blind estimation of an approximated likelihood ratio in
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Blind Estimation of an ApproximatedLikelihood Ratio in Impulsive Environment
Yasser Mestrahāā , Anne Savardā, Alban Goupilā , Laurent Clavierā and Guillaume Gelleā āIMT Lille Douai, Univ. Lille, CNRS, UMR 8520 - IEMN, F-59000 Lille, France
ā CReSTIC, University of Reims Champagne-Ardenne, France
AbstractāRobust communication is necessary for manywireless applications. Making a decision at the receiverrequires an evaluation of the likelihood. However, inimpulsive noise, the traditional Gaussian-based receiverexhibits a very significant performance loss. This paperproposes to approximate the likelihood ratio in a binarytransmission with a function adapted to impulsive noiseconditions but also efficient when noise is purely Gaussian.We introduce a blind estimation of the two parametersdefining the approximation and evaluate its performancewhen used as the inputs of the belief propagation decoder.Our proposal allows us not only to achieve performanceclose to the optimal decoding but also to have a simpleimplementation and to adapt to different environment,impulsive or not, independently of the underlying statisticalnoise model, without the need of a training sequence.
Index TermsāSoft iterative decoding, impulsive interfer-ence, impulsive noise, alpha-stable distribution, supervisedlearning, unsupervised learning.
I. INTRODUCTION
Low-Density Parity Check (LDPC) codes, introducedby Gallager [1] and rediscovered by MacKay are apowerful linear block coding tool, since the sparsenessof their parity check matrices guaranties a decodingcomplexity that increases quasi-linearly with the codelength. They are widely used in various standards, as forexample in wireless local area networks (WLAN IEEE802.11n), Digital video broadcasting and Digital audiobroadcasting. Usually, LDPC codes are decoded withthe Belief Propagation (BP) algorithm, relying on Log-Likelihood Ratio (LLR) and perform near the capacityunder Gaussian noise.
However, in many communication settings, noise can-not be modeled by a Gaussian distribution since itexhibits an impulsive behavior [2] [3]. Many workshave tackled the modelization question and numeroussolutions have been proposed since the first works byMiddleton [4]. Recently, stochastic geometry has beenused to address the problem in different network scenar-ios, but the analytical expression of the noise density isoften complex (infinite series) or non tractable (Ī±-stable).Consequently, designing the optimal receiver is complex,even assuming we can have a perfect knowledge of theinterference statistics.
If we use a traditional receiver, relying on the Gaus-sian noise assumption, the decoding will suffer from
severe performance degradation, due to the noise modelmismatch, not being able to handle the strong impactof impulsive samples. Several approaches have beenproposed in the literature to improve the robustness of areceiver against impulsive noise. For instance an approx-imation of the interference distribution can be proposedwhich leads to an analytical receiver design (for instancea generalized Gaussian distribution approximation in [6],a mixture of Laplacian and Gaussian in [5], a Cauchydistribution in [3]), a Normal Inverse Gaussian in [7]). Tobe less specific on a noise model, some robust metrics toestimate distance have also been used, for instance in [9]a saturation of the metric obtained in the Gaussian case(not of the received sample, unlike the soft-limiter), thep-norm in [8] or the Hubber metric in [10]. We proposedin [11] to directly approximate the LLR that will serve asinput of the BP algorithm for LDPC codes, moreover, oursolution is not limited to these codes. Choosing a familyof function for this approximation in a space defined bya small set of parameters allows both to estimate thoseparameters with a limited complexity and to adapt todifferent noise statistical properties, impulsive or not.
We propose to approximate the LLRwith parametric function chosen in the setf(x) = sign(x)min(a|x|, b/|x|). The main contributionof our paper is to propose a simple, fast and easy wayto implement a blind estimation of the parameters a andb. It brings two main advantages over the previouslyproposed approaches:
1) It avoids the use of a training sequence whichreduces the useful rate of the transmission.
2) It allows to use the full packet for the estimationwhich can be a great advantage especially whenlarge samples are not so frequent and could bemissing from the training sequence, misleading theestimation process.
The rest of this paper is organized as follows: Thesystem model is presented in Section II. Section IIIstarts, for the sake of completeness, by presenting thesupervised LLR approximation with mutual informationcriterion. We then propose our unsupervised LLR ap-proximation and provide a performance comparison withthe supervised approximation. Section IV gives somesimulation results using a regular LDPC code under
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Fig. 1. Pdfs of SĪ±S distributions (Ī³ = 0.5).
impulsive noise and finally, Section V concludes thepaper.
