The document proposes a new normalization technique for compensating the optimistic output information produced by a soft output Viterbi algorithm (SOVA) decoder in a turbo decoder. The technique counts the sign differences between the a-priori and extrinsic information to determine a normalization factor for each data block. Simulations show the proposed technique achieves about a 0.2 dB coding gain improvement on average compared to other techniques, while reducing the number of iterations needed for decoding by up to 21.

1. Normalization of Output Information for
a Turbo Decoder Using SOVA
Yi-Nan Lin and Wei-Wen Hung*
SUMMARY
It has been shown that the output information produced by the soft output Viterbi
algorithm (SOVA) is too optimistic. To compensate for this, the output information
should be normalized. This letter proposes a simple normalization technique that
extends the existing sign difference ratio (SDR) criterion. The new normalization
technique counts the sign differences between the a-priori information and the extrinsic
information, and then adaptively determines the corresponding normalization factor for
each data block. Simulations comparing the new technique with other well-known
normalization techniques show that the proposed normalization technique can achieve
about 0.2 dB coding gain improvement on average while reducing up to about 21
iteration for decoding.
Index Terms—Normalization of output information, Soft output Viterbi algorithm
(SOVA), Sign difference ratio (SDR), Coding gain improvement.
*
Corresponding author. Tel/fax : +886 02 29061780.

2. 1
1. Introduction
In recent years, considerable interest has been devoted to turbo codes [1] that achieve
near-Shannon limit performance. At the receiver, decoding can be done in an iterative way
either using the maximum a-posteriori (MAP) algorithm or the soft-output Viterbi algorithm
(SOVA). The SOVA is less complex, but has a degradation of about 0.7 dB compared to the
MAP. It has been found that the output information produced by a SOVA decoder does not
correctly predict the a posteriori probability (APP) of the hard decision for bad channels. In
fact, the output information is too optimistic, and thus a correction of the output information
is necessary. To compensate for this, it is suggested [2]-[3] to multiply the extrinsic
information at the output of a SOVA decoder by a set of constant normalization factors.
Pyndiah et al. fixed the evolution of the normalization factor with the iteration number to the
following values (referred to as Type 1)
⎥
⎦
⎤
⎢
⎣
⎡
⋅⋅⋅⋅⋅=
⎥
⎦
⎤
⎢
⎣
⎡
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅=
;0.1;0.1;0.1;8.0;6.0;4.0;2.0
;;;; )(
1
)2(
1
)1(
11
i
TypeTypeTypeType CCCC
(1)
where i is the index of iteration number. Z. Wang and K. K. Parhi [4] also indicated that the
number of matching bits between the signs of the a-priori information and the extrinsic
information for all bits within a decoding block is highly related to the computed factor for
normalizing the extrinsic information. Therefore, a simple mapping function (referred to as
Type 2) was used to compute the target normalization factor
[ ]⎪⎩
⎪
⎨
⎧
>−⋅∆+
≤
=
)(
)(
)(
)(
0
)(
)(
0)(
2
C THb
i
bTHb
i
b
THb
i
bi
Type
MMifMMC
MMifC
C
(2)
where 0C is the base value, C∆ is the increment, )(i
bM is the number of matching bits
within a data block in the i th iteration for decoding and )(THbM is a pre-determined
matching threshold. In addition, L. Papke and P. Robertson [5] used Gaussian distribution to

3. 2
approximate the probability density function of SOVA output v and compute its conditional
log-likelihood ratio (LLR). They concluded that the SOVA output v has to be multiplied by
the factor (referred to as Type 3)
[ ]2)(
)(
)(
3
2
i
v
i
vi
Type
m
C
σ
⋅
= . (3)
where )(i
vm and )(i
vσ are the expectation and standard deviation of output v in the i th
iteration, respectively.
In this letter, we shall deal with the normalization problem of output information for a
SOVA decoder and will present a new normalization technique. This letter is structured as
follows. Section 2 describes the formulation and block diagram of the proposed new
normalization technique. Simulation results are illustrated and discussed in Section 3. Finally,
conclusions are made in Section 4.
2. The SDR-Based Normalization Technique
For simplicity, we consider a turbo code that consists of two identical Recursive
Systematic Convolutional (RSC) codes with feedback. Let }1,{ Nkuk ≤≤ be a data block of
length N to be transmitted. At the decoder module, two soft-input soft-output (SISO) SOVA
decoders are employed to produce the estimates }1,ˆ{ Nkuk ≤≤ . Let s
ky , p
ky 1, and p
ky 2, be
the received systematic signal and parity signals corresponding to the transmitted bits ku ,
respectively; let cL be the channel reliability. Then, in the i th iteration, the first SOVA
decoder receives the channel sequence ),( 1,
p
kc
s
kc yLyL ⋅⋅ from the first encoder and the
a-priori information )ˆ()(
1, k
i
a uΛ provided by de-interleaving the extrinsic information
)ˆ()1(
2, k
i
e u−
Λ of the second SOVA decoder in the )1( −i th iteration, and hence it can produce an
improved a-posteriori information )ˆ()(
1 k
i
uΛ . Next, the second SOVA decoder comes into

