Equalization of Fiber Optic Channels

Transcription

1 Equalization of Fiber Opti Channels Johan Sjölander Deember 2001 IR SB EX 0118 ( ) ROYAL INSTITUTE OF TECHNOLOGY Department of Signals, Sensors & Systems Signal Proessing S STOCKHOLM KUNGL TEKNISKA HÖGSKOLAN Institutionen för Signaler, Sensorer & System Signalbehandling STOCKHOLM

2 1

3 IR-SB-EX Equalization of Fiber Opti Channels Johan Sjölander November 2001 Master Thesis Department of Signals, Sensors and Systems Royal Institute of Tehnology Optillion AB

4 Abstrat The performane of fiber opti ommuniation links is often limited by a phenomenon know as dispersion, whih auses optial pulses to broaden as they propagate through the fiber, thus giving rise to intersymbol interferene (ISI). Migration towards greater speed and longer links in fiber opti systems augment problems with dispersion in optial fibers, and dispersion ompensation is onsequently an inreasingly important issue. One method of ompensating for ISI, whih has proven to be effetive in for eample radio ommuniations, is equalization of the reeived signal. This tehnique is however yet to be implemented in fiber opti systems. This thesis investigates the possibilities of using adaptive eletroni equalization to ompensate for ISI aused by dispersion in optial fibers. Simulations are performed using realisti hardware models of various equalizer onfigurations in order to determine a feasible implementation. Both trained and blind equalizers are onsidered. The results of these simulations show onsiderable improvement of the signal quality after equalization, even with fairly restrained hardware designs. 2

5 Prefae This work was performed at Optillion AB during 2001 as my master thesis for a M.S. degree in Eletrial Engineering at the Royal Institute of Tehnology (KTH). It was supervised by Dr. Peter Händel at the Department of Signals, Sensors and Systems (S3) at KTH, and M.S. Tume Römer at Optillion whom I would like to thank for their guidane and support. Many thanks also to Dr. Peter Öhlén, M.S. Stig Nygård and Prof. Christer Svensson for sharing their epertise and giving me advie in onduting this work. Their input has been essential for this master thesis. Furthermore, I would like to aknowledge Dr. Magnus Jansson and M.S. George Jöngren at S3 and Dr. Tedros Tsegaye at Optillion for their oasional assistane. Finally I would like to thank Dr. Björn Rudberg at Optillion for taking the initiative to this master thesis. 3

6 Contents 1 Introdution Objetive The Fiber Opti Channel Dispersion Differential Mode Dispersion Group Veloity Dispersion Polarization Mode Dispersion Attenuation Non-linear Effets Noise Dispersion Compensation GVD Compensation PMD Compensation Equalization Linear Equalizers Non-linear Equalizers Sampling Issues Adaptive Algorithms Blind Equalization The Constant Modulus Algorithm Convergene and Initialization Issues of CMA Hardware Considerations Equalizer Configuration Finite Preision Effets Simulations The Fiber Opti System The Equalizer Simulation Aspets Bit Error Rate Measure of Performane Signal-to-noise-ratio Simulation Variables Results Eample of Equalizer Performane Samples Per Update Number of Filter Taps and Fiber Length Resolution and Stepsize Algorithm Sign LMS variants CMA and DD-LMS Diretly Modulated Laser Feasible Implementations Conlusions Future work Hardware Implementation Net Generation Systems A Simulation Parameters B Aronyms

7 Chapter 1 Introdution Dispersion in fiber opti systems is a phenomenon that auses optial pulses to broaden as they propagate through the fiber, thus giving rise to intersymbol interferene (ISI). This is beoming an inreasing problem as data rates beome higher and link lengths are strethed to meet the needs of modern fiber opti ommuniations. Consequently, ompensating effetively for dispersion is beoming a more and more important issue. Channel impairments in fiber optis have traditionally been ontrolled in the hannel itself rather than at the transmitter or reeiver, predominantly by using speially designed fibers to both mitigate and ompensate for dispersion. Reently however, tehniques ommonly used in for eample radio ommuniations to improve performane have found their way to fiber optis as well. One suh tehnique is equalization. Filtering the reeived signal with an equalizer has proven to be an effetive method to ombat ISI in many situations. However, it has not yet been implemented in fiber opti systems. Though reent advanes in eletroni hardware have opened up the possibility of designing eletroni equalizers to perform equalization at the high data rates that are of interest (10 Gbps and above), the ompleity of suh an equalizer is still somewhat limited by hardware onstraints. 1.1 Objetive This master thesis investigates the possibilities of using eletroni equalization at the reeiver to ompensate for dispersion in optial fibers. The objetive is to eamine the performane of various equalizer onfigurations (by simulation) in order to determine a feasible hardware implementation. The fous is on 10 Gbps single-mode links with intensity modulation (on-off keying) and diret (inoherent) detetion sine this is the prevalent onfiguration in high-performane fiber opti links today. Both trained and blind adaptive equalization tehniques are onsidered. 5

