error control using turbo codes -...
TRANSCRIPT
CHAPTER 5
ERROR CONTROL USING TURBO CODES
5.1 INTRODUCTION
To enhance reliability in WATM networks, many channel coding schemes such
as block and convolutional codes have been used. To get a better performance,
concatenated coding may be employed. In this scheme, the probability of error
decreases as decoding complexity increases. Turbo code is an improved concatenated
code which consists of two or more constituent codes. This chapter explains error
control for WATM networks using Turbo codes, Quantized Tuba codes and
Nonsystematic Turbo codes and also analyses the performance of the three schemes.
Turbo code consists of two Recursive Systematic Convolutional (RSC) codes as
the constituent codes. The term Turbo is named after the fact that the decoder uses its
processed output as a priori input in the next iteration. Computational speed of the Turbo
coding is high. Turbo codes are enor correction codes which use more bits of redundant
data in order to achieve better reliability. By adding parity bits to transmitted data, Turbo
codes allow the receiver to correct errors caused by the channel.
5 2 TURBO ENCODER
The Turbo encoder consists of the concatenation of two identical recursive
systematic convolutional encoders and an interleaver. The encoder shown in Figure 5.1
is a parallel concatenation structure because the two constituent encoders operate on the
same set of input bits. The interleaver is used ta permute the input bits in such a way that
the two encoders use the same set of input bits but result in different output sequences.
The information bits are first encoded by a systematic convolutional encoder
and then after passing through an interleaver, they are encoded by a second systematic
convolutional encoder. The code sequences are formed by the infomation bits followed
by the parity check bits generated by encoders, resulting in the transmission of three
output streams namely, dk information bits, XI output bits from the first encoder and X2 output bits from the second encoder.
.,
= Encoder1 * -
t
Interleaver
r
* Encoder 2 +
Figure 5.1 Turbo Encoder
The input bits are grouped into a M t e length sequence whose length is equal to
the size of the interleaver. The overall code rate is 1/3. The code rate can be varied by
using puncturing mechanism. The fundamental idea behind Turbo code is the use of the interleaver by which the mor perFomance can be significantly improved through the
separation of constituent encoders. The Interleaver randomizes the bursty error.
53 TURBO DECODER
The transmitted sequences dk, XI and X2 through a transmission channel, reach
the Turbo decoder at the receiver. Due to noise in the channel, the received values may
not match their transmitted counterparts. The Turbo decoder attempts to reconstruct the transmitted values through a series of decoding steps.
The Turbo decoder shown in Figure 5.2 consists of two identical component
decoders, interleaver/deintaleaver blocks and an output decision block. The interleavers
present in the receiver are identical to the interleaver in the transmitter. The
kinterleaver is used to reorder the sequences so that each decoder is properly
synchronized.
Figure 5.2 Turbo Decoder
The interleavers permute the data bits to support the error correction algorithm.
The output from Decoder 1 is fed into Decoder 2 through an interleaver/deinterleaver.
Multiple iterations are required before the decoder converges towards a final result.
After a pre-specified number of decoding iterations, the final decision is made in the
Data decision block by combining the outputs fiom both decoders.
Data Decoder2 , ,.
The two decoders are matched to the constituent encoders at the transmitter.
They share control information (also called extrinsic information) in an iterative fashion.
The extrinsic information is used as a priori information in the next decoding stage. The
Decoder 1 is associated with Encoder I and yields a soft decision. The emr bursts at
the Decoder 1 output are scattered by the interleaver. The same process takes place at
Decoder 2 also. The parallel concatenation featured in Turbo code is an attractive
scheme because both encoders and decoders use a single clock frequency.
