Performances of LDPC-Error Detecting Concatenated Codes Used in Adaptive OFDM Transmissions

1
Performances of LDPC-Error Detecting Concatenated
Codes Used in Adaptive OFDM Transmissions
Data Transmissions Laboratory, Technical
University of Cluj-Napoca
Zsolt Polgar, Vasile Bota, Mihaly Varga
2
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

3
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

Fig.1 Concatenation of an LDPC code with m
shorter error detecting codes
4
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)
• the coding rate of such a regular LDPC code is
  (1)

5
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), k1,,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 (variable node) is
• (2)
• 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 p0t
  the bit with index t equals 0 and p1t 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
• (3)
• finally, the log-likelihood ratio, log
  (p0t/p1t), is computed and transmitted to
• bit- node t

6
Message passing decoding of LDPC codes when
correct bits can be identified
Fig.2 Tanner graph associated to (dv3 dc6)
regular LDPC

• lets 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 pk01, pk10 or pk00, pk11 ? the
  messages generated by these bits are constant.
• the equation (3) assigned to the control nodes
  becomes
• 
  
  (4)

7
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.

8
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 j3 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 non-regular 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 k5, j3, p5 L(2,5) type, 0.4-rate LDPC code
9
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.

10
Message passing decoding of LDPC codes when
correct bits can be identified
11
Message passing decoding of LDPC codes when
correct bits can be identified. Simulation
results
Fig.5 Performances of a L(2,29) LDPC codes with
k28, j3, p29 parameters decoded considering
different numbers of correct bits indicated by
the ED code
12
Message passing decoding of LDPC codes when
correct bits can be identified. Simulation
results
Fig. 6 Performances of a L(2,31) LDPC codes with
k27, j3, p31 parameters decoded considering
different numbers of correct bits indicated by
the ED code
13
Message passing decoding of LDPC codes when
correct bits can be identified. Simulation
results
Fig. 7 Performances of a L(2,71) LDPC codes with
k71, j3, p71 parameters decoded considering
different numbers of correct bits indicated by
the ED code
14
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,
  i1,k,
• k separate ED codes would be used

15
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 i1,8 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.
• 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.

16
LDPC and error detecting concatenated codes
applied in adaptive OFDM downlink transmissions
17
LDPC - and error detecting concatenated codes
applied in adaptive OFDM downlink transmissions
18
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.

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LDPC and error detecting concatenated codes
applied in adaptive OFDM downlink transmissions
Fig.8 Performances of the L(2,p) LDPC codes with
different lengths and approximately the same rate

• 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 .

20
LDPC and error detecting concatenated codes
applied in adaptive OFDM downlink transmissions
Fig.9 The probabilities to have x correct chunk
in a 8-chunk 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

21
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.

Fig. 11 8-chunk packet error rates of the tested
coding methods
Fig. 10 Bit error rates of the tested coding
methods
22
LDPC and error detecting concatenated codes
applied in adaptive OFDM downlink transmissions
Fig.12 Spectral efficiencies of the tested
coding methods

• Conclusions
• 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 Rc0.5 and Rc0.75
• the spectral efficiency is also high it is
  approximately equal with the value ensured by a
  high rate (?0.94) LDPC code with large codeword
  length

23
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
  chunk-level coding.

24
LDPC and error detecting concatenated codes
applied in adaptive OFDM downlink transmissions
Fig.14 8-chunk packet-error rates of non-coded,
chunk-coded and packet-coded configurations
Fig.13 Bit error rates of tested 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.

25
LDPC and error detecting concatenated codes
applied in adaptive OFDM downlink transmissions
Fig.15 Spectral efficiencies of the tested
non-coded, chunk- coded and packet-coded
configurations

• 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
• can fulfill the requirements imposed by
  different type of services.

26
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)
• where n is the codeword length and k is number of
  information bits pn-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, i18), 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.

27
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.

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