Automatic Modulation Classification of Real Signals in AWGN Channel Based on Sixth-Order Cumulants

. Automatic modulation classification (AMC) represents an important integral part of modern communication systems. While novel AMC algorithms based on complex neural network structures showed significant performance improvements, in practical applications low algorithm complexity of AMC algorithms based on higher-order cumulants still makes them very attractive. AMC algorithm based on sixth-order cumulants showed very good performance in this context, especially when it comes to distinguishing Binary Phase Shift Keying (BPSK) signals from complex constellations. Still, no further analysis of expected performance with other real constellations was presented for this algorithm so far. In this paper, the performance was explored in a wider context of real signals classification, by observing various Pulse Amplitude Modulation (PAM) constellations, whose statistical features are presented for the first time. Their classification performance was tested via Monte Carlo simulations, and explained through the presence of bias under conditions of strong additive white Gaussian noise channel, reported in this paper for real signals for the first time. One new approach in AMC is proposed, which ensures improvement in the classification of real signal constellations. Achieved improvement is confirmed in many Monte Carlo experiments, where the proposed new AMC scheme is tested versus the most popular standard higher-order cumulants-based algorithms.


Introduction
Automatic modulation classification (AMC) is a technique commonly connected with wireless systems and applications, standing for modulation format recognition process and further demodulation of a priori unknown signals at the receiver side. It has found importance in electronic warfare, surveillance, and countermeasures, but also in many civilian communication applications, like software-defined radio, spectrum management, cognitive radio, smart reconfigurable transceivers, intelligent modems, and Internet of Things (IoT), [1][2][3][4][5].
There are two approaches in AMC algorithms development widely used by many authors: likelihood-based (LB) methods, which are leading to optimal solutions at the price of high computations, and feature-based (FB) methods, characterized with lower computational complexity, but still able to show performance close to optimal when properly designed [6]. FB algorithms are based on pattern recognition approach, extracting various instantaneous features of the received communication signal, and very frequently using complex classifiers for extended performance, such as neural networks [7], or deep-learning methods [3], [8]. Many modern algorithms perform extraction of several different features simultaneously and combine their classification properties.
Higher-order statistics -cumulants and moments, are especially popular as features of interest in FB AMC schemes. Simple structures of fourth-order cumulants [9] or sixth-order cumulants [10] are still considered as thestate-of-the-art of AMC [11]. Sixth-order cumulants showed significantly better performance than fourth-order cumulants [12]. They also show to be superior in low complexity, memory requirements, and inference time when compared with other up-to-date AMC algorithms, like neural networks (with difference measured in several orders of magnitude) -but requiring additional performance improvements to remain competitive with those algorithms [5]. Published research of cumulant-based AMC algorithms are mostly focused on complex signal constellations, like various Quadrature Amplitude Modulation (QAM) schemes. When it comes to real signal constellations, those are quite rarely considered in research, with some exceptions: Binary Phase Shift Keying (BPSK) signals are included in many research works; Pulse Amplitude Modulation (PAM) formats from PAM-4 to PAM-64 were considered in classical work of Swami [13] under fourthorder cumulants AMC, and also in [2] under complex network classifier; PAM-4 signals were considered in [14] under deep learning aided classification method, and [15] under fusion neural network classifier; BPSK and PAM-4 in [16] under LB classifier, and [17] under Convolution Neural Network (CNN) classifier; PAM-4 and PAM-8 in [18] under CNN classifier, and [19] under multi-layer support vector machine classifier.
While none of the research published so far considers a wider set of PAM signals classification under sixth-order cumulants (or other, higher-order cumulant structures), these constellations are still being of interest in modern communication systems, like IEEE 802.3 gigabit Ethernet [20][21][22], digital television [23], gigabit optical fiber links [24], [25], or free-space optical systems [26]. PAM signal constellations are presented in Fig. 1.
