Modulation Classification based on Modified Kolmogorov-Smirnov Test Ali Waqar Azim, Syed Safwan Khalid, Shafayat Abrar ENSIMAG, Institut Polytechnique de Grenoble, 38406, Grenoble, France Email: ali-waqar.azim@ensimag.grenoble-inp.fr Electrical Engineering Department, COMSATS Institute of Information Technology Islamabad, 44000, Pakistan Email: {aliwaqarazim, safwan khalid, sabrar}@comsats.edu.pk Abstract A modulation classification method based on modified Kolmogorov-Smirnov (K-S) test is proposed. Unlike modulation classification based on K-S test, the proposed method evaluates the sum of the difference between the empirical distribution obtained from the features extracted from received signals and the hypothesized distribution for each modulation candidate in pool. The results have been evaluated using Quadrature Amplitude Modulation (QAM) for AWGN channel using Monte Carlo simulations. Extensive simulation results demonstrate that the proposed modulation classification method based on modified K-S test offers superior performance as compared to the method based on K-S test. I. INTRODUCTION Modulation Classification is a signal processing technique used to detect the modulation type of a received signal that has been corrupted by noise and other possible interferences. Once the modulation type is correctly identified, other operations, such as signal demodulation and information extraction, can be subsequently performed. The identification of the modulation type becomes a challenging task in the absence of a priori knowledge of the transmitted signal. Moreover the presence of a time-varying, frequency selective multi-path environment can further complicate the problem of identification []. Modulation classification plays a key role in many applications which include civilian applications such as network traffic administration, intelligent modems, software defined radio, frequency spectrum monitoring and management, interference identification etc., and military applications such as electronic surveillance, electronic warfare and threat analysis [, 2]. Modulation classification techniques can be divided into two broad categories; likelihood-based methods and the feature-based methods [, 3]. Likelihood-based methods are based on computing the likelihood ratio of the received signals for each modulation class and then comparing the ratio against a certain threshold for decision making. Generally the likelihood-based methods are based on Generalized Likelihood Ratio Test (GLRT), Average Likelihood Ratio Test (ALRT) [4] and Maximum Likelihood (ML) criterion etc. Solutions obtained by likelihood-based methods provide an optimal solution i.e. they provide minimum probability of false classification and maximum probability of correct classification []. However, these methods suffer from computational complexity and their performance degrades drastically in the presence of various channel impairments [4]. On the other hand, in the feature-based methods, the classifiers make decisions based on several features extracted from the received signals. The methods based on featurebased approach are computationally less intense and easy to implement, however on the negative side, the classification performance methods based on feature based approach are suboptimal [, 4]. The features used by feature-based methods can be instantaneous time domain features such as constellation shape, zero-crossing, instantaneous amplitude, phase or frequency, transform based features such as Fourier transform, wavelet transform and statistical features such as higher-order statistics and cyclostationarity. In this paper, we propose a modulation classification method based on modified version of Kolmogorov-Smirnov (K-S) Goodness of Fit (GoF) test. The GoF tests or statistical methods based on empirical distribution measure the compatibility of a random sample with a hypothesized distribution. In order to perform modulation classification using modified K-S test the empirical cumulative distribution of a certain feature (i.e. phase, magnitude etc.) of the received signal is computed using N samples. It is then compared with the hypothesized distribution of each candidate modulation scheme. The decision is made in favor of the candidate scheme which provides the minimum of sum of distances between the hypothesized distribution and the empirical distribution, whereas, the classifier proposed by Wang et al. in [4] and the one presented in [5] takes decision in favor of modulation candidate which gives minimum of maximum distance between the hypothesized distribution and empirical distribution. Through extensive simulations we proved that the proposed modulation classification method based on modified K-S test outperforms modulation classification method based on K-S test proposed in [4] and [5]. The remainder of the paper is organized as follows. General system model and a brief overview of GoF testing is provided in Section II. Section III provides an overview on classification of QAM signals, the section further explains the types of classifiers that can be opted for classification of QAM signals. An overview of modulation classification method based on K-S test is presented in Section IV. Section V explains the proposed modulation classification method along with formulations. Simulation results and comparison is provided in Section VI. Finally, based on the results obtained in Section VI, conclusions are drawn in Section VII.
