Data Transmissions Laboratory,
Technical University of Cluj-Napoca
Zsolt Polgar, Vasile Bota, Mihaly Varga
Performances of LDPC-Error Detecting
Concatenated Codes Used in Adaptive
OFDM Transmissions
Overview
• LDPC-error detecting code concatenation
• Message passing decoding of LDPC codes when
correct bits can be identified
• LDPC and error detecting concatenated codes applied
in adaptive OFDM downlink transmissions
• BCH error detecting codes adapted to the considered
OFDM transmission scheme
• Questions for further study
2
LDPC code concatenated with error detecting codes
• consider an n’-bit long LDPC code, n’ high (several hundred at least)
concatenated with m error detecting codes (ED), of different lengths, as
shown in figure 1 – filled segments represent the ED codes control bits.
• the information bits are first LDPC coded (external code);
• the obtained codeword, n’-bit long, is divided into m equal or non-equal
groups, G1 to Gm;
• each group is coded with a high rate ED code;
• the final code word is obtained by concatenating the m code words
generated by codes G1 to Gm;
G1 – C1
G2 – C2
Gk – Ck
Gk+1 – Ck+1
Gm – Cm
1 1 0 1 1 0 0 11 1 0 1 1 0 0 0 0 11 0 1 1 1 1 0 1 01 1 1 0 0 0 01 0 11 0 1 0 1
n bits length concatenated code : LDPC code C + C1 – Cm error det. codes
3
Fig.1 Concatenation of an LDPC code with m shorter error detecting codes
Message passing decoding of LDPC codes when correct
bits can be identified
• using the concatenation method described previously in fig. 1, the LDPC
coded bits located in each of the m distinct groups can be identified as
correct or incorrect bits
• Questions: if the LDPC coded bits located in some groups are identified
as correct bits, how could this information be used by the LDPC-message
passing decoding and what is the impact of this information upon the
LDPC code performances ?
Brief analysis of the message-passing decoding algorithm for regular
binary LDPC codes
• the message passing decoding is based on the Tanner graph associated to the
LDPC code; the graph is composed of bit nodes (the bits of the codeword) and
check nodes (the check equations associated to the code) – see fig. 2
• for a regular LDPC code, each bit node (or variable node) is connected to dv
check nodes (the order of the bit node) and each check node is connected to dc
bit nodes (the order of the check node)
dv
• the coding rate of such a regular LDPC code is: R c  1 
(1)
dc
4
Message passing decoding of LDPC codes when correct
bits can be identified
• the decoding process tries to adjust the a posteriori probabilities p0 and p1 of each bit
node, based on the messages exchanged between the bit and control nodes
• if mk = log (pk0/pk1), k=1,…,dv are the log-likelihood ratios of conditional a posteriori
probabilities of a given bit value, conditioned by independent random variables (the
value of the check nodes in this case), the message generated by a bit node
d v 1
(variable node) is:
v (m 0 , m1 ,  , m d v 1 )   m i
(2)
i 0
m0 being the initial log-likelihood ratio
• a check node receives dc log likelihood ratios, log (pk0/pk1), from the bit nodes
connected to it and computes the conditional probabilities p’0t – the bit with index t
equals “0” and p’1t – the bit with index t equals “1”; these probabilities are conditioned
by the fulfilling of the considered check equation and by the probabilities of the other
bit nodes connected to this equation:
p'0t  p'1t 
 p
dc
k
0
 p1k

k 1;k t
– finally, the log-likelihood ratio, log (p’0t/p’1t), is computed and transmitted to
5
bit- node t
(3)
Message passing decoding of LDPC codes when correct
bits can be identified
bit nodes
check nodes
Fig.2 Tanner graph associated to (dv=3 ; dc=6) regular LDPC
•
•
•
let’s consider that a number of code bits can be identified as correct bits and
these bits are uniformly distributed in each group of dc bits connected to a check
node; then, in each group of dc bits connected to a check equation, dn bits can be
correctly decided (without LDPC decoding).
