A Blind Pre-Processor for Modulation Classification Applications in Frequency-Selective Non-Gaussian Channels

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1 A Blind Pre-Processor for Modulation Classification Applications in Frequency-Selective Non-Gaussian Channels SaiDhiraj Amuru, Student Member, IEEE, and Claudio R. C. M. da Silva, Senior Member, IEEE Abstract This paper presents a new pre-processing stage that allows for the reliable classification of digital amplitudephase modulated signals in a practical scenario where the classifier has no knowledge of the timing symbol transition epochs of the received signal; the noise added in the channel is non-gaussian; and 3 the fading experienced by the signal is frequency-selective. The proposed pre-processor, which is based on the Gibbs sampling algorithm, is used to acquire timing information and to estimate the channel state and noise distribution parameters blindly; that is, without knowledge of the received symbol sequence and the modulation scheme used. With the obtained estimates, in a second processing stage, the signal is then classified by using an appropriate likelihood- or feature-based classification algorithm. To quantify the performance of the proposed pre-processor, the probability of correct classification obtained by using the pre-processor with different classification algorithms is presented. It is shown that, by using the proposed pre-processor, modulation classification algorithms can perform well compared to clairvoyant classifiers assumed to be symbol synchronous with the received signal and to have perfect knowledge of the channel state and noise distribution. I. INTRODUCTION Modulation classification can be defined as the process of determining the modulation scheme of a noisy signal from a given set of possible schemes. This process has many applications in wireless communications, including in autonomous multi-mode and software-defined radios [], []. In most scenarios of interest, the difficulty in performing modulation classification is due primarily to the fact that classifiers operate with no or incomplete knowledge of the fading experienced by the signal and the distribution of the noise added in the channel. This is because a receiver typically has to first classify the received signal before it can successfully acquire symbol timing and estimate the channel state. As a result, the impractical assumption that the received signal is acquired and equalized by the radio front-end before classification is often made in the design of modulation classification algorithms [3], [4]. Manuscript received August 6, 3; revised January, 4, April, 4, July, 4, September, 4 and November, 4; accepted November 5, 4. The editor coordinating the review of this paper and approving it for publication was S. Gezici. This work was supported in part by InterDigital Communications LLC. SaiDhiraj Amuru is with the Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 46 USA adhiraj@vt.edu. C. R. C. M. da Silva was with the Bradley Department of Electrical and Computer Engineering and is now with Mobile Solutions Lab, Samsung, San Diego, CA 9 USA claudio.silva@ieee.org. Digital Object Identifier The modulation classification operation can typically be divided into two stages. In the first, referred to as the preprocessing stage, the signal is acquired and different channel state and noise distribution parameters are estimated. With the obtained estimates, the signal is then classified in the second stage using an appropriate classification algorithm. The acquisition of symbol timing by the pre-processor is needed because it is infeasible for classification to be reliably performed when the received signal s eye diagram is closed by inter-symbol interference. Also, the estimation of channel parameters is necessary for the classifier to mitigate the effects of frequency-selective fading. While it is possible for modulation classification to be performed blindly to certain unknown channel parameters, such as the noise power, such approaches are not as reliable as when unknown parameters are first estimated by a pre-processor at the possible cost of increased complexity. In this paper, we present and analyze a pre-processor that allows for the reliable classification of amplitude-phase modulated signals ASK, PSK, and QAM when the receiver has no knowledge of the timing symbol transition epochs of the received signal, the noise added in the channel is non- Gaussian, and the unknown fading experienced by the signal is frequency-selective. We assume that the additive noise is non-gaussian because various studies have shown that most radio channels experience both man-made and natural noise, and that the combined noise is impulsive. See, among others, [5]- [8] and references therein. As discussed in [5], signal processing algorithms designed for optimal performance in Gaussian noise typically perform significantly worse when non-gaussian noise is present. This is due, in part, to the lack of robustness of linear and quadratic type signal processing procedures to many types of non-gaussian statistical behavior [6], [9]. Although different classifiers have been developed for the case when the additive noise is non-gaussian see, for example, []- [5] the problem of modulation classification in frequency-selective non-gaussian channels remains largely unexplored. The proposed pre-processing stage is built upon the Gibbs sampling-based algorithm proposed by Drumright and Ding in [6], which assumes perfect synchronization and Gaussian noise. More specifically, our main contributions in the development of the pre-processing stage are: we propose a griddy Gibbs sampling algorithm to acquire symbol timing information; we generalize the Gibbs sampling-based algorithm to allow for the estimation of the parameters of a Gaussian

