Postprint (accepted version)
The final publication is available at http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6255761
DOI: 10.1109/LSP.2012.2210873
S. Sugiura, A. Ichiki, and Y. Tadokoro, "Stochastic-resonance based iterative detection for serially-concatenatedturbo codes," IEEE Signal Processing Letters, vol. 19, no. 10, pp. 655-658, Oct. 2012.
© 2012 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all otheruses, in any current or future media, including reprinting/republishing this material for advertising or
promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of
any copyrighted component of this work in other works

1

Stochastic-Resonance Based Iterative Detection
for Serially-Concatenated Turbo Codes
Shinya Sugiura, Senior Member, IEEE, Akihisa Ichiki, and Yukihiro Tadokoro, Member, IEEE
Abstract—In this letter, the concept of multiple seriallyconcatenated codes is invoked in the context of stochastic resonance (SR), where the achievable performance is improved byincreasing noise power. More specifically, the receiver’s iterative
decoding process is characterized with the aid of extrinsic information transfer (EXIT) charts, such that the SR effect induced
by a non-linear component is taken into account. Our simulation
results demonstrate that although the SR demodulator’s loglikelihood ratio (LLR) outputs are not Gaussian distributed, thecorresponding EXIT trajectory matches the prediction from theassociated inner- and outer-codes’ EXIT curves, while reaching
the perfect convergence point in terms of mutual information.
Therefore, an infinitesimally low bit-error ratio (BER) is attainedby appropriately designing the channel code’s parameters basedon EXIT charts.
Index Terms—Comparator, EXIT chart, non-linear systems,
irregular codes, iterative detection, stochastic resonance.

I. I NTRODUCTIONTOCHASTIC resonance (SR) is a physical phenomenon,
which has the potential of improving the receiver’s detection performance, upon increasing the power of the additivenoise contaminating the signal. Typically, such an SR effectcan be observed in non-linear (NL) systems [1]–[3]. For thesereasons, the SR is especially useful in a scenario, wherea small signal is corrupted by additive noise and the NL
effects imposed by the receiver are intrinsically unavoidable.
However, owing to the presence of a NL component, it isa challenging task to theoretically characterize its achievable
performance. The seminal studies [1], [2] focused on thesignal-to-noise ratio (SNR) metric in order to evaluate the SR
effect in a relatively simple manner. Furthermore, in [4], [5]it was also found that SR improves the mutual information(MI) exchange between the input and output signals. Considering that the specific definition of SNR depends on the SR
scenarios considered and that as mentioned in [3], the SNR is
not directly related to the achievable detection performance,
especially for non-Gaussian noise scenarios, it can be said
that the MI is a more suitable performance metric for the
scenario considered. Note that in order to practically attaina near-capacity performance, which may be characterized bynear-unity MI, it is necessary to incorporate a channel code
into the system.
On the other hand, to the best of our knowledge, theprevious SR related studies [6]–[10] only considered uncodedscenarios. However, practical systems typically employ a powerful channel coding scheme, such as turbo [11] and low-S

Copyright (c) 2012 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
The authors are with Toyota Central R&D Laboratories, Inc., Nagakute,
Aichi, 480-1192, Japan (e-mail: sugiura@ieee.org).

density parity-check (LDPC) codes [12], which are capable ofincreasing MI between the input/output signals, upon iterating
between multiple soft decoders. Furthermore, in [13] thesemi-analytical tool of extrinsic information transfer (EXIT)
charts was proposed for the sake of accurately predicting theconvergence behavior of the iterative decoding process, whichallows us to optimize multiple concatenated codes.
Against this background, the novel contributions of thisletter are as follows. We first propose a channel-encoded
SR system, which is capable of attaining a near-capacity
performance. More specifically, a serially-concatenated turbo
code, employing the NL component of a single-comparator,
is optimized with the aid of EXIT charts. Furthermore, we
prove that an infinitesimally low bit error ratio (BER) is
achievable by appropriately designing the channel code as
well as by providing a sufficient number of iterations amongst
multiple decoders. Another important finding is that our EXITchart aided convergence prediction is also useful even in the
scenario, where the system includes a NL component as wellas additive non-Gaussian noise.
II. S YSTEM D ESCRIPTION
A. System Model
Consider the transmitter structure of Fig. 1, supporting
multiple serially-concatenated codes, namely a channel encoder and a precoder. At the transmitter, B information bitsbi (i = 1, · · · , B) are firstly channel encoded by a convolutional encoder, having a code rate of RCC . The C = B/RCC
channel-encoded bits ci (i = 1, · · · , C) are then interleaved bya random interleaver Π. Furthermore, the interleaved bits are
encoded by a unity-rate convolutional (URC) code1 . The coded
bits are finally mapped to binary phase-shift keying (BPSK)
symbols si ∈ {+1, −1} (i = 1, · · · , C), which are transmitted
to the receiver over additive noisy channels.
(n)
At the receiver, N signals ri (n = 1, · · · , N ) are sampled
per transmitted symbol si , which are represented by
(n)

ri

(n)

= si + ηi .