In this paper we discuss the classification performance of classical sixth-order cumulants-based AMC algorithm for a wide set of PAM constellations, present their error variances and parameters for efficiency in distinguishing particular constellations, for the first time. These parameters are compared with those corresponding with fourth-order cumulants-based AMC algorithm, under the same set of PAM constellations, and tested in Monte Carlo simulations. Very good performance of sixth-order cumulants in distinguishing PAM constellations from complex signal constellations is further explained with the presence of bias under strong AWGN conditions, which is also reported for the first time. A novel AMC scheme is then proposed, which provides both: good performance in distinguishing PAM constellations from complex signal constellations, and mutual distinguishing of various PAM constellations, simultaneously. This novel scheme is based on the convenient manipulation over the basic formula for the real signal cumulant value estimation (unlike the one based on the modulation order reduction [5], which is not applicable to real signals due to their properties presented in this paper). In comparison with complex state-of-the-art AMC algorithms, our resulting solution remains superior in low complexity, memory requirements, and inference time.
The rest of the paper is organized as follows: in Sec. 2 we present standard sixth-order cumulant-based AMC algo-

AMC Algorithm Based on Sixth-Order Cumulants
In a standard communication system model where the received signal is corrupted by AWGN only during the propagation (as explained in [5], for instance), the received signal sequence y(n) can be represented by: where x(n) stands for transmitted symbols of an unknown modulation, and g(n) represents AWGN with a zero mean and variance of σ g 2 . For zero-mean random variable x, associated with transmitted data sequence x(n), the secondorder cumulant C 21,x = cum(x, x * ) is given by: The sixth-order cumulant of the same random variable x C 63,x = cum(x, x, x, x * , x * , x * ) is given by [10]: while the self-normalized sixth-order cumulant is derived from the previous two equations as: The cumulants of random variable y, associated with received sequence y(n), can further be expressed in the following manner: Consequently, the following relation is established between the cumulants' values: While the noise power σ g 2 can be measured at the receiving point, the calculation of cumulants of a received signal in practice is executed via the calculation of mean-values over an ensemble of collected signal samples, which is implementable quite easily. If the number of samples is represented with N, equation (7) in its practical realization takes the form of: In practice, estimated values of normalized cumulants are always shaped with some portion of dispersion around expected (theoretical) values. This phenomenon was explored and described originally in [13], for fourth-order cumulants, which are calculated as: The similar conclusion was reported for sixth-order cumulants in [27], where the variance of C 63,x for complex signals with N samples, is given with: Reported error variances are directly proportional with sample size N and take different values for different modulation formats. While variances of the sample estimates of fourth-order cumulants for various PAM constellations were published in [13], we calculate variances of the sample estimates C 63,x for the first time in this paper. In Tab. 1 the theoretic cumulant values and corresponding variances for various modulation constellations are shown.
The decision-making process for modulation recognition is based on a comparison of obtained values of normalized cumulant estimates with predefined thresholds. It was shown, on the basis of intensive computer simulations, that optimal comparison threshold values are positioned at the middle of intervals between expected (theoretical) values corresponding with particular modulation formats [5].

Performance in Distinguishing Real Constellations from Complex Constellations
In order to compare the expected classification performance of fourth and sixth-order cumulants-based algorithms, in [27] a ratio of standard deviation and mutual distance of nearby values was used, for both C 42,x and C 63,x , as a measure of the algorithm's selectivity. Based on this parameter's value, distinguishing BPSK from QPSK was reported to be better with sixth-order cumulants criteria, and confirmed with the simulations through 2000 Monte Carlo trials and N = 250 received data samples collected for AMC in each trial. An algorithm with C 63,x features was simulated along with an algorithm based on C 42,x features in non-dispersive channel conditions (AWGN only), and with noise power σ g 2 considered to be known. Correct classification probability P CC was calculated versus SNR, with two scenarios of modulation candidates considered from the set {BPSK, QPSK}. These results are presented in Fig. 2.
