AUTOMATIC modulation classification is a procedure

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1 2324 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 200 Fast and Robust Modulation Classification via Kolmogorov-Smirnov Test Fanggang Wang and Xiaodong Wang, Fellow, IEEE Abstract A new approach to modulation classification based on the Kolmogorov-Smirnov (K-S) test is proposed. The K-S test is a non-parametric method to measure the goodness of fit. The basic procedure involves computing the empirical cumulative distribution function (ECDF) of some decision statistic derived from the received signal, and comparing it with the CDFs or the ECDFs of the signal under each candidate modulation format. The K-S-based modulation classifiers are developed for various channels, including the AWGN channel, the flat-fading channel, the OFDM channel, and the channel with unknown phase and frequency offsets, as well as the non-gaussian noise channel, for both QAM and PSK modulations. Extensive simulation results demonstrate that compared with the traditional cumulantbased classifiers, the proposed K-S classifiers offer superior classification performance, require less number of signal samples (thus is fast), and is more robust to various channel impairments. Index Terms Automatic modulation classification, Kolmogorov-Smirnov test, fading, OFDM, frequency offset, non-gaussian noise. I. INTRODUCTION AUTOMATIC modulation classification is a procedure performed at the receiver based on the received signal before demodulation when the modulation format is not known to the receiver. It plays a key role in various tactical communication applications. It also finds applications in emerging wireless communication systems that employ interference cancelation techniques in order to demodulate and cancel the unknown interfering user s signal, its modulation format needs to be classified first. In general, there are two classes of modulation classification techniques, the likelihood-based methods and the featurebased methods [], [2]. The likelihood-based methods compute some forms of the likelihood function of the received signal for each candidate modulation, by treating the data symbols as unknown nuisance parameters. They include the average likelihood ratio test (ALRT) [3] and the generalized likelihood ratio test (GLRT) [4]. However these methods are generally computationally expensive and moreover, they Paper approved by M. R. Buehrer, the Editor for Cognitive Radio and UWB of the IEEE Communications Society. Manuscript received August 4, 2009; revised November 23, 2009 and March 2, 200. F. Wang was supported by a scholarship from the China Scholarship Council (CSC). This work was supported in part by the U.S. National Science Foundation (NSF) under grant CCF , and in part by the U.S. Office of Navel Research (ONR) under grant N F. Wang is with the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, Beijing, China. X. Wang is with the Dept. of Electrical Engineering, Columbia Univ., New York, NY 0027 ( wangx@ee.columbia.edu). Digital Object Identifier 0.09/TCOMM become ineffective in the presence of various channel impairments, such as phase or frequency offset, channel fading, or impulsive noise. The feature-based modulation classification methods typically have a lower computational complexity than the likelihood-based ones [5] [9]. The most widely used feature is the cumulant. For example, the fourth-order cumulant can be used to classify various low-order QAM modulations. For classifying higher-order QAM or PSK modulations, a higherorder cumulant is needed. An accurate estimate of the higherorder cumulant of the signal requires a large number of signal samples. Most of the existing works on modulation classification focus on the additive white Gaussian noise (AWGN) channel. A few works have considered fading and multipath channels [2], [0]. On the other hand, effective modulation classification in the presence of non-gaussian noise remains a challenge. In this paper, we propose to employ the Kolmogorov- Smirnov (K-S) test [] for modulation classification. The K-S test is a non-parametric statistical method to measure the goodness of fit. From the received signal, we compute the empirical cumulative distribution function (cdf) of certain decision statistic. A priori we also compute the cdf or the empirical cdf of the same decision statistic under each candidate modulation format. The modulation format that results in the minimum of the maximum distance between its cdf and the observed empirical cdf is the final decision. We develop various K-S classifiers based on different decision statistics for both QAM and PSK modulations, under different channel models. The computational complexity associated with the K- S test is comparable with that for calculating the cumulant. We provide extensive simulation results to demonstrate the performance gain of the proposed K-S classifiers over the cumulant-based classifiers. The remainder of this paper is organized as follows. In Section 2 we provide some background on modulation classification and on the K-S test. In Section 3, we develop the K-Sbased modulation classifiers for various channels. Simulation results are provided in Section 4. Section 5 concludes the paper /0$25.00 c 200 IEEE II. BACKGROUND A. Automatic Modulation Classification Consider the following discrete-time additive white noise channel model y n = x n + w n, n =,,N, ()

