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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 | |
42746590 | 408 | \item \SI{2.5}{dB} penalty at $\text{BER} = 10^{-3}$ |
f9a73e9e AIL |
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 | |
42746590 | 448 | \item<1-> For QPSK the phase estimate can be off by $\pi/2$ |
f9a73e9e AIL |
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} |