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A robust metric for soft-output detection in the presence of class-A noise

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A robust metric for soft-output detection in the presence of class-A noise
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  36 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 1, JANUARY 2009 A Robust Metric for Soft-Output Detectionin the Presence of Class-A Noise Dario Fertonani,  Student Member, IEEE,  and Giulio Colavolpe,  Member, IEEE   Abstract —Digital communications over channels impaired byimpulse noise are considered. We  Þ rst address the problem froman information-theoretical viewpoint, discussing the performancelimits imposed by the channel model. Then, we describe andcompare a couple of practical communication schemes employingpowerful channel codes and iterative decoding, with focus on avery simple and robust detection scheme that does not requirethe estimation of the statistics of the impulse noise.  Index Terms —Impulse noise, soft-output detection, achievableinformation rate. I. I NTRODUCTION T HE power delivery networks and some mobile radioscenarios are often characterized by interferences thatexhibit a signi Þ cant impulsive nature. Among the variousstatistical models for these phenomena,generally referred to as“impulse noise”, the most widely used in the literature is theclass-A model [1], which is adopted also in this letter. Theperformance of such systems is generally studied under theassumption of ideal knowledge of the statistical properties of the impulse noise [2]–[4]. These statistics, which are actuallyunknown to the receiver, can be effectively estimated [5], butthe estimation algorithms unfortunately affect the complexityof the system and cannot properly cope with time-varyingchannels [5]–[8]. A blind approach, based on detection metricsthat do not require the knowledge of the channel parametersnor their estimation, is thus of great interest.We  Þ rst resort to information-theoretical arguments and dis-cuss the ultimate performance limits imposed by the channel,then we consider practical communication schemes employingpowerful codes and iterative decoding [9], [10]. In particular,we propose a detection scheme that does not require to knownor to estimate the statistics of the impulse noise, and compareit with an ideal receiver that perfectly knows such statisticsand with the soft-limiting receivers [11], which are usuallyconsidered as a reference benchmark for robust detection overclass-A channels. These comparisons prove the effectivenessof the proposed solution, which performs practically as theideal one and much better than the classical soft-limitingreceivers.II. C HANNEL  M ODEL A sequence  c K  1  =  { c k } K k =1  of   M  -ary complex-valuedsymbols, possibly obtained by properly encoding a sequence Paper approved by T. M. Duman, the Editor for Coding Theory andApplications of the IEEE Communications Society. Manuscript received April12, 2007; revised October 8, 2007.The authors are with the Department of Information Engineering, Uni-versity of Parma, Viale G. P. Usberti 181/A, 43100 Parma, Italy (e-mail:fertonani@tlc.unipr.it; giulio@unipr.it).This work was presented in part at the IEEE Global CommunicationsConference (GLOBECOM’07), Washington, DC, USA, November 2007.Digital Object Identi Þ er 10.1109/TCOMM.2009.0901.070041 of information bits, is linearly modulated and transmitted overan additive white Gaussian noise (AWGN) channel that alsointroduces impulse noise. 1 Assuming ideal synchronizationand absence of intersymbol interference, we can write thereceived samples as [1] y k  =  c k  +  n k  , k  ∈{ 1 , 2 ,...,K  }  (1)where  n K  1  is a sequence of independent and identicallydistributed noise samples. At each time epoch  k , the statisticalproperties of the sample  n k  are completely de Þ ned by thechannel state  s k , which belongs to the set of the non-negativeintegers  N , and assumes the value  i ∈ N  with probability [1] P  i  =  e − A A i i !  (2)where  A  is a positive parameter characterizing the channel,generally referred to as “impulsive index”. In particular,the sample  n k  is a complex circularly-symmetric Gaussianrandom variable with variance depending on  s k , so that theprobability density function (PDF) of   n k  conditioned to  s k can be written as [1]  p ( n k | s k  =  i ) = 12 πσ 2 i exp  −| n k | 2 2 σ 2 i   , i ∈ N  (3)where  σ 2 i  is the variance per component of the noise sampleswhen  s k  =  i . Hence, the PDF of the generic sample  n k  results  p ( n k ) = ∞  i =0 P  i  p ( n k | s k  =  i ) = ∞  i =0 P  i 2 πσ 2 i exp  −| n k | 2 2 σ 2 i   . (4)The variances  { σ 2 i }  can be written as σ 2 i  =  1 +  iA Γ  σ 20  , i ∈ N  (5)where  σ 20  can be interpreted as the variance per component of the backgroundGaussian noise, while  Γ  is a positive parameterdescribing the power of the impulse noise [1]. Namely, sincethe average power of the noise samples is E {| n k | 2 } = 2 ∞  i =0 P  i σ 2 i  = 2 σ 20  + 2 σ 20 Γ  (6)the channel introduces, in addition to the background Gaussiannoise with average power  2 σ 20 , an impulsive contribution withaverage power  2 σ 20 / Γ .By properly setting the values of the parameters  A  and  Γ ,a large variety of channels with different statistical proper-ties can be described [1], [2]. In this work, we focus onscenarios where the presence of impulsive noise, that is theevent  { s k  >  0 } , is relatively infrequent with respect to the 1 For any sequence  { v k } , we denote the subsequence  { v k } n 2 k = n 1 by  v n 2 n 1 .0090-6778/09$25.00 c   2009 IEEE  FERTONANI and COLAVOLPE: A ROBUST METRIC FOR SOFT-OUTPUT DETECTION IN THE PRESENCE OF CLASS-A NOISE 37 0.10.20.30.40.50.60.70.80.91.01.11.21.310 -3 10 -2 10 -1 10 0 10 1 10 2 10 3    S   N   R   [   d   B   ] Γ  A  = 5 · 10 -2  A  = 10 -1  A  = 5 · 10 -2 , CSI  A  = 10 -1 , CSIAWGN channel Fig. 1. Signal-to-noise ratio required to achieve an information rate of 1 bitper channel use. presence of background noise only, that is the event { s k  = 0 } .Hence, we assume that  P  0  >  1 / 2 , or, equivalently, that theparameter  A  satis Þ es the inequality A <  log e (2) ≃ 0 . 693  .  (7)On the other hand, no particular restriction on the value of   Γ is assumed.III. T HEORETICAL  P ERFORMANCE  L IMITS Before describing practical communication systems, weanalyze the ultimate performance limits imposed by thechannel. In particular, we are interested in evaluating themaximum number of information bits that can be transmittedper channel use, on average, to achieve an arbitrarily smallbit-error rate (BER) when no upper limit on the length of   c K  1 is imposed. This corresponds to evaluating the informationrate  I  ( C,Y   )  between the sequences  c K  1  and  y K  1  [12]. Wewill restrict ourselves to the case of a stationary source andsymbols  c K  1  belonging to an  M  -ary phase-shift keying (PSK)alphabet. In this case, by resorting to the arguments in [13]or simply by exploiting the symmetry of both the channel andthe alphabet, it is easy to prove that a memoryless sourcethat emits equally likely symbols achieves the maximumallowed information rate. Such a source is thus consideredhereafter. Although, for the considered system, the informationrate  I  ( C,Y   )  cannot be written in a closed-form expression,we can easily evaluate it by numerical integration, exploitingthe memoryless nature of both the source and the channel [12].Some signi Þ cant outcomes of such computations are reportedand discussed in the following.In Fig. 1, it is shown how the value of the signal-to-noiseratio (SNR) required to achieve an information rate of 1 bitper channel use varies when different channels are consideredand a quaternary PSK (QPSK) modulation is adopted. In thisletter, we de Þ ne the SNR as  | c k | 2 / (2 σ 2 ) , that is with respectto the background Gaussian noise only, so that the impulsenoise is not involved in the de Þ nition. Together with the curvesrelated to class-A channels, we also reported the correspondingcurves related to the AWGN channel and to a system with idealchannel-state information (CSI), that is a system that knows 5.05.56.06.57.07.58.08.59.09.510.010 -3 10 -2 10 -1 10 0 10 1 10 2 10 3    S   N   R   [   d   B   ] Γ  A  = 5 · 10 -2  A  = 10 -1  A  = 5 · 10 -2 , CSI  A  = 10 -1 , CSIAWGN channel Fig. 2. Signal-to-noise ratio required to achieve an information rate of 1.75 bits per channel use. the actual realization of the state process  s K  1  underlying theclass-A channel. 