(1)More specifically, we assume that the real-valued additive(n)noise ηiis an identically and independently distributed(IID) variable, which obeys the Gaussian-mixture distribution
having the probability density function (PDF) of [7]f (u, σ) =√12σ 2π(1 − β 2 )
{
[
]
[
]}
(u/σ + β)2(u/σ − β)2
× exp −
+
exp
−
, (2)
2(1 − β 2 )2(1 − β 2 )1 The role of the URC code is to impose an infinite impulse response (IIR),
which improves the achievable iterative decoding performance by efficiently
spreading the extrinsic information, as detailed in [14].

2

SourceConvolutional
encoder

Outer code

Convolutional
decoder

URC
decoder

URC
encoderModulator

Inner code

ChannelsSR
demodulatorComparator

and RSC decoders4 . To be more specific, the URC decoder
outputs the extrinsic information LURC
in the form of LLRsewith the aid of the SISO MAP detector’s outputs LMAP
as welle
as of the a priori LLRs LURC
apr , which are fed back from theconvolutional decoder to the URC decoder. The extrinsic LLRs
LURC
are then input to the RSC decoder after deinterleaving.
e
The extrinsic LLRs LRSC
are calculated at the RSC decoder,eoutput and are interleaved again, before being passed back to
the SISO URC decoder as the a priori information LURC
apr .
Finally, the RSC decoder outputs the estimated bits b̂i after I
number of iterations. We note that since there is no a priori
information during the first iteration, the initial values of LURC
apr
are set to zero.

Fig. 1. The iteratively-detected transceiver structure of our RSC-coded andunity-rate precoded SR system, employing a single comparator.

B. The Optimal MAP Detector

where σ represents the standard deviation, while the value of
β is in the range of 0 < β < 12 . Here, as β approaches
zero, f (u, σ) becomes similar to the zero-mean unit-variance
Gaussian distribution. Moreover, the corresponding cumulative
distribution function (CDF) can be formulated by
1
F (u, σ) =
2
[ (
)
(
)]
1
u/σ + βu/σ − β+erf √+ erf √. (3)42(1 − β 2 )2(1 − β 2 )(n)Then, the sampled signals ri are passed onto the comparatorof Fig.[ 1, which has a ]threshold of θ. The comparator’s outputs(1)(N )xi = x i , · · · , x i
are as follows:3{(n)
1 (ri ≥ θ)
(n)
xi =
.
(4)
(n)
0 (ri < θ)
Next, the binary sequence xi is input to the soft-input softoutput (SISO) SR demodulator of Fig. 1, where the extrinsic
information Le expressed in the form of log-likelihood ratios(LLRs) is calculated under the assumption of perfect knowledge of the standard deviation σ. Although there are several
algorithms, which may be employed by the SISO detector,
in this letter we adopt optimal maximum a posteriori (MAP)
detection, which is described later in Section II-B.
Assuming that in this letter a recursive systematic convolutional (RSC) code is used for the convolutional encoder/decoder of Fig. 1, the receiver employs iterative detection between the two SISO decoders [11], [16], i.e. the URC

2 Note that additive Gaussian mixture noise has been widely adopted in thediverse SR-related studies [3], [7]–[9]. More specifically, one of its explicit
examples is constituted by multiuser interference and impulsive noise in codedivision multiple access (CDMA) systems [15], while this Gaussian-mixture
framework may also be applicable to several other impulsive electromagnetic
interference scenarios, such as automobile ignition noise, military mobile
channels and indoor interference induced by mechanical switching.
3 In order to expound a little further, we employed a comparator as a NL
component. However, other NL effects, such as particular NL channels andbistable-potential systems [10], can be readily applicable to our proposedarchitecture.

