Performance Analysis of Steepest Descent Decoding Algorithm for LDPC Codes

Regular Paper
Int. J. on Recent Trends in Engineering and Technology, Vol. 9, No. 1, July 2013

Performance Analysis of Steepest Descent Decoding
Algorithm for LDPC Codes
Finney Daniel Shadrach S 1, Michaelraj Kingston R 2, and Dr. Ramesh J 2
1

PSG College of Technology/Department of ECE, Coimbatore, India
Email: finneydaniels@gmail.com
2
PSG College of Technology/Department of ECE, Coimbatore, India
Email: king.pane@gmail.com, jramesh60@yahoo.com
Abstract - Among various hard decision based Bit Flipping (BF)
algorithms for decoding Low-Density Parity-Check (LDPC)
codes such as Weighted Bit Flipping (WBF), Improved
Reliability Ratio Weighted Bit Flipping (IRRWBF) etc., the
Steepest Descent Bit Flipping Algorithm (SDBF) achieves
better error performance. In this paper, the performance of
the Steepest Descent Algorithm for both single steepest
descent and Multi steepest descent modes is analysed. Also
the performance of IEEE 802.16e standard is analysed using
Steepest Descent Bit Flipping (SDBF) decoding algorithm.
SDBF requires fewer check node and variable node operations
compared to Sum Product Algorithm (SPA) and Min Sum
Algorithm (MSA). The SDBF achieves a coding gain of 0.1 ~
0.2 dB compared to Single-SDBF without requiring complex
log and exponential operations.
Index Terms —Bit Flipping Algorithms, Hard Decision
Decoding, LDPC Codes, Steepest Descent, IEEE 802.16e.

I. INTRODUCTION
Low density parity check codes were introduced by
Gallager [1] in 1962. Mackay and Neal reintroduced LDPC
code in 1996[2]. Due to their excellent error performance,
LDPC codes have achieved significant attention and have
been adopted by many recent communication standards such
as 10 Gigabit Ethernet (10GBASE-T), digital video
broadcasting (DVB-S2) and WiMax (802.16e). The IEEE
WiMax standard [6] covers a large range of wireless
transmission applications. Compared to Wi-Fi (or Wireless
LAN), it can support high throughput over larger distances,
even with higher mobility involved. The upcoming IEEE
WiMax 802.16e standard, also referred to as Wireless-MAN
[7], is the next step toward very high throughput wireless
backbone architectures, supporting up to 70Mbps. The
WiMax standard features LDPC codes as an optional channel
coding scheme.
LDPC codes belongs to a family of linear block code in
which non-zero entries (1’s) will be sparse. LDPC codes can
be decoded by an iterative decoding algorithm. It has been
shown that these codes achieve a remarkable performance
with iterative decoding that is very close to the Shannon
limit [1], Mackay et.al [2]. Consequently, these codes have
become strong competitors to turbo codes for error control
in many communication and digital storage systems where
high reliability is required. The decoding is an iterative
process which exchanges information between two types of
nodes. The WiMax LDPC code was designed with respect to
112
© 2013 ACEEE
DOI: 01.IJRTET.9.1.18

