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Dr. Shurjeel Wyne delivered this lecture at COMSATS Institute of Information Technology, Attock for Digital Communication Systems course. In this he discussed: Convolutional, State, Diagram, Trellis, Representation, Maximum, Likelihood, Soft, Hard, Decisions
Typology: Slides
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State diagram
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State S 0 00
S 1 01
S 2 10
S 3 11
m
u 1
u 2
u 1 u 2
Input bit 0 Input bit 1
(K-1) = Numb. Of memory elements
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Encoder Trellis Diagram (at TX)
t (^) 1 t (^) 2 t (^) 3 t (^) 4 t 5 t 6
1 0 1 0 0
11 10 00 10 11
Input bits
Output bits
Tail bits
S 0 00
S 1 01
S 3 11
Input bit 0 Input bit 1
(view code symbol sequence against time, for a given message sequence)
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If the possible input message sequences are equally likely, the optimum decoder which minimizes the probability of error is the Maximum likelihood decoder.
The ML decoder, chooses a particular codeword U ( m ´)^ as the transmitted sequence if the likelihood p ( Z| U ( m ´)^ ) is greater than the likelihoods of all the other possible transmitted sequences, where Z is the demodulated (still encoded) sequence input to the decoder
overall
ML decoding rule:
codewords to search for an L-bit message sequence
Mathematically: 2^ L
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ML decoding for memory-less channels
The path metric up to time index , is called the partial path metric.
1 1
() 1
() () , ,...,,... 1 2 ( (^ )) ( , ,..., ,...| ) ( | ) ( | ) (^12) i
n j
m ji ji i
m i i m zz z i p m^ p Z Z Z U pZ U pz u Z| U i
1 1
() 1
( ) log ( (^ )) log ( | ()) log ( | ) i
n
j
m ji ji i
m i i m p Z|U m^ pZ U pz u U Path metric Branch metric (^) Bit metric
ML decoding rule: Choose the path with maximum metric (likelihood) among all the paths in the trellis.
This path is the “closest” path to the transmitted sequence, in other words the ML decoding rule chooses, as its estimate of the transmitted sequence, the path that has minimum distance to the received sequence
" i "
Note: read articles 7.3 --- 7.3.4 from Sklar textbook
zji =demodulator output symbol corresponding to j -th code symbol of i -th branch word
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The demodulator does not assign a ‘0’ or a ‘1’ to each received bit, zji , rather each value of z(T) is quantized to more than two levels. The demodulator provides the decoder with some side information together with the decision, where the side information provides the decoder with a measure of confidence for the decision. The demodulator outputs, quantized to more than two levels, are called soft-bit
Soft and hard decision decoding…
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(3-bit quantization)
When the demodulator sends a soft binary decision, quantized to eight levels, it sends the decoder a 3-bit word describing an interval along z(T)
When the demodulator sends a hard binary decision to the decoder, it sends the decoder a single binary symbol
Soft and hard decision decoding –
Cont’d
Example: BPSK modulation with hard decision and 3-bit quantized soft decision decoding
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It processes the demodulator outputs in an iterative manner. At each step in the trellis, it compares the metric of all paths entering each state, and keeps only the path with the largest metric, called the survivor, together with its metric. It proceeds in the trellis by eliminating the least likely paths.
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For a data block of L bits, form the trellis. The trellis has L+K-1 sections, it starts at time and ends up at time Label all the branches in the trellis with their corresponding branch metric. For each state in the trellis at the time which is denoted by , define a parameter
t i S ( ti ) { 0 , 1 ,..., 2 K ^1 } ^ S^ ( ti ), ti
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t (^) 1 t (^) 2 t (^) 3 t (^) 4 t 5 t 6
0
S ( ti ), ti 11 00 10 10 01
Z
S 0 00
S 1 01
S 2 10
S 3 (^) 11
Branch metric