since these bits are correctly decided, the a posteriori probabilities assigned to
them will be constant: pk0=1, pk1=0 or pk0=0, pk1=1  the messages generated by
these bits are constant.
the equation (3) assigned to the control nodes becomes :
 dc
p'0t  p'1t   p  p  sgn   p0m  p1m
k 1; k  t
 md n
dc d n

k
0
k
1


 ;  p

m
0

 p1m  1 (4)
6
Message passing decoding of LDPC codes when correct
bits can be identified
•
equation (4) shows that, if in each group of dc bits the same number of correct
bits can be identified and if these bits generate appropriate constant messages,
the check-node order decreases and, consequently, a virtual decrease of the
coding rate occurs, see eq. (1), while the actual coding rate remains the same.
•
Question: if group of consecutive LDPC bits are controlled by an ED code, what
are the types of LDPC codes for which the correctly decided bits are uniformly
distributed in groups of dc bits? How could these codes be generated ?
• Structural properties of L(m,q) regular LDPC codes [5] [6]
•
•
•
•
L(m,q) codes have a codeword length of N = qm, where q is a prime number or a
power of a prime number and m is a natural number.
the control matrix H of such a code is generated by removing a number of rows,
according to the desired coding rate, out of a square matrix M (qm  qm).
each row of matrix M is composed of q square sub-matrices of (qm-1  qm-1) elements
each; these sub-matrices are obtained by permutations from a basis sub-matrix.
the above mentioned properties show that each control equation has a single
connection with a group of qm-1 code bits and that each control equation is connected
to each group of qm-1 code bits.
7
Message passing decoding of LDPC codes when correct
bits can be identified
•
•
fig. 3 presents the M matrix associated to an L(2,5) regular LDPC code, and the H
matrix associated to a 0.4-rate code obtained from the M matrix by retaining k = 5
columns and j=3 lines of basic sub-matrices.
the mentioned properties of this type of codes show that, if groups of Kqm-1 LDPC
bits are controlled by error detecting codes, then the number of correctly decided
bits connected to each control node will be the same, equaling K.
Note: if the lengths of
the error detecting
codes are not Kqm-1,
some kind of nonregular LDPC codes
are obtained
Fig.3 The M matrix
associated
to
an
L(2,5) regular LDPC
code and the H matrix
associated to a k=5,
j=3, p=5 L(2,5) type,
0.4-rate LDPC code
8
Message passing decoding of LDPC codes when correct
bits can be identified
• Decoding the concatenated LDPC-ED codes. Simulation results
•
•
•
•
•
•
the initial a posteriori probabilities of each bit are computed using the received levels
and the noise distribution; these probabilities are stored;
the received bits are decided, employing these probabilities (Bayes criterion) and the
ED code words are decoded  the correct bits are identified and the ED check bits
are discarded;
the LDPC code is decoded using the initial a posteriori probabilities, for the bits that
were not included in correctly received ED code words, and the information related to
the correct bits, indicated by the ED code;
The bloc diagram of such a decoder is presented in fig. 4
some simulation results obtained for tree L(2,q) codes with different code lengths and
different number of identified correct bits are presented in fig. 5 – fig. 7.
tables 1 – 3 show the main parameters of the codes employed.
9
Message passing decoding of LDPC codes when correct
bits can be identified
ED 2
encoder
LDPC
codeword
LDPC
encoder
LDPC - ED
concatenated
codeword
LDPC
codeword
segmentation
ED k
encoder
concatenation
of ED
codes
modulator+
channel
ED m
encoder
decoded
information
sequence
log-likelihood ratios
log-likelihood ratios
assigned to demodulated
assigned to LDPC code bits
bits
demodulator+
ED control bits
demapper
suppression
ED 1
decoder
received
signal
Bayes decision of
the received bits
received
sequence
segmentation
Figure 4 Coding and decoding of the proposed
LDPC-ED concatenated codes
modified LDPC
MP decoder
suppression
ED k
decoder
ED m
decoder
10
Message passing decoding of LDPC codes when correct
bits can be identified. Simulation results
2
3
1
4
5
6
Fig.5 Performances of a L(2,29) LDPC codes with k=28, j=3, p=29 parameters decoded
considering different numbers of correct bits indicated by the ED code
Curve
no.
