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Channels, Coding, and Capacity Spectral E ciency Linear Block Codes Hard and Soft Decoding Fiber Capacity Estimate
Fundamentals of Coding and Modulation
Tutorial at ECOC 2009, Vienna, Austria
Gerhard Kramer
Department of Electrical Engineering - Systems
University of Southern California
Los Angeles, CA, USA
September 2009
Gerhard Kramer Tutorial on Coding and Modulation (ECOC 2009) 1 / 45 Channels, Coding, and Capacity Spectral E ciency Linear Block Codes Hard and Soft Decoding Fiber Capacity Estimate
Outline
1 Part 1: Channels, Coding, and Capacity
2 Part 2: Spectral E ciency
3 Part 3: Linear Block Codes
4 Part 4: Hard and Soft Decoding
5 Part 5: Fiber Capacity Estimate
Acknowledgment: Several graphics were borrowed from R.-J. Essiambre
and M. Magarini. USC students provided feedback on presentation.
Gerhard Kramer Tutorial on Coding and Modulation (ECOC 2009) 2 / 45 Channels, Coding, and Capacity Spectral E ciency Linear Block Codes Hard and Soft Decoding Fiber Capacity Estimate
One goal: review (classic communications) concepts that led to the
following plot from our OFC 2009 tutorial.
Gerhard Kramer Tutorial on Coding and Modulation (ECOC 2009) 3 / 45 Channels, Coding, and Capacity Spectral E ciency Linear Block Codes Hard and Soft Decoding Fiber Capacity Estimate
Channels, Coding, and Capacity
1. Channels, Coding, and Capacity
Gerhard Kramer Tutorial on Coding and Modulation (ECOC 2009) 4 / 45 Channels, Coding, and Capacity Spectral E ciency Linear Block Codes Hard ...
Voir
Fundamentals of Coding and Modulation
Tutorial at ECOC 2009, Vienna, Austria
Gerhard Kramer
Department of Electrical Engineering - Systems
University of Southern California
Los Angeles, CA, USA
September 2009
Gerhard Kramer Tutorial on Coding and Modulation (ECOC 2009) 1 / 45 Channels, Coding, and Capacity Spectral E ciency Linear Block Codes Hard and Soft Decoding Fiber Capacity Estimate
Outline
1 Part 1: Channels, Coding, and Capacity
2 Part 2: Spectral E ciency
3 Part 3: Linear Block Codes
4 Part 4: Hard and Soft Decoding
5 Part 5: Fiber Capacity Estimate
Acknowledgment: Several graphics were borrowed from R.-J. Essiambre
and M. Magarini. USC students provided feedback on presentation.
Gerhard Kramer Tutorial on Coding and Modulation (ECOC 2009) 2 / 45 Channels, Coding, and Capacity Spectral E ciency Linear Block Codes Hard and Soft Decoding Fiber Capacity Estimate
One goal: review (classic communications) concepts that led to the
following plot from our OFC 2009 tutorial.
Gerhard Kramer Tutorial on Coding and Modulation (ECOC 2009) 3 / 45 Channels, Coding, and Capacity Spectral E ciency Linear Block Codes Hard and Soft Decoding Fiber Capacity Estimate
Channels, Coding, and Capacity
1. Channels, Coding, and Capacity
Gerhard Kramer Tutorial on Coding and Modulation (ECOC 2009) 4 / 45 Channels, Coding, and Capacity Spectral E ciency Linear Block Codes Hard ...
September 2009
Gerhard Kramer
Department of Electrical Engineering - Systems University of Southern California Los Angeles, CA, USA
Fundamentals of Coding and Modulation Tutorial at ECOC 2009, Vienna, Austria
C2CO(Eon451/9)00
Acknowledgment: Several graphics were borrowed from R.-J. Essiambre and M. Magarini. USC students provided feedback on presentation.