II. SYSTEM MODEL AND GOODNESS OF FIT TESTING A. System Model Following [4], in our system model we consider a discrete time additive white Gaussian noise (AWGN) channel. The observed symbols at the receiver are represented as: y n = x n +w n n =,, N () where n represents the total number of observations made at the receiver, x n represents the complex-valued transmitted symbol at time n, y n represents complex-valued received signal corrupted by complex noise w n at time n. It is further assumed that the noise samples are independent and identically distributed (i.i.d) and follow complex Gaussian distribution CN(0,σ 2 ); i.e. the real and imaginary components of w n are independent and have same Gaussian distribution N(0, σ2 2 ). It is considered that the complex-valued i.i.d transmitted symbols {x n } N n= are drawn from an unknown constellation set M {M,M 2,,M k }. The problem of modulation classification is to determine the constellation M k of the transmitted signal among all possible candidate constellations based on received symbols {y n } N n=. B. Goodness of Fit Testing The GoF testing is a statistical testing process to determine the distribution of data. GoF tests are used to determine how well a selected distribution fits to a given data. The test statistic for GoF tests can be applied to certain sequence of features {z n } N n= extracted from received signal samples {y n} N n=, which can be magnitude, phase etc. In order to compute the empirical distribution, consider {z n } N n= containing N samples of a feature of the received signal organized in order. Let F(z) denote the empirical distribution obtained from the sequence of features extracted from received symbols. F(z) can be represented as: F(z) = {n : z n z, n N} N where for any set U, U denotes the cardinality of U. The above equation can also be written as: F(z) N I (zn z) (2) n= where I (.) is the indicator function, which equals to one if the input is true and equals to zero otherwise. Lets consider that hypothesized distribution is represented as S k (z). Now we define modulation classification problem as a hypothesis testing problem with a null hypothesis H 0, and the general alternative hypothesis H i.e. H 0 : H : F(z) = S k (z) F(z) S k (z) The intuitive understanding of the hypothesis H 0 is that {z n } N n= is an i.i.d sequence generated by the hypothesized distribution function F(z) against alternative H which states that {z n } N n= is not an i.i.d sequence generated by the hypothesized distribution function F(z). Under the null hypothesis, F(z) will be close to S k (z) when N is large enough. Different goodness of fit tests and their modifications have been proposed in mathematical statistics based on the definition of distances between the two distributions F(z) and S k (z). The distance between the two distributions indicates the fit between the empirical distribution and the hypothesized distribution. Extensively used goodness of fit tests include Kolmogorov-Smirnov (K-S) test, Cramér Von Mises (CVM) test and Anderson-Darling (AD) test. For the scope of this paper we will focus our attention on just K-S test. III. CLASSIFICATION OF QAM SIGNALS We considered that the problem of modulation classification is to distinguish between quadrature amplitude modulation (QAM) signals with 4-QAM, 6-QAM and 64-QAM signals in a pool as possible modulation candidates. Without loss of generality, we considered constellations with unit variance obtained by normalizing the signal constellations. The constellation points for modulation candidates are given by M 4 QAM = { 2 (a + bj) a,b =,}, M 6 QAM = { 0 (a + bj) a,b = 3,,,3} and M 64 QAM = { 42 (a+bj) a,b = 7, 5, 3,,,3,5,7}, where j =. There can be number of possible ways to extract a sequence of signal features {z n } N n= which can be used directly for modulation classification. The ones presented in [4] are discussed below: A. Magnitude based Classifier In magnitude based classifier, the sequence of signal features {z n } N n= is obtained by taking the magnitude of the received signals {y n } N n=. z n = y n = (R{y n }) 2 +(I{y n }) 2 n =,, N (3) The empirical distribution of z = x + w, where w CN(0,σ 2 ) is given by ( 2 x F(z) = Q σ, 2z ) z R + (4) σ where Q(a, b) is the Marcum Q-function. Now considering all the signal points in constellation as equiprobable, the hypothesized distribution of z n = y n under modulation candidate M k is given by [4]: S k (z) = M k x M k Q z R +, k =,,K ( 2 x σ, 2z ) σ where K is the total number of modulation candidates in pool. B. Phase based Classifier Another possible classifier for QAM signals is phase based classifier. The phase of QAM signals also contains information about the type of modulation of the received signals, thus we can use the phase of the received signals as a feature to compute the decision statistic. Mathematically phase of the received signals is evaluated as: z n = (y n ) = tan ( I{yn } R{y n } ) (5) n =,, N (6)