4. 3
operation. It uses the interleaved channel sequence ),~( 2,
p
kc
s
kc yLyL ⋅⋅ from the second
encoder and the a-priori information )ˆ()(
2, k
i
a uΛ derived by interleaving the extrinsic
information )ˆ()(
1, k
i
e uΛ of the first SOVA decoder to calculate the a-posteriori information
)ˆ()(
2 k
i
uΛ . Above iterative process continues, and on average the BER of the decoded bits
decreases as the number of decoding iterations increases. It is shown in [6] that
)ˆ()ˆ()ˆ( )(
1,
)(
1,
)(
1 k
i
ek
i
a
s
kck
i
uuyLu Λ+Λ+⋅=Λ (4)
)ˆ()ˆ(~)ˆ( )(
2,
)(
2,
)(
2 k
i
ek
i
a
s
kck
i
uuyLu Λ+Λ+⋅=Λ . (5)
The iterative process is implemented by setting
[ ])ˆ()ˆ( )1(
2,
)(
1, k
i
ek
i
a uDeInteru −
Λ=Λ (6)
[ ])ˆ()ˆ( )(
1,
)(
2, k
i
ek
i
a uInteru Λ=Λ (7)
where ][Inter and ][DeInter denote the interleaving and de-interleaving operations,
respectively.
As described earlier in Section 1, the output information of a SOVA decoder need to be
normalized in order to obtain a more accurate LLR. Based on this fact, the new normalization
technique (referred to as Type 4) extends the existing sign difference ratio (SDR) technique [7]
to compensate the associated soft outputs of a SOVA decoder. The block diagram of an
iterative SOVA decoder employing the SDR-based normalization technique is shown in
Figure 1. First, we compute the values of SDR function (SDRF) of the k th estimated bit kuˆ
in the i th iteration for the SOVA decoder 1 and 2
⎪⎩
⎪
⎨
⎧
<Λ⋅Λ
=
elsewhere
uuif
NuSDRF k
i
ek
i
a
k
i
0
0)ˆ()ˆ(
1
)ˆ(
)(
1,
)(
1,)(
1
(8)
and

5. 4
⎪⎩
⎪
⎨
⎧
<Λ⋅Λ
=
elsewhere
uuif
NuSDRF k
i
ek
i
a
k
i
0
0)ˆ()ˆ(
1
)ˆ(
)(
2,
)(
2,)(
2
(9)
where N is the block length, )ˆ()(
1, k
i
a uΛ , )ˆ()(
1, k
i
e uΛ and )ˆ()(
2, k
i
a uΛ , )ˆ()(
2, k
i
e uΛ represent the
normalized versions of )ˆ()(
1, k
i
a uΛ , )ˆ()(
1, k
i
e uΛ and )ˆ()(
2, k
i
a uΛ , )ˆ()(
2, k
i
e uΛ respectively. Then,
the normalization factors )(
4,1
i
TypeC and )(
4,2
i
TypeC associated with the data block decoded by
the SOVA decoders 1 and 2 in the i th iteration can be formulated as
∑
=
−=
N
k
k
ii
Type uSDRFC
1
)(
2
)(
4,1 )ˆ(0.1 (10)
and
∑
=
−=
N
k
k
ii
Type uSDRFC
1
)(
1
)(
4,2 )ˆ(0.1 . (11)
Based on the observations made from repeated simulations, Yufei Wu et al. [7] speculated that
the terms ∑
=
N
k
k
i
uSDRF
1
)(
1 )ˆ( and ∑
=
N
k
k
i
uSDRF
1
)(
2 )ˆ( tend toward zero for a “good” (easy to
decode) data block whereas stay high for a “bad” (hard to decode) data block as the decoding
proceeds. Apparently, the normalization factors we employed for compensating the output
information of SOVA decoders are highly related to the underlying channel conditions and
will be updated from iteration to iteration. To incorporate the normalization factors )(
4,1
i
TypeC
and )(
4,2
i
TypeC into the architecture of a SOVA decoder, the iterative process described in
equations (6) and (7) should be rewritten as
[ ])ˆ()ˆ( )1(
2,
)1(
4,1
)(
1, k
i
e
i
Typek
i
a uDeInterCu −−
Λ⋅=Λ (12)
[ ])ˆ()ˆ( )(
1,
)(
4,2
)(
2, k
i
e
i
Typek
i
a uInterCu Λ⋅=Λ (13)
with the initial condition 5.0)0(
4,1 =TypeC . This initial value is determined by the experimental
results made from repeated simulations. In addition, the soft channel inputs s
kc yL ⋅ , p
kc yL 1,⋅