8 Chapter 2 The Fiber Opti Channel The apaity of fiber opti links has inreased enormously over the last few deades. Major breakthroughs were the inventions of single-mode fibers and optial amplifiers to ope with dispersion and loss respetively. More reently, wavelength division multipleing (WDM) has revolutionized the industry, where different wavelengths are used for different hannels (similar to frequeny division multipleing in radio). The development ontinues to evolve; improvements in all parts of the system assure a steady inrease in apaity for years to ome. The pot ential apaity of optial fibers is far from being fully eploited. This hapter investigates the harateristis of fiber opti hannels. In order to ompensate for hannel impairments, it makes sense to have some understanding of how the hannel behaves. The following is a brief and somewhat simplified overview of fiber opti hannels, fousing on the properties that are of interest from an equalization point of view rather than the underlying mehanisms that auses various phenomena. It should be noted that some hannel aspets are not onsidered here at all due to lak of relevane for this report. A more in depth analysis an be found in referenes [1], [2] unless other is noted. 2.1 Dispersion Dispersion in optial fibers an be ategorized into three different types, namely differential mode dispersion (DMD), group veloity dispersion (GVD) and polarization mode dispersion (PMD). They differ in what phenomenon auses them and in what properties they possess, whih has to be taken into onsideration when designing an equalizer. Beause DMD is only of onern for multimode links and PMD an usually be negleted for 10 Gbps links, the fous in this report is on GVD ompensation Differential Mode Dispersion DMD (also alled multipath dispersion or intermodal dispersion) is present only in multimode fibers and is due to that different modes (whih an be seen as different rays) travel at different speeds (take different paths) through the fiber, thus arriving at different times at the reeiver. This is somewhat similar to multipath fading in radio ommuniations as illustrated in Figure 2.1. Sine this report onerns single - mode links, DMD will not be onsidered any further. Cladding Core Figure 2.1. Differential mode dispersion in (step inde) multimode fiber. 6

9 2.1.2 Group Veloity Dispersion GVD (ommonly referred to as hromati dispersion, intramodal dispersion or simply dispersion) originates from the fat that the group veloity (the speed) of light in an optial fiber is frequeny dependent sine the refrative inde and guiding properties of the fiber vary with frequeny. Hene, different spetral omponents travel at different speeds through the fiber resulting in broadening of a transmitted pulse at the reeiver as shown in Figure 2.2 Original pulse Dispersed pulse Figure 2.2. Pulse broadening resulting from hromati dispersion. The amount of GVD is thus related to the spetral width of a transmitted pulse, whih in turn depends not only on the pulse shape but also on the spetral width of the light soure and possibly also on a phenomenon known as hirp. Eah of these parameters is desribed below. Pulse Shape A well-known fat following from Fourier analysis is that any information-arrying signal must by neessity have a spetral width larger than zero. The more rapid the hanges in the signal, the wider the spetrum. Thus, higher data rates aentuate hromati dispersion not only beause the transmitted symbols are spaed loser and hene more suseptible to dispersion, but also sine this inreases the spetral width of the signal, whih in turn auses more dispersion. Spetral Width No light soure is apable of produing perfetly monohromati light - the frequeny spetrum always ontains a ertain range of frequenies. For optial fibers, the two ommon light soures are light emitting diodes (LEDs) and semiondutor lasers. LEDs have muh wider spetral width than lasers, and this is one of the reasons why lasers are the only hoie for high -performane systems. But the spetral width also varies notably between different lasers. For light soures with narrow spetrum, the initial spetral width of the pulse has to be taken into aount when onsidering dispersion, whereas it an usually be negleted when the spetrum of the light soure is large in omparison. Chirp An optial pulse is referred to as hirped if its arrier frequeny varies with time. Chirp an arise in optial fibers due to non-linear effets desribed in the net hapter, but essentially, the amount of hirp depends on what type of laser is used, in partiular if the laser is diretly or eternally modulated. Diretly modulated lasers differ from eternally modulated in that the light intensity is varied diretly by altering the driving urrent of the laser instead of using an eternal devie to modulate a ontinuous wave (CW) laser. Diretly modulated lasers in general ehibit signifiantly higher hirp than eternally modulated lasers. The advantage of diretly modulated lasers is that they are heaper to manufature than eternally modulated lasers, and one motive of using dispersion ompensation is the possibility of replaing eternally modulated lasers with diretly modulated lasers. 7

10 GVD is also wavelength dependent, i.e. the amount of hromati dispersion a pulse eperienes depends on the enter wavelength of the pulse. Within the infrared region used in fiber optis, eah fiber has a ertain wavelength λ 0 for whih light travels the fastest, thus above and below this wavelength, photons move slower. However, this derease in speed is not linear with respet to wavelength, and as a onsequene, hromati dispersion varies with wavelength. Hene, GVD is related to the (non-onstant) derivative of the group-veloity with respet to wavelength. This an be realized by noting that two wavelengths with ertain spaing would drift further apart for some frequenies (orresponding to a large slope) than for others (small slope). A ommon measure for GVD is the dispersion parameter D epressed in units of ps/nm/km (the amount of broadening in ps that a pulse of bandwidth 1 nm would eperiene over 1 km of fiber). D is zero at λ 0, and this wavelength is often referred to as the zero or minimum dispersion wavelength. This however does not mean that GVD disappears altogether for a pulse entered at λ 0 sine zero dispersion does not apply to the entire spetrum of the pulse. But around this point, the hange in group veloity with wavelength is small, and using λ 0 as the arrier wavelength minimizes hromati dispersion. For a standard fiber, longer wavelengths travel faster than shorter above λ0, and this is referred to as positive dispersion (positive D). Below λ 0, the opposite holds and this is known as negative dispersion. A very important aspet of hromati dispersion from an equalization point of view is that while hromati dispersion in itself is linear and does not depend on the intensity of the light, its effet auses the fiber opti hannel to be non-linear Polarization Mode Dispersion PMD is a potential problem in single-mode fibers, whih despite the name, atually arries two orthogonally polarized modes that travel at different speeds through a non-ideal fiber, and as in the ase for DMD, results in a multipath output. The basi onept is illustrated in Figure 2.3. Figure 2.3. Polarization mode dispersion. However, the properties of PMD are not quite the same as for DMD, the most important differene being that PMD annot be onsidered as time invariant whih DMD an. PMD is due to variations in the shape of the ross setional area (deviations from perfet irular symmetry) of a fiber, whih auses the polarization modes to travel at different speeds. These irregularities in the fiber result from manufaturing imperfetions and stress on the fiber. From the simple model of PMD desribed above, it seems that PMD would result in just a two-path output. The multipath output aused by PMD is due to that the perturbations in the fiber ause eah polarization mode to ouple some of its energy to the other mode. The result of this is that an initial pulse will multiply along the fiber, giving rise to a multipath output rather than just a two-path output. The time variane of PMD follows from that stress on the fiber hanges due to vibrations and temperature variations. The differential group delay (DGD) of PMD is a measure of the amount of time a pulse has dispersed. It follows a Mawellian distribution, and is proportional to the square root of the fiber length. PMD is usually measured in pioseonds (of DGD) per square root kilometer (ps/ km). Measurements have shown that signifiant hanges in PMD an our within a time span of less than 10ms [9]. 8

