Transcript EEE436 - Universiti Sains Malaysia

EEE436
DIGITAL COMMUNICATION
Coding
En. Mohd Nazri Mahmud
MPhil (Cambridge, UK)
BEng (Essex, UK)
[email protected]
Room 2.14
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Error-Correcting capability of the
Convolutional Code
The error-correcting capability of a convolutional code is determined by its constraint
length K = L + 1 where L is the number of bits of message sequence in the shift
registers and also the free distance, dfree
The constraint length of a convolutional code, expressed in terms of message bits is
equal to the number of shifts over which a single message bit can influence the
encoder output.
In an encoder with an L-stage shift register, the memory of the encoder equals L
message bits and K=L+1 shifts are required for a message bit to enter the shift
register and finally come out. Thus the constraint length = K.
The constraint length determines the maximum free distance of a code
Free distance is equal to the minimum Hamming distance between any two codewords in
the code.
A convolutional code can correct t errors if dfree > 2t
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Error-correction
The free distance can be obtained from the state diagram by splitting node a into ao and a1
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Rules
A branch multiplies the signal at its input node by the transmittance
characterising that branch
A node with incoming branches sums the signals produced by all of
those branches
The signal at a node is applied equally to all the branches outgoing
from that node
The transfer function of the graph is the ratio of the output signal to
the input signal
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D on a branch describes the Hamming weight of the encoder output
corresponding to that branch
The exponent of L is always equal to one since the length of each branch is
one.
Let T(D,L) denote the transfer function of the signal flow graph
Using rules 1, 2 and 3 to obtain the following input-output relations
b = D2Lao + Lc
c= DLb + DLd
d=DLb + DLd
a1=D2Lc
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Solving the set of equations for the ratio a1/a0, the transfer function of the graph is
given by
T(D,L) = D5L3 / 1-DL(1+L)
We can use the binomial expansion and power series to get the expression for the
distance transfer function (with L=1) as
T(D,1)=D5 + 2D6 + 4D7 + ……..
Since the free distance is the minimum Hamming distance between any two
codewords in the code and the distance transfer function T(D,1) enumerates the
number of codewords that are a given distance apart, it follows that the exponent
of the first term in the expansion T(D,1) defines the free distance=5.
Therefore the (2,1,2) convolutional encoder with constraint length K = 3, can only
correct up to 2 errors.
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Error-correction
Constraint Length, K
Maximum free distance,
dfree
2
3
3
5
4
6
5
7
6
8
7
10
8
10
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Turbo Codes
A relatively new class of convolutional codes first introduced in 1993
A basic turbo encoder is a recursive systematic encoder that employs two
convolutional encoders (recursive systematic convolutional or RSC in parallel, where
the second encoder is preceded by a pseudorandom interleaver to permute the
symbol sequence
Turbo code is also known as Parallel Concatenated Codes (PCC)
Systematic
bits, xk
Message
bits
RSC
encoder 1
Interleaver
Parity
bits, y1k
RSC
encoder 2
Puncture
&
MUX
To
transmitter
Parity
bits, y2k
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Turbo Codes
The input data stream is applied directly to encoder 1 and the pseudorandomly
reordered version of the same data stream is applied to encoder 2
Both encoders produce the parity bits.
The parity bits and the original bit stream are multiplexed and then transmitted
The block size is determined by the size of the interleaver for example 65, 536 is
common)
Puncture is applied to remove some parity bits to maintain the code rate at 1/2 . For
example, by eliminating odd parity bits from the first RSC and the even parity bits
from the second RSC
Systematic
bits, xk
Message
bits
RSC
encoder 1
Interleaver
Parity
bits, y1k
RSC
encoder 2
Puncture
&
MUX
To
transmitter
Parity
bits, y2k
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RSC encoder for Turbo encoding
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RSC encoder for Turbo encoding
0011
0001
0001
0001
0001
1111
Recursive
Non-recursive
More 1’s for recursive gives better error performance
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Turbo decoding
Turbo decoder consists of two maximum a posterior (MAP) decoders and a feedback
path
Decoding operates on the noisy version of the systematic bits and the two sets of
parity bits in two decoding stages to produce estimate of the original message bits
The first decoder takes the information from the received signal and calculates the
A posterior probability (APP) value
This value is then used as the a priori probability value for the second decoder
The output is then fedback to the first decoder where the process repeats in an iterative
fashion with each iteration producing more refined estimates.