From Tab. 1 it can be clearly noted that cumulant values of QPSK signals, among all complex constellations, are the closest to cumulant values of all considered real constellations. Thus, the ability of both fourth-and sixth-order cumulants-based algorithm to distinguish a particular real constellation from complex signals can be evaluated through Again, the algorithm with C 63,x features is simulated along with an algorithm based on features C 42,x in AWGN channel conditions, with noise power considered to be known.     As it can be noticed from Fig. 8 An explanation for this effect can be found in the structure of the used C 63,x formula (3). While that formula, originally proposed in [10], is being used by some authors, other authors also report and use different structures for C 63,x in their work: for example, in [3,5,18,28] all mutually different formulas were used. In fact, these formulas mostly represent an approximation, simplified forms of the theoretical cumulant formula, which are correct for some constellations but not in the general case.
The exact sixth-order cumulant formula of random variable x can be expressed from the joint cumulant formula [29]: where  runs through the list of all partitions of {1,…,n}, and B runs through the list of all blocks of the partition  . For zero-mean random variable x, strict developing of all members in the sum of (15) for the cum(x, x, x, x * , x * , x * ) case, leads to the following formula derived here for general and unbiased cumulant: For complex signals, E(x 2 ) = 0 stands and equation (16) gets reduced to the form of (3). Similarly, other forms used by authors who focused on applications with complex signals only are derived. However, it is of interest to cover applications of real signals as well. When equation (16) is used instead of (3) in the AMC algorithm given by equations (4)-(8), the appearance of bias in estimated  63, _ UNB x C values for PAM constellations under low SNR conditions are avoided, but error estimates and mean values of cumulants are much less convenient for AMC application. This is illustrated in Fig. 9, where the situation equivalent to the one presented in Fig. 8 is shown, now corresponding with the application of C 63,x_UNB instead of C 63,x . While for complex constellations data presented in Tab. 1 are also standing with C 63,x_UNB , in Tab. 3 the theoretic C 63,x_UNB values of considered real constellations are shown, since they're different.
Although the performance in distinguishing real constellations from QPSK (and other complex constellations) is clearly better with C 63,x formula, the presence of reported bias makes mutual distinguishing among various PAM constellations practically impossible with this feature: all PAM constellations are affected with bias, so using fixed threshold values based on theoretical cumulants for decision -making in AMC is pointless. On the other side, the structure C 63,x_UNB is relaxed from this issue. Thus, it would be of interest to inspect possibilities for a solution which will benefit from both: 1) bias of C 63,x which provides superior separation of real signals from complex signals, and 2) unbiased C 63,x_UNB structure for the mutual separation of real constellations.

Novel Two-Stage AMC Scheme
The formula given in (16) can be rewritten in the following manner: where C 63,x is given by (3), whose simple expansion with the values of offset O 63,x results in the conversion of biased into unbiased sixth-order cumulants. In practical implementation normalized unbiased sixth-order cumulant can be expressed as: where  63, x C is given by (8). From (18) novel two-stage scheme AMC can be derived, as presented in Fig. 10, which is based on exploiting the benefits of both classical (biased) and general (unbiased) sixth-order cumulant structures.
As it can be noticed from Fig. 10, in the proposed AMC algorithm, within Stage 1, classical sixth-order cumulants of the received signal are estimated and used for distinguishing between real and complex constellations. In this manner, all the excellent properties reported in Sec. 3 of this paper are exploited, and very good classification performance in this step is provided. As a comparison threshold, the "middle of the interval" value between QPSK and PAM-64 (Tab. 1) was used, since these two constellations represent the border cases of two constellation clusters.