11 Like GVD, PMD is also wavelength dependent, i.e. the average PMD value varies with frequeny, but ontrary to GVD, variation ours in a random fashion. This wavelength dependene is referred to as seond order PMD. The effet is that different spetral omponents of a pulse will suffer from different PMD values, and thus different DGD, whih ontributes to pulse broadening in a manner similar to that aused by GVD. Seond order PMD also varies with time, i.e. the PMD value for a ertain wavelength varies with time. Thus, PMD varies with time at a given wavelength and also varies with wavelength at a given time. When the PMD value for a fiber is alulated it is either averaged over all wavelengths or averaged over time at a fied wavelength. Beause PMD is not really an issue for 10 Gbps links, it is not inluded in the simulations onsidered in this report. But sine the effet of PMD is essentially ISI, not very different from that aused by GVD, it seems fair to assume that an adaptive equalizer that is able to ompensate for hromati dispersion would be able to ompensate for PMD as well. And sine a oherene time of 10 ms orresponds to 10 8 bits in a 10 Gbps bitstream, an adaptive equalizer should be able to trak PMD variations rather easily. 2.2 Attenuation The seond major limiting fator in fiber optis, apart from dispersion, is fiber loss. Signals propagating through the fiber are attenuated mainly due to absorption and sattering, whih of ourse leads to degradation of reeiver performane. As shown in Figure 2.4, attenuation in optial fibers is highly wavelength dependent, and ertain wavelengths are thus better suited for ommuniation than others. Attenuation Wavelength (nm) Figure 2.4. Attenuation in optial fibers. The three most ommonly used wavelength regions are entered around 850, 1310 and 1550 nm. These regions are sometimes referred to as the first, seond and third window. (Newly developed fiber has nearly eliminated the peak around 1400 nm, whih opens up a new region to be utilized in WDM systems.) The 850 nm region is used elusively for multimode fibers and this was the first region to be used in ommerial fiber opti systems. Though it ehibits high loss it is still used today beause of the relatively low prie of omponents for this region. 9

12 1310 nm is used both for single and multimode fibers. This window shows signifiantly lower loss than 850 but slightly higher loss than 1550 nm. Cost-wise, 1310 nm also falls in between 1310 nm multimode links ost more than 850 nm, and 1310 nm single-mode links are heaper than 1550 nm. For standard single-mode fiber, the minimum dispersion wavelength is about 1310 nm, and this along with omparatively low ost is why single-mode 1310 nm is the leading tehnology for shorter links without need for amplifiation. By far the most popular hoie for long-haul appliations is 1550 nm sine this region offers minimum attenuation. However, a standard single-mode fiber ehibits onsiderable dispersion in the 1550 region and speially designed dispersion-shifted fibers are usually deployed together with dispersion ompensating devies (see Chapter 3), whih inrease osts. Also, beause of this, hirp is more ritial and eternally modulated lasers have to be used in most ases sine they ehibit muh lower hirp than, onsiderably heaper, diretly modulated lasers whih an usually be utilized for 1310 nm. Still, these etra osts are saved in long-haul appliations sine fewer amplifiers have to be used thanks to the low loss. 2.3 Non-linear Effets Dispersion and loss in fiber optis are linear effets, i.e. they do not depend on the intensity of the light. However, there are several effets that our in optial fibers that are intensity-dependent and hene are referred to as non-linear effets. These are due to either intensity dependent refrative inde or intensity dependent sattering, and as in any other physial system, these non -linear effets result in new frequenies being produed. The non-linear effets that are due to intensity dependent sattering however, suh as stimulated Raman sattering (SRS) and stimulated Brillouin sattering (SBS), do not have muh relevane for equalization and are thus not onsidered here. There are three major non-linear effets that are due to intensity dependent refrative inde: self-phase modulation (SPM), ross-phase modulation (XPM), and four-wave miing (FWM). SPM results in hirp similar to that aused by modulation of lasers sine the intensity dependene of the refrative inde auses the phase of a pulse to be modulated. The outome of this is spetral broadening whih in ombination with GVD an result in pulse broadening in the time domain. XPM is similar to SPM and arises in WDM systems where the non-linear phase shift of one hannel is not only dependent on the power of that hannel, but of the other annels as well. FWM is also of onern for WDM systems and ours when several frequenies (hannels) ombine to generate other frequenies. The resulting frequenies are ombinations of sums and differenes of the other hannel frequenies (ω = ω 1 ± ω 2 ± ω n ), the most prominent sum being of the form ω = ω 1 + ω 2 - ω3. Thus, in a WDM system with equal hannel spaing, the resulting frequenies often fall on some other hannel resulting in interhannel rosstalk, and FWM is indeed a serious problem for WDM systems. From a dispersion ompensation point of view, this is of interest beause hromati dispersion helps to alleviate the problem with FWM [7], and therefore some amount of hromati dispersion is atually desirable in WDM systems. 2.4 Noise Noise in fiber opti systems arises mainly in transmitters, reeivers and optial amplifiers. Eah part is onsidered below. Reeivers An optial reeiver is essentially a photodiode that onverts optial power to eletrial urrent. The two fundamental noise mehanisms in an optial reeiver are shot noise and thermal noise. Shot noise is aused by random flutuations of harge arriers in a ondutor and thermal noise is due to the random thermal motion of eletrons in a resistor. Both shot noise and thermal noise an be approimated as additive white Gaussian noise, however it should be noted that shot noise is atually proportional to the 10