hardware constraints, however, the high number of defined
codes (code rates and codeword Sizes) specified by different
members of the WiMax consortium imposes significant
challenges on an LDPC decoder realization.
LDPC codes can be decoded using various decoding
schemes such as hard -decisions, soft -decisions and hybrid
decoding schemes.
Bit flipping algorithms is a hard decision scheme which
possesses a good trade -off between error-correcting
performance and decoding complexity compared to Belief
Propagation Algorithm (BPA) and its variants such as SumProduct and Min-Sum algorithms (MSA). The highcomplexity Sum-Product algorithm (SPA) was shown to
achieve a near - capacity performance. However, the Steepest
Descent bit-flipping algorithm strikes a good trade -off
between the associated decoding complexity and the
achievable performance. The attractive property of this
algorithm is it significantly lower decoding in comparison to
the SPA.
LDPC codes can be described in two forms [10]: the matrix
form and the graphical form. The graphical form was
introduced by Tanner and called as tanner graph. Fig.2
represents the tanner graph for corresponding parity check
matrix shown in Fig.1. The Tanner graph is a bipartite graph
with two types of nodes called variable nodes and check
nodes. The connection between check nodes and variable
node is done using the H matrix, in such a way that columns
of H is considered as the variable nodes and rows of H is
considered as check nodes. In any location there is a ‘1’ in
the matrix, the related variable and check nodes will be
connected by an edge. The important parameter that is
considered in tanner graph is its girth. Girth is the length of
the shortest loop that begins and ends in the same node.
Usually longer girth in tanner graph provides efficient code.
So, avoiding the girth of length 4 and 6 is essential. LDPC
code can be of two types regular and irregular matrix. A LDPC
code is called regular if wc is constant for every column and
wr = wc · (n/m) is also constant for every row. The example
matrix shown in Fig1 is regular with wc = 2 and wr = 4. It’s
also possible to see the regularity of this code while looking
at the graphical representation. There is the same number of
incoming edges for every variable node and also for all the
check nodes. If H is low density but the numbers of 1’s in
each row or column aren’t constant the code is called an
irregular LDPC code.

Regular Paper

Fig.1: Example of a regular (8, 4) matrix

Fig.3: General Block Diagram of Decoder

assumption is that the MI at the output of a block is a function
of the MI at the input, regardless of the actual statistical
distribution of the messages. The decoding process can thus
be described as a recursive computation of the MI between
variable node blocks (LDPC VND) and check node blocks
(LDPC CND).

Fig.2: Tanner graph representation for (8, 4, 2) matrix shown
in Fig 1

The rest of this paper is organized as follows. In section
II , a brief overview of decoding algorithm is explored. Section III presents the code description of IEEE802.16e. Section IV provides simulation results before concluding remarks
in Section V.

A. Bit Flipping Algorithm
In a decoding process of a BF algorithm, bit flip positions
are determined based on the values of an inversion function.
The inversion functions of Weighted Bit Flipping (WBF)
is defined by Zhang and Fossorier [3], Jiang et.al [4], and it is
given in (1) as follows;

II. LITERATURE SURVEY
Assume a binary (M,N) parity check matrix H, where M is
the number of parity check equations ie., rows and N is the
number of bits ie., columns. The non-zero elements in row i
of H N (i) and the non-zero elements in column 5ØWÜ is
(5ØWÜ). The i-th bipolar syndrome of x is then written as

(1)
The inversion function of WBF provides a measure of
the invalidness of symbol assignment on 5ØeÜ, which is
given by the sum of the weighted bipolar syndromes. It
initially finds the most unreliable message node participating
in each individual check. The Modified WBF (MWBF)
algorithm considers not only the reliability of the syndrome
sequence, but also the reliability information of the code bits.
The MWBF is given as,

. Let C be the codeword of a regular
LDPC
code of length N, defined as the null space of a sparse parity
check matrix H =
. We assume information transmission using BPSK modulation over an AWGN channel, which
maps a codeword c = (c1, c2, ..., cN) into a bipolar sequence
x =(x1, x2, ..., xN), according to
Let
y = (Y1, Y2, ..., YN) be the soft-decision received vector and
x be represent the vector of received samples and x {+1,_1}
be the hard detected bits. Let the received real value sequence is
, where
statistically independent
Gaussian random variables.
In Fig 3, a pictorial representation of the receiver shows
two different blocks: block A represents the set of all variable
nodes, or variable node detector (VND); block B represents
the set of all check nodes, or check node detector (CND).
Each message set, at the input or output of a block, can be
viewed as the output of a channel with the actually
transmitted codeword at its input: the message set is indeed
a stochastic function of the transmitted codeword. For each
message set, the Mutual Information (MI) between the
transmitted codeword and the message set can then be
computed. In Fig 3, Iv denotes the MI at the output of the
VND, and Ic the MI at the output of the CND. The underlying
© 2013 ACEEE