Let us now derive the a posteriori LLR LMAP
apo , which is
calculated based on the optimal MAP criterion as follows:[]p(xi |si = −1)P1
MAP(5)
Lapo = lnp(xi |si = +1)P0
[
]
∏N
( )
(n)
P1
n=1 p(xi |si = −1)+ ln ∏N
, (6)
= ln
(n)
P0
p(x
|s
=
+1)
i
i
n=1
| {z } |
{z
}
LMAP
apr

LMAP
e

represent the a priori and extrinsic
and LMAP
where LMAPeapr
LLRs, while P0 and P1 are the a priori probabilities, whichcorrespond to the events of si = −1 and 1, respectively.
Since the comparator’s output is binary, its conditional
probability density may be expressed as{(n)
F (θ − si , σ)(xi = 0)(n)
. (7)p(xi |si ) =(n)
1 − F (θ − si , σ) (xi = 1)
Considering that the possible number of non-zero elements,
which are contained in the comparator’s binary outputs xi , isvaried from 0 to N , the legitimate number of the potential
extrinsic LLRs LMAP
is equal to (N + 1), given the specificesystem parameters of (N, σ, θ). This implies that the related
a priori LLR values, which are input from the SR MAP
detector to the URC decoder, are not Gaussian distributedin the comparator-based scenario of Fig. 1. Nevertheless,we will show later in Section III that the corresponding
EXIT trajectory matches the inner- and outer-EXIT curves,
as expected for a Gaussian scenario.
C. EXIT-Chart Analysis
In this section, we briefly highlight the concept of EXIT
charts, which is a powerful technique used for analyzing theconvergence behavior of iterative detection aided transmissions
based on the turbo-coding principle. For further detailed
4 Since in this letter the SISO MAP detector calculates extrinsic informationLMAP
based on a binary input sequence xnei , it remains unaffected by thepotential feedback from the URC decoder, which is referred to as LMAP
apr .
On the other hand, when employing multi-level constellations are input
MAP also
to the MAP detector instead of xn
i , its extrinsic LLR values Le
become affected by LMAP
apr . In such a scenario, iterations between the SISO
MAP detector and the SISO URC decoder further improve the achievable
performance, as mentioned in [11].

3

R = RCC A · log2 M[bits/symbol],(8)

where 0 ≤ RCC ≤ 1 represents the RSC code’s rate and0 ≤ A ≤ 1 denotes the area under the inner decoder’s EXIT
curve [16]. Moreover, M is the number of constellation points,
which is fixed to M = 2 in this letter. We note that the
maximum achievable rate of Eq. (8) may be useful for the SRaided systems, since the corresponding inner decoder’s EXIT
curve takes into account the effects of NL components.
III. S IMULATION R ESULTS
In this section, we provide our performance results inorder to characterize the proposed RSC-coded and unity-rateprecoded SR scheme in the context of the single-comparatorbased NL system.
We assumed that the number of information bits per frame
was B = 105 , while considering the three different halfrate RSC codes, namely the length-two RSC(2,1,2) code, thelength-three RSC(2,1,3) code and the length-five RSC(2,1,5)
code, having the octal generator polynomials of (Gr , G) =(3, 2)8 , (Gr , G) = (7, 5)8 and (Gr , G) = (35, 23)8 , respectively. The threshold value of the receiver’s comparator was set
to θ = 1.55 . Furthermore, the Gaussian-mixture noise process
was characterized by β = 0.9. Additionally, the information
bits bi were equi-probably distributed.
Firstly, Fig. 2 shows the maximum achievable rates of ourarrangements, which were obtained according to Eq. (8). Here,the number of samples per symbol was varied from N = 1
to 5. Observe in Fig. 2 that upon increasing the standard
deviation σ from zero to approximately σ = 1.1, the maximum
achievable rate monotonically increased in each scenario,
which is achieved as the explicit merit of SR. To be more
5 Here, the thresholding value θ was determined from our extensive simulations, though it does not guarantee its optimality. The related investigation
is an open issue.

1.0
Maximum achievable limit [bits/symbol]

explanations of EXIT charts, refer to [16] and the references
therein. In turbo detection, an infinitesimally low BER may
be attained by the iterative exchange of extrinsic MI between
two SISO decoders, i.e. the inner and outer decoders. Since
the iterative decoding process is not linear, the prediction of
its convergence behavior is a challenging task.
The ingenious tool of EXIT charts was proposed by ten
Brink [13] for the visualization of the iterative decoding behavior and for the prediction of the ‘BER-cliff’ position, where
the BER suddenly drops. More specifically, the input/output
relationship of the MI at each decoder is characterized by the
EXIT chart and then their interaction assisted by the iterative
decoding process is examined without time-consuming bitby-bit Monte-Carlo simulations. Ideally, the inner and outerdecoders’ EXIT curves should not intersect before the perfectconvergence point of (IA , IE ) = (1.0, 1.0), which leads to perfect extrinsic information exchange between the two decodersof Fig. 1. The emergence of an open EXIT chart convergencetunnel enables the system to achieve an infinitesimally low
BER at the corresponding SNR.
Furthermore, the maximum achievable rate may be formulated based on the inner code’s EXIT curve as follows: [16]