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Turbo decoding uses BCJR algorithm
BCJR ( Bahl, Cocke, Jelinek and Raviv, 1972) algorithm is a
maximum a posteriori probability (MAP) decoder which minimizes
the bit errors by estimating the a posteriori probabilitities of the
individual bits in a code word.
It takes into account the recursive character of the RSC codes and
computes a log-likelihood ratio to estimate the APP for each bit.
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Low Density Parity Check (LDPC) codes
An LDPC code is specified in terms of its parity-check matrix, H that
has the following structural properties
i) Each row consists of ρ 1’s
ii) Each column consists of γ 1’s
iii) The number of 1’s in common between any two columns is no
greater than 1; ie λ= 0 or 1
iv) Both ρ and γ are small compared with the length of the code
LDPC codes are recognised in the form of (n, γ, ρ)
H is said to be a low density parity check matrix
H has constant row and column weights (ρ and γ )
Density of H = total number of 1’s divided by total number of entries in H
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Low Density Parity Check (LDPC) codes
Example (15, 4, 4) LDPC code
Each row consists of ρ=4 1’s
Each column consists of γ=4 1’s
The number of 1’s in common
between any two columns is no
greater than 1; ie λ= 0 or 1
Both ρ and γ are small compared
with the length of the code
Density = 4/15 = 0.267
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Low Density Parity Check (LDPC) codes – Constructing H
For a given choice of ρ and γ, form a kγ by kρ matrix H (where k = a positive
integer > 1) that consists of γ k-by-kρ submatrix, H1, H2,….Hγ
Each row of a submatrix has ρ 1’s and each column of a submatrix contains a
single 1
Therefore each submatrix has a total of kρ 1’s.
Based on this, construct H1 by appropriately placing the 1’s.
For 1  i  k ,the ith row of H1 contains all its ρ 1’s in columns (i-1) ρ+1 to iρ
The other submatrices are merely column permutations of H1
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Low Density Parity Check (LDPC) codes – Example Constructing H
Choice of ρ=4 and γ=3 and k=5
Form a kγ by kρ (15-by-20) matrix H that consists of γ=3 k-by-kρ (5-by-20)
submatrix, H1, H2,Hγ=3
Each row of a submatix has ρ=4 1’s and each column of a submatrix contains a
single 1
Therefore each submatrix has a total of kρ=20 1’s.
Based on this, construct H1 by appropriately placing the 1’s.
For 1  i  5 ,the ith row of H1 contains all its ρ=4 1’s in columns (i-1) ρ+1 to iρ
The other submatrices are merely column permutations of H1
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Low Density Parity Check (LDPC) codes –
Example Constructing H for (20, 3, 4) LDPC code
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Construction of Low Density Parity Check (LDPC) codes
There are many techniques of constructing the LDPC codes
Constructing LDPC codes with shorter block is easier than the
longer ones
For large block sizes, LDPC codes are commonly constructed by first
studying the behaviour of decoders.
Among the techniques are Pseudo-random techniques, Combinatorial
approaches and finite geometry. These are beyond the scope of this
lecture.
For this lecture, we see how short LDPC codes are constructed from a
given parity check matrix.
For example a (6,3) linear LDPC code given by the following H
0
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Construction of Low Density Parity Check (LDPC) codes
For example a (6,3) linear LDPC code given by the following H
0
The 8 codewords can be obtained by putting the parity-check matrix H into
this form H=[PT I In-k ]; where P=coefficient matrix and In-k = identity matrix
The generator matrix is, G = [In-k I P]
At the receiver, H=[PT I In-k ] is used to check the error syndrome.
Exercise:
Generate the codeword for m=001 and show how the receiver performs
the error checking
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