Deinterleaver
J decision " o@
Traditional Turbo codes employ two recursive systematic convolutional codes
separated by an interleaver in a parallel concatenation. Two component decoders are
used to form. the complete Turbo decoder. The decoding strategy is operated iteratively
where each component decoder uses partial information about the transmitted sequence
supplied by the other component decoder from a previous decoding step. The great
interest in this type of coding/decoding stems fkorn the fact that Turbo codes can perform
very closely to the theoretical limits developed by Shannon [lOO]. Shannon showed that
a communication chmel is capable of transmitting information with few errors, even
when the channel is subject to errors due to noise or interference, if the capacity of the
channel is not exceeded. This capacity depends on the signal-to-noise ratio.
The Turbo encoder is built using a parallel concatenation of two recursive
systematic convolution encoders, The associated decoder operates based on a feedback
decoding rule. Consider a biOnary rate, R=1/2, convolutional encoder with constraint
length K and memory M = K-1. The input to the encoder at the instant of time 'i3 is a bit
di and the corresponding codeword Ck is the binary couple (Xk, Y k ) given by,
k-1 and & = c g,,d, , mod-2 where, g2,=o, 1 (5-2)
i=O
5.4 QUANTIZED TURBO DECODER
Quantization is an important step in transforming ideal conceptual models into
practical working solution as a small size digital chip. As long as digital circuits rely on
finite precision representations of numbers, it is desirable to represent information in
compact farm, with as small a performance loss as possible. In a conventional Turbo
system, inefficiently represented extrinsic information may require a large amount of
memory for bigger codes. There is a need to represent extrinsic information in compact
form to lower memory needs and therefore directly reduce processing Ioads at the
decoder.
In a quantized Turbo decoder, the extrinsic information passed between two
decoders is quantized. It is found that the quantization scheme requires only one bit to
represent extrinsic information of binary Turbo system. In this way, only hard decisions
of the extrinsic information as well as a set of simple weighting factors which is
common to all extrinsic information are stored in memory. The proposed quantization
scheme is studied through Extrinsic Information (EXIT) analysis using a Binary
Symmetric Channel (BSC) model. EXIT analysis improves the reliability of extrinsic
information passed between decoders. It examines proper weighting factors that
maximize the performance of the quantized decoder.
The likelihood hnctions are conditional probability density functions. If all the
input message sequences are equally likely, a decoder that achieves the minimum
probability of error is one that compares all of the conditional probabilities and chooses
the maximum. Therefore in the maximum likelihood context, the decoder chooses a
particular transmitted sequence if the likelihood is greater than the likelihoods of all
other possible transmitted sequences. It is computationally more convenient to use the
logarithm of the likelihood function since it permits the summation instead of
multiplication of terns. Log Likelihood Ratio (LLR) is a useful metric. It can be defined
as a posteriori probability of a received bit, computed by the component decoder. It is a
real number representing a so& decision out of a decoder. Each decoder processes
information and passes the resulting extrinsic information to the other decoder which
uses it as a priori information called log likelihood ratio of a priori information (LLR,).
The log likelihood ratio of extrinsic idormation &L&) is obtained using the equation
given by [44],
*ere, (r2 is the variance for the channel noise and y is the received bit at the decoder.
APP DECODER 1
APP DECODER 2
Figure 5.3 Quantized Turbo Decoder
Figure 5.3 represents the main idea of a 1-bit quantization decoder. A Post-Priori
Decoder (APP) receives and processes both posteriori and priori information. The
quantized extrinsic infomation is interleaved / deinterleaved and when used by an APP
decoder as a priori information, they are multiplied by a weighting factor colIected f?om
a look-up table. This weighting factor is constant for a complete decoding path of the
component APP decoders but varies with iteration number, choice of particular
component code and the signal-to-noise ratio.
5.5 SYSTEM DESCRIPTION
The EXIT test system shown in Figure 5.4 is used to find proper weighting
factors and examine the performance of the system with single bit quantization. It
employs a BSC test-channel with crossover probability p, This setup can be used to
perform an EXIT analysis of the quantized system as follows:
Factor I I Figure 5.4 EXIT Test System
U
The encoded information sequence is transmitted in a wireless channel. At the
decoder, the noisy data is fed into the APP decoder along with a priori information.