1Part 1: Channels, Coding, and Capacity 2 SpectralPart 2: Efficiency 3 LinearPart 3: Block Codes 4 and Soft DecodingPart 4: Hard 5 Capacity EstimatePart 5: Fiber
00C2CO(E
Outline
452/9)
One goal: review (classic communications) concepts that led to the following plot from our OFC 2009 tutorial.
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Capacity
and
/45
Channels, Coding, and Capacity
1.
Channels,
Coding,
OC2009)4
Communication Problem
Information source for our purposes:bits Channel: the part of a system one isunableorunwillingto change. Idea: Everything else can be optimized. Example channels: fiber (leaving many parameters, e.g., fiber type, filters, etc.) specific fiber + filters + noncoherent detector specific fiber + filters + specific modulator + coherent soft detector Goal: transmit informationquicklyandreliablyto the destination How should we design the bits-to-waveform mapping? How should we design the waveform-to-bits mapping?
/554duMondga(EontilaairotuTrnidoCnol
Suppose the channel isbandlimitedand passes the frequencies f0−W/2 < f< f0+W/2Hertz where the center frequencyf0is much larger thanW Use Nyquist-Shannon sampling theory to represent signals byn regularly-spaced samples that areTs= 1/Wseconds apart X(t) =Pni=1Xic∙sinπ(πtWWt−−iππi) SoX(t)andY(t)are represented bydiscrete-timeandcontinuous amplitude/phasesignalsX1c, X2c, . . . , XcnandY1c, Y2c, . . . , Ync We can “cl l ” proximate these signals withdiscrete ose y ap amplitude/phasesignalsX1, X2. . . , XnandY1, Y2, . . . , Yn
Continuous to Discrete in Time and Amplitude/Phase
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Result: an essentiallyoptimaltransmission structure is: modulator: usediscrete levelsto approx. optimal input statistics waveform: use acompact spectrumthat approx. sin(x)/x signaling demodulator: (1) Samplesignal(field) in time, i.e., usecoherent detection (2) Sample signal in amplitude/phase; use many sampling levels to avoid losing information, i.e., usesoft outputs Summary: the mod/demod may as well convert thecontinuous-time waveformchannel into adigital(discrete-time/alphabet) channel
/754009)COC2on(ElatineyciLenrtlaffiEicodesHardarBlockCoC,sgnidahClenntyciecSpnd,apaCa
Noisychannel: acond. probability distrib.PY1Y2...Yn|X1X2...Xn(∙|∙). This model permits memory. We will process to remove memory. Memorylesschannel:Xrepresentsoneinput andY=f(X,Z) wheref(∙)is some function andZisnoise, e.g.,Y=X+Z Some common memoryless channels (see next page): binary erasure channel (BEC) binary symmetric channel (BSC) additive white Gaussian noise channel (AWGN channel)
naMdidgntaoidoluOC20n(EC/4509)8tiEytsmitaeFingdicoacaprCbeadraHsedeDtfoSdnineancyLckCorBloeptctiSycffieiarElCdnacapadoC,,gnianChlsne
Binary Symmetric Channel (BSC): Example: OOK with hard detection is crossover probability
Binary Erasure Channel (BEC): Example: Sudoku or crossword Puzzles δis erasure probability Δis erasure symbol
Additive white Gaussian noise (AWGN) channel: complex input/output,Zis Gaussian with varianceN
5/4)90920OC
How should weusethe noisy digital channelPY|X(∙|∙)? Supposereliabilityis paramount. we guarantee reliability? Can Simple answer:yesbyrepeatingevery symbol sufficiently often. Sophisticated answer:yes usingand with positive rate (!) bycoding
Coding
taoidolunaMdidgn0/4509)1OC20n(ECahrerKdrGalriCooneramtoTuanChlsneCdnacapadoC,,gni
Example: suppose we use the BSCntimes to transmitk=n/2 information bits (terminology: an(n, k)codewithratek/n= 1/2) There are2kinformation-bit vectors and2nlength-nbit vectors. Map each information-bit vector to a unique length-ncodevector. Table: the fraction of code vectors becomestinyasngets large Intuitive idea: the“distance”between code vectors becomes large, thereby ensuring reliability