(2)
Although the inversion function of MWBF has a similar
form to the inversion function of the WBF algorithm, the
inversion function of the MWBF algorithm consists of an
additional term
where is a positive real number and
is the information of the corresponding bit received from the
channel.
Both the WBF and the MWBF algorithms consider only
the specific check-node based information, which relies on
the message node having the lowest soft-value . The
Reliability Ratio Weighted Bit Flipping (RRWBF) is defined
as Guo and Hanzo [5] ,
(3)
113

Regular Paper
Where

is the reliability ratio. The reliability ratio algorithm

normalizes the value of
at each bit. The Gradient Descent
algorithm is composed of the correlation between the hard
decision and the received soft value of the corresponding bit
plus the syndrome sum as follows Wadayama et.al [6]:

3. Parity Check :
If

for all i ,then stop decoding and

output x
4. Bit Flipping :
Initially set
and
, where and are
the previous and present value of the objective function.

(4)
B. Steepest Descent Bit Flipping Algorithm
The Maximum Likelihood (ML) decoding problem for the
binary AWGN channel is equivalent to the problem of finding
a (bipolar) codeword which gives the largest correlation to a
given received word. This objective function has been
inspired by the correlation MLD rule. Based on this correlation
decoding rule, we define the following objective function f(x)
as follows:

Here if
and
, then flip all the
bits. Else flip a single bit .
5. Update the algorithm :
Check the no.of iterations. If no.of iterations is greater
than the maximum iterations then stop and exit. Else repeat
the decoding process.
III. IEEE 802.16E CODE DESCRIPTION

(5)

The LDPC code is based on a set of one or more
fundamental LDPC codes. Each of the fundamental codes is
a systematic linear block code. Each LDPC code in the set of
LDPC codes is defined by a matrix H of size m-by-n, where n
is the length of the code and m is the number of parity check
bits in the code. The number of systematic bits is k=n-m. The
matrix H is defined as

Here the first term corresponds to the correlation between
a bipolar codeword and the received word, and the second
term is the sum of the bipolar syndromes of x. Here the correlation term should be maximized. The inversion function can
be obtained from the objective function. The inversion function of the steepest descent bit flipping algorithm is given
by,

(6)
In order to reduce the computational time we use either
single or multi bit flipping. With the single bit flipping the
objective function is gradually ascended but much faster
convergence could be expected with a large step size. In
order to obtain the faster convergence the multi BF is used.
In single BF only one bits are flipped in each iterations based
on the inversion function, due to which computational time
may be large. But in the case of multi BF parameter such as
inversion threshold is used based on which all bits can be
flipped at the same time. Multi BF may not sound good for
smaller codeword length and in such cases single BF can be
most preferred. The inversion threshold is given as ,

where Pi,j is one of a set of z-by-z permutation matrices or a zby-z zero matrix. The matrix H is expanded from a binary base
matrix Hb of size mb-by-nb, where n  z  n b and m  z  m b ,
with z an integer  1. The base matrix is expanded by replacing
each 1 in the base matrix with a z-by-z permutation matrix, and
each 0 with a z-by-z zero matrix. The base matrix size nb is an
integer equal to 24. The permutations used are circular right
shifts, and the set of permutation matrices contains the zz
identity matrix and circular right shifted versions of the
identity matrix. Because each permutation matrix is specified
by a single circular right shift, the binary base matrix
information and permutation replacement information can be
combined into a single compact model matrix Hbm. The model
matrix Hbm is the same size as the binary base matrix Hb, with
each binary entry (i,j) of the base matrix Hb replaced to create
the model matrix Hbm. Each 0 in Hb is replaced by a blank or
negative value (e.g., by –1) to denote a zz all-zero matrix,
and each 1 in Hb is replaced by a circular shift size p(i,j)0.
The model matrix Hbm can then be directly expanded to H. Hb
is partitioned into two sections, where Hb1 corresponds to