Expression of the absolute value of extrinsic information in a I-bit quantized system as a
hnction of p, can be given as,
BER Evaluation
where, ye is the received test signal and
U is the transmitted signal.
BER
w n
Eq.(5.4) represents a weighting factor with which the hard decision extrinsic
information fi-om a previous APP decoder has to be multiplied as shown in Figure 5.3.
Turbo Encoder
Ye LLR,
A binary mdulator which maps the coded symbols ik = { 0 ,I) into the channel
symbols bk = {- 1, 1 } is considered. An n-bit quantizer with L = 2" quantization levels is
also considered. Let (%,al ,, . .,aL) with a, = -a and a, = ao be the boundaries of the
quantization intervals. Let i = O,l, .... L-1 be the quantization intervals and
qo,q12.. . . . ..,q~1 be the possible quantized values within each of the L quantization
intervals. Then the transition probabilities in Figure 5.5 are given by,
Decoder
BSC
X Y
C
b
LLRdl * APP
p (,I.) denotes the conditional probability density function for the received data, given
that a 0/ 1 was transmitted.
Figure 5.5 Quantized Channel Model
5.6 DECODING ALGORITEIMS
Two well-developed and widely used algorithms for determining the state
sequence of a trellis encoder are the Viterbi Algorithm (VA) and the Maximum a
Posteriori (MAP) algorithm. The Viterbi Algorithm is widely used for decoding
convolutional codes but variants of the MAP algorithm are more suitable for Turbo code
decoding for the following reason. Given the received observation sequence y, the
Viterbi dgoritbm determines the most probable state sequence (denoted as s) which is
given by,
This hplies that the states estimated by the Viterbi algorithm will always form a
connected path through the trellis, The MAP algorithm, on the other hand, attempts to
determine each state transition without regard to the overall sequence of the trellis as, A
si = ms7 PIS I Y I (5 -7)
On the basis of performance compafisons, it has been observed that the VA
results in minimizing Frame E m Rate (FER) while the MAP algorithm minimizes the
Bit Error Rate (BER). The cwent work investigates Turbo coded error control using the
Maximum A Posteriori (MAP), Quantized MAP (QMAP), Soft Output Viterbi
Algorithm (SOVA) and Quantized SOVA (QSOVA) algorithms. QMAP and QSOVA
are the quantized versions of MAP and SOVA algorithms respectively.
5.6.1 MAP Algorithm
The maximum a posteriori (MAP) algorithm calculates the a posteriori
probability of each message bit/symbol that must have been transmitted at the encoder,
given the noisy obsewations of symbols produced by a Markov process. l h i n g the
decoding sequence, the APPs obtained after every iteration are put into log likelihood
ratio form for manipulation by the next decoder and hard decisions are only made after
the last iteration. In order to find the a posteriori probability for each message bit from
the given noisy observations, the MAP algorithm first calculates the probability of each
state transition as,
The imporbnt state transition probabilities which are used to find log likelihood
ratio can be listed asa (si), P(si), and y(sj -) Si+,) [124]. The term a (si) gives the state
transition probability during the fomard recursion and is given by,
where, A is defined as Ehe set of states si_l connected to si. Conversely, given a particular
trellis state si at a specific instant of quantization interval, P(si) gives the state transition
probability at the backward recursion as given by,
whm, A is defined as the set of states S i t 1 connected to sj. The term y( si+ si+,) is a
trellis tree branch metric parameter which represents the chances of making a transition
from state sj to si+l where the states si and sf+l are not directly connected on the trellis
diagram. It can be calculated as,
where, mi is the message bit at si
After determining the a posteriori probability for each state transition, the
probabilities for the message bits/symbols are determined by,
and
where, so = (si+si+, : mi =O) is the set of dl state transitions associated with
transmitting a 'O', and sl = {s~+s*~: mi =1) is the set of all state transitions associated
with transmining a 'l '.