Coding /
decoding type
No.
correct bits
1
2
3
4
5
6
Uncoded
Coded / normal
Coded / modified
Coded / modified
Coded / modified
Coded / modified
0
0
162
324
486
648
Tabel 1 Curve parameters of figure 5
Act. length /
virtual length
(bits)
810 / 810
810 / 648
810 / 486
810 / 324
810 / 162
Act. rate /
virtual rate
1/1
0.885 / 0.885
0.885 / 0.856
0.885 / 0.808
0.885 / 0.712
0.885 / 0.425
Coding gain (dB)
(pbe=1e-4) /
(pbe=1e-6)
0/0
3.5 / 4.5
4.25 / 4.75
4.75 / 5
5 / 5.25
6/6
11
Message passing decoding of LDPC codes when correct
bits can be identified. Simulation results
2
3
1
4
5
6
Fig. 6 Performances of a L(2,31) LDPC codes with k=27, j=3, p=31 parameters decoded
considering different numbers of correct bits indicated by the ED code
Curve
no.
Coding /
decoding type
No.
correct bits
1
2
3
4
5
6
Uncoded
Coded / normal
Coded / modified
Coded / modified
Coded / modified
Coded / modified
0
0
162
324
486
648
Tabel 2 Curve parameters related to fig. 6
Act. length /
virtual length
(bits)
810 / 810
810 / 648
810 / 486
810 / 324
810 / 162
Act. rate /
virtual rate
1/1
0.885 / 0.885
0.885 / 0.856
0.885 / 0.808
0.885 / 0.712
0.885 / 0.425
Coding gain (dB)
(pbe=1e-4) /
(pbe=1e-6)
0/0
3.5 / 4.5
4.25 / 4.75
4.75 / 5
5 / 5.25
6/6
12
Message passing decoding of LDPC codes when correct
bits can be identified. Simulation results
2
3
1
4
5
6
Fig. 7 Performances of a L(2,71) LDPC codes with k=71, j=3, p=71 parameters decoded
considering different numbers of correct bits indicated by the ED code
Curve
no.
Coding /
decoding type
No.
correct bits
1
2
3
4
5
6
Uncoded
Coded / normal
Coded / modified
Coded / modified
Coded / modified
Coded / modified
0
0
2911
3976
4331
4615
Tabel 3 Curve parameters related to fig. 7
Act. length /
virtual length
(bits)
5041 / 5041
5041 / 2130
5041 / 1065
5041 / 710
5041 / 426
Act. rate /
virtual rate
1/1
0.957 / 0.957
0.957 / 0.9
0.957 / 0.8
0.957 / 0.7
0.957 / 0.5
Coding gain (dB)
(pbe=1e-4) /
(pbe=1e-6)
0/0
3/4
3.75 / 4.5
4.5 / 4.75
4.75 / 5
5.5 / 5.5
13
LDPC and error detecting concatenated codes applied in
adaptive OFDM downlink transmissions
•
•
•
the proposed code concatenation was experimented in the downlink scheme
proposed in the WINNER project.