C. Quadrature based classifier In square QAM, the real and imaginary parts are of a distortion free constellation are independent and identically distributed. Similarly, the real and imaginary parts of complexvalued noise w n are also independent of each other. So it is feasible to use both the in-phase and quadrature components of the received signal simultaneously, as features to calculate the decision statistic. As discussed in [4], we can form a sequence of 2N samples for decision statistic form N received signal samples. Mathematically, this can be represented as: z 2n = R{y n }, z 2n = I{y n }, n =,,N (7) As the real and imaginary components of are independent and have same Gaussian distribution, we can say that z n N(0, σ2 2 ). Thus the hypothesized distribution for each modulation candidate M k can be evaluated as: ( ) 2(z x) S k (z) = Q g (8) Mk σ x R{M k } z R +, k =,,K where K is the total number of modulation candidates, Q g (a) is Gaussian Q function and R{M k } represents the set of real components of signal points in modulation candidate M k. In this work we have employed the magnitude based classifier during our simulations. IV. K-S TEST BASED MODULATION CLASSIFICATION Wang et al. in [4] proposed a modulation classification method based on K-S test, the same results were also reproduced in [5]. In modulation classification method based on K-S test, the distance between the hypothesized distribution S k (z) for each modulation candidate M k and the empirical distribution F(z) of i.i.d data samples {z n } N n= is computed and the modulation type of the candidate which gives the minimum distance obtained by the maximum difference between the hypothesized distribution and the empirical distribution is selected form the pool of candidates as the modulation of the received signals. In order to perform modulation classification using K-S test, firstly a sequence of features {z n } N n= is obtained from the received signals {y n } N n= using any of the classifier described in previous section, however in our case we used the magnitude based classifier. Secondly, the empirical distribution is evaluated from the received data samples using the following equation: F(z) N I (zn z) (9) n= where I (.) is the indicator function. Thirdly, for each modulation candidate M k we obtain the hypothesized distribution Ŝ k (z). The test statistic for K-S test which computes the maximum difference between the two distribution is given as: D sup F(z) S k (z) (0) z R where sup is the supremum of the set of distances, and D is the distance obtained. When the hypothesized distribution S k is not available, the hypothesized distribution Ŝk is obtained by an i.i.d sequence {ξ n } N0 n= drawn from S k. Mathematically this can be represented as: Ŝ k (ξ) N 0 I (ξn ξ) () N 0 n= So, practically the following test statistic for K-S test is computed: D = max F(z) Ŝk(z) (2) n N The decision is made in favor of that modulation candidate that provides the minimum distance D among all the candidates is the pool. Thus the decision rule of modulation detection based on K-S test is following: K = arg min D k (3) n N where D k denotes the distance evaluated for each modulation candidate M k. In [4, 5], the authors also associate a significance level α k = P(D > D k M k) with each K-S statistic D k. The significance level can be evaluated using the following equation: ([ ] ) N α = P(D > D 0. ) = Q +0.2+ D N with Q(x) 2 ( ) m e 2m2 x 2 m= (4) K-S test is used to test the null hypothesis i.e. H 0 : F(z) = S k (z), it is important to note that the null hypothesis is rejected if at a significance level α if α = P(D > D ) < α. V. PROPOSED MODULATION CLASSIFICATION BASED ON MODIFIED K-S TEST The test statistic for proposed modulation classification method is presented in [6]. Proposed method is similar to K-S based method in a sense that it also computes the distance between the hypothesized distribution S k (z) for each modulation candidate M k and the empirical distribution