| 1 | % presentation |
| 2 | %\documentclass{beamer} |
| 3 | %\setbeameroption{hide notes} |
| 4 | % notes |
| 5 | \documentclass[handout]{beamer} |
| 6 | %\usepackage{pgfpages} |
| 7 | \setbeameroption{show only notes} |
| 8 | \setbeamercolor{note page}{bg=white} |
| 9 | \setbeamercolor{note title}{bg=white} |
| 10 | \setbeamertemplate{note page}{\vspace{1em}\insertnote} |
| 11 | % To compile, run |
| 12 | % pdflatex -jobname presentation+notes ./presentation.tex && \ |
| 13 | % pdfnup --nup 1x4 --no-landscape --a4paper presentation+notes.pdf |
| 14 | |
| 15 | \title[100 GbE PON]{100 GbE Passive Optical Access Networks} |
| 16 | \author[A.~Lam (ail30)]{Adrian Lam (ail30)} |
| 17 | \institute[]{Supervised by: Dr.~Seb Savory} |
| 18 | \date[M 2018]{Michaelmas 2018} |
| 19 | |
| 20 | \usetheme{Madrid} |
| 21 | \usepackage{lmodern} |
| 22 | \usepackage{amsmath} |
| 23 | \usepackage{siunitx} |
| 24 | \usepackage{graphicx} |
| 25 | \usepackage{epstopdf} |
| 26 | \usepackage{multimedia} |
| 27 | \usepackage{hyperref} |
| 28 | \hypersetup{colorlinks,linkcolor=,urlcolor=magenta} % https://tex.stackexchange.com/a/13424 |
| 29 | \usepackage{pgffor} |
| 30 | \usepackage{pdfpages} |
| 31 | \usepackage{blox} |
| 32 | \usepackage{makecell} |
| 33 | |
| 34 | \begin{document} |
| 35 | |
| 36 | \frame{\titlepage} % 15s |
| 37 | |
| 38 | \begin{frame} |
| 39 | \frametitle{What is a passive optical network (PON)?} |
| 40 | \begin{itemize} |
| 41 | \item Point-to-multipoint |
| 42 | \item Unpowered beam splitter |
| 43 | \item ``Everything sent to everyone'' |
| 44 | \end{itemize} |
| 45 | |
| 46 | {\centering% |
| 47 | \includegraphics[height=0.5\paperheight]{PON.png}\\ |
| 48 | \tiny Image credit: Riick\textasciitilde commonswiki, ``PON vs AON.png''. |
| 49 | CC BY-SA 3.0. \url{https://commons.wikimedia.org/wiki/File:PON_vs_AON.png}. Cropped. |
| 50 | |
| 51 | } |
| 52 | |
| 53 | \note{ |
| 54 | A passive optical network, or PON, is typically used in a |
| 55 | fibre-to-the-home setting. Optical signal from the network provider |
| 56 | is split, without amplification or selection, to all users in the same |
| 57 | network. The ONT here which sits somewhere in your home will select |
| 58 | the data related to you. |
| 59 | } |
| 60 | \end{frame} |
| 61 | |
| 62 | \begin{frame} |
| 63 | \frametitle{Need for high-speed (\SI{100}{Gb/s}) PON} |
| 64 | \begin{itemize} |
| 65 | \item Fibre-to-the-home? |
| 66 | \item Well-served by \SI{1}{Gb/s} PON, mature (mass deployment $>$ 1 decade) |
| 67 | \pause |
| 68 | \item Reuse existing fibres in other applications |
| 69 | %%\item Business users |
| 70 | \item Mobile |
| 71 | \begin{itemize} |
| 72 | \item Increase density in cell sites $\rightarrow$ PON to deliver |
| 73 | backhaul |
| 74 | \item Possible 5G fronthaul: radio receivers sample RF signals and relay |
| 75 | them to centralized location for processing |
| 76 | \end{itemize} |
| 77 | \end{itemize} |
| 78 | |
| 79 | \note{ |
| 80 | * Typical usage of PON is in FTTH. However, gigabit PONs are already |
| 81 | quite sufficient and mature with low cost hardware. |
| 82 | |
| 83 | * The motivation to go up to 100 Gb/s is to share other applications along existing |
| 84 | fibres, to reduce the cost of installing new fibres. |
| 85 | |
| 86 | %% For example, business users. |
| 87 | A particularly interesting application is in mobile networks. |
| 88 | As people get more addicted to their smart phones, mobile network |