2 All following considerations qualitativelyhold irrespectively of the values of the impulsive index and thetarget information rate, but, as shown in Fig. 2, the differencesbetween the various performance limits are more signi Þ cantas these values increase—when the impulsive index is smallenough, all curves collapse on that related to the AWGNchannel. It is interesting to note that the curves related to class-A channels without CSI exhibit a non-monotonic behaviorwith respect to  Γ . In particular, there exists a value  Γ 0  suchthat, for  Γ  <  Γ 0 , the lower the value of   Γ  the better theperformance, up to an asymptotic value worsening as theimpulsive index and/or the target information rate increase.This behavior, which is somehow surprising since it impliesthat the system can take advantage of a larger power of theimpulsive interferers, 3 can by explained by considering thatthe gap between the CSI system and the real system is dueto a non-ideal channel-state identi Þ cation. In fact, the curvesrelated to CSI systems exhibit a monotonic behavior withrespect to  Γ , proving that the theoretical power ef  Þ ciencyworsens as the power of the impulse noise increases whenideal channel-state identi Þ cation is available, as expected.This allows us to conjecture that the bene Þ cial effect of anincreasing power of the impulse noise is due to the factthat, when it is large enough, the system can better detectthe presence of interfering impulses and, consequently, betterapproach the performance of the CSI system. Such conjecturesare con Þ rmed by the fact that the gap between real systemsand CSI systems tends to vanish as the value of   Γ  tends tozero, that is when the impulse noise is much more powerfulthan the background noise and thus is easier to detect. 2 The information rate of CSI systems equals the statistical average of theinformation rates over the channel states [13]. 3 Although similar conclusions are drawn in [4] while discussing thenumerical simulations, our results prove that the non-monotonic behavior isnot due to any particular coding scheme, but is instead an intrinsic feature of the channels affected by impulse noise.  38 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 1, JANUARY 2009 IV. P RACTICAL  C OMMUNICATION  S CHEMES  A. Optimal Detection In a typical environment, the statistics of the impulse noiseare such that, when no channel encoding is adopted,the systemis basically impulse-noise limited and an error  ß oor in theBER curve occurs [2]–[4]. We thus consider a communicationsystem that adopts a powerful channel code, such as turbo-like or low-density parity-check (LDPC) codes, and performsiterative decoding [9], [10]. For each time epoch  k  and foreach trial value  ˜ c k  belonging to the modulation alphabet, theoptimal detector should send to the decoder a likelihood mes-sage  I  k (˜ c k ) , simply referred to as “metric” in the following,such that 4 I  k (˜ c k ) ∝  p ( y k | ˜ c k ) = ∞  i =0 P  i 2 πσ 2 i exp  −| y k − ˜ c k | 2 2 σ 2 i   (8)where  p ( y k | ˜ c k )  is the PDF of the received sample  y k  condi-tioned to the transmission of the symbol  ˜ c k . In the case athand, after straightforward manipulations of (8) based on (2)and (5), we can write the generic metric  I  k (˜ c k )  as I  k (˜ c k ) = exp  −| y k − ˜ c k | 2 2 σ 20  + ∞  i =1 A i i ! A Γ( A Γ +  i ) exp  −| y k − ˜ c k | 2 2 σ 20 A Γ A Γ +  i   (9)pointing out that the  Þ rst term is exactly the optimal metricfor AWGN channels, whereas the second term can be seenas a correction term accounting for the presence of impulsenoise. The summation in (9) involves an in Þ nite series, andthus it is not suitable for practical uses. Hence, we de Þ ne theparameter  i MAX  as the minimum integer such that ∞  i = i MAX +1 P  i  <  10 − 10 (10)and, slightly abusing the notation, we refer to the metricobtained by neglecting the terms with index  i > i MAX  in (9)as “optimal metric”. When the hypothesis (7) is satis Þ ed, weget  i MAX  ≤  10  irrespectively of the value of the impulsiveindex  A .  