F(z) obtained through a sequence of i.i.d data samples {z n } N n=. However, unlike the modulation classification method based on K-S test which take decision in favor of the modulation candidate M k which gives the minimum distance obtained by the maximum difference of hypothesized distribution S k (z) and the empirical distribution F(z), the proposed modulation classification method takes decision in favor of the modulation candidate M k which gives the minimum distance obtained through sum of the difference obtained through the hypothesized distribution S k (z) and empirical distribution F(z). To perform modulation classification using proposed method, we first compute a sequence of features{z n } N n= from the received signals {y n } N n= using magnitude based classifier. Afterwards,we evaluate the empirical distribution F(z) using eq. (9) and Ŝk(z) for each modulation candidate M k. The proposed test statistic is given as: D M F(z) S k (z) (5)
Similar to the case with K-S test, the hypothesized distribution Ŝk for the proposed modified K-S test is obtained by eq. (). So, practically the following test statistic is computed: D M = F(z) Ŝk(z) (6) Finally, the decision is made in favor of that modulation candidate M k in the modulation pool M = {M 4 QAM,M 6 QAM,M 64 QAM } according to the following decision rule: K M = arg min D M k (7) n N where D M k denotes the distance evaluated for each modulation candidate M k using the proposed modulation classification scheme. VI. SIMULATION RESULTS AND DISCUSSION In this section, two sets of simulation results obtained through Monte Carlo simulations have been provided to compare the performance of proposed modulation classification method and K-S based modulation classification in AWGN channel, with noise as w n CN(0,σ 2 ). We assume that the classification task is to distinguish between M-QAM, where M = {M 4 QAM,M 6 QAM,M 64 QAM }. Here we define Signal to Noise Ratio (SNR) in decibels as SNR = 0log 0 ( Es /σ 2), where, E s is the signal power and σ 2 is the noise power. In first set of experiments we assumed a fixed sample size of N = 00 for our simulations, classification performance of the proposed classification method is compared with classification performance of K-S based method for an SNR range of 5 db to 20 db. In the second set of experiments, we evaluate the performance of both proposed classification method and K-S based classification method for different sample sizes N. The upper and lower bound for sample sizes in this case is assumed to be N = 00 and N = 000 respectively. A. Classification performance vs SNR In Fig., we have shown the classification performance of detecting M 4 QAM among the pool of available modulation candidates M = {M 4 QAM,M 6 QAM,M 64 QAM }. It is observed that the performance of the proposed modulation classification method is throughout superior to K-S test based modulation classification for the whole SNR range. The probability of correct classification for K-S test based modulation classification method in case of 4-QAM at 0 db is approximately while the probability of correct classification for the proposed classification method is approximately 4. Moreover, the classification performance of both K-S based method and the proposed method in case of M 4 QAM monotonically increase with increase in SNR. Similarly, while detecting M 6 QAM from the pool of modulation candidates, the classification performance of the proposed method outperforms the performance of K-S based method for complete SNR range as shown in Fig. 2. A similar trend i.e. the classification performance monotonically increases with increase in SNR is observed as well. In case of 6-QAM, the maximum achieved value of correct classification for K-S based method is approximately 857 while that of the proposed method maximum value of correct classification achieved is unity, whereas for 4- QAM, the maximum value of correct classification i.e. for the proposed method is achieved at 9 db. Fig. 3 provides us with a comparison of classification performance of both methods for classification of 64-QAM among the pool of modulation candidates. The classification of proposed method is better than that of K-S test based classification method for whole SNR range and a increasing trend in classification performance with increase in SNR exists for both classification methods. B. Classification performance vs Sample Size The results of classification performance of proposed