2 mixture distribution; and 3 we propose a new approach to data detection that avoids possible degenerate conditions which would negatively impact the estimation of the channel state and noise distribution parameters. The remaining parts of this paper are organized as follows. The system model, including a discussion on the use of a Gaussian mixture distribution to model the additive noise, is introduced in Section II. In Section III, we provide a basic description of Gibbs sampling and formulate the proposed pre-processor. The modulation classification stage is then discussed in Section IV. In Section V, numerical results are presented. In this section, an extension of the proposed preprocessor for the case when the received symbols suffer phase rotation due to the presence of a residual carrier frequency offset is also considered. Conclusions are drawn in Section VI. II. SYSTEM MODEL We assume that the data conveyed in the transmitted signal is mapped onto an unknown digital amplitude-phase constellation denoted by S b. The low-pass equivalent of the transmitted signal is st = m= s mgt mt, where T is the symbol interval and gt is a real-valued pulse. The random variables {s m } m=, s m S b, denote the modulated symbols and are assumed to be independent and uniformly distributed among all constellation points. Without loss of generality, the energy of gt and the average energy of S b E[ sm ] are normalized to unity. It is assumed that the transmitted signal passes through a frequency-selective fading channel with L resolvable paths. The channel is assumed to be slowly varying and, in particular, constant during the observation interval. By using a conventional tapped delay line model with tap spacing equal to T, the low-pass equivalent of the received signal is given by rt = m= l= L α l e jθ l s m gt mt lt τ + zt, where α l and θ l are the amplitude and phase of the l th multipath component, respectively, and τ is an unknown time delay. Without loss of generality, it is assumed that τ lies in the interval [, T. In, zt is the low-pass equivalent of the noise added in the channel. Using, the signal at the output of a matched filter, when sampled at t = kt, is given by y k = {rt gt} t=kt L = h l s k l+m p mt τ + w k, m= l= for k =,..., K. If the unknown time delay τ was known that is, if it was perfectly estimated and the matched filter output was sampled at t = kt + τ, we would have L y k = {rt gt} t=kt +τ = h l s k l + w k, 3 l= for k =,..., K. In and 3, h l = α l e jθ l and pt = gt gt. As previously discussed, the noise in most radio channels is known from experimental studies to be non-gaussian, due in part to the impulsive nature of man-made and natural noise [5]- [8]. For this reason, we assume that the probability density function pdf of the complex noise variable w k {zt gt} t=kt in and 3 is given by the N- term Gaussian mixture N λ n pw = exp w, 4 πσ n= n where λ n is the probability that w k is chosen from the n th term in the pdf, with N n= λ n =. This model is chosen because the Gaussian mixture pdf closely approximates Middleton s canonical Class A interference model [5], [7] and other symmetric pdfs [8]. In addition, 4 includes the Gaussian pdf as a special case. We assume that the random variables {w k } K k= are independent and identically distributed iid a common assumption made for analytical purposes see, for example, [6]- [8], [8], and [9]. In the analysis that follows, the time delay τ, the complex channel gains {h l } L l=, and the noise distribution parameters {λ n } N n= and {σ n} N n= are considered to be unknown. The pulse shape gt and the symbol interval T are assumed to be known. The formulation given in Sections III and IV also assumes that the channel length L and the number of terms in the Gaussian mixture N are known. Because in practice these two parameters would have to be estimated, the potential performance loss in the classification process resulting from over/under-estimating L and N is quantified in Section V. Also in Section V, the effect of carrier frequency offset in the classification process is evaluated by considering an extension of the proposed pre-processor. III. BLIND GIBBS SAMPLING-BASED PRE-PROCESSING STAGE In this section, we formulate a pre-processing stage that allows for the blind no knowledge of the received sequence and its modulation scheme estimation of the channel and noise distribution parameters necessary for the reliable classification of modulated signals in frequency-selective non-gaussian channels. This estimation problem is quite challenging due in part to the large number of unknown parameters involved. Specifically, in the considered system model, we have L complex channel gains, N variances, N mixture weights, and the timing offset τ. As a result, extending conventional estimation techniques such as the expectation-maximization EM algorithm [4] and cumulant-based methods [] to the The assumption that gt and T are known is valid for different applications. For example, 8. radios that make spectrum access decisions by relying on spectrum sensing must detect TV signals []. The format of TV signals can be found in their standards. For example, the digital TV standard for North America ATSC defines the pulse shape to be a raised-cosine with % excess bandwidth and a symbol interval of 93 ns []. For applications where such an assumption does not hold, such as electronic warfare systems in which T is determined by using a coarse estimate of the signal bandwidth, the unknowns should be incorporated into the classifier design as nuisance parameters. σ n

3 3 problem at hand is either not analytically possible or results in estimators likely to cause a large estimation error as, for example, cumulants of high order would have to be used. For this reason, we use a numerical Bayesian technique known as Gibbs sampling to estimate the necessary parameters. A. Short introduction to Gibbs sampling In the Bayesian framework, the unknown parameters are modeled as random variables and estimates are obtained by using the unknowns posterior pdf [3]. To evaluate the posterior pdf of the unknowns, a prior distribution is associated with each of them. Because the receiver in modulation classification applications has no or limited knowledge of the unknowns, we use conjugate and non-informative prior pdfs. In certain problems, such as the one considered here, the main challenge lies in evaluating the posterior pdf. For example, let θ = [θ,..., θ D ] T be the unknown parameter vector. The posterior pdf of a given parameter θ d, d =... D, can be written as pθ d y... py θpθdθ... dθ d dθ d+... dθ D. 5 In cases in which it is not possible to obtain a closed-form solution for 5, one must resort to numerical methods. In particular, the posterior pdf can be evaluated by using an approach in which samples that is, randomly drawn realizations of a variable with a given distribution are generated in an iterative process in such a way that, after a given number of iterations known as the burn-in period, they become distributed according to the posterior pdf [4]. More specifically, the key idea is to generate samples by running an ergodic Markov chain whose distribution after convergence