2 WANG and WANG: FAST AND ROBUST MODULATION CLASSIFICATION VIA KOLMOGOROV-SMIRNOV TEST 2325 where x n,y n and w n are respectively the complex-valued transmitted modulation symbol, the received signal, and the noise sample at time n. The transmitted symbols {x,..., x N } are drawn from an unknown constellation set M which in turn belongs to a set of possible modulation formats {M,...,M K }. The modulation classification problem refers to the determination of the constellation set M to which the transmitted symbols belong based on the received signals {y,..., y N }. There are two major approaches in the literature to solving the above modulation classification problem. In the likelihoodbased methods [], [3], some form of the likelihood for each modulation format is calculated by making certain assumption on the data sequence. The classification decision then corresponds to the modulation with the largest likelihood value. These methods are typically computationally very expensive. Moreover, they require the knowledge of the various channel parameters and become ineffective in the presence of unknown channel impairment such as fading, phase and frequency offsets, and non-gaussian interference/noise. The more popular and low-complexity approach to automatic modulation classification is based on cumulant [2], [9]. Specifically, for the system given by (), we calculate the normalized sample fourth-order cumulant of the received signal {y n } as ˆC = E{ y 4 } E(y 2 ) 2 2E 2 { y 2 } {E{ y 2 } σ 2 } 2. (2) The modulation whose theoretical cumulant [] is closest to ˆC is then the classification decision. The fourth-order cumulants can be used to classify 4-QAM, 6-QAM and 64- QAM modulations. For even higher-order modulations, the difference between the cumulants becomes small, which leads to low classification accuracy. Furthermore, the fourth-order cumulants of 4-PSK, 8-PSK and 6-PSK are the same; hence it is impossible to classify these modulations using the fourthorder cumulants. A higher-order cumulant, e.g., the eighthorder cumulant [2], can be used to classify these modulations, with a considerably increased computational complexity. B. Kolmogorov-Smirnov (K-S) Test The Kolmogorov-Smirnov (K-S) test is a non-parametric test of goodness of fit for the continuous cumulative distribution of the data samples [], [3], [4]. It can be used to approve the null hypothesis that two data populations are drawn from the same distribution to a certain required level of significance. On the other hand, failing to approve the null hypothesis shows that they are from different distributions. ) One-sample K-S Test: In the one sample K-S test, we are given a sequence of real-valued data samples z,z 2,...,z N with the underlying cumulative distribution function (cdf) F (z), and a hypothesized distribution with the cdf F 0 (z). The null hypothesis to be tested is H 0 : F = F 0. (3) The K-S test first forms the empirical cdf from the data samples ˆF (z) N I(z n z), (4) N n= where I( ) is the indicator function, which equals to one if the input is true, equals to zero otherwise. The largest absolute difference between the two cdf s is used as the goodness-of-fit statistic, given by D sup F (z) F 0 (z) ; (5) z R and in practice, it is calculated by ˆD = max ˆF (z n ) F 0 (z n ). (6) n N The significance level ˆα of the observed value ˆD is given by ˆα P (D > ˆD) ( [ N 0.] ) =Q ˆD, (7) N with Q(x) 2 ( ) m e 2m2 x 2. (8) m= The hypothesis H 0 is rejected at a significance level α if ˆα = P (D > ˆD) <α. 2) Two-sample K-S Test: When the hypothesized cdf F 0 is not available, but instead, an data sequence drawn from F 0, ξ,ξ 2,...,ξ N0 is available, similarly as in (4), we can form the empirical cdf ˆF 0 ˆF 0 (ξ) N 0 I(ξ n ξ). (9) N 0 n= The test statistic is now ˆD = max ˆF (z n ) ˆF 0 (z n ). (0) n N The significance level ˆα of the observed value ˆD is still given by (7), but with N replaced by an equivalent sample size N, given by N = NN 0. () N + N 0 3) Two-dimensional K-S Test: Since the signals in () are complex-valued, the corresponding distributions are twodimensional (2D). Consider a sequence of 2D real-valued data samples (u,v ),...,(u N,v N ). When considering the 2D K-S test, the cdf s for all four quadrants of the 2D plane are examined, i.e., F I (u, v) P (U < u,v < v), F II (u, v) P (U > u,v < v), F III (u, v) P (U > u, V > v), and F IV (u, v) P (U < u,v > v). In [5], it is suggested to calculate the four empirical cdf s using all possible combinations of the 2D data samples. On the other hand, in [6], it is proposed to use the 2D samples directly rather than all possible combinations for forming the empirical cdf s. For example, the first quadrant empirical cdf becomes ˆF (u, I v) = N I(u n <u)i(v n <v). (2) N n= The two methods are compared in [6], where it is shown that when the data components from the two dimensions are uncorrelated their performance is similar. The statistic of the 2D K-S test is the largest absolute difference of between the hypothesized cdf and the empirical cdf among all four quadrants, i.e., ˆD = max max ˆF q (u n,v n ) F q 0 (u n,v n ). (3) n N q {I,II,III,IV}