(7)
Here
is the variance of the received signal,
denotes
the number of negative inversions. Based on the inversion
threshold the multi bit flipping is performed.
Steps in Decoding Process:
1. Intialisation :
Initialise the no.of iteration .
2. Hard Decision :
© 2013 ACEEE

114

Regular Paper
version 7.9.0.529(R2009b). The simulation is performed for
binary PEGReg504x1008 [7] which has column weight 3 and
row weight 6 and it’s a regular matrix. The codeword is
transmitted through the AWGN channel using bpsk
modulation scheme. The maximum decoding iteration is set
to 20. The size of the codeword is taken as 1008 with a rate
0.5.
Fig.5 and Fig.6 shows the BER versus Eb/No analysis for
both Progressive Edge Growth and Gallager constructions
respectively. Simulated results shows that the better BER
performance of around
can be achieved with less
complexity and in less computational time when comparing
to other BF algorithms [3-6]. The Table 1 shows the SNR
value corresponding to BER considering various parameters
such as code rate, maximum iteration etc. A considerable
coding gain of around 0.1 ~ 0.2 is achieved. The IRRWBF
and Single-SDBF holds good only for smaller codeword
length. The parameters such as maximum iterations, code
length, encoding techniques etc., are analysed during
simulations. In both Gallager and progressive edge growth
constructions multi- SDBF perform well when comparing to
other algorithms.
Normally BER of around
is preferred for analysing
SNR since it optimal.
The multi-SDBF achieves better convergence for larger
codeword length and for larger error as it depends on the
inversion threshold. The single-SDBF can be best used for
smaller codeword as it provide good BER performance.

the systematic bits and Hb2 corresponds to the parity-check
bits, such that
(8)
In particular, the non-zero submatrices are circularly right
shifted by a particular circular shift value. Each 1 in Hb2 is
assigned a shift size of 0, and is replaced by a zz identity
matrix when expanding to H. The two 1s located at the top
and the bottom of hb are assigned equal shift sizes, and the
third 1 in the middle of hb is given an unpaired shift size.
The Fig.4 shows the base matrix for the Wi-Max system.
A base model matrix [8] [9] is defined for the largest code
length (n=2304) of each code rate. The set of shifts {p(i,j)} in
the base model matrix are used to determine the shift sizes for
all other code lengths of the same code rate. Each base model
matrix has nb=24 columns, and the expansion factor zf is equal
to n/24 for code length n. Here f is the index of the code
lengths for a given code rate, f=0, 1, 2, … 18. For code length
n=2304 the expansion factor is designated z0=96. For code
rates 1/2, the shift sizes {p(f, i, j)} for a code size corresponding
to expansion factor zf are derived from {p(i,j)} by scaling
p(i,j) proportionally,

(9)

where x denotes the flooring function which gives the
nearest integer towards . The LDPC code flexibly supports
different block sizes for each code rate through the use of an
expansion factor. Each base model matrix has nb=24 columns,
and the expansion factor (z factor) is equal to n/24 for code
length n. In each case, the number of information bits is equal
to the code rate times the coded length n.

Fig.5: Bit Error Rate comparison of the proposed algorithm
Adaptive Multi-SDBF and Single-SDBF against IRRWBF using
PEGReg504x1008
Fig.4: Base parity check matrix for rate ½ mobile Wi-Max system

The decoding performance of IEEE 802.16e using SDBF
algorithm obtained from computer simulations are discussed
further. The simulation is done in MATLAB version
7.9.0.529(R2009b). The simulation is analysed for binary
337x672 IEEE 802.16e standard matrix with sub matrix size 28.
The resultant matrix is in non-systematic form. The codeword
is transmitted through the AWGN channel using bpsk