B. Suboptimal Detection According to (9), the evaluation of the optimal metrics re-quires that the receiver knows the values of   σ 20 ,  A  and  Γ . Sincethese parameters are actually unknown to the receiver, theoptimal metric can be just considered as an ideal solution. Apossible practical approach consists of estimating the statisticsof the noise by means of proper algorithms [5]. Since the esti-mation of the power of the background Gaussian noise is notcritical, a perfect knowledge of the value of   σ 20  is assumed inthis letter. On the other hand, the algorithms for the estimationof the values of   A  and  Γ , besides increasing the complexity of the receiver, cannot properly cope with impulse noise whosestatistics are signi Þ cantly time-varying [5]. Hence, the designof detection schemes that do not require the knowledge of the 4 We use the proportionality symbol  ∝  when the sides can differ by apositive multiplicative factor irrelevant for the decoding process. 10 − 7 10 − 6 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 10 0  0 2 4 6 8 10 12 14        I        k        (     c     ~       k        ) |  y k  − c ~k  |/  σ 0 Γ  = 10 − 1 Γ  = 10 0 AWGN channel A = 10 − 1 A = 10 − 2 Fig. 3. Optimal metric for different parametrizations of the impulse noise. values of   A  and  Γ  nor their estimation is of great interest fortheir simplicity and robustness [6]–[8]. A couple of possiblesolutions are compared in the following. 5 We  Þ rst consider the classical “soft limiting” (SL) met-ric [11] I  SL k  (˜ c k ) = exp  −| z k − ˜ c k | 2 2 σ 20   (11)where, at each time epoch  k , the sample  z k  is obtained byproperly cutting the amplitude of the received sample  y k .Formally, given a suitable threshold value  V  T   >  0 , we canwrite z k  =  f  ( ℜ{ y k } ) +  jf  ( ℑ{ y k } )  (12)where the non-linear function  f  ( · )  is de Þ ned as f  ( x ) =  x  if   | x |≤ V  T  V  T   if   x > V  T  − V  T   if   x < − V  T  .  (13)The rationale of the SL metric is discussed in [11].In this letter, we propose an alternative solution, basicallyextending our proposal in [7]. In Fig. 3, the behavior of theoptimal metric  I  k (˜ c k )  is shown as a function of   | y k  −  ˜ c k | ,for different values of   A  and  Γ . Under the hypothesis (7), thedominant term in (9) is the  Þ rst one when the value of theEuclidean distance  | y k  − ˜ c k |  is low, so that all curves matchthe classical AWGN metric I  AWGN k  (˜ c k ) = exp  −| y k − ˜ c k | 2 2 σ 20   (14)at the left side of Fig. 3, irrespectively of the actual statisticsof the impulse noise. On the other hand, at the right sideof Fig. 3, that is for large values of   | y k  −  ˜ c k | , the secondterm is the dominant one in (9) and the curves thus differdepending on the statistics of the impulse noise. Hence, in thatregion, any metric that is blind with respect to the values of   A and  Γ  cannot approximate with accuracy the ideal metric. Weexploit the fact that the knowledge of the values of   A  and  Γ  is 5 Other solutions addressed to a different model of the impulse noise arepresented in [6], but they are not considered here since they are not suitablefor receivers employing iterative decoding [7], [8].  FERTONANI and COLAVOLPE: A ROBUST METRIC FOR SOFT-OUTPUT DETECTION IN THE PRESENCE OF CLASS-A NOISE 39 required only for the description of the “tails” of the metric,and propose the following threshold approximation I  k (˜ c k ) = max  I  AWGN k  (˜ c k ) , ∆   (15)where  ∆  ∈  [0 , 1]  is a design parameter discussed later. Therationale of the approximation (15), which is just a saturationof the AWGN metric (not of the received sample, unlike the SLmetric) to a constant threshold and thus is very simple from acomputational viewpoint, is explained in the following. First,when the Euclidean distance  | y k  −  ˜ c k |  is low, the proposedmetric matches the AWGN metric, as an effective metricshould de Þ nitely do according to Fig. 3. On the other hand,when the received sample is far away from the constellation,that is when the Euclidean distance  | y k − ˜ c k |  is very large forall possible values of the modulation symbol  ˜ c k , the proposedmetric is saturated to the minimum threshold  ∆  irrespectivelyof the value of   ˜ c k , so that the detector produces the so-called“erasure” decision. In