3 2326 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 200 As in the D test, for a given significance level, using ˆD in (3) we can then test whether the data samples are drawn from the hypothesized distribution F 0. Furthermore, similarly as before, when F 0 is unknown but samples from F 0 are available, the two-sample 2D K-S test can be performed. III. K-S-BASED MODULATION CLASSIFICATION A. Modulation Classification in AWGN Channels Consider the signal model in (). In this section, we assume that the noise samples follow the complex Gaussian distribution, i.e., w n N c (0,σ 2 ); that is, the real and imaginary components of w n are independent and have the same Gaussian distribution N (0, σ2 2 ). To classify the modulation based on the received signals {y n } using the K-S test, we first form a sequence of decision statistics {z n } from {y n }, where z n can be either the magnitude, or the phase, or the real and imaginary components of y n, and then compute the corresponding empirical cdf ˆF. In the meantime, for each possible modulation candidate M k, we can obtain either the exact cdf F0 k, or the empirical cdf k ˆF 0,for{z n }.TheK-S statistic is then calculated by max ˆD ˆF (z n ) F0 k(z n), one-sample test n N k = max ˆF (z n ) ˆF k, 0 (z n ), two-sample test n N k =, 2,,K. (4) The decision on the modulation is given by the minimum K-S statistic, i.e., ˆk =arg min ˆD k. (5) k K Moreover, recall that associated with each K-S statistic ˆD k, there is a significance level ˆα k P (D > ˆD k M k ),computed by (7). The normalized {ˆα k } can be used to give a soft decision on the modulation, that is, the probability that the modulation M k is used is approximately q k ˆα k / K k= ˆα k, k =,...,K. In what follows, we discuss different choices of the decision statistics {z n } for different modulation formats, and the corresponding cdf F 0. ) Classification of QAM Signals: We first consider the quadrature amplitude modulation (QAM) formats, e.g., 4- QAM, 6-QAM, and 64-QAM. The set of signal points of unit-energy constellations { for these modulations } are given by M 4 QAM = 2 (a + bȷ) a, b =,, M 6 QAM = { } 0 (a + bȷ) a, b = 3,,, 3, and M 64 QAM = { } 42 (a + bȷ) a, b = 7, 5, 3,,, 3, 5, 7, where ȷ =. Magnitude-based K-S classifier: For QAM signals, one possible choice of the decision statistic is the signal magnitude, i.e., z n = y n = (R{y n }) 2 +(I{y n }) 2, n =,...,N. (6) For fixed x, sincew N c (0,σ 2 ),thecdfofz = x + w is given by ( ) 2 x 2z F (z) = Q σ,, z R +, (7) σ where Q (a, b) is the Marcum-Q function. Assuming all signal points in the modulation constellation are equiprobable, the cdf of z n = y n under modulation M k is then given by F0 k (z) = M k ( 2 x σ, ) 2z Q, σ x M k z R +, k =,...,K. (8) We can then perform the one-sample K-S test in (4)-(5) using ˆF and {F k 0 } on the samples {z,...,z N }. Quadrature-based K-S classifier: Since for QAM input signals, the real and imaginary components of the received signal y n are independent and have identical distributions, we can also use them directly as the decision statistics. That is, from the N received signals samples y,...,y N,weform a sequence of 2N samples of decision statistic z 2n = R{y n }, z 2n = I{y n }, n =,...,N. Then we have z n N(0, σ2 2 ). Hence the cdf under modulation M k is given by ( ) F0 k 2(z x) (z) = Q 0, Mk σ x R{M k } z R, k =,...,K, (9) where Q 0 (a) is the Gaussian-Q function, and R{M k } denotes the set of real components of the signal points in M k.the one-sample K-S test in (4)-(5) can be performed using ˆF and {F0 k} on the samples {z,...,z 2N }. 2) Classification of PSK Signals: We next consider the classification of M-ary phase-shift keying (PSK) modulations. The signal constellation set is given by M M PSK = {e ȷ 2π M (m ), m =,...,M}. Phase-based K-S classifier: Since the information is embedded in the signal phase, we can use the phase of the received signal as the decision statistic, i.e., z n = (y n )=tan ( I{yn } R{y n } ), n =,...,N. (20) When the phase θ is transmitted, the pdf of φ = (e ȷθ + w), with w N c (0,σ 2 ), is given by [7] f(φ θ) = ( ) πσ 2 λ exp [λ2 + 2λcos(φ θ)] 0 σ 2 dλ. (2) Hence the cdf of z n = (y n ) under the M-ary PSK modulation is given by F0 M (z) = M z ( f φ θ m = 2π ) (m ) dφ, M m= π M z [0, 2π). (22) We can then perform the one-sample K-S test on the samples {z,...,z N } using ˆF and {F0 M}. On the other hand, since the exact cdf F0 M given by (2)- (22) involves numerically evaluating M double integrals, it is computationally expensive to implement the one-sample K-S test. Instead, we can obtain the empirical cdf ˆF 0 M for each