6. 5
for decoder 1 and s
kc yL ~⋅ , p
kc yL 2,⋅ for decoder 2 also need to be normalized. Thus, the
equations (4) and (5) used to compute the a-posteriori information )ˆ()(
1 k
i
uΛ and )ˆ()(
2 k
i
uΛ
should be modified as
)ˆ()ˆ(
2
0.3
)ˆ( )(
1,
)(
1,
)1(
4,1)(
1 k
i
ek
i
a
s
kc
i
Type
k
i
uuyL
C
u Λ+Λ+⋅⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
=Λ
−
(14)
)ˆ()ˆ(~
2
0.3
)ˆ( )(
2,
)(
2,
)(
4,2)(
2 k
i
ek
i
a
s
kc
i
Type
k
i
uuyL
C
u Λ+Λ+⋅⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
=Λ . (15)
3. Simulation Results and Discussions
In this section, we conduct a series of experiments to evaluate the effectiveness of the
SDR-based normalization technique we proposed for SOVA decoding. In the simulation, a
data block of 1000 bits (N=1000) and 2000 bits (N=2000) are both considered and 50,000
data blocks are transmitted. Two kinds of encoding schemes are under investigation. One is
“two 8-state (memory size M=3) RSC constituent encoders with generator polynomials
(13,15)” and another is “two 4-state (memory size M=2) RSC constituent encoders with
generator polynomials (7,5)”. The encoders are linked together by a pseudorandom interleaver.
The overall code rate is 21 . The coded bits are modulated using binary phase shift keying
(BPSK) and white Gaussian noise with a double-sided power spectral density of 20N is
added to the modulated signal. At the decoder, only eight iterations are carried out, as no
significant improvement in performance is obtained with a higher number of iterations. In
addition, the parameters used for “Type 2” normalization are
(N=1000, 980)( =THbM , 8.00 =C , 01.0=∆C ) and
(N=2000, 1960)( =THbM , 8.00 =C , 005.0=∆C ).
The results of the simulations are shown in Figures 2 ~ 4 and Figures 5 ~ 7. Figures 2
and 3 show the BER and FER plotted versus 0NEb for a SOVA decoder employing various

7. 6
normalization techniques and the encoding parameters are M=3, N=1000. The performance of
the case without normalization is also plotted for reference. As can be seen that for low
0NEb region ( dBNEb 4.20 < ), the BER and FER of a SOVA decoder can be improved by
means of various normalization techniques we discussed. Above all, the proposed SDR-based
approach performs better than the other techniques for normalization. When the channel is
less distorted ( dBNEb 4.20 > ), there is no any difference in performance between the cases
with and without normalization. Figure 4 shows the average number of iterations plotted
versus 0NEb for the same cases illustrated in Figures 2 and 3. In this figure, the decoding
is terminated when the number of sign changes between the a-priori information and the
extrinsic information within a data block is less than ( N×01.0 ). From this figure we can
observe that for most of the 0NEb region we simulated, the presented technique requires
less number of iterations than the other techniques. Also, the new technique uses only N
binary additions of sign bits and a counter no longer than N to compute the target
normalization factor. Obviously, the complexity of our approach for normalizing the output
information of a SOVA decoder is significantly less than the complexity of extra iterations for
decoding. Figures 5 and 6 show the BER and FER plotted versus 0NEb for a SOVA
decoder employing various normalization techniques in which M=2 and N=2000. Figure 7
shows the average number of iterations plotted versus 0NEb for the different
normalization schemes with M=2 and N=2000. By comparing the figures illustrated in
Figures 2~4 (for the case M=3, N=1000) and Figures 5~7 (for the case M=2, N=2000), we
can observe that all the normalization techniques we evaluated have similar relative
performances even with different encoding schemes and block sizes. Apparently, the proposed
SDR-based normalization technique is effective and easy to implement under various
conditions.

8. 7
4. Conclusion
In this letter, we proposed a new normalization technique and compared it with some
widely used normalization techniques. The idea of our approach is first to check the sign
consistency between the a-priori information and the extrinsic information, and then compute
the corresponding sign difference ratio so as to normalize the output information of a SOVA
decoder. It has been shown by simulations that the SDR-based normalization technique
achieves better performance in terms of BER, FER and the average number of iterations than
the other normalization techniques we discussed.