N=1
N=2
N=3
N=4
N=5

0.8

0.6

0.4

0.5 bits/symbol

0.2

0.0
0

1

2

3

4

Noise RMS value σ

Fig. 2. The maximum achievable rates of our RSC-coded and URC-precoded
SR system, employing a single comparator. Here, the number of samples per
symbol was varied from N = 1 to 5.

specific, this SR effect is especially beneficial for low-SNR
scenarios, albeit this is achieved by sacrificing the achievable
performance in the low-noise regime, as shown in Fig. 2.
By contrast, in the high-noise range of σ > 1.1, an increasein the value of σ degraded the achievable performance, in asimilar manner to its linear-system counterpart. Furthermore,as expected, a higher number of samples N leads to a higher
maximum achievable rate, which is a benefit of having a higher
sampling frequency. In order to provide further insights, sincea half-rate (RCC = 0.5) channel code was employed in our
simulations, the scenarios of N = 3, 4 and 5 have the potential
of attaining a near-error-free performance around the noiseroot-mean-square (RMS) value of σ = 1, as shown in Fig. 2.
Furthermore, the inner- and outer-codes’ EXIT curves weredrawn in Fig. 3, where we varied the standard deviation fromσ = 0.7 to 1.3 with the step of 0.2, while the number ofsamples per symbol was N = 3. Furthermore, we plotted
the outer code’s EXIT curves, which correspond to the three
aforementioned half-rate RSC codes, as well as to the EXIT
trajectory associated with σ = 1.1 and the RSC(2,1,2) code. It
was found from Fig. 3 that an open tunnel emerged between
the inner- and outer-codes’ EXIT curves for the scenario of
σ = 1.1 and the RSC(2,1,2) code. More specifically, the
EXIT trajectory of σ = 1.1 reached the perfect convergencepoint of (IA , IE ) = (1, 1), while reasonably matching thecorresponding inner- and outer-codes’ EXIT curves.6
Finally, Fig. 4 compares the achievable BER of the channel
coded system corresponding to the trajectory of Fig. 3, where
the number of iterations I was given by I = 0, 5, 10, 20 and
35. As predicted from the EXIT charts of Fig. 3, a near-errorfree performance was achieved around σ = 1.1 with the aid
6 The application of EXIT chart is based on the assumption that the a priori
LLR values are uncorrelated and that their PDF is Gaussian distributed [16].
Our SR-based turbo-coded system of Fig. 1 does not apply to this, since
the legitimate number of the MAP detector’s extrinsic LLR values LMAPise
(N +1). However, Fig. 3 shows that the effects of the non-Gaussian distributed
a priori LLR are not severe in the simulated scenarios, which still enables us
to predict the convergence behavior of the iterative process.

4

1.0

using a single comparator [6], which acts as a 1-bit quantizer.
However, the proposed system can be readily extended to othernonlinear systems, such as a multiple-comparator based one[17], which may further improve the SR performance, although
the detailed investigations will be left for our future study.

σ = 0.7
σ = 0.9σ = 1.1
σ = 1.3

0.8

CC, RSC(2,1,2)
CC, RSC(2,1,3)
CC, RSC(2,1,5)
trajectory, σ = 1.1

IV. C ONCLUSIONSIn this letter, we proposed the serially-concatenated channelencoded arrangement of Fig. 1 in the context of SR. Morespecifically, the concept of EXIT charts was invoked foranalyzing the convergence behavior of iterative decoding
process, while taking into consideration the effects of the
NL component and the non-Gaussian noise. Our simulation
results showed that an infinitesimally low BER performanceis achievable in our arrangement, as predicted from the associated EXIT charts. It was also implied that an adaptive
channel coding scheme based on the irregular code may be
useful for SR-aided NL systems, similarly to its linear-systemcounterpart [16].

IE

0.6

0.4

0.2

0.0
0.0

0.2

0.4

0.6

0.8

1.0

IA

Fig. 3. EXIT charts of our RSC-coded and URC-precoded SR system,
employing a single comparator. Here, the standard deviation of Gaussianmixture noise was varied from σ = 0.7 to 1.3 with the step of 0.2, while
the number of samples per symbol was N = 3. Furthermore, the outer EXIT
curve of half-rate length-two RSC(2,1,2), length-three RSC(2,1,3) and lengthfive RSC(2,1,5) codes were considered. We also plotted the EXIT trajectory,
which was associated with σ = 1.1, the interleaver length of C = 2 × 105
and the RSC(2,1,2) code.