IV. SIMULATION RESULTS
In this section the decoding performance of SDBF algorithms
and IEE802.16e using SDBF obtained from computer
simulations are discussed.
For the analysis of SDBF the simulation is done in MATLAB
© 2013 ACEEE

115

Regular Paper

Fig.6: Bit Error Rate comparison of the proposed algorithm using
Adaptive Multi-SDBF and Single-SDBF against IRRWBF using
Gallager Construction

Fig.7: Bit Error Rate comparison of Multi-SDBF, Single-SDBF and
IRRWBF for IEEE802.16e

TABLE I :PERFORMANCE ANALYSIS OF DIFFERENT ALGORITHM

TABLE II: PERFORMANCE ANALYSIS OF DIFFERENT ALGORITHM FOR
IEEE 802.16 E

Hmatrix
Size

Encoding
Techniqu
e

Progressi
ve Edge
Growth
(Regular)
504x10
08

IRRWBF

20

1008

0.5

10- 5

SNR
(dB)
(Req
uired
to
achie
ve the
BER)
6.9

Single –
SDBF

20

1008

0.5

10- 5

6.8

Multi –
SDBF
IRRWBF

20

1008

0.5

10- 5

6.6

20

1008

0.5

10- 2

9

Single –
SDBF
Multi –
SDBF

20

1008

0.5

10- 2

5.7

20

1008

0.5

10- 2

5.7

Decoding
Techniqu
e

Maxim
um
Iterati
ons

Cod
e
lengt
h

Co
de
Rat
e

BE
R

SNR required to achieve BER of 10- 5
Decoding
Algorithm

n (bits)

z
factor
R=1/2
Max
iterations
10

Max
iterations
20

Max
iterations
30

576

7.7

672

28

7.5

~8

7.5

32

7.2

~8.5

7.4

36

7.4

~8

7.6

960

40

7.6

8.1

>8

576

24

>8

>8.5

>8.5

672

28

>8

>8.5

~8.3

768

32

>8

~8.3

~8.3

864

36

>8

>8.5

>8.5

960

40

>8

~8.3

~8.3

576

24

6.4

6.3

~6.3

672

28

6.3

6.3

6.28

32

6.3

~6.1

~6.1

864

36

6.4

~6.2

~6.1

960

© 2013 ACEEE

8.1

864

modulation scheme. The maximum decoding iteration is set
to 10. The size of the codeword is taken as 672 with a rate 0.5.
Normally better BER performance of around
can be
achieved in SPA and MSA algorithm. But the complexity involved in it is very high. SDBF algorithm has very less complexity when comparing to other BF algorithms as well as
other soft decision algorithms. Fig 7 shows BER versus SNR
for IEEE802.16e for z-factor 28. Simulated results shows a
considerable coding gain of around ~0.3 is achieved in the
case hard decision algorithms. The SDBF holds good in terms
of complexity, thereby providing a tradeoff between complexity and BER performance.

7.5

768

Single- SDBF

24

768

Gallager
Construct
on

40

6.3

~6.3

~6

IRRWBF

Multi- SDBF

116

Regular Paper
REFERENCES

TABLE III: PERFORMANCE ANALYSIS OF DIFFERENT ALGORITHM FOR
IEEE 802.16 E WITH Z-FACTOR AS 28

Decoding
Algorithm

n (bits)