4 WANG and WANG: FAST AND ROBUST MODULATION CLASSIFICATION VIA KOLMOGOROV-SMIRNOV TEST 2327 modulation format, by generating a large number of samples of the form with z i = ( x i + w i ), i =,..., N, x i uniform(m M PSK ), w i N c (0,σ 2 ), and then perform the two-sample K-S test in (4)-(5) using ˆF and { ˆF 0 M } on the samples {z,...,z N }. Quadrature-based 2D K-S classifier: Since for the PSK signal x n, the real and imaginary components of the received signal y n are not independent, in theory it is no longer valid to employ the D K-S classifier using the real and imaginary components of the received signal as decision statistics (although in practice this approach still seems to give good performance). Instead, we can perform the 2D K-S test introduced in Section II-B3 based on the real and imaginary pairs of the received signals, i.e., (u n,v n )=(R{y n }, I{y n }), n =,...,N. Not that the 2D K-S test typically require a large signal sample size N to obtain good performance. 3) Complexity Analysis: We next consider the computational complexity of the K-S classifier, and compare it with that of the cumulant-based classifier. For QAM modulations, the K-S detector involves evaluating the Marcum-Q function or the Gaussian-Q function for each signal sample, which is computationally more extensive than calculating the cumulant. For PSK modulations, the K-S detector is performed by first generating a set of signal samples based on which we form the empirical cdf and then apply the two-sample test. This approach involves random number generation (that can be done offline) and sorting (to form cdf) and therefore does not require sophisticated calculations. Note that this method can also be applied to QAM modulation to avoid evaluating the Q functions. A complexity analysis of the two-sample K-S ( test reveals that the quadrature-based classifier involves O 6 N(log 2 N ) +2)+2N(log 2N +3) real additions and no multiplications; ( and the magnitude-based classifier involves O 3 N(log N ) +9)+N(log N +4) real additions and O(9 N +4N) multiplications. On the other hand, the cumulant method requires O(6N) real multiplications and O(6N) real additions. Hence the two methods have comparable complexities. B. Modulation Classification in Fading Channels We now consider modulation classification in flat-fading channels, where the signal model is given by y n = Hx n + w n, n =,,N, (23) where H is a complex-valued channel fading gain that is assumed unknown, and w n N c (0,σ 2 ). In what follows, we show that for QAM modulations, the magnitude-based K- S classifier discussed in Section III-A together with a simple channel magnitude estimator, can still perform modulation classification in the presence of unknown channel gains. On the other hand, for PSK modulations, we develop a K-S classifier based on the phase differences between adjacent received signals. Note that since the channel gain H is unknown, the quadrature-based K-S classifiers discussed in the previous section are no longer effective. ) Magnitude-based Detector for QAM Signals: For classification of QAM signals in fading channels, the decision statistic z n is the magnitude of the received signal, given by (6). The cdf of the magnitude of the signal in (23), z n = y n = Hx n + w n under modulation M k is now given by F0 k ( H x (z) = Q, z ), M k σ σ x M k z R +, k =,...,K. (24) Hence we need to first obtain an estimate of the channel magnitude H. From (23), we have E{ y n 2 } = H 2 E{ x n 2 } + σ 2 = H 2 + σ 2, (25) since the modulation symbols are assume to have unit energy, i.e., E{ x n 2 } =. Therefore a channel magnitude estimator is given by H ˆ = N y n N 2 σ 2. (26) n= Substituting (26) into (24) we obtain the cdf s {F0 k},which together with the empirical cdf ˆF, are used to perform the one-sample K-S test on the samples {z,...,z N }. 2) Phase-difference-based Detector for PSK Signals: We now consider the classification of PSK signals in fading channels. Since the channel fading gain H in (23) is unknown, we use the phase difference between two adjacent received signals as the decision statistic, i.e., z n = (y n ) (y n+ ), n =,...,N. (27) Denote Δθ as the phase difference between two adjacent symbols, i.e., Δθ = (x n ) (x n+ ).SinceforM-ary PSK modulation, (x n ) takes values uniformly in { 2π M (m ),m=,...,m}, wehave 2π <Δθ <2π. By taking the modulo 2π operation we will have 0 (Δθ mod 2π) < 2π. Furthermore, it can be verified that (Δθ mod 2π) also takes values uniformly from same phase set { 2π M (m ),m=,...,m}. In [8] the distribution of the phase difference between two vectors perturbed by Gaussian noise is analyzed. Given the phase difference Δθ between the symbols x n and x n+,using the result from [8], the cdf of the phase difference z between the corresponding received signals y n and y n+ is given by F (z Δθ) = sin(δθ z) 4π π 2 π 2 e ρ[ cos(δθ z)cost] cos(δθ z)cost dt, z [0, 2π), (28) where ρ = H 2 σ is the SNR of the received signal. Hence 2 the cdf of z n = (y n ) (y n+ ) under the M-ary PSK modulation is given by F0 M (z) = M ( F z Δθ m = 2π ) (m ), M M m= z [0, 2π). (29)