is the desired posterior distribution [5]. Such numerical methods are referred to as Markov Chain Monte Carlo MCMC techniques. The minimum mean square error MMSE estimate of the unknown parameters can then be obtained by taking the mean of their corresponding samples after the burn-in period [4]- [6]. In this paper, the posterior pdfs, and thus parameter estimates, are obtained by using a MCMC technique known as Gibbs sampling. The Gibbs sampling algorithm iterates as shown in Algorithm, where θ d a is the estimate of θ d at the ath iteration. After a number of iterations, the Markov chain generated by Gibbs sampling is irreducible and aperiodic [4], and the samples generated from each marginal posterior pdf pθ d y, θ,..., θ d, θ d+,..., θ D are approximately distributed as pθ d y. Details about the convergence of Gibbs sampling can be found in [4]. In wireless The property that both the posterior and the prior pdfs belong to the same family is termed conjugacy. A conjugate prior is usually chosen such that the posterior pdf has a simple closed-form expression and is easy to generate. Also, the prior pdf is chosen such that it has minimal impact on the posterior pdf. Such priors are known as non-informative in the sense that the information available in the observed data is only minimally affected by the prior external information [6], [4]. That is, with a judicious choice of a non-informative prior and with a reasonable sample size, the chosen prior pdf will only have a minor effect on the posterior pdf and thus the resultant estimate obtained by combining the prior pdf with the pdf of the observed data. As discussed in this section, the prior pdfs used in the paper are extensively adopted in the broad area of Bayesian statistics and, in particular, in wireless communications. Algorithm Gibbs sampling algorithm : Generate initial values θ = [θ,..., θ D ]. : Initialize a to. 3: Obtain θ a + by sampling pθ θ a,..., θ D a, y. 4: Obtain θ a + by sampling pθ θ a +, θ 3 a,..., θ D a, y.. 5: Obtain θ D a + by sampling pθ D θ a +,..., θ D a +, y. 6: Increment a and return to step 3. communications, this technique was used in the development of algorithms for channel estimation [7], equalization [8], and modulation classification [9], among others. In the proposed pre-processor, in addition to h = {h,..., h L } T, σ = {σ,..., σn }, λ = {λ,..., λ N }, and τ, the observed symbol sequence S = {s M+L,..., s K+M } and its corresponding constellation S b are also estimated by the Gibbs sampling algorithm. Let H b be the hypothesis that the modulation scheme that is, the amplitudephase constellation of the received signal is S b. Thus, the unknown parameter vector can be written as {h, σ, λ, τ, H b, S}. To estimate this vector, the marginal posterior pdfs ph y, σ, λ, τ, H b, S, pσ y, h, λ, τ, H b, S, pλ y, h, σ, τ, H b, S, pτ y, h, σ, λ, H b, S, and ph b, S y, h, σ, λ, τ, where y = {y,..., y K } T is the matched filter output vector, are used. The prior pdfs used in this formulation are given in Section III-C and the corresponding marginal posterior pdfs are obtained in Section III-D. But first, in Section III-B, we introduce the concept of superconstellation [6]. B. Superconstellation The pre-processor uses the concept of superconstellation proposed in [6]. Let the superconstellation B be the set with all constellation points of all possible modulation formats; that is, B = B b= S b. Estimates of the modulated symbols {s i }, denoted by {x B i }, are obtained by sampling the discrete posterior pdf of the superconstellation B in each iteration. It should be noted that the timing offset τ is not known by the pre-processor. Therefore, the output of the matched filter is given by and the samples {y k } K k= are correlated due to inter-symbol interference. By taking the pulse shape pt to be zero for t MT, the samples {y k } K k= are a function of the symbols {s i } K+M i= M+L. This truncation is reasonable because the tails of practical pulses, such as the raised cosine, approach zero. Using, the received signal vector y is written as y = X B h + w, 6 where w = {w,..., w K } T and X B is the K L matrix defined as M X B = p mt τx m, 7 m= M

4 4 with x B +m L... x B +m [X m ] K L = x B K+m L... x B K+m Given that w has a Gaussian mixture distribution, using 4, the conditional pdf of the received signal is py h, σ, λ, τ, H b, X = K k= n= N λ n N X B h k, σn, 8 where the notation N X B h k, σ n represents a Gaussian pdf with mean value X B h k and variance σ n, and X B h k is the kth term in the K vector X B h. The conditional pdf given by 8 is necessary to evaluate the posterior pdfs of the unknowns. To simplify its form and reduce the complexity of the Gibbs sampling algorithm, we introduce an indicator variable to represent the mixture component to which w k belongs [4], which is defined as I k = n, if w k belongs to the n th component of the Gaussian mixture, for k =,,..., K. With this definition, the unknown vector becomes {h, σ, λ, τ, I, H b, X}, where I = {I,..., I K }, and the conditional pdf of the received signal y is written as py h, σ, λ, τ, I, H b, X exp y X B h H Λ y X B h, 9 where the kth element of the diagonal matrix Λ = diagσ I,..., σ I K is the variance of the mixture component to which w k belongs. C. Prior pdfs for the unknown parameters The following conjugate and non-informative pdfs are assumed for the different unknown parameters [3], [4], [3]: For the unknown channel h, a Gaussian prior distribution is used. More specifically, ph N, Σ, where is an all-zeros vector with length L and Σ is an L L covariance matrix. As the channel coefficients are assumed to be i.i.d, Σ is a diagonal matrix. The variances of all elements of h are taken to assume large values [6]. The variance of each Gaussian mixture component is assumed to have an Inverse Gamma distribution pσn IGα n, β n, where α n is taken to be. and β n is large [3]. 