If the constellation is initially recognized as belonging to a cluster of real signals, Stage 2 follows where estimates of sixth-order cumulants are first converted into unbiased estimates form by adding the value of estimated offset When compared with the structure of standard sixthorder cumulants AMC algorithm, the one presented in At this point it is of interest to highlight the difference between the novel AMC algorithm proposed here and the one described in [5]: although both solutions provide im- provements in cumulant-based recognition via two-step procedure, their nature and applicability is completely different. Algorithm described in [5] represents an ad-hoc solution for QAM constellations recognition, based on modulation order reduction. Important precondition for its application lays in absence of any bias in cumulants' estimates. On the other hand, the presence of strong bias reported for PAM signals in this work makes the solution [5] being useless in classification of PAM constellations. Instead of any manipulations over the signal structure, here we propose the method based on smart switching between the fundamental formulas for sixth-order cumulants calculation, dedicated for PAM signals exactly, as described previously in this section. It should be also noted that neither the algorithm proposed in this paper makes any improvement in the classification of QAM signals, meaning that two methods remain contributing to completely different classes of signal constellations.

Simulations and Performance Analysis
We carried out the simulations through 2000 Monte Carlo trials and N = 250 received data samples were collected for AMC in each trial, where the algorithm proposed in Sec. 4 was tested in AWGN channel conditions, in scenarios with modulation candidates considered from the set: can be found at [30].
In order to provide a fair comparison of achieved performance with other comparable algorithms, the AMC algorithm based on standard fourth-order cumulants [9] was simulated as well, under the same set of modulation candidates and sample size N = 250, along with the AMC algorithm based on standard (biased) sixth-order cumulants [10]. The value of N was selected to match directly with the one used in simulations described in [9] and represents the main controlling parameter from the aspect of AMC performance. It should be also noted that the algorithm [9] is unbiased in its nature (i.e. for all simulated signal constellations). The AWGN channel was simulated with noise power σ g 2 considered to be known. For each particular SNR value, and within every particular Monte Carlo trial, the same set of N samples was processed by all simulated algo-rithms, thus providing the fair comparison under the exactly same channel conditions and over the exactly same input data.
Comparison thresholds for sixth-order cumulants AMC algorithm are having the values as previously described in this paper, while comparison thresholds for the algorithm [9] were selected as the "middle of interval" values between the theoretical fourth-order cumulants of considered modulation formats (Tab. 1).
Correct classification probability P CC was calculated versus SNR, and Figure 11 illustrates the results of simulation in scenario (i); results in scenario (ii) are presented in Fig. 12, while Figure 13 illustrates the results of simulation in scenario (iii).
As can be confirmed from Fig. 11 and Fig. 12, the proposed novel two-stage AMC algorithm based on sixthorder cumulants shows better performance than the algorithm based on standard fourth-order cumulants, and this difference in performance is more significant as the SNR value is getting lower. Performance of standard sixth-order cumulants is generally poor, due to the presence of bias, as described in this paper, but its robustness in the classification   of complex (QPSK) signal remains obvious even at low SNR. Achieved classification performance of the proposed algorithm comes mainly from this, well-exploited, excellent classification of complex (QPSK) signals, while moderate performance in recognition of real constellations is provided at the same time -very similar to the one given by the standard fourth-order cumulants. Also, it should be noted that as the number of constellations under test rises (four constellations in scenario (ii), in comparison with three constellations in scenario (i)) general performance of all simulated algorithms gets lower in terms of P CC . This is especially illustrated in Fig. 13, where a wide set of seven constellations was considered in simulation, i.e. all real constellations discussed in this paper, along with QPSK.
From Fig. 13 it can be noted that at SNR = 20 dB the most competitive algorithms under the test of seven constellations show the performance of around P CC = 0.55, while for the same SNR value in a scenario with four constellations ( Fig. 12) their performance was approximately P CC = 0.85. Nevertheless, Figure 13 confirms again that the proposed novel AMC algorithm based on sixth-order cumulants outperforms the classical algorithms based on higher-order cumulants, even in the problem with the widest set of modulation candidates.
In order to explore the impact of sample size N on reported classification performance, we repeat simulations in scenario (iii), now with N = 2000 received data samples for AMC. The results are presented in Fig. 14.