this case, the presence of impulsivecontributions in addition to the background noise is verylikely, and a receiver that does not know the statistics of theimpulse noise cannot produce any more reliable decision thanan erasure. To better realize these statements, it is useful toconsider the results reported in Fig. 4, which refer to a binaryPSK (BPSK) modulation with alphabet { 1 , − 1 } and a class-Achannel characterized by  A  = 10 − 1 ,  Γ = 10 − 1 , and an SNR of 0 dB. In Fig. 4, it is shown how the log-likelihood ratio (LLR),that is the natural logarithm of the ratio between the metriccorrespondingto the hypothesis  ˜ c k  = 1  and that correspondingto the hypothesis  ˜ c k  = − 1 , varies with respect to the receivedsample  y k  when different metrics are considered. The behaviorof the optimal metric exhibits two key points, namely the needfor producing very low-magnitude LLRs when the receivedsample is far away from the constellation, that is when thepresence of interfering impulses is very likely, and the needfor exploiting the imaginary component of   y k —unlike theAWGN channels, the real and imaginary components of theclass-A noise are not independent. Fig. 4 de Þ nitely provesthat the proposed metric, here implemented with  ∆ = 10 − 3 ,approximate this behavior much better than the consideredalternatives. In particular, both the AWGN metric and the SLmetric, here implemented with  V  T   = 1 . 3 , cannot exploit theimaginary component of   y k  (thus not reported in Fig. 4) anddramatically fail in producing low-magnitude LLRs when thepresence of interfering impulses is very likely. In conclusion,it is easy to predict that the proposed metric will signi Þ cantlyoutperform them when systems requiring high-quality soft-output detection are considered. On the other hand, the gener-ation of erasure decisions (that is null LLRs in Fig. 4) makesthe proposed metric less suitable for hard-output detection.We now discuss the choice of the threshold parameter  ∆ ,which is crucial for the performance of the proposed metric.Following the arguments in [8], one can derive a rule of thumb for choosing the value of   ∆  given the statistics of the impulse noise. On the other hand, extensive computersimulations, some of which are reported in Section V, showthat the proposed metric is very robust, and that values of   ∆ in the order of   10 − 3 result effective irrespectively of theactual statistics of the impulse noise, provided that a powerful  0 2 4 6 8 10 0 2 4 6 8    L   L   R        k Re{Im{Im{  y y y k k k  }}=0}=2 AWGN metricSL metricProposed metricOptimal metric Fig. 4. Log-likelihood ratios for a channel characterized by  A  = 10 − 1 , Γ = 10 − 1 , and SNR  = 0  dB. channel code is adopted. Such a robustness is due to the factthat the value of   ∆  does not affect the key features requiredfor good soft-output detection, namely the generation of low-magnitude LLRs when the presence of interfering impulsesis very likely, and the capability of exploiting that the realand imaginary components of the received samples are notindependent.V. S IMULATION  R ESULTS In this section, the performance of the described detectionschemes is assessed by means of computer simulations. Thereported results refer to QPSK transmissions over channelscharacterized by different statistics of the impulse noise. A (3 , 6) -regular LDPC code of rate  1 / 2  is applied to sequencesof   2000  information bits. At the receiver side, the LDPCdecoder performs  40  self-iterations before producing the de-cisions on the information bits [10]. The iterative process canalso stop before the  40 th iteration if, by checking the codesyndrome, a valid codeword is found.We  Þ rst consider a channel with impulse noise characterizedby  A  = 10 − 1 and  Γ = 10 − 1 . Fig. 5 shows the performanceof the system when different metrics are used, in terms of BER versus SNR. As a comparison, the performance over anAWGN channel is also reported. Although, according to (6),the impulse noise increases the power of the overall noise of about  10  dB with respect to the AWGN channel, the optimalmetric provides a performance degradation lower than  1 . 