3 The probability of each Gaussian mixture component is assumed to have a Dirichlet conjugate prior, pλ k Dγ k, with γ n = [3]. It should be noted that P I k = n λ = λ n. 4 The timing offset τ is assumed to be uniformly distributed in the interval [, T. 5 The possible modulation formats of the received signal are assumed to be equally likely. D. Marginal posterior pdfs Because the marginal posterior pdfs of the unknowns {h, σ, λ, τ, H b, X} are a function of the indicator variable vector I, the proposed Gibbs sampling algorithm is implemented in two steps as follows. Step : The marginal posterior probability of the indicator variable I k I can be written as P I k = n I [ k], y, h, σ, λ, τ, H b, X = pi, h, σ, λ, τ, H b, X y/ pi [ k], h, σ, λ, τ, H b, X y }{{} not a function of I k pi, h, σ, λ, τ, H b, X y py I, h, σ, λ, τ, H b, XpI k = n, I [ k], h, σ, λ, τ, H b, X py I, h, σ, λ, τ, H b, XP I k = n I [ k], h, σ, λ, τ, H b, X pi [ k], h, σ, λ, τ, H b, X }{{} not a function of I k where I [ k] = [I,..., I k, I k+,..., I K ] and I = [I k, I [ k] ]. Given that I k is only dependent on the Gaussian mixture proportion λ k, we have P I k = n I [ k], y, h, σ, λ, τ, H b, X py I, h, σ, λ, τ, H b, XP I k = n λ σn exp σn ȳ k P I k = n λ, where ȳ = y X B h. The estimate of I k is obtained by sampling the normalized posterior probability given by. The estimate of Λ, defined in 9, is updated with each received symbol y k. Step : Using the indicator variable vector estimate, the marginal posterior pdfs of the unknown parameters h, σ, λ, τ, H b, and X are obtained as described next. Gaussian mixture proportions: The prior Dirichlet pdf used for the mixture proportion is a conjugate prior of the multinomial distribution [4]. Therefore, pλ y, h, σ, τ, I, H b, X = pλ I P I λpλ = Dγ + k,..., γ N + k N, where k n is the number of samples of the received signal y that belong to the nth mixture component which is estimated in step and N n= k n = K. Channel gains: Assuming that the channel gains are independent, we have [6] ph y, σ, λ, τ, I, H b, X = ph, σ, λ, τ, I, H b, X y/ pσ, λ, τ, I, H b, X y }{{} not a function of h exp h ĥh ˆΣ h ĥ, where ˆΣ = X B H ˆΛ X B + Σ and ĥ = ˆΣ X B H ˆΛ y. 3 Gaussian mixture variances: The posterior pdf of the Gaussian mixture variances can be shown to be given by an Inverse Gamma pdf [4]. Therefore, pσ n y, h, λ, τ, I, H b, X IGα n, β n, 3

5 5 where α n = α n + k n +, β n = [ K k= ȳ k. {Ik =n} + β n ], and {Ik =n} is the indicator function such that {Ik =n} = if I k = n and {Ik =n} = if I k n. The variable ȳ k is the kth element of the vector ȳ = y X B h. 4 Timing offset: Given the assumption that the timing offset τ is uniformly distributed in [, T, pτ y, h, σ, λ, I, H b, X can be written as pτ y, h, σ, λ, I, H b, X = ph, σ, λ, τ, I, H b, X y/ ph, σ, λ, I, H b, X }{{} not a function of τ ph, σ, λ, τ, I, H b, X y py h, σ, λ, τ, I, H b, Xpτ exp ȳ H ˆΛ ȳ T I T, 4 where I T is an indicator function equal to when τ < T and otherwise. Unfortunately, to the best of the authors knowledge, it is not possible to generate random variables with pdf given by 4. For this reason, we propose to use a grid-based approximation known as griddy Gibbs sampling to estimate τ. Griddy Gibbs sampling is an approach to generate random variables from pdfs that cannot be readily expressed in a standard format [33], [34]. In this algorithm, for the problem at hand, a grid of points in the interval [, T is first selected. If a resolution factor of OS is used, the points selected are T {, OS, T OS,..., OS T OS }. The posterior pdf given by 4 is then evaluated at all grid points and an approximate inverse pdf is evaluated using the obtained values as described in [33]. An estimate for the timing offset τ is then obtained by using the inverse transformation technique [4]. It is worth noting that the Metropolis-Hastings algorithm is not effective in this case because its convergence is significantly affected when implemented inside a Gibbs sampling algorithm [35]. 5 Signal constellation: In this case, we have ph b y, h, σ, λ, τ, I, X = ph b X = p H b {x B i } K+M i= M+L c b, 5 where c b is the proportion of the symbols in {x B i }K+M i= M+L that belong to H b, and is evaluated for all modulations that constitute the superconstellation [36]. If all elements of X belong to H b, then the conditional probability ph b X is equal to for X H b and to for X / H b. If the current estimate of the received signal s modulation format is incorrect for example, due to a wrong estimate in one of the Gibbs sampling iterations during the burn-in period, the probability of sampling from a modulation format other than the current estimate is zero. Therefore, the obtained Markov chain will be rendered reducible as the possibility that the received signal has a different modulation format will not be explored. Since irreducibility is a necessary property for the Markov chain to converge to a stationary distribution [4], this problem can lead to a failure in the estimation process. To avoid this problem, we propose an alternative sampling procedure in which at least c th % of the generated samples at every Gibbs sampling iteration correspond to each possible modulation scheme. For example, if c th = % and the possible modulation schemes are BPSK and QPSK, at least % of the generated samples will correspond to BPSK and at least % of the generated samples will correspond to QPSK. The modified sampling procedure is as follows: Determine the posterior probability c b of each modulation scheme by using 5. Without loss of generality, assume that c c... c B. 3 Find the minimum index b min such that c bmin c th. 4 Assign c = c =... = c bmin = c th. 5 For all b b min, the new probabilities ĉ b are calculated as ĉ b = c b b min c th. 6 B c b b=b min 6 Modulated symbols: Let X = [X [ i], x B i ]. Using this definition, the posterior pdf of the symbol x B i can be written as P x B i y, h, σ, λ, τ, I, H b, X [ i] = py h, σ, λ, τ, I, H b P x B i h, σ, λ, τ, I, H b, X [ i] py h, σ, λ, τ, I, H b, X [ i] }{{} not a function of x B i exp ȳ H ˆΛ ȳ P x B i h, σ, λ, τ, I, H b, X [ i]. 7 In 7, the conditional probability of each symbol of the superconstellation is given by P x B i = a j b h, σ, λ, τ, I, H b, X [ i] = ĉ b H b, 8 where a j b is the jth symbol of the constellation H b. Using 7 and 8, the normalized posterior probability of each superconstellation symbol is given by P x B i = a j b h, y, σ, λ, τ, I, H b, X [ i] exp ȳ H ˆΛ ȳ ĉ b H b. 9 The ith symbol x B i is estimated by sampling 9. The estimate of the ith symbol is then used in the estimation of the subsequent symbols, and this procedure is repeated for all symbols. As a last comment on the estimation of the modulated symbols, other procedures could have been used. For example, multiple Gibbs sampling chains could have been run in parallel, one for each modulation scheme. However, the complexity of this procedure might become prohibitive if the number of possible modulation schemes is large. A reversible-jump MCMC algorithm could also have been considered, but the number of tuning parameters and distributions used in this procedure would also be relatively large [35].