The log likelihood ratio (LLR) is thus defined as,
Numerical stability problems can arise in the computations of %(.I ,a(.) and p(.,
due to the large dynamic ranges of these quantities. This problem can be solved by
performing computations in the logarithmic domain, Although this technique solves
computational complexity and dynamic range issues, it still requires a large memory.
5.6.2 QMAP Algorithm
The QMAP algorithm works in the following way: Let the state of the
constituent recursive systematic convolutional encoder at time k be sk, which takes
values from (0, 1, 2, . . ., 2M -1) and let M be the number of memory elements in the
encoder. The received sequence is denoted by qk = [y, ,y2 ,......, yk 1 with k=l, 2, . . . , N,
where N is the block length. It is to be noted that typically the elements of this sequence
can be vector valued as they denote the receiver output(s) at a time. Hence, for a rate I/2
RSC which produces 2 coded bits every time instant, the elements of the received
sequence can be written as y, = [Y;, y : ] , where yi and y,P are the receiver outputs
corresponding to the symbol and parity bits transmitted at a time. This sequence is then
quantized and the quantizer output is denoted by qil = [qq1,,4 .r12,........ 7 ~ [ q , k ] with b l ,
2,. . ., N. The QMAP algorithm, like the MAP algorithm, provides an estimate of the log
likelihood ratio (LLR) for each bit. The LLR is the logarithm of the ratio of the a
posteriori probability of the bit being 1 to the a posteriori probability of it being 0 and is
given by,
which can be fbrther expressed as,
where, a,(s,) and P,(s,) are dehed as follows:
and Ti = P, (ii I S I ,sk)P, (ylq1, I sk-1, sk )J>, ( ~ ~ - 1 , sk
The forward recursion for the calculation of ak (s, ) is givm by,
and the backward recursion for the calculation of ,8" (s, ) is given by,
where the initial conditions are pN (s,) = 1 and bN ) = o .
The properties of the channel and the code are used in the formulation of yi . If a rate 1/2 RSC is considered, each information bit d, , produces two coded bits [ii , i[ 1,
where ii = d, . These coded bits get mapped into binary symbols bk = [b," , b , ~ ] and a h r
being transmitted over the channel are received as y, = [pi, y,P ] and then quantized. It is
to be noted that due to the presence of a quantizer, the input to the decoder is constrained
to a set of discrete vaIues as decided by the choice of the quantizer. Let us refer to these
quantized values of yiandy,P as ykv and y&ikrespectively 11241. Since the
transitionsk-, -+ sk corresponds to the coded bit^^^;,^;.], the state transition probability
becomes,
Using Eq.(5.2 1) and Eq.(5.22), the branch transition probabilities of the code trellis are
given bx
Thus for implementation purpases, QW approach differs h m the usual
implementation of the MAP algorithm simply in the calculation of they,(.). Equations
(5.19) and (5.20) would be used by a decoder in the QMAP algorithm.. The MAP-based
decoder must compute the required minimum distances in real time, while the QMAP
based decoder simply uses the precompiled channel transition probabilities.
5.63 SOVA Algorithm
Decoder complexity can be reduced further by using the Soft Output Viterbi
Algorithm (SOVA). The traditional Viterbi aigorithm is a forward dynamic
programming solution to the decoding of convolutional codes. In the conventional
Viterbi algorithm, the path is selected at each node by maximizing over all possible
paths. Usually, the competing path determines the reliability of information. The SOVA
algorithm computes only one competing path at every decoding step. Thus, for each
information bit, it considers only one survivor of the Viterbi algorithm.
Like Max-Log-MAP, SOVA processing is also performed in the logarithmic
domain, while executing the same operations as in the Viterbi algorithm. The SOVA
path-update process generates reliability information about the bits along the survivor
paths as we1 as hard outputs. A fundamental difference between the MAP algorithm and
the SOVA is that the MAP algorithm minimizes bit error probability, while the SOVA
minimizes sequence error probability.