5  dB,canceling out the greatest part of the impulse noise thanksto the powerful coding scheme. We notice that the proposedmetric, here implemented with  ∆ = 10 − 3 , ensures the sameperformance as the optimal one, resulting the most convenientperformance/complexity tradeoff. Let us point out that thesaturation to the minimum threshold  ∆ , which actually is theonly difference between the proposed metric and the AWGNmetric, provides a gain of more than  12  dB at the expense of a practically null increase in complexity. On the other hand,the simulation results also con Þ rm the conjectures carriedout in Section IV-B on the ineffectiveness of the SL metricwhen employed in systems requiring high-quality soft-outputdetection. In fact, although the threshold value  V  T   has been  40 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 1, JANUARY 2009 10 − 6 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 10 0  0 2 4 6 8 10 12 14 16    B   E   R SNR [dB] AWGN metricSL metricProposed metricOptimal metricAWGN channel Fig. 5. Performance over a class-A channel characterized by  A  = 10 − 1 and  Γ = 10 − 1 . 2.02.53.03.54.010 − 3 10 − 2 10 − 1 10 0 10 1 10 2 10 3    S   N   R   [   d   B   ]  a   t   B   E   R  =   1   0    −    5 Γ Proposed metricOptimal metric A = 10 − 2 A = 10 − 1 Fig. 6. Performance over channels with different parametrizations of theimpulse noise. optimized for each value of the SNR, we notice that the SLmetric exhibits a performance degradation larger than  2  dBwith respect to the proposed one.Fig. 6 shows the values of the SNR corresponding to a BERequal to  10 − 5 when different statistics of the impulse noise areconsidered. The performance of the optimal metric and that of the proposed one working with  ∆ = 10 − 3 are compared. Weremark the robustness of the proposed metric, which providesa negligible performance degradation with respect to the idealbenchmark irrespectively of the statistics of the impulse noise.In practice, when the proposed metric is adopted, we can justset  ∆ = 10 − 3 and there is no need for information on thevalues of   A  and  Γ .This fact, together with the very low computational com-plexity, de Þ nitely makes the proposed solution the most conve-nient one. The same conclusion holds even when a simpli Þ edmodel is assumed for the impulse noise [7]. Moreover, it isworth to notice that the results reported in Fig. 6 greatly agreewith the theoretical discussion carried out in Section III.In particular, it is con Þ rmed that the impulse noise, irrespec-tively of its power, can be practically canceled out when theimpulsive index is small enough (with the same LDPC code,over an AWGN channel, a BER equal to  10 − 5 is achievedwhen the value of the SNR is about  2  dB), and that thereexists a value  Γ 0  such that, for  Γ  <  Γ 0 , the system can takeadvantage of a larger power of the impulse noise.VI. C ONCLUSIONS The performance of communication systems over channelsimpaired by impulse noise has been analyzed. We havediscussed the ultimate performance limits of these systems byexploiting information-theoretical arguments, and presented adetection metric that, besides being characterized by a minimalcomputational complexity, does not require the knowledgeof the statistics of the impulse noise. When combined withpowerful channel codes, the proposed scheme has been shownto perform practically as the ideal one, much better thanthe classical soft-limiting detectors, and fairly close to thetheoretical limit.R EFERENCES[1] D. Middleton, “Statistical-physical model of electromagnetic interfer-ence,"  IEEE Trans. Electromagn. Compat. , vol. 19, no. 3, pp. 106-126,Aug. 1977.[2] A. D. Spaulding and D. Middleton, “Optimum reception in an impul-sive interference environment–part I: coherent detection,"  IEEE Trans.Commun. , vol. 25, pp. 910-923, Sept. 1977.[3] D. Umehara, H. Yamaguchi, and Y. Morihiro, “Turbo decoding inimpulsive noise environment," in  Proc. IEEE Global Telecommun. Conf. ,Dallas, Texas, Nov./Dec. 2004.[4] H. Nakagawa, D. Umehara, S. Denno, and Y. Morihiro, “A decodingfor low density parity check codes over impulsive noise channels," in Proc. IEEE Intl. Symp. on Power Line Commun. Its Appl. , Vancouver,Canada, Apr. 2005.[5] S. M. Zabin and H. V. 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