6 6 Algorithm Proposed Gibbs sampling algorithm : Obtain Ia+ by sampling using y and the following estimates from the previous iteration: Xa, ha, σ a, λa, and τa. : Obtain λa + by sampling using y, Xa, ha, σ a, τa, and Ia +. 3: Obtain σ a + by sampling 3 using y, Xa, ha, λa +, τa, and Ia +. 4: Obtain ha + by sampling using y, Xa, σ a +, λa +, τa, and Ia +. 5: Obtain τa + by sampling 4 using y, Xa, σ a +, λa +, ha +, and Ia +. 6: Obtain Xa + by sampling 6 and 9 using y, σ a +, λa +, ha +, Ia +, and τa +. 7: Increment a and repeat the algorithm until convergence is achieved. E. Summary and a note on complexity Based on the derivation presented in this section, the proposed Gibbs sampling algorithm is summarized in Algorithm. In Algorithm, Xa, Ia, ha, σ a, λa, and τa denote the samples of X, I, h, σ, λ, and τ at the ath iteration, respectively. The initial values of h, σ, λ, and τ are randomly generated using their prior distributions. After the algorithm converges, the obtained samples are used to calculate the MMSE estimates of the unknown parameters. The complexity of the proposed algorithm is approximately OK 3 per iteration, and is dominated by the matrix inversion in. The complexity of generating a random variable with a given posterior pdf is relatively small when compared with matrix inversion, for example. For example, the complexity of the Cholesky decomposition a common approach to generate complex Gaussian variables is approximately OL 3. It is expected that in typical applications, the number of observed samples K would be larger than the number of multipath components L. Furthermore, it should be noted that the complexity of the proposed algorithm is much lower than that of a maximum likelihood estimator, which is exponential in K [6]. IV. MODULATION CLASSIFICATION Using the estimates of the different channel and noise distribution parameters obtained by the pre-processor, the signal can be reliably classified by using an appropriate modulation classification algorithm in a second processing stage. For example, a feature-based classifier which exploits one or more modulation dependent features of the signal, such as cumulants see, among others, [], [], and [37], could be employed. Alternatively, a likelihood-based classifier, which is optimal in the Bayesian sense as it minimizes the probability of classification error [3], could also be used. It is important to note that while feature-based classifiers are generally easier to implement than the likelihood-based one, they are sub-optimal. To quantify the performance of the proposed pre-processor, we will present in Section V numerical results for the probability of correct classification obtained when the proposed preprocessor is used in conjunction with two examples of classification algorithms. The first one, referred to as numerical classifier, is the actual Gibbs sampling algorithm, a numerical Bayesian technique, formulated in Section III. As previously discussed, as part of the iterative process given by Algorithm, the observed symbol sequence S and its corresponding constellation S b are estimated, as shown in sub-sections III.D.6 and III.D.5, respectively. Recall that this process corresponds to a numerical implementation of the MMSE estimator and, therefore, it is an approximated implementation due to numerical errors of a well-established procedure. The second modulation classification algorithm to be considered is an approximated version of the likelihood-based classifier, and is referred to as approximated likelihood classifier. Maximum-likelihood modulation classification is a multiple hypothesis testing problem in which the hypothesis that maximizes the log-likelihood of the received signal {y k } K k= is chosen [38], Ĥ = arg max log py,... y K H b. H b For the system model defined in Section II, p y,..., y K H b is a function of the modulated symbols, τ, {h l } L l=, {λ n} N n=, and {σ n} N n=. Because the likelihood function is not completely known, following a procedure known as hybrid likelihood ratio test HLRT approach [3], in the approximated likelihood classifier, the unknowns are first estimated by the pre-processor and then the signal is classified using a maximum-likelihood procedure. Because the channel and noise distribution parameters are estimated by the pre-processor, the unknowns except for the modulated symbols are substituted into the likelihood function as if they were known. Therefore, using 3, 4, and the Total Probability Theorem, py,... y K H b is given by, where H b is the modulation order of H b and the averaging summation is performed over all possible values the symbols {s i } K i= L may assume. Note that the evaluation of requires KN + H b K+L additions and K multiplications. Since it can be expected that K >> L in practical applications, the implementation complexity of the optimal likelihood-based classifier for frequency-selective non-gaussian channels is high. As such, we are forced to consider approximated solutions, such as the numerical classifier and the approximated likelihood classifier. To reduce the computational complexity of the likelihood function, in the formulation of the approximated likelihood classifier, we make the assumption that the received signal is sampled with an interval that is greater than the delay spread of the channel. Specifically, we take the sampling interval to be LT. In this case, each sample {y k L } K k= is a function of different sets of symbols. For example, y is a function of the symbols {s L, s L 3,..., s } and y L+ is a function of the symbols {s, s 3,..., s L+ }. Thus, the samples {y, y L+,..., y K L+ } are statistically independent. With this approximation, py, y L+,..., y K L+ H b is given