5.6.4 QSOVA Algorithm
The QSOVA algorithm is similar to SOVA algorithm except that the extrinsic
information is quantized before giving the next decoder as the prior information. The
QSOVA algorithm computes only one competing path at every decoding steq. Thus for
each information bit, it considers only one survivor of the Viterbi algorithm.
5.7 COMPARISON OF DIFFERENT TURBO CODING ALGORITHMS
The MAP and the SOVA algorithms work with the same rnetrics. For had
decisions, they are identical. But they behave in different ways whiie computing the
information of the decoded bit 4- The SOVA considers only one competing path per
decoding step. That is, for each bit dk, it does not consider all the competing paths but
only the survivors of the Viterbi algorithm.
The MAP looks at two paths per step: the best with bit zero and the best with bit
one at transition j; it then outputs the difference of the log likelihoods. However from
step to step, though these paths can change, the calculated path will always be the
maximum-likelihood (ML) path. The SOVA algorithm will always conectly find one of
these two paths, but not necessarily the other, since it may have been eliminated before
merging with the ML path.
The difference between the MAP and SOVA is illustrated in Figure 5.6. The
continuous and dotted lines indicate zero bit path and one bit path respectively in the
trellis diagram. Thick lines in the diagram show the paths considered for calculation. The
MAP takes all paths into its calculation, but splits them into two sets: those that have
information bit one at step 'j' and those that have a zero. Then it returns the LLR of
these two sets. Due to the Markov properties of the trellis, the computation can be done
relatively easily. Finally, the comparison of MAP and SOVA algorithms yields that
MAP algorithm provides a better performance.
Figure 5.6 Comparison between MAP and SOVA
The number of operations in the MAP is about twice the number of operations in
the SOVA. However, the former is more suited for parallel processing. So it may be
concluded that the MAP is particularly suitable for decoding Turbo codes. The Turbo
code is capable of handling a very low signal to noise ratio signal, but a small number of
states and therefore the additional complexity is less pronounced. As the MAP algorithm
achieves better performance than the SOVA algorithm, it is suggested that the MAP
algorithm is a suitable choice for decoding Turbo codes.
5.8 RESULTS OF QUANTIZED TURBO CODES
To ilIustrate the principle, a Turbo code with code rate of 1/3 is employed and it is
formed by two identical four-state recursive convolutional codes with generating
polynomials of (1 + D*) and (1 + D + D~). The code rate is chosen as 113 where every
source data bit is mapped into 3 coded symbols. Figure 5.7 and Figure 5.8 shows the BER
performance for the MAP and QMAP algorithms respectively using four and eight
iterations. Simulations are carried out for Coherent QPSK modulation system at a data rate
of 1 Mbps. The size of the WATM cell considered for simulation is 56 bytes. The wireless
channel is assumed as the multipath fading channel.
5.8.1 3-Bit Quantized SOVA AND MAP Algorithms
The extrinsic information is reduced to a 3-bit quantized data in the QMAP
algorithm. In the MAP algorithm, no quantization takes place. Figures from 5.7 to 5.10 show
the BER performance for the four and eight iterations using MAP, QMAP, SOVA and
QSOVA algorithms respectively. From these figures, it is undmtood that as the number of
iterations increases, the BER comes down. Therefore, the Turbo decoding process can be
improved by increasing the number of iterations.
Figures 5.1 1 and 5.12 explain the comparisons between the effects of using 3-bit
quantized data as the extrinsic information for different algorithms. To store the weighting
Figure 5.7 S N R Vs BER for the MAP Algorithm
Figure 5.8 SNR Vs BER for 3-bit QMAP Algorithm
Figure 5.10 S N R Vs BER for 3-bit QSOVA Algorithm
Figure 5.11 Comparison of M A P and QMAP for 3-bit EXIT information
---8-- 8 iteration
i
Figure 5-12 Comparison of SOVA and QSOVA for
- -.
I I I I -.-..A 1.5 2 2.5 3 3.5 4 4.5 5
SNR (dB)
Figure 5.13 SNR Vs BER for 1-Bit QMAP
factor, less than one kilobits of the buffer size is required. From the simulation results, it is
understood that the quantized extrinsic values provide almost the same infomation as the
unquantized extrinsic information.