5 2328 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 200 Given the received signals {y n },wefirst estimate the channel amplitude using (26) to obtain the received SNR ˆρ. We can then perform the one-sample K-S test using ˆF and {F0 M } on the samples {z,...,z N }. On the other hand, to avoid numerically computing the integral (29), we can obtain the empirical cdf ˆF 0 M for each modulation format, by generating a large number of samples of the form z i = ( ˆH x i + w i ) ( ˆH x i+ + w i+ ), i =,..., N, with x i uniform(m M PSK ),and w i N c (0,σ 2 ),and then perform the two-sample K-S test using ˆF and { ˆF 0 M} on the samples {z,...,z N }. C. Modulation Classification with Phase and Frequency Offset Next we further extend the modulation classification techniques in fading channels to the case when there is unknown phase and frequency offset present. The signal model becomes y n = He εn+φ x n + w n, n =,,N, (30) where φ is an unknown phase shift and ε is an unknown frequency offset. Note that the magnitude-based classifier for QAM signals discussed in Section III-B can be directly applied here since the term e εn+φ in (30) has a unit magnitude. On the other hand, the phase-difference-based classifier for PSK signals discussed in Section III-B2 is not applicable here since the phase difference between two adjacent noise-free received signal is now of the form Δθ n + ε where Δθ n = (x n ) (x n+ ) is the phase difference between two adjacent symbols. In order to remove the unknown frequency offset term ε, we can use the difference between the adjacent phase differences as the decision statistic, i.e., D. Modulation Classification in OFDM Systems In general, the wireless channels exhibit frequency-selective or multipath fading. The orthogonal frequency-division multiplexing (OFDM) technique effectively transforms a multipath channel into a set of parallel flat-fading channels and has been chosen as the air interface for several key emerging wireless systems, such as the LTE cellular systems [9] and the WiMax systems [20]. In this section we consider modulation classification in orthogonal OFDM systems. The received signal at the l-subcarrier of the n-thofdmwordisgiven by Y l (n) =H l X l (n)+w l (n), l =,...,P; n =,...,N, (32) where P is the total number of subcarriers; X l (n),y l (n), and W l (n) denote respectively the transmitted symbol, the received signal, and the noise sample at the l-th subcarrier of the n-th OFDM word, with W l (n) N c (0,σ 2 ); H l is the channel gain at the l-th subcarrier, which is assumed to remain the same for N OFDM words. Typically the adjacent subcarriers have similar channel gains and therefore they will employ the same modulation format. Here we assume that a group of p adjacent subcarriers, i.e., l =(l )p +,...,lp, will employ the same modulation format, l =,..., P p. In order to classify the P/p modulation formats within the N OFDM words, we perform the following steps. Estimate the channel magnitude H l for each subcarrier from the N received signal samples {Y l (n),n =,...,N} using (26), to obtain { ˆH l,l=,...,p}. Smooth the channel estimates by performing, e.g., polynomial fitting on { ˆH l,l =,...,P}, to obtain smoothed channel estimates { H l,l=,...,p}. Perform per-subcarrier equalization using the estimated channel magnitudes, i.e., Y l (n) =Y l (n)/ H l e ȷ (Hl) X l (n)+ W l (n), l =,...,P; n =,...,N, (33) where W l (n) N c (0,σ 2 / H l 2 ). Perform modulation classification for the l-th group (l =,..., P p ) of subcarriers based on the signals { Y l (n),l= (l )p +,...,lp, n =,...,N}, by approximating W l (n) N c (0, σ 2 ), with σ 2 = σ 2 / z n =[ (y n ) (y n+ )] [ (y n+ ) (y n+2 )], p H l 2, (34) l=(l )p+ n =,...,N 2. (3) and using the amplitude-based K-S classifier discussed As before, we first form the empirical cdf ˆF in Section III-B for QAM modulations, or the phasedifference-based K-S classifier discussed in Section for the above decision statistics {z n }. Then for each PSK modulation format III-B2. M M PSK, we generate a large number of samples of the form ψ i = ( ˆH x i + w i ) ( ˆH x i+ + w i+ )], i =,..., N When there is frequency-offset present in the OFDM system, we can first estimate and compensate for it by exploiting, the cyclic prefix [2]. The residual frequency-offset will with x i uniform(m M PSK ), w i N c (0,σ 2 ) and z i = ψ i ψ i+, i =,..., N 2, and form the empirical cdf ˆF introduce a small amount of intercarrier interference, which 0 M can be simply treated as noise. using { z i }. Finally we perform the two-sample K-S test using ˆF and { ˆF 0 M} on the samples {z,...,z N 2 }. E. Modulation Classification in Non-Gaussian Noise lp So far we have assumed that the ambient channel noise is Gaussian distributed. On the other hand, non-gaussian noise can arise in many communication systems due to the impulsive nature of the various electromagnetic interference. The highorder cumulant-based modulation classification methods in the two-term Gaussian mixture noise is considered in [9], where a clipper is employed to suppress the impulsive noise. In [22] a modulation classification method based on cyclic cumulant is proposed, and its performance in Poisson impulsive noise is analyzed.