II. SYSTEM MODEL AND BACKGROUND
In this paper, we consider the transmission of abinary message X in presence of interference. Let Ydenotes the received message modeled by Y = X +N ,where N denotes the interference, that is assumed to beindependent of X . Throughout the paper, we assume thatthe information source X belongs to a simple binaryphase-shift keying (BPSK) constellation where X = Ā±1with equal probability and N follows a symmetric alphastable (SĪ±S) distribution.
The characteristic function of a SĪ±S random variableis given as ĻSĪ± (t) = exp (ā|Ī³t|Ī±) , where (0 < Ī± ā¤ 2)is the characteristic exponent and Ī³ the dispersion. Inwireless context, Ī± is directly associated with the pathloss exponent of the radio channel [12].
Fig. 1 shows the pdf of various Ī±-stable distributionsassociated with different values of Ī±. Remark that thesmaller the value of Ī±, the heavier the tail of thepdf, which increases the likelihood of having impulseswith large amplitudes and far from the center location.The dispersion measures the spread of the noise and isconsidered as a scale parameter, similarly to the variancefor Gaussian distribution, which is a special case of SĪ±Sdistribution with Ī± = 2 and Ī³ = Ļ/
ā2.
In this paper, we assume that X is encoded usingLDPC codes, whose parity check matrix H , of dimen-sion mĆn, is sparse, i.e. has a low density of ones. Theirusually associated decoder, the Belief Propagation (BP)algorithm, expects the input to be given as LLRs. Thus,in order to use the BP decoder, one has to transform thechannel output Y into LLR, which is thereafter calleddemapping. Consequently, working on this demappingallows our solution to be compatible with all types ofdecoders. The decoder inputs are given as:
LLR(y) = log
(Pr[Y = y|X = +1]
Pr[Y = y|X = ā1]
)=log
(fN [y ā 1]
fN [y + 1]
),
where fN (Ā·) is the pdf of the noise N .
Fig. 2. LLR demapper for Ī±=1.4, Ī³ = 0.5, and its approximation.
If Ī± = 2, the studied channel reduces to the AWGNchannel. In this case, one has access to a closed-formexpression of the noise pdf. The decoder inputs are givenas LLRGauss(y) = 2y
Ļ2 = yĪ³2 . This is the widely used
LLR demapper, which is a linear function of y, whichslope depends only on the channel conditions.
However, in the general case, SĪ±S distributions doesnot have a closed-form expression for the pdf. We cannevertheless overcome this impediment by numericalcalculation of the LLR e.g. by numerical integration ofthe inverse Fourier transform of the characteristic func-tion, as fĪ±,Ī³(x) = 1
2Ļ
āā«āā
exp(ā|Ī³t|Ī±)eāitx dt. However,
this induces a generally prohibitive computational bur-den as well as requiring the knowledge of the noiseparameters. Fig. 2 illustrates the non-linearity of the LLRfunction for the channel output y when the noise is Ī±-stable with Ī± = 1.4 and Ī³ = 0.5. Except for Ī± = 2,this shape is similar for every value of Ī±. Fig. 2 clarifiesthat, as the output channel increases, the LLR decreasesmeaning that the received sample becomes less reliable.Moreover, two specific parts can be observed: whenthe output is close to zero, the LLR is almost a linearfunction of y, whereas when y is larger the LLR presentsa power-law decrease. This observation is the startingpoint of our supervised approximated demapper used forLDPC codes under additive impulsive noise [11]. Evenif Fig. 2 delineates a special impulsive noise model, theoverall appearance of the LLR stays the same for theother noise models.
In the next section, we start by presenting our su-pervised LLR approximation as proposed and we thenextend it yielding a novel unsupervised approximatedLLR demapper.
III. PROPOSED DEMAPPER BASED ONAPPROXIMATED LLR
To avoid the complexity induced by the Ī±-stableassumption about the noise and to have a solution robustto a model mismatch, we propose to implement a simpleapproximated demapper.
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Based on the presence of the two aforementioned partsin the LLR, we proposed in [11] the following LLRapproximation:
LĪø(y) =
{ay if |y| <
āb/a ,
b/y otherwise.(1)
which requires the knowledge of the parameter Īø =(a, b), a>0 and b>0, to tune the receiver, and thus tomatch to the channel situation. In [13] we proposed toestimate Īø in a supervised manner, which can drasticallydecrease the throughput, because of the needed learningsequence. In this paper, we propose to optimize Īø in anunsupervised manner.
In [13], we proposed three methods solving the su-pervised optimization : two of them were dependent onthe noise model, whereas the last one was based on amaximization of the mutual information (MI) between alearning sequence and the approximated demapper. Weshowed that the later yields the best performance in termsof bit error rate (BER) in various impulsive noise types.