this scheme uses a special data structure, which is adapted to the frequency
selectivity (due to the multipath propagation) and fast fading (due to the motion of
the user) characteristic to a radio channel;
– this structure is called chunk and it is composed of 8 OFDM subcarriers and 12
OFDM symbols – OFDM sub-carrier separation = 39,062Hz ; OFDM symbol
period (with guard interval) = 28.8s; only 81 QAM-symbols are payload
symbols, out of the 96 included in a chunk;
– adaptive QAM modulation is used in each chunk according to the predicted
SNR; adaptive coded modulation could also be used in each chunk
the LDPC codes are used to code the bit sequence designated to a certain user;
the encoded sequence is loaded onto several chunks; for each possible chunk
load (depending on the number of bits/symbol of the selected QAM modulation), a
different ED code is used;
– this code has as information bits the bits generated by the LDPC coding;
– the length of this ED code should match the number of bits loaded in the
chunk; if adaptive QAM modulations are used with i bits/symbol, i=1,…k, 14
k separate ED codes would be used;
LDPC and error detecting concatenated codes applied in
adaptive OFDM downlink transmissions
•
•
•
•
•
•
•
•
Simulation conditions
adaptive non-coded modulation in each chunk with i bits/symbol i=1,…8; the SNR thresholds
associated to the modulations used are presented in table 4;
Nus = 1 user/site;
ED codes were not used in these simulations; the error detection was done by comparing the
received and transmitted bits;
WP5 macro urban model was employed for the channel simulation;
the background SNR ranges between 1dB and 19dB, with a 3dB step;
for each background SNR a different high-rate L(2,q)-type LDPC code was used, with different
length that match closely the average length of a 8-chunk data packet - to simplify the results
interpretation.
– the codes used for each background SNR are presented in table 6.
– the 8-chunk long data packet is proposed in the WINNER project as a possible MAC
access packet.
the simulations were also performed, for each background SNR, with different constant rate
LDPC codes, with a non-modified MP decoding; the length of the code is constant for a given
background SNR and is related to the average number of bits/ QAM symbol – see table 5 for
these parameter and other parameters related to the non-coded transmission
– the proposed modified LDPC decoding method is equivalent to a virtual decrease of the
coding rate according to the number of correct bits detected.
– it is of interest to compare the performances ensured by the proposed modified LDPC
decoding, with the performances obtained by using constant rate codes, decoded with
classical MP decoding.
15
– the constant rate of these codes is selected to mach in average the virtual rate of the
LDPC code decoded with correct bits, i.e. modified MP decoding.
LDPC and error detecting concatenated codes applied in
adaptive OFDM downlink transmissions
no. bit/symbol
threshold (dB)
1
-2
2
8.3
3
13.2
4
16.2
5
20.2
6
23.6
7
26.6
8
29.8
Table 4 Parameters of the non-coded modulations employed
background
SNR (dB)
average no.
of bit/symbol
average
packet length
bit error prob.
bin error
prob.
packet error
prob.
1
4
7
10
13
16
19
3.2179
4.0308
4.8479
5.7611
6.7219
7.5793
7.9478
2085.1992
2611.9584
3141.4392
3733.1928
4355.7912
4911.3864
5150.1744
0.9718e-3
0.14347
0.6647e-3
0.12248
0.3745e-3
0.08471
0.2721e-3
0.07425
0.2462e-3
0.07378
0.1724e-3
0.05949
0.0818e-3
0.02994
0.7103
0.6484
0.5074
0.4606
0.4584
0.3878
0.2159
Table 5 Parameters of the non-coded adaptive transmission
16
LDPC - and error detecting concatenated codes applied in
adaptive OFDM downlink transmissions
no. of correct bins
?