9. 8
Acknowledgment
This research has been partially sponsored by the National Science Council, Taiwan, ROC,
under contract number NSC-95-2221-E-131-015-. Moreover, The authors would like to thank
Dr. Erl-Huei Lu for his very helpful suggestions and comments.
Authors
Yi-Nan Lin received his B.S. degree from the Electrical Engineering Department of National Taiwan Institute
of Technology in 1989, and the M.S. degree in Computer Science & Engineering from the Yuan Ze University in
2000. He joined the Department of Electrical Engineering at Mingchi University of Technology, Taishan, Taiwan,
in 1990. He is now a lecturer in the Department of Electronic Engineering. He is also a Ph.D. candidate in the
Electrical Engineering Department of Chang Gung University, Taoyuan, Taiwan. His current research interests
include error-control coding, and digital transmission systems.
Wei-Wen Hung received his B.S. degree from the Electrical Engineering Department of Tatung Institute of
Technology in 1986, and the M.S. and Ph.D. degrees in electrical engineering from the National Tsinghua
University in 1988 and 2000, respectively. He joined the Department of Electrical Engineering at Mingchi
University of Technology, Taishan, Taiwan, in 1990. He was the Vice Dean of Student Affairs from 2000 to 2002.
He was also the Chairman of Department of Electronic Engineering in 2003. He is now a professor in the
Department of Electronic Engineering. His current research interests include speech signal processing, wireless
communication and embedded system design.

10. 9
References
[1] C. Berrou and A. Glavieux, “Near-optimum error-correcting coding and decoding :
Turbo codes,” IEEE Trans. Commun., vol. 44, pp. 1261-1271, Oct. 1996.
[2] R. M. Pyndiah, “Near-optimum decoding of product codes : block turbo codes,” IEEE
Trans. Commun., vol. 46, no. 8, pp. 1003-1010, Aug. 1998.
[3] D. W. Kim, T. W. Kwon, J. R. Choi and J. J. Kong, “A modified two-step SOVA-based
turbo decoder with a fixed scaling factor,” in Proc. IEEE Int. Symposium on Circuits and
Systems (ISCAS), pp. IV-37 ~ IV40, May 2000.
[4] Z. Wang and K. K. Parhi, “High performance, high throughput turbo/SOVA decoder
design,” IEEE Trans. Commun., vol. 51, no. 4, pp. 570-579, Apr. 2003.
[5] L. Papke and P. Robertson, “Improved decoding with the SOVA in a parallel
concatenated (turbo-code) scheme,” in Proc. IEEE Int. Conf. Communication, pp.
102-106, 1996.
[6] J. Hagenauer, E. Offer, and L. Papke, “Iterative decoding of binary block and
convolutional codes,” IEEE Trans. Inform. Theory, vol. 42, pp. 429-445, Mar. 1996.
[7] Y. Wu, B. D. Woerner, and W. J. Ebel, “A simple stopping criterion for turbo decoding,”
IEEE Commun. Letters, vol. 4, no. 8, pp. 258-260, Aug. 2000.

11. 10
Figure Captions
Fig. 1. Block diagram of an iterative SOVA decoder employing the SDR-based normalization
technique.
Fig. 2. BER versus 0NEb for a SOVA decoder employing various normalization techniques
(encoding parameters M=3, N=1000).
Fig. 3. FER versus 0NEb for a SOVA decoder employing various normalization techniques
(encoding parameters M=3, N=1000).
Fig. 4. Average number of iterations versus 0NEb for a SOVA decoder employing various
normalization techniques (encoding parameters M=3, N=1000).
Fig. 5. BER versus 0NEb for a SOVA decoder employing various normalization techniques
(encoding parameters M=2, N=2000).
Fig. 6. FER versus 0NEb for a SOVA decoder employing various normalization techniques
(encoding parameters M=2, N=2000).
Fig. 7. Average number of iterations versus 0NEb for a SOVA decoder employing various
normalization techniques (encoding parameters M=2, N=2000).

12. 11
Fig. 1
s
kC yL ⋅
p
1,kC yL ⋅
p
2,kC
yL ⋅
s
kC y~L ⋅
)uˆ( k
)i(
2,aΛ
)uˆ( k
)i(
2Λ )uˆ( k
)i(
2,eΛ
)uˆ( k
)i(
1,eΛ
)1i(
4Type,1
C −
2
C3.0 )1i(
4Type,1
−
−
2
C0.3 )i(
4Type,2−
∑−
=
N
1k
k
)i(
1
)uˆ(SDRF1
)uˆ( k
)i(
1Λ
)uˆ( k
)i(
1,aΛ
kuˆ
)i(
4Type,2C
∑−
=
N
1k
k
)i(
2
)uˆ(SDRF1
Fig. 2

13. 12
Fig. 3
Fig. 4

14. 13
Fig. 5
Fig. 6

15. 14
Fig. 7