| 89 | operators would need to increase the density in cell sites, making |
| 90 | PON a good candidate to deliver the cell site backhaul. |
| 91 | |
| 92 | As 5G mobile develops, PON may also be useful in the fronthaul. |
| 93 | RF receivers relay the signals back to a centralized location |
| 94 | for processing, which is seen as a way to reduce operating costs. |
| 95 | } |
| 96 | \end{frame} |
| 97 | |
| 98 | \begin{frame} |
| 99 | \frametitle{Direct detection vs coherent receivers} |
| 100 | \begin{itemize} |
| 101 | \item |
| 102 | Direct detection receiver |
| 103 | \begin{itemize} |
| 104 | \item Single photodiode with amplifier |
| 105 | \item Rx current $\propto$ received optical power (Phase information lost) |
| 106 | \item On-off keying (mainly) |
| 107 | \end{itemize} |
| 108 | \item |
| 109 | Coherent receiver |
| 110 | \begin{itemize} |
| 111 | \item $\propto$ real and imaginary parts of received electrical |
| 112 | field |
| 113 | \item Polarization-division multiplexing |
| 114 | \item Phase-shift keying, quadrature amplitude modulation |
| 115 | \end{itemize} |
| 116 | \end{itemize} |
| 117 | {\centering% |
| 118 | \fbox{\includegraphics[height=.3\paperheight,clip,trim=40mm 167mm 40mm 55mm]{coherentRx.pdf}} \\ |
| 119 | {\tiny Image credit: Seb J.~Savory, ``Digital filters for coherent |
| 120 | optical receivers,'' 2008.} |
| 121 | |
| 122 | } |
| 123 | \note{ |
| 124 | To achieve a high data rate, coherent receivers are to be used. |
| 125 | This is different from the direct detection receivers which you may |
| 126 | have come across last year in 3B6, where the receiver photocurrent |
| 127 | is proportional to the optical power. In a coherent receiver, as illustrated |
| 128 | by this example here, %the light, after passing through a single-mode fibre, |
| 129 | %is first split by polarization, |
| 130 | %and then the real and imaginary parts of each polarization are detected |
| 131 | both the real and imaginary parts of two orthogonal polarizations are detected. |
| 132 | This allows different modulation schemes, similar to |
| 133 | radio transmissions, such as PSK and QAM. |
| 134 | } |
| 135 | \end{frame} |
| 136 | |
| 137 | \begin{frame} |
| 138 | \frametitle{Aims of this project} |
| 139 | \begin{itemize} |
| 140 | \item Build simulation models for optical networks with coherent receivers |
| 141 | \item Use DSP to correct for fibre effects |
| 142 | \item Simulate different options for achieving \SI{100}{Gb/s} |
| 143 | \item Experimentally validate simulation results |
| 144 | \item Evaluate feasibility to use in PONs |
| 145 | \end{itemize} |
| 146 | \note{ |
| 147 | In this project, simulation models for optical networks will be |
| 148 | built using MATLAB, with digital signal processing to correct |
| 149 | for fibre effects. Different options for achieving the target |
| 150 | data rate of 100 Gb/s will be simulated and compared, and |
| 151 | later experimentally validated. The results will be evaluated |
| 152 | in terms of suitability to use in a PON. |
| 153 | } |
| 154 | \end{frame} |
| 155 | |
| 156 | \begin{frame} |
| 157 | \frametitle{Simulations performed thus far} |
| 158 | QPSK with symbol rate \SI{25}{GBd} over AWGN channel |
| 159 | |
| 160 | \begin{itemize} |
| 161 | \item Chromatic dispersion |
| 162 | \item Adaptive equalizer |
| 163 | \item Phase noise (laser linewidth) |
| 164 | \end{itemize} |
| 165 | |
| 166 | {\centering% |
