Chapter 7: Channel coding: Convolutional codes · I Viterbi proposed an ML decoding by...

OutlineChannel coding

Convolutional encoderDecoding

Chapter 7: Channel coding:Convolutional codes

Vahid Meghdadi

University of [email protected]

Reference : Digital communications by John Proakis;Wireless communication by Andreas Goldsmith

Vahid Meghdadi Chapter 7: Channel coding:Convolutional codes



Channel coding

Convolutional encoderEncoder representation

DecodingMaximum Likelihood (ML) decodingViterbi algorithmError probability of CC




Communication system

ChannelEncoder

Interleaver Modulator

Channel

ChannelDencoder

Deinterleaver Demodulator

Data

Output

The data is first coded then interleaved. The interleaving is usedto avoid the channel presenting burst errors. Then it is transmittedthrough a kind of modulation over a noisy channel. At thereceiver, all the operations must be inversed to estimate thetransmitted data.




Channel coding

Channel coding can be classified in two categories:I block coding

I Cylclic codesI BCHI Reed-SolomonI Product Turbo codeI LDPC

I trellis codingI Convolutional codingI TCM (Trellis code modulation)I Turbo codes (SCCC or PCCC)I Turbo TCM

Here, we are concentrating on convolutional coding.




Encoder representation

Convolutional encoder

ChannelEncoder


Channel

ChannelDencoder


Data

Output

cv(1)

v(2)

This is the simplest convolutional encoder. The input sequence is

c = (..., c−1, c0, c1, ..., cl , ...)

where l is the time index. The output sequences are:

v(1) = (..., v(1)−1 , v

(1)0 , v

(1)1 , ..., v

(1)l , ...)

v(2) = (..., v(2)−1 , v

(2)0 , v

(2)1 , ..., v

(2)l , ...)





Convolutional code characteristics

I The constraint length of a CC is the number of input bitinvolved to generate each output bit. It is the number of thedelay elements plus one. For the previous example, theconstraint length is 3.

I Once all the output are serialized and get out of the coder, kright shift occurs.

I The code rate is defined as Rc = k/n where n is the numberof output.

I For the previous example n = 2, k = 1, and Rc = 0.5.





Describing a CC by its generator

I In the previous example, assuming all-zero state, the sequence

v(1)1 will be [101] for a 1 at the input (impulse response).

I At the same time the sequence v(2)1 will be [111] for a 1 at the

input.

I Therefore, there are two generators g1 = [101] and g2 = [111]and the encoder is completely known.

I It is convenient to represent the encoder in octal form of as(g1, g2) = (5, 7).

Exercise: Draw the encoder circuit of the code (23,35).





State diagram representation

ChannelEncoder


Channel

ChannelDencoder


Data

Output

cv(1)

v(2)

State01

State00

State10

State11

00

11

00

10

01

10

11

01

input=0

input=1

Code (7,5)

Using this diagram one can easily calculate the output of the coder.Exercise: What is the coded sequence of [10011]?





Trellis representation

00

00

1111

10

10

0101

00

00

1111

10

10

0101

00

00

1111

10

10

0101

00

11

10

01

00

11

00

11

10

01

input 0 input 1

We assume that the trellis begins at the state zero. The bits on

the diagram show the output of the encoder (v(1)t , v

(2)t ).

Exercise: Using above diagram what is the coded sequence of the[1 0 1 1 0 1]?




Maximum Likelihood (ML) decodingViterbi algorithmError probability of CC

Modulation and the channel

The coded sequence is denoted by C. The output of the encoder ismodulated, then sent to the channel.We assume a BPSK modulation with 0→ −1 volt and 1→ +1volt. The channel adds a white Gaussian to the signal.The RX receives a real sequence R. The decoder should estimatethe transmitted sequence by observing the channel output.For the i-th information bit, corresponding to the i-th branch inthe trellis, the code sequence of size n is denoted by Cij where0 < i < n− 1. Assuming a memoryless channel, the received signalfor the i-th information bit is:

rij =√

Ec(2Cij − 1) + nij





Maximum Likelihood decoding

Given the received sequence R, the decoder decides that the codedsequence C∗ was transmitted if

p(R|C∗) ≥ p(R|C) ∀C

It means that the decoder finds the maximum likelihood path inthe trellis given R. Thus, over an AWGN channel and for a code ofrate 1/n and for an information sequence of size L, the likelihoodcan be written as:

p(R|C) =L−1∏i=0

p(Ri |Ci ) =L−1∏i=0

n−1∏j=0

p(Rij |Cij)





Log Likelihood function

Taking logarithm:

log p(R|C) =L−1∑i=0

log p(Ri |Ci ) =L−1∑i=0

n−1∑j=0

log p(Rij |Cij)

The expression Bi =∑n−1

j=0 log p(Rij |Cij) is called the branchmetric. Maximizing the likelihood function is equivalent tomaximizing the log likelihood function.The log likelihood function corresponding to a given path in thetrellis is called the path metric, which is the sum of the branchmetrics in the path.





