bandwidth.
2.1.4.2 The measured magnitude responses
The outcome of the network analyzer is illustrated in Figure2.7, where the three colored curves represent the measured electrical magnitude responses for the cascade of LED, PD and POF in three lengths, respectively. In addition, the magnitude response of a theoretical Gaussian low-pass filter with 3 dB bandwidth of 85 MHz is plotted by the black curve, corresponding to the magnitude response of the theoretical modulation transfer function of a 15 m POF given by (2.3). Note that the term bandwidth always refers to the electrical 3 dB bandwidth, which is defined as the frequency where the electrical level of a sine-modulated signal has dropped by 3 dB.
It can be observed that the measured magnitude response for the 15 m POF is well con-sistent with the theoretical Gaussian low-pass shape up 160 MHz. Above this frequency,
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 −48 −45 −42 −39 −36 −33 −30 −27 −24 −21 −18 Frequency [MHz] Magnitude Response [dB] 1m POF 3m POF 15m POF Gaussian function (3dB 85MHz) Gaussian function (3dB 110MHz) 3dB = 85MHz 3dB = 85MHz 3dB = 105MHz 3dB = 110MHz
Figure 2.7: The measured magnitude responses between LED input and PD
output usingdierent lengths ofPOF
the Gaussian low-pass filter is no longer a good approximation for the measured chan-nel magnitude response but decays much faster than the measured result. The result implies, the Gaussian approximation of the POF magnitude response in (2.3) is rather accurate up to about twice the 3 dB bandwidth. It also implies that the overall channel bandwidth is limited by POF rather than by optical transceivers.
Moreover, the magnitude responses for 1 or 3 m POF are not really Gaussian shaped. The reason is that now the LED transmitter somehow dominates the magnitude response of the cascade. Also it has to be pointed out that significant measurement errors occurred for frequencies beyond about 180 MHz, mainly due to external interference which is present even without an incident light.
2.2 Overview of the Transmission System
In the previous section, the POF based optical channel in the continuous time domain is described. By taking into account that the continuous- and discrete-time domain can be interchanged without information loss, and the objective is to transfer digital signals, we want to develop a comprehensive digital transmission system in this section, from in-troducing the modulation technique and error-correction coding, through exploring the
major channel impairments including noise and inter-symbol-interference (ISI), to de-duce a performance upper bound for the minimum output BER if ISI can be completely removed.
2.2.1 The dis rete system model
By including DAC and ADC in Figure 2.1 for the channel modeling, the channel im-pulse response ha(t) given by (2.4) can then be converted into its discrete-time version
h[k], as illustrated by Figure2.8. For a 10 m POF, the channel impulse response ha(t)
is plotted in Figure2.2. We select t0 and tesuch that the very small values of ha(t) for
t < t0 and t > tecan be ignored. We then sample ha(t) with T to get the discrete-time
channel impulse response h[k] for the cascade of DAC, the optical link and the ADC: h[k] =
(
T · ha(t0+ kT ), 0 ≤ k ≤ Lc− 1,
0 k < 0; k ≥ Lc
(2.28)
where t0 is the sampling delay, k ∈ Z and Lc = ⌈te−tT 0⌉ + 1 is the length of the
discrete channel. The sampling duration T considered in the dissertation equals either the symbol duration Tsor Ts/2 . To compensate for the sampling effect on the numerical
expression, T should be multiplied with ha(t) in (2.28) for converting a continuous-time
system to its discrete-time counterpart. Note that the effects of non-ideal DAC and ADC conversions are ignored here, because the sampling rates as well as the bandwidths of DAC and ADC are much higher than the channel bandwidth and the quantization step size is selected sufficiently small. Neither is the impulse shaping considered, because the channel itself can be considered as a Gaussian-pulse shaper.
DAC ha(t), Ha(f ) h[k] ADC x(t) y(t) n(t), σ2 n n[k] x[k] y[k] x[k] y[k] + +