| 167 | \begin{tikzpicture} |
| 168 | \scriptsize |
| 169 | \bXInput[$x_n$]{input} |
| 170 | \bXBlocL[3]{p}{\makecell[c]{Pulse shaping\\$p(t)$}}{input} |
| 171 | \bXSumb*[9]{AWGN}{p} |
| 172 | \bXLink[$x(t)$]{p}{AWGN} |
| 173 | \path (AWGN) ++(0,-1) node (noise) {$n(t)$}; |
| 174 | \bXLink{noise}{AWGN} |
| 175 | \bXBloc[3]{sim1}{?}{AWGN} |
| 176 | \bXLink{AWGN}{sim1} |
| 177 | \bXOutput[3]{y}{sim1} |
| 178 | \bXLink[$y(t)$]{sim1}{y} |
| 179 | \end{tikzpicture} |
| 180 | |
| 181 | \addvspace{1em} |
| 182 | |
| 183 | \begin{tikzpicture} |
| 184 | \scriptsize |
| 185 | \bXInput{yt} |
| 186 | \bXBloc[3]{q}{\makecell[c]{Matched filter\\$q(t)=p(-t)$}}{yt} |
| 187 | \bXLink[$y(t)$]{yt}{q} |
| 188 | \bXBloc[3]{sampler}{\makecell[c]{Sample\\$T_s=1/R_\text{sym}$}}{q} |
| 189 | \bXLink[$r(t)$]{q}{sampler} |
| 190 | \bXBloc[3]{sim2}{$?'$}{sampler} |
| 191 | \bXLink[$r_n$]{sampler}{sim2} |
| 192 | \bXBlocL[3]{decision}{Decision}{sim2} |
| 193 | \bXOutput[3]{xhatn}{decision} |
| 194 | \bXLink{decision}{xhatn} |
| 195 | \path (xhatn) ++(0.2,0) node {$\hat{x}_n$}; |
| 196 | \end{tikzpicture} |
| 197 | |
| 198 | } |
| 199 | \note{ |
| 200 | The overall model has this structure, with root-raised cosine |
| 201 | pulses used. |
| 202 | The channel is modelled as |
| 203 | additive white Gaussian noise, followed by the simulated physical effect. |
| 204 | |
| 205 | On the receiver side, after analog-to-digital conversion, the |
| 206 | received complex symbols $r_n$ are further processed to compensate |
| 207 | for channel effects before decision and demodulation. |
| 208 | |
| 209 | Currently I have based by simulations on a quadriphase-shift keying |
| 210 | modulation scheme with |
| 211 | $25\times 10^9$ symbols per second. QPSK gives two bits per symbol, |
| 212 | giving 50 Gb/s. Adding in polarization-division multiplexing, |
| 213 | which I haven't done yet, would reach the target of 100 Gb/s. |
| 214 | } |
| 215 | \end{frame} |
| 216 | |
| 217 | \begin{frame}[t] |
| 218 | \frametitle{Chromatic dispersion} |
| 219 | \onslide<1->{\begin{itemize} |
| 220 | \item Group speed of light varies with wavelength |
| 221 | \item Modelled as linear system, impulse response: |
| 222 | \[ |
| 223 | g(z, t) = \sqrt{\frac{c}{\mathrm{j} D \lambda^2 z}} |
| 224 | \exp{\left(\mathrm{j} \frac{\pi c}{D \lambda^2 z} t^2\right)} |
| 225 | \] |
| 226 | \end{itemize}} |
| 227 | |
| 228 | {\centering% |
| 229 | \includegraphics<2>[height=0.4\paperheight]{chromaticDispersionTest.eps} |
| 230 | \includegraphics<3-4>[width=4cm]{qpsk_clean.eps} |
| 231 | \includegraphics<3>[width=4cm]{cd_qpsk_noiseless_Dz17.eps} |
| 232 | \includegraphics<4>[width=4cm]{cd_qpsk_noiseless_Dz85.eps} |
| 233 | |
| 234 | } |
| 235 | |
| 236 | \note{ |
| 237 | * The first effect investigated was chromatic dispersion. |
| 238 | This effect occurs as a result of the speed of light being |
| 239 | slightly different at different wavelengths, and lasers have a |
| 240 | wavelength band that, although small, still makes a large impact |
| 241 | over long distances. |
| 242 | |
| 243 | Literature has shown that chromatic dispersion can be modelled |
| 244 | as a linear system, with D being the dispersion coefficient and |
| 245 | z the distance travelled. |
| 246 | |
| 247 | * In direct detection receivers, this effect can be seen as a pulse- |
| 248 | broadening effect. However, in coherent receivers, we are more interested |
| 249 | in the changes to the complex constellation symbols. |