In this paper, we propose to perform the blind opti-mization of the demapper using the MI based method.
The channel capacity has been well studied in theliterature [14]. For memoryless binary input symmetric-output channel (MBISO) that corresponds to our channelmodel, the capacity is given by the MI between the inputX and the channel output Y as C = I(X,Y ), wherethe binary input is uniformly distributed. As a type ofMBISO, the mutual information over the additive SĪ±Snoise channel can be expressed as:
IL(X,Y ) = 1ā E[log2(1 + eāXL(Y ))
], (2)
where L denotes the LLR. The MI with approximatedLLR LĪø is thus given as:
ILĪø (X,Y ) = 1ā E[log2(1 + eāXLĪø(Y ))
]. (3)
Authors in [15] proved that (3) reaches its maximumwhen the pdf of LĪø is equal to the pdf of true L.
Theoretically, to find back the optimum LLR from theMI we must maximize ILĪø (X;Y ). In order to narrow thesearch space of the best function that fits to the optimumL, we look for the parametrized function LĪø as we justproposed. Thus, to fit the optimal L our goal is to findĪø that maximizes the mutual information as:
Īøā = argmaxĪøILĪø (X;Y ) (4)
The expectation operator in (3) relies on the noisedistribution knowledge. Since we donāt make any as-sumption on the noise model, the knowledge of the noisedistribution is missed, but a good estimation is authorizedby replacing the expectation operator by an empirical av-erage with large values of N . Our optimization problemcan thus be rewritten as:
Īøā ā argmaxĪø
1ā 1
N
Nān=1
log2
(1 + eāxnLĪø(yn)
)ā argmin
Īø
1
N
Nān=1
log2
(1 + eāxnLĪø(yn)
)ļøø ļø·ļø· ļøø
fopt(xn; yn)
,(5)
where xn and yn are samples that represent the inputand output of the channel respectively.
The minimization of fopt(Ā·) will be tackled in ourimplementation via simplex method based algorithm[16]. Eventually, the goal is to optimize the parameterĪø, which allows us to find back the best approximatedLLR that fits the optimum L.
To get the samples xn and yn two ways are con-sidered: the supervised way, where xn is a learningsequence and yn the output of the channel given theinput xn and the unsupervised way, where only yn isknown to the receiver which needs thus to rebuilt fromyn a corresponding input sequence xn.
In the following, we will start by briefly presenting thesupervised optimization as performed in [11], and then,we will present our proposed unsupervised optimizeddemapper.
A. Supervised
šApproximated LLR
Demapper
š
a b
Mutual Information
Noise Channel N = noise +
Interferenceš BP-Decoder
šæšš(Y)
Fig. 3. Supervised LLR demapper.
If the optimization is performed in a supervised man-ner, the value of the parameter Īø = (a, b) is triggeredafter receiving the channel output Y by maximizing theMI between the learning sequence X and channel outputY as shown in Fig. 3.
The major drawback of this method is the large neededlength of the learning sequence, inducing a tediousoverload for each packet transmission. Because of theimpulsive nature of the noise, the estimation is morecomplex than in the Gaussian case. Indeed, the learningsequence has to be long enough to guaranty the presenceof large impulse amplitude, else the estimation of Īø wonātlead to a good LLR estimation.
In the next subsection, we propose to overcome theseissues by providing a blind estimation of a and b, whichfocuses directly on the output of the channel, withoutany prior on the input.
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B. Unsupervised
nOfIter
BP-DecoderApproximated LLR
Demapper
š
a b
Mutual Information
Noise Channel N = noise +
Interferenceš
šæšš(š)
ą·©šµ
ą·¦šš
š
Fig. 4. Unsupervised LLR demapper.
Unsupervised optimization is notably attractive sinceit does not rely on any overload.
Since the MI based method requires the noise knowl-edge in order to optimize Īø, we propose to estimate noisesamples directly from the received samples Y using asign detector as N = Y ā sign(Y ). Once the noisesamples are extracted, one can simulate a new channelfor which the input sequence X is known and follows ani.i.d. BPSK independent of Y and N . The output of thisnew channel will be given as Y = X + N . Since thisknown sequence is designed at the decoder, the methodwill not suffer from a throughput loss: even if X can beseen as a learning sequence used to mimic the supervisedoptimization, it is not sent over the channel. The optimalvalue of the parameter Īø is obtained by maximizing theMI between the known input X and the output of thedesigned channel Y using approximated LLRs. Oncethe optimal parameter Īøā is obtained over the designedchannel, it is used to build the approximated demapperover the real channel as LĪøā(Y ).