classification method as compared with K-S test based classification method as a function of sample size and is presented in Fig. 4. It is evident from the results that the performance of proposed classification method is superior to K-S test based classification method for all sample sizes i.e. 00 N 000. It is also important to note that the classification performance of both methods monotonically increases with increase in sample size N. For sample size N = 00, the probability of correct classification for K-S test based modulation classification is 25 while that of the proposed modulation classification method, probability of correct classification is 9. Also, the probability of correct classification of proposed classification method for N = 000 is approximately 5 as compared of the probability of correct classification for K-S based modulation classification is about. It is also evident that classification performance of proposed method is better than the classification performance of K-S test based modulation classification method for all sample size analyzed. The modulation scheme adopted for second set of experiments is 4-QAM. C. Analysis of proposed classification method It is important to note that the classification performance of proposed method is superior as compared to K-S test based method because the proposed method does not compute the distance based on the maximum difference between the empirical cumulative distribution function F(z) and the hypothesized distribution S k (z) for each modulation candidate M k, rather it evaluates the sum of the differences between the two distribution functions. By doing so, each data sample contributes towards computing the distance between the hypothesized distribution and empirical distribution, which virtually increases the data samples for computing the decision statistic using (7). VII. CONCLUSION In this paper, we have presented a modulation classification method based on modified Kolmogorov-Smirnov (K-S) test for QAM modulations with 4-QAM, 6-QAM and 64-QAM as candidate modulation schemes. The proposed method exploits the distance properties between the empirical distribution function obtained from the features extracted from received signals and the hypothesized distribution function for each modulation candidate in the pool. We propose to use the sum of difference between the two distributions rather than using the maximum of difference between the distributions as in case of modulation classification based on K-S test. Compared with modulation classification method based on K-S test, the proposed method
Proposed 0.3 Proposed Fig.. Performance comparison of detecting M 4 QAM among modulation candidates M k = {M 4 QAM,M 6 QAM,M 64 QAM } with N = 00. Fig. 3. Performance comparison of detecting M 64 QAM among modulation candidates M k = {M 4 QAM,M 6 QAM,M 64 QAM } with N = 00. 0.3 Proposed 5 5 5 Proposed 00 200 300 400 500 600 700 800 900 000 Sample Size Fig. 2. Performance comparison of detecting of M 6 QAM among modulation candidates M k = {M 4 QAM,M 6 QAM,M 64 QAM } with N = 00. has a superior performance for classification of all candidate modulation schemes. It is also evident from the results that the classification performance increases with increase in number of samples. Fig. 4. Effect of varying sample size on the probability of classification with SNR = 2dB. [5] F. Wang, R. Xu, and Z. Zhong, Low complexity Kolmogorov-Smirnov modulation classification, IEEE Wireless Communications and Networking Conference (WCNC), 20, pp. 607-6, Mar. 20. [6] J. R. Green and Y. A. S. Hegazy, Powerful Modified-EDF Goodnessof-Fit Tests, Journal of the American Statistical Association, vol. 7, no. 353,, pp. 204-209, Mar. 976. REFERENCES [] O. A. Dobre, A. Abdi, Y. Bar-Ness, and W. Su, Survey of automatic modulation classification techniques: classical approaches and new trends, IET Commun., vol., no. 2, pp. 3756, Apr. 2007. [2] M. Zaerin and B. Seyfe, Multiuser modulation classification based on cumulants in additive white Gaussian noise channel, IET Signal Processing, vol. 6, no. 9, pp. 85-823, Dec. 202. [3] H.C. Wu, M. Saquib, and Z. Yun, Novel automatic modulation classification using cumulant features for communications via multipath channels, IEEE. Trans. Wireless Commun., vol. 7, no. 8, pp. 3098305, 2008. [4] F. Wang and X. Wang, Fast and robust modulation classification via Kolmogorov-Smirnov test, IEEE Trans. Wireless Commun., vol. 58, no. 8, pp. 23242332, Aug. 200.