Fundamentals of Coding and Modulation - Tutorial at ECOC 2009 ...

Gerhard Kramer

Description

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 ...

Sujets

• Modulation d'amplitude en quadrature
• Tutoriel
• Dentelles du Cygne
• Estimated Time of Arrival
• WAVEform audio format
• Gaussian minimum-shift keying

Informations

• Publié par: Ummo
• Nombre de lectures: 108
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

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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

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Acknowledgment: Several graphics were borrowed from R.-J. Essiambre and M. Magarini. USC students provided feedback on presentation.

1 Part 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

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Outline

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One goal: review (classic communications) concepts that led to the following plot from our OFC 2009 tutorial.

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Capacity

and

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Channels, Coding, and Capacity

1.

Channels,

Coding,

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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?

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Suppose the channel isbandlimitedand passes the frequencies f 0 −W/2 < f< f 0 +W/2Hertz where the center frequencyf 0 is much larger thanW Use Nyquist-Shannon sampling theory to represent signals byn regularly-spaced samples that areT s = 1/Wseconds apart X(t) = P n i=1 X ic ∙ sin π (πtW Wt− − iπ πi) SoX(t)andY(t)are represented bydiscrete-timeandcontinuous amplitude/phasesignalsX 1 c , X 2 c , . . . , X c n andY 1 c , Y 2 c , . . . , Y nc We can “cl l ” proximate these signals withdiscrete ose y ap amplitude/phasesignalsX 1 , X 2 . . . , X n andY 1 , Y 2 , . . . , Y n

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

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Noisychannel: acond. probability distrib.P Y 1 Y 2 ...Y n |X 1 X 2 ...X n (∙|∙). 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)

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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

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How should weusethe noisy digital channelP Y|X (∙|∙)? Supposereliabilityis paramount. we guarantee reliability? Can Simple answer:yesbyrepeatingevery symbol sufficiently often. Sophisticated answer:yes usingand with positive rate (!) bycoding

Coding

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Example: suppose we use the BSCntimes to transmitk=n/2 information bits (terminology: an(n, k)codewithratek/n= 1/2) There are2 k information-bit vectors and2 n length-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