13 signal urrent. The most ommonly used photodiodes in optial systems are p-i-n photodiodes followed by avalanhe photodiodes (APD). APDs are more sensitive than p-i-n diodes but they also ehibit higher shot noise, whih in general an be negleted for p-i-n reeivers in whih thermal noise dominates. Transmitters Flutuations in the optial power from the transmitter add to the noise in the reeiver when the light is onverted to eletrial urrent. Suh flutuations are referred to as relative intensity noise (RIN) and are due to spontaneous emission and eletron-hole reombination in lasers. For diret detetion systems, RIN is usually negligible. Amplifiers Noise in optial amplifiers is due to amplified spontaneous emission (ASE) and an be onsidered as white and additive. It is naturally inherent in optial amplifiers and annot be eliminated. Despite relatively low noise figures of modern amplifiers, amplifier noise is often the limiting fator for long-haul links. 11

14 Chapter 3 Dispersion Compensation Various tehniques have been proposed and developed to improve dispersion harateristis of the fiber opti hannel, either by modifying the fiber itself or by inserting some sort of optial dispersion ompensating devies along the link. This hapter onerns aspets of dispersion ompensation for both GVD and PMD, and some of the eisting and proposed ompensation tehniques (to whih eletroni equalization ould be a viable alternative) are briefly reviewed. The fous is again on GVD sine PMD is not of great signifiane for 10 Gbps links, not many tehniques have yet been fully developed to ope with it. 3.1 GVD Compensation Sine the spetral width inreases and the spaing between pulses dereases as the data speed beomes higher, the tolerane of hromati dispersion at the reeiver dereases with the square of the bit rate. Beause of this, even though effetive ompensating methods for GVD eist, it ontinues to be a potentially limiting fator in fiber opti systems as bit rates inrease. The best way to redue dispersion is obviously to seek to minimize the ourrene of it. For hromati dispersion, this an be done by areful design of the fiber, and by optimizing the hirp of the optial signal. As noted earlier, standard fibers, also known as non-dispersion shifted fibers (NDSF) have minimum dispersion in the seond window around 1310 nm. In the third window around 1550 nm, whih is used espeially for long-haul links for whih dispersion is more ritial, dispersion is typially about ps/nm/km. Unfortunately, a large portion of the installed fibers today are of this type and onsequently requires effetive dispersion ompensation. To deal with this problem, so-alled dispersion shifted fibers (DSF) were developed whih had zero dispersion around 1550 nm. However, as WDM systems emerged, the low dispersion of these fibers in the third window aused problems with rosstalk due to FWM (see setion 2.3). Instead, so alled non-zero dispersion shifted fibers (NZDSF) were invented, whih have low but not zero dispersion in the 1550 nm region, typially around 4-8 ps/nm/km. A higher dispersion value allows for loser hannel spaing in WDM systems but requires more dispersion ompensation. As disussed in setion 2.1.2, the best way to ontrol hirp is to use an eternally modulated laser. Suh a laser is not hirp-free however, but in general ehibits signifiantly lower hirp than a diretly modulated. Another advantage of eternally modulated lasers is that it is possible to ahieve negative transient hirp as opposed to diretly modulated lasers, whih ehibit positive transient hirp. A pulse is said to have positive or negative transient hirp depending on if its leading edge has lower or higher frequeny than the trailing edge respetively. The benefit of aquiring negative transient hirp is that a pulse would initially ompress when launhed on a fiber with positive dispersion beause the leading edge (lower frequeny) of the pulse would move slower than the trailing edge (higher frequeny). Sine a standard fiber ehibits positive dispersion for wavelengths above 1310 nm, some amount of negative hirp is preferred for most systems, espeially long-haul systems operating in the third window around 1550 nm. The proess of deploying negative hirp on a transmitted pulse to ombat dispersion is known as pre-hirping. Sine the ourrene of dispersion annot be avoided altogether, espeially not in WDM systems, it must be ompensated for if high performane is demanded. The most established method for GVD ompensation is dispersion ompensating fiber (DCF). In onventional DCFs, the zero dispersion wavelength has been shifted towards longer wavelengths than those that are to be ompensated. This results in negative dispersion in the region that is to be ompensated, thus aneling the positive dispersion that has aumulated in the fiber prior to the DCF. A problem for WDM systems is that with this 12

