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Lecture 9b ConvolutionalCoding/Decoding and Trellis

Code modulation

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Convolutional Coder Basics

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Coder State Diagram

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Coder Tree

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Viterbi Decoding

For Simplicity assume Binary Sym.Channel Encoder has Constraint length 3, Rate

A trellis represents the decoder

Trellis transitions are labeled with branchmetrics (hamming distance between branch

code word and received codeword If two paths merge the path with larger

metric is eliminated

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Decoder Trellis

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Free Distance

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Free Distance and Coding Gain

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Introduction to TCM

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What is TCM?

TCM schemes achieve coding gains without using additional

bandwidth

The technique employs non-binary modulation in conjunction

with a finite state encoder

The encoder dictates the selection of modulation waveformsfor generating a sequence of coded waveforms

At the receiver the noisy signals are detected/decoded by a soft

decision maximum-likelihood decoder

TCM is another technique in the evolution of coding methods

that contributes toward the fulfillment of shannon;s coding gain

prediction

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Today TCM can be used to privde a coding of 3 dB withrelative ease

6 dB coding gain can be provided with additional

complexity

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Evolution of TCM

First Proposed by Ungerboeck and Csajkain 1976 (IBM Research in Swiss)

A more detailed publication in 1982

received the information theory best paperaward

In 1984 TCM with 4 dB coding gain wasadopted by the CCITT for use in high speedvoice band modem

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Structured Sequence Coding

Channel coding can be partitioned into twostudy areas: Structured sequence coding(parity bit design) and Waveform coding

Waveform coding attempts to find betterwaveforms which provide improveddistance properties

Waveform coding can include redundantwaveforms as in the case of TCM

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Reasons for Disappointing results of

conventional coding methods when thechannel is band-limited

With hard-decision decoding, irreversible errors can be made by the

demodulator prior to decoding With soft decision decoding (using Euclidean distance), the following

problem becomes apparent

For a code, the optimized for hamming distance, the mapping of code

symbols into non-binary modulation waveforms doe not guarantee a

good Euclidean distance structure

Generally one cannot find a monotonic relationship between hamming

and Euclidean distances Squared Euclidean and hamming distances are equivalent only in the

case of binary modulation or 4 phase modulation

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Coding gain when bandwdith

expansion is allowed

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Theory of TCM

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Theory of TCM

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Set Partitioning Rules for 8-PSK

All signals should occur with equal frequency and

with a fair amount of regularity and symmetry

Transitions originating from the same state are

assigned signals either from subset B0 or B1 Transitions joining in the same state are assigned

signals either from subset B0 or B1

Parallel transitions receive signals either from

subset C0 or C1 or C2 or C3

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Partitioning (contd)

Rule 1 follows our intuition that good codes

should have a regular structure

Rules 2,3,4 guarantee that bit sequences are

assigned to waveforms so that the free distancewill exceed the free distance of the uncoded 4

PSK reference modulation by at least 3 dB

Parallel transitions refer to the branch wordsresulting from the transmission of uncoded bits

along with coded bits.

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TCM Code Construction First a suitable trellis structure is selected. This can be

done without any particular encoder in mind If k bits are to be encode per modulation interval then there

must be 2^k possible transitions from each state to a

successor state More than one transition (Parallel transitions) may occur

between pairs of states

Next from an extended set of 2^k+1 modulation signals,assignments of signals to trellis transitions will be made so

as to maximize the free Euclidean distance

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Modulation Signal Sets

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4-TCM Encoder

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TCM Trellis

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4 State Trellis with parallel Paths

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8 state trellis with partitioning

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16 QAM Partitioning

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16-QAM Trellis

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