# Random Graph-Based M-QAM Classification for MIMO Systems

Authors: Mubashar Sarfraz; Sheraz Alam; Sajjad A. Ghauri; Asad Mahmood; M. Nadeem Akram; M. Javvad Ur Rehman; M. Farhan Sohail; Teweldebrhan Mezgebo Kebedew
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

Automatic modulation classification (AMC) has been identified to perform a key role to realize technologies such as cognitive radio, dynamic spectrum management, and interference identification that are arguably pivotal to practical SG communication networks. Random graphs (RGs) have been used to better understand graph behavior and to tackle combinatorial challenges in general. In this research article, a novel modulation classifier is presented to recognize M-Quadrature Amplitude Modulation (QAM) signals using random graph theory. The proposed method demonstrates improved recognition rates for multiple-input multiple-output (MIMO) and single-input single-output (SISO) systems. The proposed method has the advantage of not requiring channel/signal to noise ratio estimate or timing/frequency offset correction. Undirected RGs are constructed based on features, which are extracted by taking sparse Fourier transform (SFT) of the received signal. This method is based on the graph representation of the SFT of the 2nd, 4th, and 8th power of the received signal. The simulation results are also compared to existing state-of-the-art methodologies, revealing that the suggested methodology is superior.


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## Body

## 1. Introduction

Automatic modulation classification (AMC) is utilised by the observation of received data sample, received data samples which are noise-related, and different fading effects to automatically determine the modulation type of the broadcast signal. There are numerous military and civilian applications of AMC such as electronic warfare, signal surveillance and spectrum sensing, and cognitive radio. The modulation classification procedure takes place after preprocessing prior to demodulation of the incoming signal [1–7]. The likelihood-based techniques are based on the signal received probability, while the feature-based methods contain two modules: function extraction and structure of the classifier [8–15].Random graphs (RGs) have been used to model a multitude of random-like networks, including the unpredictable growth of the Internet’s web graph, page ranking, and neural networks. The concept of a random graph was developed by Erdos and Rényi in 1959 as an extension of the probabilistic approach for discovering the existence of certain graph features [16]. They introduce two closely related models Gn,m and Gn,P, where Gn,m is the ensemble of graphs having exactly m edges and n vertices with equal probability and Gn,P is obtained by introducing each of the available chart edges individually with probability of P. The diagram is considered to not be directed, such that no direction is linked to the edges if these pairings are unordered. Moreover, if some randomness is involved in its formation, a graph is said to be random.QAM signals are used in microwave digital radio for transmission with enhanced spectral efficiency over limited channel bandwidths. It is used widely in many modern domestic applications such as digital cable TV, cable modems, power-line Ethernet, and microwave backhaul systems. In particular, it is used in almost all the 802.11 standards including common a, b, g, and n to latest ac, ah, and ad. On the other hand, one technology has revolutionized wireless communication technologies as a key enabler to provide high data rates in MIMO antenna systems. The combination of M-QAM and MIMO is used in IEEE 802.11 ac and ad standards. This is the reason for choosing M-QAM signals in this research work in combination with different antenna configurations such as SISO, SIMO, MISO, and MIMO for detailed analysis.


### 1.1. Contribution of the Research

In this paperGn,P model is considered for the construction of undirected RGs [17]. The main contribution of our research work is to classify the M-QAM signals using random graphs which to the best of our knowledge is not yet to be done in the literature. Random graph theory is seldom explored for modulation classification, in particular for M-QAM signals that are widely used with MIMO technology. The features are extracted by using sparse Fourier transform (SFT) and RGs for the classification. The most striking feature of the proposed solution is that it neither required channel/SNR estimation and the performance is much better for SISO and MIMO on different channel conditions and in the presence of additive white Gaussian noise.


### 1.2. Organization of the Article

The rest of the paper has the following structure: Section2 provides a short overview of related work for AMC. The system model and the major phases of the algorithm suggested are explained in Section 3 and Section 4, respectively. In Section 5, simulation parameters and results are shown. The article is then completed by comparing the suggested method with current state-of-the-art procedures in various future areas in Section 6.