6 WANG and WANG: FAST AND ROBUST MODULATION CLASSIFICATION VIA KOLMOGOROV-SMIRNOV TEST 2329 K S D K S 2 D Cumulant 8th order Hellinger distance K S D K S 2 D Cumulant 8th order Fig.. QAM modulation (4-QAM, 6-QAM, 64-QAM) classification performance in AWGN channels. The number of samples N = 00. Although several non-gaussian noise models exist in the literature, such as the Middleton Class-A model [23] and the symmetric alpha-stable (SαS) model [24], in general it may not be feasible to accurately characterize the impulsive noise environment of interest a priori using a trackable analytical distribution function. With the framework of the two-sample K-S test, we propose the following training-based modulation classification scheme. Training stage: For each possible modulation format M k, transmit a sequence of T symbols equiprobable from M k, and collect the corresponding received signals samples, from which form the empirical cdf ˆF 0 k. Modulation classification: During the communication stage, collect the received signals samples y,...,y N corresponding to the transmitted symbols from the unknown modulation, and form the corresponding empirical cdf of the decision statistic ˆF. Perform the two-sample K-S test to decide on the modulation format. IV. SIMULATION RESULTS In this section, we provide simulation results to compare the performance of the proposed K-S-based modulation classifiers with that of the cumulant-based ones in various channels. For the QAM modulations, we will consider the set {4-QAM, 6-QAM, 64-QAM}; and for the PSK modulations, we will consider the set {4-PSK, 8-PSK, 6-PSK}. AWGN channels: The classification performance of various classifiers in AWGN channels for QAM modulations and PSK modulations is shown in Fig. and Fig. 2, respectively. The channel model is given by () with w n N c (0,σ 2 ).The signal-to-noise ratio (SNR) is defined as /σ 2. The 4-th order and 8-order cumulants are used for QAM and PSK modulations, respectively. The number of received signal samples used is N = 00. In Fig. we also show the performance of the Hellinger-distance-based classifier [25] which has a very high complexity. It is seen that for such a small sample size, at high SNR, the cumulant-based methods exhibit a ceiling on the classification probability around for both QAM Fig. 2. PSK modulation (4-PSK, 8-PSK, 6-PSK) classification performance in AWGN channels. The number of samples N = 00. TABLE I MISCLASSIFICATION PROBABILITIES FOR SNR = 0dB. K-S / Cumulant Actual (QAM) 4-QAM 6-QAM 64-QAM 4-QAM.000 / / / Estimated 6-QAM / / / QAM / / / 2 K-S / Cumulant Actual (PSK) 4-PSK 8-PSK 6-PSK 4-PSK 75 / / / Estimated 8-PSK / / / PSK / / / 55 and PSK. However, the K-S classifiers monotonically improve the classification performance as the SNR increases and they significantly outperform the cumulant-based classifiers at high SNR. Moreover, for QAM signals, the quadrature-based K- S classifier outperforms the magnitude-based one, since the sample size for the former is 2N and for the latter is N. The Hellinger-distance-based classifier performs worse than the cumulant method in the low SNR region and in the high SNR region it performs worse than the K-S quadrature method. For PSK signals, the phase-based D K-S classifier and the quadrature-based 2D K-S classifier have similar performance. The classification performance for QAM modulations as a function of the sample size is shown in Fig. 3. The misclassification probability matrices under both QAM and PSK modulations are shown in Table I for SNR = 0dB. Each entry contains the probabilities P (Estimmated modulation Actual modulation) by both the K-S method and the cumulant method. It is seen that most classification errors occur between higher-order modulations, e.g., between 6-QAM and 64-QAM, or 8-PSK and 6-PSK. This is because the pdfs of the high-order modulations are more similar. Since both the proposed K-S detector and the cumulantbased detector require the knowledge of the channel noise variance σ 2, we illustrate the robustness of both detectors

7 2330 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST K S quadrature K S magnitude Cumulant Sample size Fig. 3. QAM modulation (4-QAM, 6-QAM, 64-QAM) classification performance versus sample size in AWGN channels. SNR= 4dB. K S w/ known channel K S w/ estimated channel Cumulant Cumulant w/ freq offset K S w/ est. chan. & freq offset Fig. 5. QAM modulation (4-QAM, 6-QAM, 64-QAM) classification performance in flat-fading channels. The number of samples N = KS Cumulant SNR mismatch (db) K S with ideal chan esti K S with esti chan ampli Cumulant 8th order Cumulant 8th order with freq offset K S with esti chan ampli & freq offset Fig. 4. QAM modulation (4-QAM, 6-QAM, 64-QAM) classification performance versus SNR mismatch. True SNR= 5dB. Fig. 6. PSK modulation (4-PSK, 8-PSK, 6-PSK) classification performance in flat-fading channels. The number of samples N = 00. against mismatched σ 2. Fig. 4 shows the performance of both detectors over a mismatch range from 3dB to 3dB. It is seen that the K-S detector still performs significantly better than the cumulant method in the presence of noise variance uncertainty. Flat-fading channels (with frequency offset): We next consider the modulation classification