background
SNR (dB) 
1
average pack.=
2085.1992 bits
code length:
2106=26*81 bits
4
average pack.=
2611.9584 bits
code length:
2673=33*81 bits
7
average pack.=
3141.4392 bits
code length:
3159=39*81 bits
10
average pack.=
3733.1928 bits
code length:
372146*81 bits
13
average pack.=
4355.7912 bits
code length:
4374=54*81 bits
16
average pack.=
4911.3864 bits
code length:
4941=61*81 bits
19
average pack.=
5150.1744 bits
code length:
5184=64*81 bits
0
2
3
4
5
6
7
8
shortened
k=45;j=3;p=47
Rc=0.933
virtual Rc
=0.90984
virtual Rc
=0.8915
virtual Rc
=0.86476
virtual Rc
=0.8196
virtual Rc
=0.7295
virtual Rc
=0.459
virtual Rc
=0
j=4
Rc=0.91
virtual Rc
=0.9127
j=5
Rc=0.888
virtual Rc
=0.902
j=6
Rc=0.866
virtual Rc
=0.878
j=8
Rc=0.821
virtual Rc
=0.837
j=12
Rc=0.732
virtual Rc
=0.756
j=24
Rc=0.464
virtual Rc
=0.513
j=44
Rc=0.018
virtual Rc
=0
j=4
Rc=0.92
virtual Rc
=0.924
j=5
Rc=0.9
virtual Rc
=0.9098
j=6
Rc=0.88
virtual Rc
=0.8873
j=8
Rc=0.841
virtual Rc
=0.849
j=12
Rc=0.762
virtual Rc
=0.774
j=24
Rc=0.524
virtual Rc
=0.5492
j=50
Rc=0.008
virtual Rc
=0
j=4
Rc=0.925
virtual Rc
=0.9128
j=5
Rc=0.9066
virtual Rc
=0.8954
j=6
Rc=0.8879
virtual Rc
=0.8692
j=8
Rc=0.85
virtual Rc
=0.8257
j=12
Rc=0.775
virtual Rc
=0.7385
j=24
Rc=0.5517
virtual Rc
=0.4771
j=50
Rc=0.010
virtual Rc
=0
j=5
Rc=0.918
virtual Rc
=0.9179
j=6
Rc=0.901
virtual Rc
=0.901
j=8
Rc=0.8688
virtual Rc
=0.8769
j=10
Rc=0.836
virtual Rc
=0.8359
j=16
Rc=0.7377
virtual Rc
=0.7538
j=32
Rc=0.4754
virtual Rc
=0.5077
j=60
Rc=0.0163
virtual Rc
=0
j=5
Rc=0.923
virtual Rc
=0.922
j=6
Rc=0.908
virtual Rc
=0.9074
j=8
Rc=0.877
virtual Rc
=0.8845
j=11
Rc=0.831
virtual Rc
=0.8458
j=15
Rc=0.7549
virtual Rc
=0.7687
j=32
Rc=0.5098
virtual Rc
=0.5374
j=64
Rc=0.0196
virtual Rc
=0
j=5
Rc=0.928
virtual Rc
=0.9244
j=6
Rc=0.9137
virtual Rc
=0.9092
j=8
Rc=0.885
virtual Rc
=0.8866
j=11
Rc=0.8419
virtual Rc
=0.8488
j=16
Rc=0.770
virtual Rc
=0.7732
j=32
Rc=0.540
virtual Rc
=0.5464
j=69
Rc=0.0085
virtual Rc
=0
j=5
Rc=0.9295
j=6
Rc=0.9155
j=8
Rc=0.8873
j=11
Rc=0.8451
j=16
Rc=0.7746
j=32
Rc=0.5493
j=71
Rc=0.0002
shortened
k=51;j=3;p=53
Rc=0.940
shortened
k=54;j=3;p=59
Rc=0.943
K=61;j=4;p=6
1
Rc=0.9344
shortened
k=65;j=4;p=67
Rc=0.9387
shortened
k=70;j=4;p=71
Rc=0.9425
shortened
k=72;j=4;p=73
Rc=0.9436
17
Table 6 Parameters of the tested coded transmissions using constant rate normally decoded LDPC codes and high rate
LDPC codes decoded considering the correct bits
LDPC and error detecting concatenated codes applied in
adaptive OFDM downlink transmissions
•
•
•
three parameters were measured for each background SNR and each code
employed – 9 cases were considered:
– a high rate code (0.94 ) decoded considering the correct bits.
– 8 codes normally decoded with the same length and different coding rates,
matching in average the virtual coding rates of the LDPC code decoded
considering the correct bits – see table 6.