Table 5.1 Weighting factor used in the simulations
For every frame, 'p,' (cross over probability of the channel) was evaluated for each
iteration and component dec&. The weighing factors may be found by averaging 'pa' over
Iterationi.
4
8
APP
1
2
1
2
1dB
6.96
7.12
7.32
7.56
2dB
7.89
8.08
8.17
3 dB
8.48
9.12
9.36
4 dB
9.68
9.86
10.32
10.45
5 dB
10.52
1 1.69 -
12.12
12.38 8.24 9.48
~imulated fi-ames. The weighing factors used are given in Table 5.1 and are listed for
different values of S N R and iteration. As an example, the signal S N R of 2 dB at decoder 2 of
iteration 4 is to be multiplied by 8.08.
5.8.2 QMAIP Algorithm Using 1-Bit Quantization
Figure 5.13 shows the performance of S N R Vs BER for different iterations using
QMAP algorithm. The extrinsic information is quantized into 1-bit information before
passing to the next decoder. Results show that the BER is decreased as the number of
iterations increases.
5.83 SOVA Algorithm Using I-Bit Quantization
Figure 5.14 shows S N R Vs BER for QSOVA algorithm by varying the number of
iterations in the Turbo decoding process. In the QSOVA algorithm, extrinsic information is
quantized into 1 bit before giving to the next decoder. The quantized extrinsic information is
multiplied by the corresponding weighting factor using the lookup Table 5.1.
- - - -- . '"i t.5 i 2:s d 3:. 1 4:s 5
SNR (dB)
Figure 5-14 SPJR Vs BER using 1-bit QSOVA
5.8.4 Comparison of different Turbo coding Algorithms
In Figure 5.15, Turbo code using QSOVA achieves almost the same performance of
sOVA. In fact, the match between the two algorithms is maintained at all iterations. So,
QsOVA is a better approach than SOVA to achieve better BER as it requires less number of
extrinsic information bits. The computational complexity is reduced in QSOVA than SOVA.
In Figure 5.16, QMAP achieves practically the same performance as MAP. The
match between the two algorithms is well maintained for all iterations. This indicates that
QMAP exhibits almost the same performance when compared to MAP algorithm, Therefore
QW is better than MAP algorithm as it requires less number of extrinsic information bits.
The computational complexity is reduced in the QMAP algorithm compared to the MAP
algorithm. It is also to be observed that the QMAP algorithm performs better than the
QSOVA algorithm.
SNR (dB)
Figure 5.15 Comparison between SOVA and QSOVA
lo4-- I I I I I I 1 1.5 2 2.5 3 3.5 4 4.5 5
SNR (dB)
Figure 5.16 Comparison between MAP and QMAP
5.9 NONSYSTEMATIC TURBO CODES
Nonsystematic Twbo codes can achieve lower error rates than Systematic Turbo
codes because of their effective free distance [45] properties. Nonsystematic Turbo
codes have greater values of effective fkee distance than the systematic Turbo codes of
the same constraint length. The effective fiee distance of a Turbo code can be defined as
the minimum distance of an input sequence where weight-two codeword is present
[125]. The weight of a code word can be defined as the number of non-zero elements
present in a code word. Since the error corxecting capabilities of nonsystematic Turbo
codes largely depends on the free distance, nonsystematic Turbo code can exhibit better
performance than other codes of the same complexity.
Figure 5.17 shows a Nonsystematic Turbo encoder block diagram. The first
encodes has a rate R=1/2 nonsystematic feedback conv01utio~al encoder with generator
matrix Gl(D) = [(g,'"(~) l g,'0'(D))(g,'2'(~) l gl(oO"(~))]. This encoder has a total of 2" states,
where u is the constraint length of the encoder. If g , ( ' ) ( ~ ) = g,(o"~) or
g , ( 2 ) ( ~ ) = g,(0)(D) then the rate R=1/3 systematic Turbo code results. n is the
interleaver which is used to randomize the burst error.