7 7 p y, y,..., y K H b = H b K+L... s L s K K N λ n πσ k= n= n exp y k L l= h ls k l σn py, y L+,..., y K L+ H b = K k= H b L... s k L+ N λ n πσ s k L+ n= n exp y k L+ L l= h ls k L+ l σn Ĥ = arg max H b { K log H b L... s k L+ k= s k L+ n= N ˆλ n πˆσ n exp y k L+ L l= ĥls k L+ l ˆσ n } 3 by 3. In, KN + H b L additions and K multiplications are necessary to evaluate the likelihood function, which is significantly less in comparison to recall that K >> L. However, this is possible only at the expense of an increased observation window size KL in when compared to K in. Thus, there is a tradeoff between the computational complexity of the likelihood-based approach and the length of the observation window which can be chosen based on the requirements. Substituting into, the approximated likelihood classifier is given by 3. It should be noted that because the unknowns are substituted into the likelihood function as if they were known, h l, {λ n } N { } n=, and σ N n were replaced with their corresponding estimates n= } N {ĥl} L l= {ˆλn, and n=, {ˆσ n} N, respectively, in 3. For n= the same reason, the matched filter output given by 3, which corresponds to the case in which τ is known, is used in 3. The matched filter output given by, which corresponds to the case when inter-symbol interference exists due to time offset, was used in the formulation of the pre-processing stage in Section III. As a last comment on the two modulation classification approaches considered here, we would like to emphasize that the signal is sampled with an interval that is greater than the delay spread of the channel approximation is made only in the development of the approximated likelihood classifier. This approximation is not used in the pre-processor formulation given in Section III nor in the derivation of the numerical classifier. V. NUMERICAL RESULTS In the results that follow, the complex channel gains {h l } L l= are assumed to be independent zero-mean circularly symmetric Gaussian random variables with L l= E[ h l ] equal to. The multipath intensity profile is assumed to be exponential. The pulse pt is taken to be a square-root raisedcosine with 65% excess bandwidth and is truncated to have a 3 It should be noted that the likelihood function does not reduce to that corresponding to a flat-fading channel with the approximation used sampling interval to be LT. In a flat-fading channel, y would be a function of s only, y would be a function of s only, and so on. In the proposed formulation, due to multipath propagation, each of the samples used in the likelihood function depends on L transmitted symbols. Samples of h, h, τ, λ imagh Timing Offset realh Impulsive fraction realh imagh Iterations Fig.. Realization of the samples obtained in the estimation process of h, h, τ, and λ. The dotted lines represent the true values of the parameters being estimated and the bold lines are the samples obtained in each Gibbs sampling iteration. time duration of 4T. Therefore, the truncation parameter M is equal to. Also, the number of terms in the Gaussian mixture distribution is taken to be N =. This case is often used in the literature as a model for impulsive noise see, for example, [6], [7], and [4]. In this model, the first and second terms of the mixture represent the thermal noise component with variance σ and proportion λ and the impulsive noise component with variance σ σ and proportion λ = λ, respectively. The values of λ and of the ratio of the variances σ/σ are set to. and, respectively. We define the received SNR as the ratio of the average signal power and the thermal noise power. Unless otherwise noted, in the results that follows, the resolution factor OS and the parameter c th used in the preprocessing stage are equal to and., respectively. Also, the numbers of symbols observed and of iterations used by the pre-processing stage are 5 and 3, respectively. A. Pre-processing stage The performance of the proposed pre-processing stage is shown in Figs. -7. In these figures, the channel length is

8 8 Correlation among the samples of σ and of the real part of h σ realh Lag Samples of τ True Value OS = 3 OS = OS = Iterations Fig.. Correlation among the samples of σ square and among the samples of the real part of h circle after the burn-in period. taken to be, the received symbols are BPSK modulated, and the superconstellation consists of the BPSK and QPSK constellations. Higher-order modulation schemes and larger superconstellations are considered later in this section. To demonstrate the convergence of the proposed Gibbs sampling algorithm, a given realization of the samples obtained in the estimation of different channel and noise distribution parameters is shown in Fig.. In our simulations, we observed that the burn-in period that is, the number of iterations required for convergence is approximately 3 for moderate to high SNR values and around 7 for low SNR values. For this reason, we take the burn in period to be iterations. It is important to note that, as discussed in [4] and [35], among others, the convergence of the Gibbs sampling algorithm can be ensured by using a large number of iterations 4. It is known in the Gibbs sampling literature that if the samples generated in the estimation of different parameters are correlated, the obtained estimates will likely be biased [4]. For this reason, as an example, we plot in Fig. the correlation of the thermal noise variance σ samples and of the real part of h samples after the burn-in period. It is observed in this figure that the correlation among samples taken with a lag of approximately 5 iterations is very small. This result is also valid for the other parameters considered. Therefore, in a process known as thinning [4], [4], we take the MMSE estimate of a given parameter to be the mean of samples that are generated every 5th iteration after the burn-in period. Fig. 3 shows the impact of using different resolution factor OS values in the estimation of the timing offset τ. It is seen in this figure that, in this particular run of the Gibbs sampling algorithm, the algorithm converges when OS is 5 and, and does not converge to the true parameter value when OS is set to 3. While using a larger resolution factor increases 4 Alternatively, several Gibbs sampling chains with random initialization could be run in parallel and the convergence assessed by using a heuristic potential scale reduction factor PSRF [39]. As previously discussed, the complexity of running multiple Gibbs sampling chains in parallel could be prohibitive. Fig. 3. Realization of the