| 250 | |
| 251 | * Here is a result of a simulation with 17 ps/(nm km) dispersion, with |
| 252 | 1 kilometre of fibre, in the absence of any additive noise. |
| 253 | You can still cleanly decode the symbols without much difficulty. |
| 254 | |
| 255 | But when we go slightly longer to 5 km... |
| 256 | |
| 257 | * we get this mess. |
| 258 | |
| 259 | So clearly the receiver needs to do something to mitigate the effects |
| 260 | of chromatic dispersion. A linear filter can be used. How do we design |
| 261 | this filter? Well, we know the impulse response of the dispersion |
| 262 | model, so if we invert the sign of D here... |
| 263 | } |
| 264 | \end{frame} |
| 265 | |
| 266 | \begin{frame}[t] |
| 267 | \frametitle{Chromatic dispersion compensation} |
| 268 | \[ |
| 269 | g_\text{c}(z, t) = \sqrt{\frac{c}{\mathrm{j} (-D) \lambda^2 z}} |
| 270 | \exp{\left(\mathrm{j} \frac{\pi c}{(-D) \lambda^2 z} t^2\right)} |
| 271 | \] |
| 272 | |
| 273 | {\centering% |
| 274 | \includegraphics<2>[height=.6\paperheight]{cd_qpsk_Dz3400.eps} |
| 275 | \includegraphics<3>[height=.6\paperheight]{cd_qpsk_Dz34.eps} |
| 276 | |
| 277 | } |
| 278 | |
| 279 | \note{ |
| 280 | * We get the impulse response of the dispersion compensating filter. |
| 281 | |
| 282 | * Additive noise was added back to the channel, and a million bits were |
| 283 | sent through the channel. |
| 284 | This is a plot of the bit-error rate, or the probability of decoding |
| 285 | a bit incorrectly, against a measure of the signal-to-noise ratio, |
| 286 | at a simulated transmission distance of 200 km. |
| 287 | We can see that while the magenta curve, that is without any compensation, |
| 288 | doesn't do any better than chance, |
| 289 | The red curve, which is the simulation result of |
| 290 | the compensating filter, does a very good job at approaching the |
| 291 | theoretical blue curve of an ideal AWGN channel. |
| 292 | |
| 293 | * An interesting behaviour was observed when the transmission distance |
| 294 | was reduced, in this case, to 2km. We can see that the compensation |
| 295 | filter actually does worse, which is counter-intuitive. |
| 296 | This is due to |
| 297 | a reduced number of filter taps when converting this continuous-time |
| 298 | filter to a discrete-time filter. We can, of course, add an extra filter |
| 299 | to try to correct for this, which brings us to the next topic: |
| 300 | } |
| 301 | \end{frame} |
| 302 | |
| 303 | \begin{frame} |
| 304 | \frametitle{Adaptive equalizer} |
| 305 | \begin{itemize} |
| 306 | \item Error of previous symbol fed back to change filter tap weights |
| 307 | \item Can correct for static and time-varying effects |
| 308 | \item Constant modulus algorithm (CMA) |
| 309 | \item For PSK, magnitude of transmitted symbols is constant (unity) |
| 310 | \item Error signal is distance of received signal from unit circle |
| 311 | \end{itemize} |
| 312 | |
| 313 | \note{ |
| 314 | Adaptive equalization. Here, the error of the previous symbol is used |
| 315 | to update the filter taps, which can correct for static as well as |
| 316 | slowly varying effects. In the following simulations, the constant |
| 317 | modulus algorithm was implemented. This relies on the fact that, while |
| 318 | the receiver doesn't know what symbol was transmitted, it knows, for |
| 319 | PSK, that the symbols must lie on a unit circle. The distance from the |