Figure 5.17 Nonsystematic Turbo Encoder
To obtain a nonsystematic Turbo code, one of the first constituent generators is
chosen as,
l l or g,'l'(~) - g,'2'(~) -- do' (Dl g?' (Dl g?) (Dl do) (0
Table 5.2 shows that the nonsystematic Turbo codes can achieve larger effective
fiee distances than the systematic Turbo codes. However, for some nonsystematic
encoders, the iterative APP decoding algorithm provides poor extrinsic estimates of the
information bits after the first iteration of decoding, so that decoding may not converge
to the maximum- likelihood solution.
A Turbo encoder is called a feed forward encoder. The weight of feed forward
inverse is a useel metric in classifjling nonsystematic Turbo codes. Table 5.3 presents
the weight of the feed forward inverse for different Turbo encoders. A nonsystematic
~onvolutional encoder that has a weight-two feed forward inverse is known as Quick
~ o o k In (QLI) encoder. In contrast to nonsystematic encoders, systematic feedback
encoders have a feed forward inverse of one. Since nonsystematic QLI encoders have
inverses of weight two, they can be considered as "almost sysfernatic." Encoders with
feed forward inverses of weight three are called Easy Look In (ELI) and encoders whose
weight is greater than three are called Nearly Quick Look In (NQLI). But catastrophic
encoders do not possess a feed forward inverse as it is infinite. Thus the nonsystematic
feedback QLI encoders exhibit good performance when used as constituent encoders in
the Turbo coding scheme.
Table 5.2 Maximum Effective Free distances of Rate R=1/3 Turbo codes
Table 5.3 Weight of the Feed forward Inverses of the Rate R=1/2 Convolutional
Encoders
The symbol-by-symbol Logarithmic Maximum a Posteriori algorithm (Log-
M.AP) [126-1273 is optimal for iterative decoding. Log- MAP algorithm executes MAP
algorithm in the logarithmic domain. To avoid the high complexity of the optimal
Encoder
SYS
QLI
ELI
NQLI
CAT
Weight of the
feed forward
inverse
I
2
' 3
>3
a
algorithm, sub-optimal algorithms such as the log-MAP, max-log-MAP and constant-
log-MAP, are used in practice to achieve the compromise between performance and
complexity. The MAP parameters are approximated in the max-log-MAP algorithm by
the maximization operation. The correction function is a constant for the constant-log-
MAP decoder. The log-MAP algorithm computes the MAP parameters by utilizing a
correction h c t i o n to compute the logarithm of the sum of numbers.
5.10 RESULTS OF NONSYSTEMATIC TURBO CODES
A set of simulations are performed to investigate the influence of nonsystematic
Turbo codes on WATM networks. The BER performance of the eight-state
nonsystematic turbo code is simulated with aE =16 and GI@) = [ 6/15 13/15 ] and
Gz@) = [13/15] for the AWGN channel and the Rayleigh fading channel by considering
the WATM environment. Simulations are canied out for Coherent QPSK modulation
system at a data rate of 1 Mb/s. The size of the WATM cell considered for simulation is
56 bytes. All the simulations are run for a maximum of 8 iterations.
The BER performance curves of the nonsystematic Turbo codes for the max-
log-MAP, constant-log-MAP and log-MAP decoding algorithms under the AWGN
channel are shown in Figures 5.18, 5.19 and 5.20 respectively. Similarly the set of BER
performance curves under the Rayleigh fading channel are shown in Figures 5.2 1, 5.22
and 5.23. Figures 5.24 and 5.25 shows the BER performance comparison of different
algorithms for AWGN and Rayleigh Fading channels respectively. It can be observed
that BER performance is better in AWGN than in Rayleigh fading channel.
SNR (dB)
Figure 5.18 BER Performance of max-log-MAP in AWGN channel
AWGN, constant-logMAP
-.. '.. --. --.