15 method, dispersion an only be fully ompensated for at one wavelength. Longer wavelengths will be underompensated and lower overompensated sine the slope of the dispersion urve (with respet to wavelength) is of the same sign (positive) as for regular fiber (though usually flatter). Compensating equally over a wide range of wavelengths would require a DCF with negative slope, whih is diffiult to ahieve. One way to deal with this problem is to derease the dispersion slope of the fibers, but this also results in inreased problems with non-linear effets [7], [8]. Reent tehnial advanements have made other tehniques suh as fiber Bragg gratings inreasingly popular. Though only DCFs are reviewed here, most of the novel ompensating tehniques work aording to the same priniple, i.e. to delay ertain frequeny omponents of the optial arrier signal 3.2 PMD Compensation PMD has previously not been of major onern in the development of fiber opti ommuniation systems. The reason is, as previously mentioned, that its effets are negligible at low data rates over short distanes. But as data rates and link lengths inrease, the effets of PMD an no longer be ignored. The tolerane of PMD at the reeiver dereases linearly with bit rate. Sine PMD is time-variant and an deviate onsiderably from the average value, the PMD dispersion tolerable in a system is only a fration of a bit slot. Modern fibers usually have PMD values below 0.1 ps/ km and for suh fibers PMD is not really a problem for bit rates up to 10 Gbps, even for quite long links. For eample, a 1000 km link over a fiber with PMD of 0.1 ps/ km would eperiene an average DGD of just 3 ps. While this is not a big deal for 10 Gbps systems, it ould be a problem for a 40 Gbps system where a bit slot is only 25 ps. Hene, PMD in modern fibers is essentially a problem for future generation systems operating at 40 Gbps and above. However, a small part of installed fibers around the world have signifiantly higher PMD of up to 2 ps/ km. For these fibers, PMD an be a problem for 10 Gbps systems as well. The time varying nature of PMD imposes problems in the design of ompensating devies. Though PMD an be within reasonable limits most of the time, etreme onditions ould ause outage of a link. Thus, in order for a ompensator to work satisfatory, it should be able to adapt to hanges in the hannel, i.e. some sort of feedbak system must be implemented. Moreover, sine PMD an vary signifiantly within just a few milliseonds, these devies should be able to adapt within reasonable time. Several methods have been proposed for PMD ompensation, both in the optial and eletrial domain, but sine PMD is not really a major problem yet, no partiular method has been ommerially deployed thus far. The idea behind most proposed ompensation methods is to separate the power of the two polarization modes and delaying one relative to the other in either the optial or eletrial domain [10]. The feedbak information an be based on either the eletrial spetrum of the reeived signal or on the degree of polarization (DOP) [11]. These methods are however only able to ompensate for first order PMD. Other proposed methods, whih are able to overome the effets of seond order PMD as well, eploit the wavelength dependeny of PMD by swithing between different wavelengths [12], similar to frequeny diversity swithing in radio systems. 13

16 Chapter 4 Equalization As previously mentioned, equalization is a ommon way to deal with intersymbol interferene in other ommuniation systems, but it is yet to be implemented for fiber opti systems. There is no prinipal differene between a fiber opti hannel and e.g. a radio hannel in terms of ISI; the reeived baseband signal is distorted in a similar manner in both systems, i.e. symbols spread out over neighboring symbols as they propagate through the hannel. Consequently, equalizer tehniques used for radio and other systems should in essene be vi able for fiber opti links as well. However, one important differene is that while a radio hannel an usually be onsidered as linear, a fiber opti hannel ehibits nonlinear harateristis. This hapter presents an overview of the equalizer onfigurations and algorithms used in the simulations. General information on equalizers an be found in referenes [4], [15] and [16]. 4.1 Linear Equalizers In the following analysis, the disrete time baseband representation of the system is modeled aording to figure 4.1. Channel n[n] Equalizer Slier s[n] H(z) + [n] C(z) y[n] s[n] Figure 4.1. Baseband representation of a system with a linear equalizer. Here, s[n] represents the transmitted symbols that are sent into the hannel with frequeny transfer funtion H(z), n[n] is additive white Gaussian noise (AWGN), [n] is the input to the linear equalizer with frequeny transfer funtion C(z) and y[n] is the filtered equalizer output. The basi idea of a linear equalizer is simply to filter the reeived signal through a filter that approimates the hannel inverse, i.e. C ( z) = 1 H ( z) The equalizer is usually implemented as a finite impulse response (FIR) filter, i.e. the system is given by y [ n] k [ n k] = N k= N where { k } are the M = 2N+1 filter oeffiients. Suh an equalizer is often alled a feed forward equalizer (FFE) or a transversal filter. 14

17 If the taps are adjusted to approimate the hannel inverse, the effet the hannel has on the signal is minimized by the equalizer (in fat it is ompletely eliminated in the ideal ase if a suffiiently long filter is used). An equalizer that has this property is alled a zero-foring (ZF) equalizer sine it fores the ISI to be zero at the sampling instants. However if noise is present, an equalizer based on the ZF riterion is no longer ideal. This is beause the equalizer will have large gain at frequenies for whih the magnitude of H(f) is small, and the noise will thus be signifiantly amplified for these frequenies. To put it in another way, a ZF equalizer performs poorly in the presene of noise if the hannel has spetral nulls. A better solution when noise is present is to adjust the taps aording to the minimum mean square error (MMSE) riterion, i.e. to minimize the error between the desired (transmitted) signal s[n] and the equalizer output y[n] in the least squares sense. The mean square error (MSE) to be minimized is thus given by MSE = E {( s[ n] y[ n] ) } 2 where E{. } denotes statistial epetation. This funtion is also referred to as a ost funtion and is ommonly denoted by J. If the filter oeffiients and the input samples are written in vetor format as = M0 1 M 1 = [ n] [ n 1] M [ n M + 1] the ost funtion an be epressed as J {( ) } ( ) = E s[ n] [ n] 2 2 E s [ n] T T { } E s[ n] [ n] = 2 { } T { } + E [ n] [ n] Defining the rossorrelation vetor between the input vetor [n] and the transmitted symbols s[n] r { } T [ n] E s[ n] [ n] = s and the autoorrelation matri of [n] yields R J { } T [ n] = E [ n] [ n] 2 ( ) E s [ n] T { } T [ n] + [ ] = 2 r n s R Minimizing this epression gives the MMSE oeffiients by solving the normal equations aording to T 1 = r sr Adapting the filter oeffiients in this manner redues the problem with noise enhanement for hannels with spetral nulls ompared to a ZF equalizer. However the problem essentially remains, only to a lesser degree. 15