| 320 | received signal to the unit circle is thus used as a measure of error. |
| 321 | } |
| 322 | \end{frame} |
| 323 | |
| 324 | \begin{frame} |
| 325 | \frametitle{Adaptive equalizer: convergence} |
| 326 | {\centering% |
| 327 | \foreach \x in {1,2,3,4,5,6,7,8,9} {%,10,11,12} {% |
| 328 | \includegraphics<\x>[height=0.75\paperheight]{adaptEqAni_\x.eps}% |
| 329 | } |
| 330 | |
| 331 | \includegraphics<10>[height=0.75\paperheight]{CD+CMA_fin.eps} |
| 332 | |
| 333 | } |
| 334 | \note{ |
| 335 | * Here is an animation of how the adaptive equalizer converges, |
| 336 | again with a small dispersion but without additive noise. |
| 337 | %At first it doesn't do much, and the symbols are still quite |
| 338 | %widely spread. |
| 339 | Initially the symbols are quite widely spread, |
| 340 | but as the algorithm runs, /just click through the slides/ |
| 341 | |
| 342 | we can see the equalizer brings the symbols close to 4 single points. |
| 343 | |
| 344 | * The overall effect can be seen with additive noise added back in, |
| 345 | here with the green curve very closely agreeing with the theoretical |
| 346 | blue curve. |
| 347 | } |
| 348 | \end{frame} |
| 349 | |
| 350 | \begin{frame} |
| 351 | \frametitle{Laser phase noise} |
| 352 | \begin{itemize} |
| 353 | \item Laser linewidth: deviations from the nominal wavelength |
| 354 | \item Instantaneous change in wavelength (frequency) $\rightarrow$ |
| 355 | change in phase |
| 356 | \item $\phi[k]$ modelled as \textit{one-dimensional Gaussian random walk} |
| 357 | \[ |
| 358 | \phi[k] = \phi[k-1] + \Delta\phi |
| 359 | \] |
| 360 | \[ |
| 361 | \text{where}\quad\Delta\phi \sim \mathcal{N}(0, 2 \pi \Delta\nu T_s) |
| 362 | \] |
| 363 | \end{itemize} |
| 364 | {\centering% |
| 365 | \includegraphics[height=.5\paperheight]{../phasenoise_linewidths.eps} |
| 366 | |
| 367 | } |
| 368 | \note{ |
| 369 | The next effect investigated was laser phase noise. This arises as the |
| 370 | true laser wavelength is a slight deviation from the nominal central |
| 371 | wavelength within the linewidth. |
| 372 | This instantaneous change in wavelength can be modelled as |
| 373 | a phase shift. Literature suggests a one-dimensional Gaussian |
| 374 | walk model, where each sample of the phase differs |
| 375 | from the previous sample with a random delta phi drawn from a Gaussian |
| 376 | distribution, with zero mean and whose variance is proportional to both the linewidth |
| 377 | delta nu and the sampling time $T_s$. |
| 378 | } |
| 379 | \end{frame} |
| 380 | |
| 381 | \begin{frame} |
| 382 | \frametitle{Laser phase noise} |
| 383 | \begin{itemize} |
| 384 | \item Rotates symbol constellation |
| 385 | \item Problematic, e.g.~for QPSK, rotation by $\pi/2$ gives another |
| 386 | constellation with all symbols decoded wrongly |
| 387 | \end{itemize} |
| 388 | {\centering% |
| 389 | \includegraphics[height=.6\paperheight]{../phasenoise_rotation.eps} |
| 390 | |
| 391 | } |
| 392 | \note{ |
| 393 | The effect of phase noise is a rotation of the constellation symbols |
| 394 | by an arbitrary amount, which is very problematic for PSK modulation |
| 395 | as this would mean completely incorrect decoding. |
| 396 | } |
| 397 | \end{frame} |
| 398 | |
| 399 | \begin{frame}[t] |
| 400 | \frametitle{Laser phase noise: solutions} |
| 401 | \begin{columns}[t] |
| 402 | \begin{column}{0.5\textwidth} |
| 403 | \begin{itemize} |
| 404 | \item Differential PSK |