– NOTE: the use of different code lengths for different background SNR has a
negligible effect on overall performances because the coding gain remains
relative constant with the code length if this length is relatively large – see fig.8.
the measured parameters are: bit error, 8-chunk packet error rate, spectral efficiency
the analysis of the results requires the evaluation of the probability to have a given
number of correct chunks in an 8-chunk packet received – this parameter is related
to the number of correctly received bits.
– the average performance of the proposed decoding method depends on the
average number of correct bits detected in a LDPC-codeword, parameter directly
related to the probability to have a given number of correct chunks within a
packet – see figure 9 which shows these probabilities.
18
LDPC and error detecting concatenated codes applied in
adaptive OFDM downlink transmissions
2
3
1
4
Fig.8 Performances of the L(2,p) LDPC codes with different lengths and approximately the same rate
Curve no.
1
2
3
4
k
29
31
73
47
j
3
3
6
4
p
29
31
73
47
Rc
0.8965
0.9032
0.9178
0.9148
Code length
841
961
5329
2209
Table 7 Curve parameters related to fig. 8
•
the LDPC codes considered in figure 8, with approximately the same rate and
different lengths, have the coding gain located in a range of 0.5dB; the SNR
variation step used in simulations also equaled 0.5dB .
19
LDPC and error detecting concatenated codes applied in
adaptive OFDM downlink transmissions
8 correct chunks
7 correct chunks
6 correct chunks
0
lg(p_x)
-2
5 correct chunks
-4
4 correct chunks
3 correct chunks
-6
-
-8
2 correct chunks
-10
0 correct chunks
-12
-14
0
2
4
6
8
10
12
14
16
Fig.9 The probabilities to have x correct chunk in a 8-chunk
18
20
SNR(dB)
packet vs. the background SNR; x= 0, 2, 3, 4, 5, 6, 7, 8
•
the occurrence of 6, 7 and 8 correct chunks within an 8-chunk packet are the
most probable – these cases establish practically the spectral efficiency of the
transmission in which an ARQ protocol is used to manage the 8-chunk packets
20
LDPC and error detecting concatenated codes applied in
adaptive OFDM downlink transmissions
•
the bit error rates, 8-chunk packet error rates and the spectral efficiencies of the
transmissions using the proposed coding method and the LDPC coding, with
different rates and classical decoding, are presented in figures 10 - 12.
-2.5
-4.5
lg(pack_er)
lg(bit_er)
-3
-3.5
Rc  0.94
Rc  0.91
Rc  0.9
Rc  0.94
-4
Rc  0.87
Rc  0.91
Rc  0.9
Rc  0.87
-4.5
-5
-5.5
-6
Rc  0.83
-
-5
Rc  0.76
Rc  0.83
-5.5
-6.5
-7
0
2
4
6
8
10
12
14
16
18
SNR(dB)
Fig. 10 Bit error rates of the tested coding
methods
Rc  0.76
-6
Rc - variable
Rc0.51&0.01-6.5
0
20
Rc - variable
Rc  0.51
2
4
6
8
10
12
14
16
20
18
SNR(dB)
Rc0.01
Fig. 11 8-chunk packet error rates of the tested
coding methods
21
LDPC and error detecting concatenated codes applied in
adaptive OFDM downlink transmissions
6
Rc – variable
& Rc  0.94
Rc  0.91
Rc  0.9
Rc  0.97
Rc  0.83
Rc  0.76
Rc  0.51
spec_ef (bps/Hz)
5
4
3
Fig.12 Spectral efficiencies of
the tested coding methods
-
2
1
0
0
2
4
• Conclusions
6
8
10
12
14
16
18
20
SNR(dB)
Rc0.01
• the proposed coding-decoding method ensures a low average bit error and packet
error probabilities; both probabilities are located between the values ensured by
normally decoded codes with Rc=0.5 and Rc=0.75;
• the spectral efficiency is also high; it is approximately equal with the value ensured
22
by a high rate (0.94) LDPC code with large codeword length;
LDPC and error detecting concatenated codes applied in
adaptive OFDM downlink transmissions
• comparison between the performances of the proposed coding method and adaptive
coded modulation used at the chunk level, which is an alternative option, worth to be
analyzed.