performance in flat-fading channels. The channel model is given by (23) with w n N c (0,σ 2 ). The sample size is N = 00. The classification performance is averaged over different realization of the channel H N c (0,σH 2 ), The signal-to-noise ratio (SNR) is defined as σh 2 /σ2. The performance of various classifiers for QAM and PSK modulations is shown in Fig. 5 and Fig. 6, respectively. As is the case for AWGN channels, the cumulantbased classifiers exhibit classification probability ceilings of and for QAM and PSK, respectively; whereas the K-S classifiers do not have such ceilings. For QAM signals, the performance of the magnitude-based K-S classifier is robust against the unknown channel and frequency offset, and is significantly better than that of the cumulant-based classifier at high SNR. For PSK signals, the phase-difference-based K-S classifier is again robust against the unknown channel. When there is frequency offset present, the K-S classifier is then based on the double phase difference, which incurs a loss compared with that based on the phase difference, due to the noise enhancement effect. Nevertheless, note that the cumulant-based classifier is virtually unusable for detecting PSK signals in fading channels, with or without frequency offset; whereas the K-S classifiers offer excellent performance. OFDM channels: We now consider modulation classification in OFDM systems. The model is given in (32). The number of subcarriers is P = 52 and the number of OFDM symbols for which the channel remain static is N =0.We assume that within a group of p =6adjacent subcarriers the same QAM modulation format is employed. The 3GPP channel model is used to generate the time-domain multipath channel response [26], [27]. The classification performance is shown in Fig. 7. It is seen that similar to the flat-fading

8 WANG and WANG: FAST AND ROBUST MODULATION CLASSIFICATION VIA KOLMOGOROV-SMIRNOV TEST 233 K S w/ known channel K S w/ estimated channel Cumulant Cumulant w/ freq offset K S w/ esti. chan. & freq offset 2 sample K S quadrature 2 sample K S magnitude Theoretical cumulant Esti cumulant Cumulant w/ clipping Fig. 7. QAM modulation (4-QAM, 6-QAM, 64-QAM) classification performance in OFDM channels. The total number of subcarriers P = 52, group size p =6; and number of OFDM symbols is N =0. case, the magnitude-based K-S classifier is robust against the unknown channels and significantly outperforms the cumulantbased classifier. When there is frequency offset present, we can estimate and compensate for it using, e.g., the method in [2] by exploiting the cyclic prefix. With this approach the mean squared error (MSE) of the frequency offset estimate is around 0 4 to 0 3 with less than 20 sample cyclic prefix atsnr=0db. And the normalized residual frequency offset is around 0.0 to [2]. Here we consider the worse case value of We simply treat the intercarrier interference caused by the residual frequency offset as noise. The modulation classification performance in this case is also shown in Fig. 7. It is seen that there is little attendant degradation in performance. Impulsive noise channels: Finally we consider modulation classification in the presence of non-gaussian or impulsive noise. The non-gaussian noise samples are generated using the toolbox [28]. We compare the performance of the trainingbased two-sample K-S classifier outlined in Section III-E with that of the cumulant-based classifier. In the K-S classifier, N 0 = 00 training samples are used for each modulation format, and N = 00 received signal samples are used for modulation classification. For the cumulant-based classifier, we either calculate the sample cumulant for each candidate modulation using the corresponding training signal (estimated cumulant), or use the theoretical cumulant of each modulation. To classify the modulation, we then compare the sample cumulant of the received signal with the (either estimated or theoretical) cumulants of each modulation format, and pick the modulation whose cumulant is closest to that of the received signal. We also consider the clipping technique in [9], where a data-adaptive zero-memory nonlinearity is applied to the received signal before the cumulant is computed. In Fig. 8, the noise distribution follows the Middleton Class A model. And in Fig. 9, the noise distribution follows the SαS model, which has an even heavier tail than the Middleton Class A model. It is seen that in both cases the K-S classifier significantly outperform the cumulant-based classifier with or Fig. 8. QAM modulation (4-QAM, 6-QAM, 64-QAM) classification performance in Middleton Class A noise. The number of training samples and signal samples N 0 = N = sample K S quadrature 2 sample K S magnitude Theoretical cumulant Esti cumulant Cumulant w/ clipping Generalized Fig. 9. QAM modulation (4-QAM, 6-QAM, 64-QAM) classification performance in symmetric alpha stable noise. The number of training samples and signal samples N 0 = N = 00. The generalized SNR is defined as the ratio of the signal power and the dispersion parameter of the symmetric alpha stable distribution. without clipping. Moreover, with the training sample size N 0 = 00, the performance using the estimated cumulant from the training signal is worse than that using the theoretical cumulant. V. CONCLUSIONS We have proposed a new modulation classification technique based on the Kolmogorov-Smirnov (K-S) test, for classifying both QAM and PSK modulation formats. The basic procedure involves computing the ECDF of some decision statistic derived from the received signal, and comparing it with the CDFs or the ECDFs of the signal under each candidate modulation format. Compared with the popular cumulant-based modulation classifiers, with a comparable computational complexity, the proposed K-S classifiers offer faster (i.e., requiring less number of signal samples) and