samples obtained in the estimation process of τ for different resolution factor OS values. The dotted line represents the true value of τ. OS is set to 3 dashed, 5 dash-dot and bold. Normalized estimation error variance σ σ SNR db Fig. 4. Average normalized variance of the error in the estimation of σ and σ. Number of observed symbols equal to circle, 3 square, and 5 diamond. The average normalized variance in the estimate ˆX of X is defined as V ar[x ˆX]/X. the accuracy of the estimation process of τ, it also increases the overall estimator complexity. Furthermore, using small resolution factor values can negatively impact the estimation of τ and, consequently, the performance of the pre-processing stage and that of the modulation classifier. Thus, the value of OS should be carefully chosen depending on the constraints of the application scenario being considered. As previously stated, we set OS to. The average normalized variance of the error in the estimation of the thermal and impulsive noise variances σ and σ, respectively is shown in Fig. 4 for different observation lengths. As expected, it is seen that the performance of the proposed algorithm improves with increasing SNR and/or as the number of observed symbols increases. It is also seen that the variance of the estimates of σ, for a given SNR value and

9 9 Normalized mean squared error NMSE Impulsive Noise σ Thermal Noise σ BPSK QPSK 8 PSK 6 QAM 3 QAM 64 QAM Samples of h, h, h realh 3 realh SNR db Fig. 5. Normalized MSE in the estimation of σ and σ when the modulation scheme of the received symbols is known and is either BPSK, QPSK, 8 PSK, 6 QAM, 3 QAM, or 64 QAM. The average normalized MSE in the estimate ˆX of X is defined as E[X ˆX ]/X. observation length, is always greater than the variance of the estimates of σ. This is because, in the considered simulation scenario, impulsive noise is present in only % λ =. of the observed symbols, and thus the number of actual samples used in the estimation of σ is smaller on average than in the estimation of σ. In order to evaluate the performance of the proposed preprocessing stage when the observed symbols belong to different constellations, we show in Fig. 5 the normalized meansquared error MSE in the estimation of σ and σ when the modulation scheme of the received symbols is known and is either BPSK, QPSK, 8 PSK, 6 QAM, 3 QAM, or 64 QAM. It can be seen that the performance of the proposed estimator is approximately the same for all modulation schemes in the estimation of σ. Recall that the estimates of σ are noisier due to the low probability of occurrence of impulsive noise. The gap in the estimator performance for the different modulation schemes is more noticeable, but still relatively small, in the estimation of σ. Similar behavior specifically, comparable performance for different modulation schemes is seen in the estimation of all other channel and noise distribution parameters. The effects of over-estimating the number of channel taps L and the number of terms in the Gaussian mixture N can be seen in Figs. 6 and 7, respectively, for the case when the true values of L and N are two but they are taken to be equal to three. As seen in these figures, the estimates given by the Gibbs sampling algorithm tend to zero for the multipath and the noise components that do not exist. As a result, over-estimating the values of L and N only leads to a small performance loss in the classification process. This result is presented in the next sub-section. Additionally, we also show in the next sub-section that the performance loss of the classifier can be noticeable when the values of L and/or N are under-estimated. In order to further evaluate the performance of the proposed.3 realh Iterations Fig. 6. Realization of the samples obtained in the estimation process of the real part of h, h, and h 3. The dotted lines represent the true values of the parameters being estimated and the bold lines are the samples obtained in each iteration. Samples of σ, σ, σ3.3.. σ Iterations Fig. 7. Realization of the samples obtained in the estimation process of σ, σ, and σ 3. The dotted lines represent the true values of the parameters being estimated and the bold lines are the samples obtained in each iteration. pre-processor, we now present numerical results for the probability of correct classification obtained when the pre-processor is used in conjunction with two examples of classification algorithms. We consider the numerical classifier in sub-section V.B, and the approximated likelihood classifier in sub-section V.C. B. Numerical classifier The probability of correct classification of the numerical classifier is shown in Fig. 8 for different numbers of observed symbols. For reference, the performance of a clairvoyant Gibbs sampling-based classifier assumed to be symbol synchronous with the received signal and to have perfect knowledge of the channel state and noise distribution is also presented in this σ σ 3

10 Probability of Correct Classification Probability of correct classification.3 Clairvoyant Classifier 75 symbols symbols 5 symbols SNR db Fig. 8. Probability of correct classification of the numerical classifier for different numbers of observed symbols 75,, and 5. Clairvoyant classifier uses 5 symbols. Set of possible modulation schemes: BPSK, QPSK, 8 PSK, and 6 QAM. L = 3..3 c th =. c th =. c th = SNR db Fig. 9. Probability of correct classification of the numerical classifier for different values of c th. Number of observed symbols: 5. Set of possible modulation schemes: BPSK, QPSK, 8 PSK, and 6 QAM. L =. figure. It is seen that the classifier performance improves as the number of observed symbols increases, as expected. It can also be observed that the numerical classifiers performance closely approaches that of the clairvoyant classifier as the number of observed symbols increases. In Fig. 9, the performance of the numerical classifier is shown for different values of c th used in the estimation of the signal constellation by the Gibbs sampling algorithm. It is seen that the performance of the classifier when c th is nonzero. or. is better than the conventional approach