ML and hard decoding

In hard decoding, the sequence R is demodulated and containsonly 0 and 1. This is called hard decision. The probability of errorin hard decision depends to the channel state and is represented byp. If R and C are N symbol long and differ in d places (i.e.Hamming distance), then

p(R|C) = pd(1− p)N−d

log p(R|C) = −d log1− p

p+ N log(1− p)

The second term is independent of C. Therefore only the first termshould be maximized. Conclusion: the sequence C with minimumHamming distance to the received sequence R corresponds to theML sequence.





Soft decoding

In an AWGN with a zero mean Gaussian noise of varianceσ2 = N0/2,

p(Rij |Cij) =1√2πσ

exp

[−

(Rij −√

Ec(2Cij − 1))2

2σ2

]ML is equivalent to choose the Cij that is closest in Euclideandistance to Rij .Taking logarithm and neglecting all scaling factor and commonterms to all Ci , the branch metric is:

µi =n∑

j=1

Rij(2Cij − 1)





Viterbi algorithm (1967)

I There are a huge number of paths to test for even moderatevalues of L. The Viterbi algorithm reduces considerably thisamount.

I Viterbi proposed an ML decoding by systematically removingpaths that cannot achieve the highest path metric.

I When two paths enter into a given node, the path with lowestlikelihood cannot be the survivor path: it can be removed.





Viterbi algorithm

ML path

P1

P2

P3

N

Bn+1

Bn+2

Bk

Survivor path

I P1 is the survivor path because the partial path metric P1 isgreater than P2 and P3 where Pi =

∑n−1k=0 B i

k .

I The ML path starting from node N to infinity has the pathmetric of Pi +

∑∞k=n Bk (i = 1, 2, 3).

I Thus, the paths P2 and P3 cannot complete the path from nto ∞ and give a ML path. They will be discarded.





Trace back

ML path

P1

P2

P3

N

Bn+1

Bn+2

Bk

Survivor path

Trace back. All the paths arrive to a common stem

Common stem

I Theoretically, the depth of the trace back operation is ∞.However, it is observed that all the surviving paths mergefinally to one common node, giving the decoding delay.

I Statistically speaking, for sufficiently large trellis’ depth(typically five times the constraint length), all the pathsmerge to a common node. Therefore a decoding delay of 5Kis considered.





Distance properties (example CC(7,5))

ML path

P1

P2

P3

N

Bn+1

Bn+2

Bk

Survivor path

Trace back. All the paths arrive to a common stem

Common stem

t0 t1 t2 t3 t4 t50 0 0 0 0

2 2

1 11

0

22 2

11

00

01

10

11

00

00

1111

10

10

0101

00

00

1111

10

10

0101

00

00

1111

10

10

0101

00

11

10

01

00

11

00

11

10

01

input 0 input 1

Survivor Path

tk tk+1 tk+2

path 1

path 2

path 3

Bt+1 Bt+2

00

00

1111

10

10

0101

I The error occurs when the survivor path is to be selected.

I If the Hamming distance between two sequences entering to anode is large, the probability of making error becomes small.

I The free distance of CC(7,5) is 5 corresponding to the path00-10-01-00.





Good codes

Testing all the possible combinations of encoders, the best codesfor a given constraint length are proposed in tables.

Constraint Length Generaor in octal dfree

3 (5,7) 54 (15,17) 65 (13,35) 76 (53,75) 87 (133,171) 10

Rate 1/2 maximum free distance CC





Good codes

Constraint Length Generaor in octal dfree

3 (5,7,7) 84 (13,15,17) 105 (25,33,37) 126 (47,53,75) 137 (133,145,175) 15

Rate 1/3 maximum free distance CC





Error probability

I Convolutional code is linear so the Pe is calculated bysupposing all zero sequence is sent.

I Coherent BPSK over AWGN is assumed.I The energy per coded bit is related to the energy per

information bit by Ec = RcEb.I The probability of mistaking all zero sequence by another

sequence with Hamming distance of d is:

Pe(d) = Q

(√2Ec

N0d

)= Q (2γbRcd)

I This probability is called pairwise error probability.I There is no analytic expression for bit error rate but just some

upper bounds.





Error probability


Date post:	12-Jan-2020
Category:	Documents
Upload:	others
View:	19 times
Download:	0 times

Chapter 7: Channel coding: Convolutional codes · I Viterbi proposed an ML decoding by...

Documents