| 405 | \item Information encoded as the difference in phase with previous |
| 406 | symbol |
| 407 | \item Difference in phase noise between consecutive symbols small |
| 408 | \item \SI{2.5}{dB} penalty at $\text{BER} = 10^{-3}$ |
| 409 | \end{itemize} |
| 410 | \end{column} |
| 411 | \begin{column}{0.5\textwidth} |
| 412 | \begin{itemize} |
| 413 | \item Estimate phase noise by Viterbi-Viterbi algorithm |
| 414 | \item Taking average over a small block of samples |
| 415 | \end{itemize} |
| 416 | |
| 417 | {\centering% |
| 418 | \includegraphics[height=.5\paperheight]{../phasenoise_estimation.eps} |
| 419 | |
| 420 | } |
| 421 | \end{column} |
| 422 | \end{columns} |
| 423 | \note{ |
| 424 | We will consider two methods to mitigate this effect. The first is to |
| 425 | use a differential PSK scheme, where the information is encoded not as |
| 426 | the absolute phase, but as a difference in phase with the previous |
| 427 | symbol. This works relying on the assumption that the phase noise between |
| 428 | two consecutive symbols is sufficiently smaller than 90 degrees. |
| 429 | However, by considering two |
| 430 | symbols together, the effect of phase noise is enhanced, |
| 431 | translating to a penalty in the signal-to-noise ratio. |
| 432 | |
| 433 | Another method is to estimate the amount of phase noise in each symbol |
| 434 | and to rotate the symbols back by the estimated amount. |
| 435 | The Viterbi-Viterbi algorithm, very briefly, |
| 436 | works by taking an average over |
| 437 | a block of samples to estimate the average phase noise in that block. |
| 438 | } |
| 439 | \end{frame} |
| 440 | |
| 441 | \begin{frame}[t] |
| 442 | \frametitle{Cycle slips} |
| 443 | \begin{columns}[t] |
| 444 | \begin{column}[t]{0.5\textwidth} |
| 445 | \begin{itemize} |
| 446 | \item<1-> At large linewidths, Viterbi-Viterbi algorithm can |
| 447 | make mistakes |
| 448 | \item<1-> For QPSK the phase estimate can be off by $\pi/2$ |
| 449 | \item<1-> All subsequent symbols will be decoded incorrectly\\[1em] |
| 450 | \item<3-> Solution: Differential encoding |
| 451 | \end{itemize} |
| 452 | |
| 453 | \end{column} |
| 454 | |
| 455 | \begin{column}[t]{0.5\textwidth} |
| 456 | |
| 457 | \includegraphics<1>[width=\textwidth]{../cycleslip.eps} |
| 458 | |
| 459 | \includegraphics<2-3>[width=\textwidth]{../phasenoise_ult_sansDE.eps} |
| 460 | |
| 461 | \includegraphics<4>[width=\textwidth]{../phasenoise_ult.eps} |
| 462 | \end{column} |
| 463 | \end{columns} |
| 464 | \note{ |
| 465 | This estimation algorithm does not always work though. Sometimes, due to noise, |
| 466 | a cycle slip would occur, which results in a systematic error of 90 degrees. |
| 467 | This means that all the following symbols would actually be decoded |
| 468 | incorrectly. |
| 469 | |
| 470 | * We can see in this simulation result that, when a cycle slip does not |
| 471 | occur, the performance is much closer to the ideal curve than using |
| 472 | differential PSK, but when it does occur the result is disastrous. |
| 473 | |
| 474 | Can we get the best of both worlds, with a small penalty but without |
| 475 | any cycle slips? |
| 476 | |
| 477 | * The answer is yes, with a differential *encoding*. Note that this is |
| 478 | different from differential PSK. Differential encoding is an operation |
| 479 | on the bits, and the receiver would perform differential decoding |
| 480 | on the bits *after* choosing the closest constellation symbols. |