proposed in [6] which corresponds to c th equal to zero. This is because, as previously discussed, it is possible that the data symbols could be incorrectly estimated and consequently the Markov chain would go into a degenerate state when the threshold is zero. It is also observed that the probability of correct classification is approximately the same when c th is equal to. and.. This is because c th is used only to prevent the Markov chain from becoming irreducible and does not have an impact on the limiting distribution of the chain. The resultant performance loss of the numerical classifier due to the over- or under-estimation of the values of L and N is seen in Fig.. As previously discussed, over-estimating the values of L and N leads to a small performance loss since the same set of symbols would be used for estimating a larger number of parameters, which are not necessary. At the same time, the performance loss can be noticeable when the values of these parameters are under-estimated. In the example shown, the performance resultant of the case when an algorithm designed for frequency-flat channels L = is used in a frequency-selective channel L = is seen to be quite poor, as expected. Similarly, when N is under-estimated, significant performance loss is observed. Note that the example used in this figure corresponds to the case when a classifier designed assuming Gaussian noise N = is used in an Probability of correct classification N=4, L= N=, L=4 N=3, L= N=, L= N=, L=3 N=, L= N=, L= SNR db Fig.. Probability of correct classification of the numerical classifier for the case when the values of L or N are over- or under-estimated. The correct values of L and N are. Number of observed symbols: 75. Set of possible modulation schemes: BPSK and QPSK. impulsive noise N = environment. C. Approximated likelihood classifier The performance of the approximated likelihood classifier obtained by using the proposed pre-processor is shown in Figs. -3. The clairvoyant classifier used in this case is given by 3 with the true values of h l, {λ n } N n=, and { σ n} N n= used. The performance of the approximated likelihood classifier is shown in Fig. for different numbers of symbols used for estimation and classification. Similar to the behavior seen

11 Probability of correct classification.3.. Clairvoyant classifier, Classification Sym Clairvoyant classifier, Classification 75 Sym Estimation 5 Sym, Classification Sym Estimation 5 Sym, Classification 75 Sym Estimation 3 Sym, Classification Sym Estimation 3 Sym, Classification 75 Sym SNR db Probability of correct classification.3 Clairvoyant classifier, Classification Sym Estimation Sym, Classification Sym Estimation 5 Sym, Classification Sym SNR db Fig.. Probability of correct classification of the approximated likelihood classifier for different numbers of symbols used by the pre-processing stage 3 or 5 and by the classifier K=75 or K=. L = 3. in Fig. 8 for the numerical classifier, it is seen that the performance of the approximated likelihood classifier improves as the number of symbols used for estimation and/or classification increases, as expected. It is also seen that the classification algorithm s performance closely approaches that of the clairvoyant classifier as the number of symbols used increases. The trends seen in Fig. also hold true when the set of possible modulation schemes increases. For example, the performance of the approximated likelihood classifier for the case when the set of possible modulation schemes is BPSK, QPSK, 8 PSK, 6 QAM, 3 QAM, and 64 QAM is shown in Fig.. The proposed pre-processing stage can thus be used in a wide range of scenarios. Lastly, the performance of the approximated likelihood classifier for different values of β roll-off factor of the raised cosine pulse shape is shown in Fig. 3. It is seen that the classifier performance is approximately the same for the different values of β used in the simulation. This fact implies that, for the considered simulation scenario, the effects of intersymbol interference resultant from errors in the sampling time offset estimation is approximately the same for the different values of β. In cases of severe inter-symbol interference such as when the number of symbols available for synchronization is small, it is expected that the classifier performance would be more sensitive to the value of β. D. Carrier frequency offset The approach taken in this paper for the derivation of a pre-processor for modulation classification applications in frequency-selective non-gaussian channels can be extended to other scenarios. The Gibbs sampling-based pre-processing stage derived in Section III and the HLRT formulation presented in Section IV allow, for example, for other unknowns to be included in the system model and for other channel models and noise distributions to be considered. Fig.. Probability of correct classification of the approximated likelihood classifier for different numbers of symbols used by the pre-processing stage 5 or and a fixed number of symbols used for classification K=. Set of possible modulation schemes: BPSK, QPSK, 8 PSK, 6 QAM, 3 QAM, and 64 QAM. L = 3. Probability of correct classification β = 5 β = β = SNR db Fig. 3. Probability of correct classification of the approximated likelihood classifier for different values of β. Number of symbols used for both estimation and classification is equal to. Set of possible modulation schemes: BPSK, QPSK, 8 PSK, and 6 QAM. L =. A case of great practical interest that can be addressed by extending the proposed pre-processor is when the received symbols suffer phase rotation due to the presence of a residual carrier frequency offset. In this scenario, the signal at the output of the matched filter is given by y k = e jπkδ f m l h ls k l+m p mt τ + w k, where δ f is the normalized carrier frequency offset. In addition to the unknowns considered in this paper namely, τ, {h l } L l=, {λ n} N n=, and {σ n} N n=, δ f could be incorporated as an unknown and be estimated at the preprocessing stage. This can be accomplished by modeling δ f as a uniform random variable whose domain is chosen such that