As can be concluded from Fig. 14, bigger sample size N directly contributes to the classification performance of algorithms under test: at SNR = 20 dB the most competitive algorithms reach the value of P CC = 0.65 with N = 2000 samples, which represents an obvious improvement in comparison with the same scenario and N = 250 samples from Fig. 13. Again, the achieved classification performance of the proposed new AMC algorithm based on sixth-order cumulants is confirmed as better than of the classical algorithm based on fourth-order cumulants. The algorithm based on standard sixth-order cumulant structure is outperformed by the others in all considered scenarios.
Reported performance stands in direct correspondence with the ability of a particular algorithm to distinguish complex from real signals effectively. Simulations showed that distinguishing complex (QPSK) signals from real constellations is errorless with fourth-order cumulants and N = 250 samples at SNR > 7 dB, while with sixth-order cumulants and the same N value distinguishing complex from real constellations is errorless even around SNR = 2 dB. With N = 2000 samples, fourth-order cumulants achieve QPSK signal classification without any errors at SNR > 4 dB, while the proposed sixth-order cumulants algorithm provides an errorless classification of QPSK signals even around the values low as SNR = -1 dB. Reported numerical values are stated directly in an observed performance presented in Fig. 15.
Although the only channel disturbance source considered (and simulated) in these experiments is the noise, the nature of cumulants provides that all presented considerations will also stand in the case of the flat-fading channel (this property results directly from the definition of self-  normalized cumulants, as presented in Sec. 2). For more complex channel models additional channel estimation procedure needs to be introduced in simulated algorithms, like [9], [10], representing one of the most important tasks for future work and considerations of proposed novel twostage AMC, along with the conditions of interference and other channel imperfections.
Achieved performance improvement with the AMC algorithm proposed in this paper comes at the cost of some added numerical complexity: (N + 1) new multiplications and (N + 1) new additions are introduced for the calculation of offset and conversion of the standard into the unbiased form of sixth-order cumulants, in comparison with classical sixth-order cumulants, as described with (18) and Fig. 10. Still, those introduce no practical change in the overall complexity of the AMC algorithm, meaning that it remains very competitive in terms of numerical and memory resources, and inference time [5]. Finally, the algorithm proposed in this paper can be expanded further with more complex classifiers (like various neural network structures), in applications where introducing more significant overall computational complexity represents an acceptable cost for providing even higher classification performance.

Conclusion
In this paper, a novel scheme for improving the performance of AMC based on sixth-order cumulants, in the form of a simple two-stage feature extraction structure is presented, which exploits benefits from both classical and theoretical (unbiased) sixth-order cumulant structures. Statistical properties of classical sixth-order cumulants of various PAM constellations were analyzed for the first time; their superior performance in distinguishing real from complex constellations was tested in computer-aided simulations, and explained through the presence of bias, which was reported for the first time for this class of signals. This presence of bias was also identified as an obstacle for mutual distinguishing of real constellations with classical sixth-order cumulants. The difference in the structure was explained between standard sixth-order cumulants and theoretical cumulants of the same order, which are relaxed from the presence of bias, thus being suitable for the classification of particular real constellations. This difference was then further used for the implementation of a simple mechanism for conversion between two cumulant structures in practical AMC. The same mechanism was incorporated in the novel two-stage AMC scheme. In computer simulations performance of a new classification algorithm was compared with the performance of a popular AMC algorithms based on standard sixth-and fourth-order cumulants, and better results with the new algorithm were reported in several simulation scenarios. Expectations in the practical use of tested algorithms were also discussed, in the context of SNR and sample size N values. Achieved new performance of the proposed algorithm is coming at the price of (some) added complexity, which was also discussed, but making no significant impact on the overall complexity of sixth-order cumulant structures application in AMC, thus providing their well-known competitiveness to remain high. The proposed algorithm can be further combined with nowadays popular complex structures for enhanced feature extraction, like neural networks or deep learning methods, for potentially achieving even better performance. For future work, the proposed algorithm should be tested in propagation conditions that are not limited to the presence of noise only, but also consider the effects of multipath propagation, interference, and other real-world conditions.