z factor

SingleSDBF
IRRWBF
MultiSDBF

672

28

672
672

28
28

[1] R. G. Gallager, “Low-density parity-check codes,” in Research
Monograph Series. Cambridge, MA: MIT Press,1963.
[2] D. Mackay et.al. “Near Shannon limit Performance of Low
density parity check Codes , “ Electronics Letters ,Vol.32
no.18 pp. 1645-1646, August 1996.
[3] J. Zhang and M. P. C. Fossorier ,” A Modified weighted bitflipping decoding of low-density Parity - check codes ,” IEEE
Communications Letters, vol. 8, no. 3, pp. 165–167, Mar.
2004.
[4] M. Jiang, C. Zhao, Z. Shi and Y. Chen, “An improvement on
the modified weighted bit flipping decoding algorithm for
LDPC codes,” IEEE Communications Letters, vol. 9, no. 9,
pp. 814– 816, Sep. 2005.
[5] F. Guo and L. Hanzo, “Reliability ratio based weighted bitflipping decoding for low-density parity-check codes,”
Electronics Letters, vol.40,no. 21, pp. 1356–1358, Oct. 2004.
[6] T.Wadayama , K. Nakamura , M. Yagita ,Y. Funahashi , S.
Usami , and I.Takumi , “ Gradient descent bit flipping
algorithms for decoding LDPC codes,” IEEE Transactions on
Communications , vol. 58,no. 6, pp.1610 – 1614, Jun.2010.
[7] D. J. C. Mackay, “ Encyclopedia of sparse Graph codes”.
[Online] , Available : http://www.inference.phy.cam.ac.uk/
mackay/odes/data.html
[8] IEEE 802.16eÔ-2005: Part 16: “Air Interface for Fixed and
Mobil Broadband Wireless Access Systems Amendment 2:
Physical and Medium Access Control Layers for Combined
Fixed and Mobile Operation in Licensed Bands and
Corrigendum 1”
[9] Daud, M. ; Suksmono, A.B Hendrawan, Sugihartono
,”Comparison of decoding algorithms for LDPC codes of IEEE
802.16e standard “ Telecommunication Systems, Services, and
Applications (TSSA), 2011 6th International Conference on
Digital Object Identifier pp. 280-283.
[10] P. Jagatheeswari , M. Rajaram, “Performance Comparison of
LDPC Codes and Turbo Codes “, European Journal of
Scientific Research , Vol.54 No.3 (2011), pp.465-472.

BER of 10- 5

SNR Required to achieve

~>7.5

R=1/2
Max
iterations
20
~8

Max
iterations
30
7.5

~>8
6.3

~>8.5
6.3

~8.3
6.28

Max
iterations 10

The Table 2 provides the comparison between different
algorithm such as IRRWBF, Single-SDBF, Multi-SDBF for
IEEE 802.16e. From the Table 2 it is evident that for a BER of
the SNR required for multi-SDBF is less among hard
decision algorithms.
The computational complexity and the processing
platform used determines the running time required by each
algorithm. Table 3 gives the analysis for different algorithms
considering z-factor 28. Table 4 shows the computational
complexity of the algorithms in terms of the number of additions
and multiplications needed per iteration. In the table, LDPC
codes are assumed to be regular. This means that the sizes of
sets of non-zero elements in any row and in any column of H
are constant as |N(i)|= N and |M( j)|= M, respectively.
TABLE IV: C OMPUTATIONAL COMPLEXITY O F EACH ITERATIONS
ALGORITHMS

NUMBER OF
ADDITIONS

NUMBER OF
MULTIPLICATIONS

IRRWBF

n(M-1)+m(N-1)

m(N+2)

Single - SDBF

nM+m(N-1)

m(N-1)+n+2nM

Multi - SDBF

nM+mN+2n-3

m(N-1)+ 2n+2nM+ 2

V. CONCLUSION
In this paper, the bit error performance of regular LDPC
for single and multi – SDBF mode is analysed. The error and
convergence performance is analysed by introducing the
additional term to the bipolar syndrome. This scheme uses a
similar error-term evaluation method to other algorithms,
hence imposing no complexity increase during the iterative
decoding process .The inversion threshold is determined by
the mean of the negative inversion and variance of the
received signal. It is evident from simulation results that MultiSDBF achieves coding gain of 0.1~0.2 dB over Single-SDBF
and IRRWBF for PEG 1008x504.

© 2013 ACEEE

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