• the adaptive coded modulations used are shown in table 8 and some parameters
characterizing the transmission scheme are presented in table 9;
• figures 13 – 15 show comparatively the bit error rate, 8-chunk packet-error rate and
spectral efficiency of the proposed coding method, at the packet-level, and of the chunklevel coding.
chunk coded configurations
code param.
nb
k
j p
nc nnc Rcfg
SNR
thr.
(dB)
8
3 11 1
0
0.59 -
8
4 23 2
0
0.43 3.3
13 3 13 2
0
0.76 6.3
9
4 37 4
0
0.55 10.1
9
5 19 2
2
0.84 12.5
10 3 17 2
2
0.84 13.9
12 4 41 6
0
0.66 16.4
12 4 29 4
2
0.76 17.9
10 3 17 2
4
0.89 20.7
8
4 41 4
4
0.74 22.8
15 3 29 4
4
0.86 25.4
10 3 17 2
6
0.92 26.8
Table 8 Parameters of chunk coded
adaptive modulations
background
SNR (dB)
average bit
no./symbol
average
packet
length
bit error
prob.
bin error
prob.
packet error
prob.
1
4
7
10
13
16
19
4.6243
5.6974
6.4736
7.5527
7.9677
7.9975
8
2996.5464
3691.9152
4194.8928
4894.1496
5163.0696
5182.38
5184
0.1255e-3
0.0652e-3
0.0334e-3
0.0205e-3
0.0123e-3
0.0074e-3
0.0035e-3
0.0131
0.0082
0.0081
0.0072
0.0065
0.0061
0.0022
0.0998
0.0637
0.0630
0.0559
0.0505
0.0478
0.0172
Table 9 Parameters of the chunk coded adaptive transmission
23
LDPC and error detecting concatenated codes applied in
adaptive OFDM downlink transmissions
-3
0
lg(bit_er)
lg(pack_er)
-3.5
non-coded
-1
-4
non-coded
chunk coding
-2
-4.5
-
-5
chunk coding -3
-5.5
packet coding
Rc  0.94
-4
packet coding
Rc  0.94
-6
-5
-6.5
-7
0
2
4
6
8
10
12
14
16
packet coding
Rc  variable -6
0
18 20
SNR(dB)
Fig.13 Bit error rates of tested non-coded,
chunk- coded and packet-coded configurations
•
packet coding
Rc  variable
2
4
6
8
10
12
14
16
18 20
SNR(dB)
Fig.14 8-chunk packet-error rates of non-coded,
chunk-coded and packet-coded configurations
the proposed packet coding method ensures lower average bit and packet error
rates, than the adaptive chunk-coded configurations and packet-coded
configurations which use high-rate LDPC codes and classical MP decoding.
24
LDPC and error detecting concatenated codes applied in
adaptive OFDM downlink transmissions
packet coding
Rc  0.94
6
spec_ef(bps/Hz)
packet coding
Rc  variable
chunk coding
5
4
3
non-coded
-
Fig.15 Spectral efficiencies of
the tested non-coded, chunkcoded
and
packet-coded
configurations
2
1
0
0
2
4
6
8
10
12
14
16
18
20
SNR(dB)
• Conclusions
•
•
the adaptive chunk-coded modulation ensures the highest spectral eficiency, but
the proposed packet-coding method provides a close spectral efficiency.
the main advantage of the proposed packet-coding is the fact that high spectral
eficiency is ensured together with low bit-error probability; the method
25
can fulfill the requirements imposed by different type of services.
BCH error detecting codes adapted to the considered
OFDM transmission scheme
• high rates error detecting codes with low undetected error probability are
required; the undetected error probability can be approximated as:
(5)
p
 2(n k)  2p
un det
where n is the codeword length and k is number of information bits; p=n-k.