18 4.2 Non-linear Equalizers Though a linear equalizer an perform satisfatory when the ISI of the hannel is not severe (flat frequeny response and short ISI span), a muh better equalizer is aomplished if a deision feedbak equalizer (DFE) is used in onjuntion with a FFE. A DFE uses past symbol deisions to remove ISI from the sueeding symbols. This is ahieved by feeding the symbol deisions through a FIR filter whose output is subtrated from the reeived signal as shown in Figure 4.2. Sine the deisions are noise free, the DFE eliminates the main problem with linear filters - that the signal and noise are proessed together. The basi idea of a DFE is very straightforward; the ISI aused by eah reeived symbol on the sueeding symbols an simply be subtrated from these if the amount of ISI is known. In prinipal, a suffiiently long DFE removes all postursor ISI (i.e. ISI from prior symbols) aused by eah symbol. Preursor ISI (due to subsequent symbols) is usually handled by a preeding feed forward filter. Though the feedbak filter is a linear filter, the DFE is nonlinear sine the symbol deision is a nonlinear operation. Channel n[n] Slier s[n] H(z) [n] y[n] s[n] - Equalizer C(z) Figure 4.2. Implementation of a deision feedbak equalizer. Sine the DFE is usually employed in onjuntion with a FFE, the term DFE often refers to an equalizer with both a feedbak and a feedforward filter. Note that when the MMSE riterion is used jointly for both the FFE and DFE, the FFE will attain different oeffiient values than when only a FFE is used. When a DFE is also used, the FFE will try to remove preursor ISI rather than both preursor and postursor as when no DFE is used. In other words, the FFE will try to shape the signal so that it beomes asual if a DFE is used. A problem with DFE is error propagation if a symbol is erroneously deteted, this error will be fed bak to the sueeding symbols, potentially ausing more errors whih will be fed bak and so on. Therefore, deision feedbak equalization only works if the input to the deision iruit has a fairly low probability of error. When filtering the signal prior to detetion, the noise from the hannel is olored by the feed forward filter. Symbol-by-symbol detetion after the equalizer does not take the resulting noise orrelation into aount, whih results in loss of performane. An equalizer that overomes this problem is the maimum likelihood sequene estimation (MLSE) equalizer, whih is another ommonly used nonlinear equalizer. It does not use any filtering, instead it works by finding the data sequene with the maimum probability of all possible reeived sequenes, based on knowledge of the hannel and the statistial distribution of the noise. Though an MLSE an be effiiently implemented by use of the Viterbi algorithm, it is still very omple ompared to DFE. It does however give better performane, but due to the etensive hardware requirements, MLSE is not onsidered any further in this report. 16

19 4.3 Sampling Issues Equalizers are ategorized as either frationally spaed (FS) or baud spaed (BS) depending on if the equalizer uses more than one sample per symbol or just one. FS equalizers have ertain advantages suh as being robust to symbol timing but will not be onsidered in this report sine sampling at a rate higher than 10 GHz is not pratially ahievable with urrent tehnology. A potential problem with BS equalizers is that if the input signal to the equalizer ontains frequenies higher than half the sampling frequeny, aliasing ours and the equalizer will ompensate for an alias distorted signal. This effet is obviously eliminated for a FS equalizer if the sampling rate is hosen suffiiently high (however aliasing still ours at the output of the FS equalizer when the signal is deimated for symbol-by-symbol detetion). 4.4 Adaptive Algorithms Adapting the filter oeffiients aording to the MMSE riterion as shown in hapter 4.2 requires knowledge of the transmitted symbols s[n]. This is usually ahieved by transmitting a training sequene known to the reeiver from whih the oeffiients are determined by minimizing the mean square error between the training sequene and the orresponding reeived equalized sequene. However, adaptation of the oeffiients by solving the normal equations is omputationally omple, espeially if the hannel is time variant and the oeffiients have to be updated regularly. Therefore, an iterative proedure known as the steepest desent method is usually employed. The idea is to update the oeffiients reursively aording to µ 2 [ n + 1] = [ n] n where µ is a onstant referred to as the step size and n is the gradient of the ost funtion at iteration n given by n J = [ n] [ n] = E {( s[ n] [ n] ) } 2 T But sine the ost funtion is not available, the gradient annot be omputed in this manner. Instead, it is approimated as n ( s[ n] [ n] ) = s[ n] [ n] T 2 ( ) [ n] = e[ n] [ n] 2 2 T where e is the error signal. [ n] = s[ n] [ n] = s[ n] y[ n] T With this approimation, the oeffiients an be updated as [ n +1 ] = [ n] + µ e[ n] [ n] This is known as the least mean square (LMS) algorithm and it thus strives to minimize the MSE ost funtion 17