9 2332 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 200 superior performance in a number of environments, including AWGN channels, flat-fading channels, OFDM channels, and channels with non-gaussian noise. Moreover, the proposed K-S classifiers are also robust against unknown phase and frequency offset. REFERENCES [] O. 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 , [2] 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 , [3] W. Wei and J. Mendel, Maximum-likelihood classification for digital amplitude-phase modulations, IEEE Trans. Commun., vol. 48, no. 2, pp , [4] N. Lay and A. Polydoros, Per-survivor processing for channel acquisition, data detection and modulation classification, in Proc. 994 Asilomar Conf. Sig., Syst. & Comp., vol. 2, 994, pp [5] Y. Yang and C.-H. Liu, An asymptotic optimal algorithm for modulation classification, IEEE Commun. Lett., vol. 2, no. 5, pp. 7 9, 998. [6] S. Soliman and S.-Z. Hsue, Signal classification using statistical moments, IEEE Trans. Commun., vol. 40, no. 5, pp , 992. [7] W. Dai, Y. Wang, and J. Wang, Joint power estimation and modulation classification using second- and higher statistics, in Proc IEEE Wireless Commun. & Networking Conf. (WCNC), vol., 2002, pp [8] P. Marchand, J.-L. Lacoume, and C. Le Martret, Multiple hypothesis modulation classification based on cyclic cumulants of different orders, in Proc Int l Conf. Acoust., Speech & Sig. Proc. (ICASSP), vol.4, 998, pp [9] A. Swami and B. Sadler, Hierarchical digital modulation classification using cumulants, IEEE Trans. Commun., vol. 48, no. 3, pp , [0] A. Swami, S. Barbarossa, and B. Sadler, Blind source separation and signal classification, in Proc Asilomar Conf. 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Commun., vol. 30, no. 8, pp , Aug [9] F. Khan, LTE for 4G Mobile Broadband. New York: Cambridge University Press, [20] J. Andrews, A. Ghosh, and R. Muhamed, Fundamentals of WiMax. Upper Saddle River, NJ: Prentice Hall, [2] M. B. P. Van de Beek, J. J. Sandell, ML estimation of time and frequency offset in OFDM systems, IEEE. Trans. Signal Process., vol. 45, no. 7, pp , July 997. [22] O. Dobre, Y. Bar-Ness, and W. Su, Robust QAM modulation classification algorithm using cyclic cumulants, in Proc IEEE Wireless Commun. & Networking Conf. (WCNC), vol. 2, Mar. 2004, pp [23] D. Middleton, Non-Gaussian noise models in signal processing for telecommunications: new methods an results for class A and class B noisemodels, IEEE Trans. Inf. Theory, vol. 45, no. 4, pp , May 999. [24] J. Gonzalez and G. Arce, Optimality of the myriad filter in practical impulsive noise environments, IEEE Trans. Signal Process., vol. 49, no. 2, pp , Feb [25] X. Huo, A simple and robust modulation classification method via counting, in Proc. 998 IEEE Int. Conf. Acoust., Speech & Sig. Proc. (ICASSP), 998. [26] MATLAB implementation of the 3GPP spatial channel model (3GPP TR ), Jan [27] Spatial channel model for multiple input multiple output (MIMO) simulations, 3GPP TR V8.0.0, Dec [28] RFI/impulsive noise toolbox.2 for Matlab, bevans/projects/rfi/software/index.html. Fanggang Wang received the B.S. degree in the School of Information and Communication Engineering from Beijing University of Posts and Telecommunications, Beijing, China in Now he is pursuing his Ph.D. degree at the same university. Since 2008 he has been a visiting Ph.D. student in the Electrical Engineering Dept., Columbia University, New York. His research interests include MIMO and OFDM techniques in wireless communications and optical communications. Xiaodong Wang (S 98-M 98-SM 04-F 08) received the Ph.D. degree in Electrical Engineering from Princeton University. He is a Professor of Electrical Engineering at Columbia University in New York. Dr. Wang s research interests fall in the general areas of computing, signal processing and communications, and has published extensively in these areas. Among his publications is a recent book entitled Wireless Communication Systems: Advanced Techniques for Signal Reception, published by Prentice Hall in His current research interests include wireless communications, statistical signal processing, and genomic signal processing. Dr. Wang received the 999 NSF CAREER Award, and the 200 IEEE Communications Society and Information Theory Society Joint Paper Award. He has served as an Associate Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS ON WIRELESS COMMU- NICATIONS, the IEEE TRANSACTIONS ON SIGNAL PROCESSING, andthe IEEE TRANSACTIONS ON INFORMATION THEORY. He is a Fellow of the IEEE and listed as an ISI Highly-cited Author.

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