• taking into account the chunk structure of the considered transmission
system (81 data symbols/chunk) and the type of used adaptive
modulations (QAM with i bits/symbol, i=1…8), the BCH codes specified in
table 10 could be used as ED codes in the proposed coding technique
based on concatenated codes.
• considering the probabilities the packet error rate and the spectral
efficiency requirements, the possible solutions are those corresponding to
Code 2 of table 10.
No.
bits/symbol
1
2
3
4
5
6
7
8
chunk
length
81
162
243
324
405
486
567
648
p
7
8
8
9
9
9
10
10
Code 1
Rc
pundet
0.913 7.81e-3
0.950 3.9e-3
0.967 3.9e-3
0.972 1.95e-3
0.977 1.95e-3
0.981 1.95e-3
0.982 9.76e-4
0.984 9.76e-4
p
14
16
16
18
18
18
20
20
Code 2
Rc
pundet
0.827 6.1e-5
0.901 1.52e-5
0.934 1.52e-5
0.944 3.81e-6
0.955 3.81e-6
0.962 3.81e-6
0.964 9.53e-7
0.969 9.53e-7
p
21
24
24
27
27
27
30
30
Code 3
Rc
pundet
0.74
4.76e-7
0.851
5.9e-8
0.901
5.9e-8
0.916
7e-9
0.933
7e-9
0.944
7e-9
0.947 9.31e-10
0.953 9.31e-10
Table 10 Possible shortened BCH codes which can be used as ED codes in the transmission scheme considered
26
Questions for further study
• Elaboration of detection algorithms of correct LDPC bits that use
the LDPC check equations [2]; the ED codes are not required this
way, avoiding the coding rate decrease due to these codes;
• Concatenation of a long LDPC code (at packet-level) with m
shorter LDPC codes (at chunk-level) used both for error
correction and error detection
– for LDPC codes decoded with the MP algorithm, the
probability to obtain after decoding a valid codeword, different
to the original codeword, is relatively low (even very low ); if
the MP decoding fails, it would not generate a valid codeword
(with very high probability)  the internal LDPC codes can do
combined error correction and error detection. The correct(ed)
bits identified this way, could be used to improve the decoding
performances of the external LDPC code.
27
References (selected)
[1] J. Hagenauer, “The Turbo principle in Communications”, Proc. of Nordic Radio
Symposium, August 2004, Oulu, Finland.
[2] M. G. Luby, M. Mitzenmacher, “Verification Base Decoding for Packet-Based LDPC
Codes”, IEEE Trans. Inform. Theory, vol. 51, No.1, Jan. 2005.
[3] ETSI, “ETSI EN 302 307 v1.1.1”, DVB-S2 standard, 2004.
[4] T. Richardson, R. Urbanke, “Capacity of Low Density Parity Check Codes Under
Message Passing Decoding”, IEEE Trans. Inform. Theory, vol. 47, Feb. 2001.
[5] J.L.Kim, U.N. Peled, I. Prepelitsa, V. Pless, S. Friedland, “Explicit Construction of
Families of LDPC Codes of girth at least six”, Proc. of 40th Allerton Conference on
Communication, Oct. 22, 2002.
[6] E. Eleftheriou, S. Olcer, “G.gen:G.dmt.bis:G.lite.bis: Efficient Encoding of LDPC
Codes for ADSL”, ITU-T, Doc. SC-064, 2002.
[7] IST-2003-507581 WINNER, “Final report on identified RI key technologies”, Report
D2.10 v1.0, 23 Dec. 2005.
[8] M. Sternad, T. Ottosson, A. Ahlen, A. Svensson, “Attaining both Coverage and High
Spectral Efficiency with Adaptive OFDM Downlinks”, Proc. of VTC 2003, 2003,
Orlando, Florida.
[9] G. Wade, Signal Coding and Processing, Cambridge University Press, 1994.
28