20 J = E {( s[ n] y[ n] ) } 2 For eah iteration, the algorithm moves in the negative diretion of the gradient towards the optimum value of in a step proportional to the step size µ. The hoie of step size is a ompromise between onvergene rate and the residual error after the algorithm has onverged a larger step size gives faster onvergene but in addition larger misadjustment, i.e. larger flutuations around the MMSE solution. The misadjustment is also proportional to the number of filter taps. Provided that the step size is suffiiently small, the LMS algorithm will onverge towards the optimum MMSE oeffiient values and remain stable. A similar algorithm that is also very popular is the reursive least squares (RLS) algorithm. The idea of this algorithm is to minimize the sum of all estimation error squares, i.e. to minimize the ost funtion given by J = [ n] e 2 The RLS algorithm onverges faster than the LMS algorithm, but at the prie of higher omputational ompleity. It is therefore not onsidered any further in this report. Sign LMS Beause of the stringent hardware requirements assoiated with equalization at 10 Gbps, it is highly desirable to use an algorithm with as low omputational ompleity as possible. One lass of the LMS algorithm that is omputationally simpler than standard LMS is so-alled sign LMS. The idea of sign LMS is to redue the omputational ompleity by replaing multipliation with addition (whih is simpler to implement in hardware) by using only the sign of the error funtion e[n] and/or the tap input signals [n] to update the oeffiients. There are a few of possible variations of this suh as sign-data, sign-error and sign-sign LMS. The sign-data and sign-error variants perform the sign operation on the data signal and error signal respetively, and the oeffiient update funtions are thus given by and [ n + 1 ] = [ n] + µ e[ n] sgn( [ n] ) [ n + 1 ] = [ n] + µ sgn( e[ n] ) [ n] Sign-sign LMS is an even oarser variant that uses the sign of both the error funtion and the input signal, i.e. [ n + 1 ] = [ n] + µ sgn( e[ n] ) sgn ( [ n] ) If the multipliation of µ is implemented as a fied bit shift (whih requires that µ is a power of two) these algorithms eliminate the need for hardware multipliations. 4.5 Blind Equalization When a training sequene is not available, probabilisti and statisti properties of the transmitted data an be eploited to adapt the equalizer oeffiients. This is known as blind equalization [5]. If the input signals are independent, identially distributed (i.i.d), the information that an be eploited by a blind equalization sheme is the power spetral dens ity (PSD) of the hannel output, the finite size alphabet of the input signals and higher order statistis (HOS) [17] of the sampled hannel output. It should be noted that these soures of information are not neessarily independent. 18

21 The elusion of a training sequene does not ome without a prie. Apart from being generally more omple, the two main disadvantages with blind equalization ompared to trained equalization are slow onvergene and possible onvergene to loal minima (i.e. the oeffiients do not always neessarily onverge towards the optimum values). There are several blind equalization algorithms, most of whih eploit HOS and/or the finite size alphabet of the soure. Blind algorithms based on PSD (seond order statistis) will not be onsidered here sine these are omputationally omple and usually require oversampling of the signal, whih is not pratially ahievable at symbol rates of 10 Gbps. However, these algorithms are often able to signifiantly redue the problems with HOS algorithms namely slow onvergene and loal onvergene. The simplest form of blind equalization is deision-direted (DD) equalization where the output from the deision devie s[n] is used to update oeffiients in the same manner as when a training sequene is utilized. The DD-LMS update of the oeffiients is thus given by [ n + 1 ] = [ n] + µ ( sˆ [ n] y[ n] ) [ n] - This obviously requires that not too many symbols are erroneously deteted. Th erefore, DD equalization is often used for slowly varying hannels after the oeffiients have already been adjusted by some other method (e.g. with a training sequene). The equalizer is then said to be swithed into deision direted mode in whih the equalizer is able to trak slow variations in the hannel without the need for a training sequene. When the initial equalizer output is suh that the probability of deision error is high, the filter oeffiients resulting from DD equalization an onverge to values that do not remove suffiient ISI. In this ase, if there is no training sequene, some other (better) blind equalization sheme has to be used to adapt the oeffiients (after whih, swithing to deision direted mode ould be possible). A blind equalizer works aording to the same priniple as a trained equalizer by minimizing a ost funtion, given by J ( ) = E{ ( y[ n] )}? where? (y[n]) is determined by the blind equalization algorithm being used. Adjusting the oeffiients aording to the method of steepest desent (see setion 4.4) then yields [ n + 1 ] = [ n] + µ ( y[ n] ) = [ n] + 'µ [ n]? If the first derivative of? is defined as? ( ) [ n] T () ψ =? ( ) the blind equalization algorithm an be written as [ n +1 ] = [ n] + µψ ( y[ n] ) [ n] whih is the same form as the LMS algorithm eept that the error funtion is now given by? (y[n]). An important aspet of equalizers in general and blind equalizers in partiular, is the onvergene properties of the algorithm, i.e. how fast and to what values the filter oeffiients onverge. As previously 19

22 noted, while trained LMS equalizers always onverge towards the MMSE oeffiient values, provided that the step size is not too large, blind equalizers might onverge to a loal minimum far from optimum. For baud-rate equalizers, this loal onvergene an be due either to the finite length of the equalizing FIR filter, so alled length dependent loal minima, or to imperfetions of the ost funtion adopted by the blind algorithm whih is known as ost-dependent loal minima. Length-dependent loal minima are inherent to all baud-rate blind equalizers while ost-dependent loal minima an be avoided by hoie of ostfuntion The Constant Modulus Algorithm The blind equalization algorithm onsidered in this report is the onstant modulus algorithm (CMA) [5], [13], [14]. The reasons for this hoie are mainly its low (LMS-like) omput ational ompleity, robustness against additive noise and absene of ost-dependent loal minima. Of all blind equalization shemes, CMA is also the most widely used. The idea of CMA is basially to fore the output of the equalizer to be of onstant amplitude. This is ahieved by penalizing equalizer output samples whose modulus (magnitude) deviates from a desired fied value?, related to the soure output as shown below. Like the LMS algorithm, the CMA is based on the method of steepest desent. The ost funtion of the CMA is given by? 1 ( y[ n] ) = y[ n] 4 2 ( γ ) 2 where γ = E E 4 { s[ n] } s[ n] 2 { } is the kurtosis [17] of the soure. The error funtion is easily derived to be ψ 2 ( ) ( y[ n] ) = y[ n] γ y[ n] and the CMA an be epressed as 2 ( ) [ n] [ n + 1] = [ n] + µ y[ n] γ y[ n] - It should be noted that the CMA also works for soures that do not have onstant modulus suh as those using multilevel pulse amplitude modulation (M-PAM). For soures with onstant modulus however,? is simplified to? = s[n] 2 = 1. Sine CMA is essentially used to redue the BER suffiiently to allow for DD-LMS to be used, the initial BER must be rather high to motivate use of CMA. Consequently, to avoid problems with error propagation, CMA is usually employed with a feedforward filter only. When the BER has been redued to aeptable levels, the equalizer is swithed to DD mode and a feedbak filter may be used. Like the LMS algorithm, CMA an also be made omputationally more effiient by using only the sign of the error signal and/or input samples when updating the oeffiients, so-alled sign-error, sign-data and sign-sign CMA. 20