C. Comparison between the supervised and unsuper-vised optimization
Fig. 5. Comparison of the mean and standard deviation evolution forparameters (a, b) as a function of the dispersion Ī³ of a SĪ±S noise withĪ± = 1.4 for the supervised and unsupervised optimization.
In order to be efficient, we can expect that the ap-proximated demapper under blind optimization performsclose to the one using a learning sequence. Thus, theoptimal values for Īø obtained under the supervised and
the unsupervised optimization must be close to eachother.
Under both supervised and unsupervised optimization,the evolution of the mean and variance of the parameterĪø = (a, b) are compared as a function of the dispersionĪ³ of a SĪ±S noise with Ī± = 1.4 as shown in Fig. 5.For each channel state, a and b are resultant of 1000experiments, furthermore, a learning sequence of 20000samples is used to optimize a and b under the supervisedcase. Such a long sequence allows to limit estimationerrors so that a and b will be obtained with a highaccuracy in the supervised approach. Whereas the gapbetween the obtained values for a, under supervisedand unsupervised optimization, is rather small, the oneobtained for b is significantly larger. Nevertheless, thiswill not have major consequences in terms of BERperformance as we will see in the next section. Fig. 6
Fig. 6. Comparison of the LLR shapes under the effect of the estimateda and b parameters with Ī³ = 0.5 and Ī± = 1.4, in the supervised LLRapproximation, the unsupervised LLR approximation and with LLRobtained by numerical integration.
compares the LLR shapes obtained under supervisedoptimization (a = 3.15, b = 4.96) and unsupervisedoptimization (a = 3.3, b = 3.75) to the true LLRobtained via numerical integration for a SĪ±S noise ofparameters Ī± = 1.4 and Ī³ = 0.5. This comparison showsthe convergence between the LLR shapes, despite theaforementioned gap between the estimated values of b.
IV. SIMULATION RESULT
For our simulation results, we propose to use a reg-ular (3,6) LDPC code of length n = 20000 over anadditive SĪ±S noise channel of parameter Ī± = 1.4 (highimpulsiveness) and Ī± = 1.8 (low impulsiveness). Foreach channel state, a learning sequence of 20000 samplesis used for the supervised optimization. In case of animpulsive environment with Ī± < 2, the second-ordermoment of a stable variable is infinite [17, Theorem 3],making the conventional noise power measurement notwell-defined. Accordingly, we present our simulation re-sults as a function of Ī³, which is used as a measurementof the strength of the Ī±-stable noise.
Fig. 7 and Fig. 8 present the obtained BER in lowand high impulsiveness respectively, as a function of
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Fig. 7. Evolution comparison of the BER as a function of thedispersion Ī³ of a SĪ±S noise in poorly impulsive environment withĪ± = 1.8, in the supervised LLR approximation, the unsupervised LLRapproximation and with LLR obtained by numerical integration.
Fig. 8. Evolution comparison of the BER as a function of thedispersion Ī³ of a SĪ±S noise in highly impulsive environment withĪ± = 1.4, in the supervised LLR approximation, the unsupervised LLRapproximation and with LLR obtained by numerical integration.
the dispersion Ī³ of the alpha-stable noise. For eachvalue of Ī³, we compare the BER obtained via LLRapproximation, either in a supervised or unsupervisedmanner to the BER obtained with the true LLR. First,note that the unsupervised method performs close tothe supervised case. One major advantage of the blindoptimization, besides the absence of a learning sequence,relies on the direct use of the noise impacting thecodewordās transmission, reducing thus the risk of a badnoise estimation, especially if the noise varies quicklyfrom one packet to another. Moreover, the gap betweenthe estimated LLR and exact LLR is close in both envi-ronments, proving the strength of our proposed method,and the ability to adapt to the change of impulsiveness.
V. CONCLUSION
In this paper, we proposed an unsupervised LLRapproximation used as the input of an LDPC decoderover an additive symmetric Ī±-stable noise channel. Ournew unsupervised approximated LLR demapper featuresan easy implementation, thanks to the simplicity of ourproposed noise extraction used in the optimization step.Moreover, the performance obtained with our solution
is very close to the one obtained using a supervisedLLR approximation or the one obtained with the trueLLR, proving the strength of our blind approximation.Indeed, a very small gap to the true LLR is achievedwithout decreasing the throughput and without requiringnumerical integration.
ACKNOWLEDGMENTS
This work was partly supported by IRCICA, CNRSUSR 3380, Lille, France.
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