| 481 | Whereas in DPSK the receiver evaluates the difference in phase between |
| 482 | the received samples directly, before converting them to bits. |
| 483 | |
| 484 | * The result is this cyan curve, with a smaller penalty than DPSK, and |
| 485 | not affected by cycle slips. |
| 486 | } |
| 487 | \end{frame} |
| 488 | |
| 489 | \begin{frame} |
| 490 | \frametitle{What next?} |
| 491 | \begin{itemize} |
| 492 | \item Integrating CD and phase noise into a single system |
| 493 | \item Adaptive equalizer |
| 494 | \begin{itemize} |
| 495 | %%\item Convergence |
| 496 | \item Decision-directed algorithms |
| 497 | \item Training sequences |
| 498 | \end{itemize} |
| 499 | \item QAM |
| 500 | \item Polarization-division multiplexing |
| 501 | \begin{itemize} |
| 502 | \item Polarization mode dispersion |
| 503 | \item Adaptive equalizer |
| 504 | \end{itemize} |
| 505 | \item Non-linear effects |
| 506 | \end{itemize} |
| 507 | \note{ |
| 508 | That's all I have done as of now. Looking forward, simulations |
| 509 | would need to integrate both effects into a single channel. |
| 510 | Further investigations into adaptive equalizers can also be done. |
| 511 | %For example, their convergence with different number of taps or |
| 512 | %with different convergence parameters, and also different algorithms |
| 513 | For example, the use of decision-directed algorithms, |
| 514 | %%such as decision-directed algorithms, |
| 515 | and to train the equalizer |
| 516 | with pre-known sequences, to increase accuracy at the cost of losing |
| 517 | some data rate. |
| 518 | |
| 519 | Other modulation schemes can also be investigated, such as |
| 520 | higher-order QAM. |
| 521 | It would require careful alterations to existing implementations, |
| 522 | for example with the constant modulus algorithm, but can reduce the |
| 523 | required symbol rate. |
| 524 | |
| 525 | The simulations performed only considered a single polarization. |
| 526 | Further investigations should be done to send data through one more |
| 527 | polarization, and the resulting effect of polarization mode dispersion |
| 528 | needs to be addressed. Adaptive equalizers for PDM signals also need |
| 529 | to be revised. |
| 530 | |
| 531 | Finally, non-linear fibre effects can also be considered. |
| 532 | |
| 533 | All these should hopefully be done before mid-Lent, leaving sufficient |
| 534 | time for experimental measurements. |
| 535 | |
| 536 | That's all I've got. Thank you. |
| 537 | } |
| 538 | \end{frame} |
| 539 | |
| 540 | \bgroup |
| 541 | \setbeamercolor{background canvas}{bg=black} |
| 542 | \setbeamertemplate{navigation symbols}{} |
| 543 | \begin{frame}[plain,noframenumbering]{} |
| 544 | \end{frame} |
| 545 | \egroup |
| 546 | |
| 547 | \begin{frame}[noframenumbering] |
| 548 | \frametitle{Viterbi-Viterbi algorithm} |
| 549 | Assume $\hat{\phi}\;\approx\; \phi[1] \approx \phi[2] \approx \cdots \approx \phi[N]$ |
| 550 | \begin{align*} |
| 551 | r[k] &= \exp\left(\mathrm j \phi[k] + \mathrm j \frac \pi 4 + \mathrm j |
| 552 | \frac {d[k] \pi} 2\right) + n[k] \\ |
| 553 | r[k]^4 &= \exp\left(\mathrm j 4 \phi[k] + \mathrm j \pi\right) + n'[k] |
| 554 | \\ |
| 555 | \sum_{k=1}^{N} r[k]^4 &\approx N \exp\left(\mathrm j 4 \hat{\phi} + \mathrm j \pi\right) + n''\\[1.5em] |
| 556 | \hat{\phi} &\approx \frac14 \arg\left( -\sum_{k=1}^{N} r[k]^4 \right) |
| 557 | \end{align*} |
| 558 | \end{frame} |
| 559 | |
| 560 | \end{document} |