23 4.5.2 Convergene and Initialization Issues of CMA Unlike the LMS algorithm, the initialization values of the oeffiients are of great importane for the CMA. Depending on the initial oeffiient values, the CMA ould onverge to a loal minimum for whih no ISI is removed, thus making the equalizer useless. In addition, the initialization values also affet how fast the algorithm onverges. This an be realized by viewing a plot of the CMA ost funtion for two oeffiient parameters as shown in Figure 4.3. The two shallow minima are undesirable loal minima while the two deeper minima are global. Figure 4.3. Eample of CMA mean ost funtion for two oeffiients 0 and 1. As an be seen, sine the algorithm moves in the negative diretion of the gradient, if the initialization values are lose to a loal minimum, the algorithm will onverge towards the orresponding undesirable oeffiient values. Furthermore, if the initialization values are lose to a saddle point, the onvergene will be slow (sine the gradient here is small). Additive white noise hanges the loations of both the loal and global minima and also makes the global minima shallower, whih results in less stable global minima. But the noise effet itself (possible large flutuations) also redues the stability of the loal undesirable minima. By proper initialization of the equalizer oeffiients, problems with bad onvergene an be avoided. Unfortunately, no method eists for determining absolutely safe initialization values. However, it has been shown that the probability of onvergene to undesirable minima is lower when the largest tap weights are onentrated around the enter taps. This suggests that a good initialization strategy is to employ soalled enter-spike initialization, whereby the enter tap of the filter is set to unity and all the other taps are set to zero. Although no rigorous analysis of how well it really works has been shown, this initialization strategy has proven to be suessful in a vast majority of its implementations. 21

24 Chapter 5 Hardware Considerations As mentioned in the Introdution, implementing an equalizer to operate on a 10 Gbps bitstream in hardware is not an easy task even with the most advaned tehnologies urrently available. Unfortunately, this severely restrits the ompleity of the equalizer down to filters with just a few taps and low resolution. 5.1 Equalizer Configuration A solution that helps alleviate this problem is to perform the filtering in the analog domain (whih is potentially faster than its digital ounterpart), with the filter oeffiients omputed digitally at a rate slower than bit rate. This is shown for a deision-direted LMS equalizer in Figure 5.1. (t) t t t µe(n) Coeffiient Update A/D µ D/A C0 C1 C2 C3 e(n) + A/D y(t) s(n) (k), s[k] µe(k) + (k-1) Coeffiient Update Coeffiient Update D/A C6 + C5 C4 T (k) = (k-1) + µe(k)(k) T T T Figure 5.1. LMS Equalizer with analog filters. This iruit uses analog FIR filters with analog multiply/adds and analog delays of length τ = T. In the feedforward filter, the time-ontinuous analog input signal (t) is sampled at eah tap one every N symbols, and the digital oeffiient update iruitry omputes the new oeffiient values from these 22

25 samples and the error signal e[k ]. The feedbak filter works in a similar manner eept that it does not need to sample the input signal s[k] sine it is already available in digital format from the deision iruit. A new set of oeffiients is thus produed at a rate N times slower than the symbol rate. For eample, if the input signal is 10 Gbps (binary) and N =16, the oeffiient update iruitry needs to run at a frequeny of 625 MHz. Reduing the frequeny at whih the oeffiients are updated in this manner should have no effet on the performane of the equalizer other than slower onvergene, provided that the hannel an be onsidered as time invariant. Another problem with the high speed requirements is that it is diffiult to get the signal in the deision feedbak filter to omplete the loop in just one bit slot (100 ps for a 10 Gbps bitstream). It would seem that the performane of the feedbak filter would deteriorate onsiderably if the filter ould not remove ISI from the losest following bit sine this bit would presumably have been affeted the most by ISI from the preeding bit. However, this of ourse depends on the hannel properties and may not neessarily be the ase. The simulations in this report are performed both with and without feedbak filter, and in addition, feedbak filters without the first tap are also evaluated. 5.2 Finite Preision Effets The hardware limitations also impose onstraints on the resolution of the digital signals in the equalizer to ahieve the required speed of the eletronis, it is neessary to represent eah sample and oeffiient by no more than a few bits. This obviously degrades the performane of the equalizer. Quantization effets on a signal an normally be modeled as additive white noise on an infinite resolution signal. However when an analog filter is used, the signal itself is never quantized and this matter an thus be negleted for the analog equalizer onsidered in this report. The quantization of the filter oeffiients on the other hand is of onern sine this affets the auray of a filter, i.e. the filter will not be able to ahieve the desired harateristis beause the oeffiients an only attain a limited number of values (presumably different from the desired ones). The error indued in this manner inreases with the number of filter taps. Another issue regarding quantization is that the adaptive algorithm will stall when the magnitude of the orretion term in the adaptation algorithm is smaller than the value of the least signifiant bit (LSB). For eample, in the ase of the LMS algorithm this riterion is µ e [ n] [ n] LSB This affets the hoie of step size when the resolution is low, a large step size must be used for the algorithm to onverge. 23