100

I=0
-1

10

I=5

I = 10

BER

10-2

10-3

I = 2010-4

I = 35
10-5
0.0

0.5

1.0

1.5

2.0

2.5

3.0

Noise RMS value σ

Fig. 4. Achievable BER performance of our RSC-coded and URC-precoded
SR system, employing a single comparator. The number of iterations wasgiven by I = 0, 5, 10, 20 and 35, while the number of samples per symbol
was maintained to be N = 3.

of a sufficiently high number of iterations I. Additionally, it
is specific to the SR scheme of Fig. 1 that a so-called ‘BER
cliff’ [16] was found not only for a low σ but also for a highσ, which is not observed in the classic-modem scenarios.
An important implication of the above-mentioned simulation results is that by adaptively matching the shape of theouter code’s EXIT curve to that of the inner code with the aidof irregular channel coding [16], an infinitesimally low BER
may be achieved over a wide range of SNRs.
In this letter, we focused our attention on the scenario of

R EFERENCES[1] P. Jung and P. Hänggi, “Amplification of small signals via stochastic
resonance,” Physical Review A, vol. 44, no. 12, pp. 8032–8042, 1991.
[2] J. Collins, C. Chow, and T. Imhoff, “Stochastic resonance without
tuning,” Nature, vol. 376, no. 20, pp. 236–238, 1995.
[3] H. Chen, P. Varshney, S. Kay, and J. Michels, “Theory of the stochastic
resonance effect in signal detection: Part I - fixed detectors,” IEEE
Transactions on Signal Processing, vol. 55, no. 7, pp. 3172–3184, 2007.
[4] I. Goychuk and P. Hänggi, “Stochastic resonance in ion channels
characterized by information theory,” Physical Review E, vol. 61, no. 4,
pp. 4272–4280, 2000.
[5] S. Mitaim and B. Kosko, “Adaptive stochastic resonance in noisy
neurons based on mutual information,” IEEE Transactions on Neural
Networks, vol. 15, no. 6, pp. 1526–1540, 2004.
[6] F. Chapeau-Blondeau, “Nonlinear test statistic to improve signal detection in non-Gaussian noise,” IEEE Signal Processing Letters, vol. 7,
no. 7, pp. 205 –207, July 2000.
[7] S. Zozor and P. Amblard, “On the use of stochastic resonance in sine
detection,” Signal Processing, vol. 82, no. 3, pp. 353–367, 2002.
[8] ——, “Stochastic resonance in locally optimal detectors,” IEEE Transactions on Signal Processing, vol. 51, no. 12, pp. 3177–3181, 2003.
[9] F. Chapeau-Blondeau and D. Rousseau, “Raising the noise to improve
performance in optimal processing,” Journal of Statistical Mechanics:
Theory and Experiment, vol. 2009, p. P01003, 2009.
[10] A. Ichiki, Y. Tadokoro, and M. I. Dykman, “Singular response of bistablesystems driven by telegraph noise,” Physical Review E, vol. 85, p.
031106, Mar. 2012.
[11] S. Sugiura, S. Chen, and L. Hanzo, “MIMO-aided near-capacity turbo
transceivers: Taxonomy and performance versus complexity,” IEEE
Communications Surveys & Tutorials, vol. 14, no. 2, pp. 421–442,
Second Quarter 2012.
[12] T. Richardson and R. Urbanke, Modern coding theory. Cambridge
University Press, 2008.
[13] S. ten Brink, “Convergence behavior of iteratively decoded parallel
concatenated codes,” IEEE Transactions on Communications, vol. 49,
no. 10, pp. 1727–1737, 2001.
[14] D. Divsalar, S. Dolinar, and F. Pollara, “Serial concatenated trellis coded
modulation with rate-1 inner code,” in IEEE Global Telecommunications
Conference, vol. 2, San Francisco, CA, November–December 2000, pp.
777–782.
[15] X. Wang and H. Poor, “Robust multiuser detection in non-Gaussianchannels,” IEEE Transactions on Signal Processing, vol. 47, no. 2, pp.
289–305, 1999.
[16] L. Hanzo, O. Alamri, M. El-Hajjar, and N. Wu, Near-Capacity MultiFunctional MIMO Systems: Sphere-Packing, Iterative Detection andCooperation. John Wiley and IEEE Press, 2009.
[17] N. Stocks, “Suprathreshold stochastic resonance in multilevel threshold
systems,” Physical Review Letters, vol. 84, no. 11, pp. 2310–2313, 2000.