Figure 5.19 BER Performance of constant-log-MAP in AWGN channel
SNR (dB)
Figure 5.20 BIER Performance of log-MAP in AWGN channel
SNR (dB)
Figure 5.21 BER Performance of constant-log-MAP in Rayleigh channel
- -- , - p--- - i
-- ---- 1
RAYLEIGH, constant-log-MAP I -4
rj
Figure 5.22 BER Performance of log-MLAP in Rayleigh channel.
Figure 5.23 BER Performance of log-MAP in Rsxleigh channel
I oO r---- - t RAYLEIGH, Iog-MAP : 3
" --, -1
A W G N . ~ ~ ; - I ~ ~ - M A P AWGN, corstant - l~MAP -, AWGN, leg-MAP _
SNR (dB)
Figure 5.24 BER Performance comparisons in AWGN channel
o0 - 7--- - _ -- - -. --
C _ _ I _ _ __-_---I _-- - E I RAYLEIGH,max-log-MAP
-- -----I RAYLEIGH, constant-10flAP 1 - - 1 0 ' ~ - -- RAYLEIGH, log-MAP -- - - --
SNR (dB)
Figure 5.25 BER Performance comparisons in Rayleigh channel
(a) Transmitted Picture @) Code Combining ARQ
(c) Concatenated Early-stop ARQ
(d) RCPC Code
(e) Turbo Code
Figure 5.26
(a) Transmitted Picture (b) Code Combining ARQ
(c) Concatenated Early-stop ARQ
(d) RCPC Code
(el Turbo Code
Figure 537
(a) Transmitted Picture (b) Code Combining ARQ
(c) Concatenated Early-stop ARQ
(d) RCPC Code
(e) Turbo Code
Figure 5.28
5.11 SUMMARY
This chapter describes the Turbo coded error control mechanism using one bit
quantized extrinsic information for the WATM networks in multipath fading channel.
The simulation has been done using Matlab software. From the obtained results, it is
confirmed that Turbo coding provides better BER in the error prone environment. From
the simulation results, it is confirmed that the pedormance is improved by increasing the
number of iterations in the Turbo decoding process. It is understood from the results,
that Turbo decoding is possible with only single bit information of the extrinsic
information and it causes only little loss. When the extrinsic information is used as a
priori information, their values are weighted by a constant stored in a look-up table
which is indexed by the iteration and current component decoder number. This table
does not require much storage. With the proposed scheme, memory requirements of
Turbo decoders are greatly reduced without significant loss in performance when
compared to infinite precision decoding. But for practical single bit implementations, a
carefid optimization of both weighting factor and interleaver design would have to be
carried out to achieve a better BER performance.
From the results, it is confirmed that the nonsystematic Turbo coding provides
better BER in the error prone environment compared to other existing coding techniques,
including systematic Turbo codes. It is amazing to obtain lo-' BER at a very small SNR
of (2.5 - 3) dB which is almost wire line BER performance. It can be concluded that
Turbo coding is a very promising technique to effectively control channel errors in
WATM networks
Many channel coding schemes for error correction is in use, such as Reed-
Solomon code, Convolutional code and Concatenated codes so as to increase the
reliability of information transmission. But no such coding is useful enough to recover
the bmty error data It is found that Turbo coding technique reduces bursty error better
other schemes described in earlier chapters. The proposed emor control schemes
namely, Code combining ARQ, Cancatenated Early-Stop ARQ, RCPC coding and
~uantized Turbo coding have been tested by sending pictures shown in Figures 5.26 (a),
5.27 (a) and 5.28 (a). The received pictures are as shown in Figures from 5.26 (6) to
5.26 (e), 5.27 (b) to 5.27 (e) and 5.28 (b) to 5.28 (e). I t can be observed that Turbo code
performs much better than other techniques. It can also be observed that Concatenated
Early-Stop ARQ and RCPC codes exhibit a moderate performance, while Code
combining ARQ shows poor performance.