## 1.1. Contribution of the Research

In this paperGn,P model is considered for the construction of undirected RGs [17]. The main contribution of our research work is to classify the M-QAM signals using random graphs which to the best of our knowledge is not yet to be done in the literature. Random graph theory is seldom explored for modulation classification, in particular for M-QAM signals that are widely used with MIMO technology. The features are extracted by using sparse Fourier transform (SFT) and RGs for the classification. The most striking feature of the proposed solution is that it neither required channel/SNR estimation and the performance is much better for SISO and MIMO on different channel conditions and in the presence of additive white Gaussian noise.


## 1.2. Organization of the Article

The rest of the paper has the following structure: Section2 provides a short overview of related work for AMC. The system model and the major phases of the algorithm suggested are explained in Section 3 and Section 4, respectively. In Section 5, simulation parameters and results are shown. The article is then completed by comparing the suggested method with current state-of-the-art procedures in various future areas in Section 6.


## 2. Related Work

The AMC literature is largely divided into two techniques: likelihood-based classification (LBC) and feature-based classification (FBC). The existing work on the likelihood-based AMC can be found in [18–23] and feature-based AMC in [24–30].Convolutional neural networks (CNN) are based on convolution instead of matrix multiplication [31]. In grid constellation matrix- (GCM-) based AMC, a contrastive fully convolutional network (CFCN) is used; this network benefits from a deep learning method and high-dimensional representations from variables via GCMs and a quick training procedure [32].The constant envelope modulation (CEM) classification is performed using a radial basis function network (RBFN) with a deep learning network and a sparse autoencoder-based neural network (SAEDNN) without any prior knowledge of the CEM or channel [33]. In curriculum learning, two neural networks are used: MentorNet for supervision and StudentNet is trained for the classification [34].Authors in [35] uses K-nearest neighbour (KNN) with several distance measuring methods, including correlation, Mahalanobis, Euclidean, and Minkowski distance in combination with HOS to identify signals. In [36], a unique feature engineering to improve the efficiency of KNN for NRZ and PAM-4-modulated signals is introduced. The authors of [37] use GP to boost the effectiveness of the KNN classifier by using super features. The Gabor filter network (GFN) was used to extract features and classify digitally modulated signals. The delta rule is used to adaptively modify GFN parameters, which are then tuned with GA using the HMM classifier [38, 39].Current approaches often adopt a single neural network category, or stack different network categories in series, and seldom properly extract diverse types of information. Softmax is employed in classification to widen the distance between classes when it gets to the output layer. A parallel hybrid network for the AMC problem is suggested in [40].In [41], the authors present a deeper learning feature fusion system that tries to fuse features from multiple domains of the input signal in order to achieve a more reliable and efficient display of signal modulation kinds. The complementarity of features which may be utilised to remove the impact of the background sounds and the wide dynamic range of (intercepted) signalling is taken into consideration.For improved feature variety, a hierarchical multifunction fusion approach is presented in [42]. Modulation features are represented by the state signals, and its process of statistical ordering characterises time on a regular basis. Initially, the authors recommended that you use various dimensional convolution filters based on two streams to fully leverage IQ signal characteristics. Parallel to each stream multidimensional convolution filters are used to better characterise individual channels and to improve the interactions between them. A small LSTM network is subsequently used to further exploit sequential data.In [43], categorization of radar transmitter signal intrapulse modulation refers to the classification of the pulse of each radar emitter to a certain model. An intrapulse modular intrapulse 1-D selective kernel convolutional neural network called 1-D SKCNN is proposed. A basic purpose is to construct a parallel hybrid structure that uses the CNN and Gate Rate Unit (GRU) to draw spatial characteristics and time characteristics separately.AMC with a feature-based two-lane capsule network based on these concerns is suggested in [44]. As channel characteristics and signal overlaps in MIMO systems lack prior knowledge, standard probability-based and feature-based methods cannot be used directly in these circumstances. Therefore, the time-frequency analysis technique based on the Fourier transform window was used to evaluate the time-frequency characteristics of the time-domain-modulated signals in order to tackle the problem of blind modulation classification in MIMO systems [45].Chang and Shih [46] proposes a radio transformer that takes instantaneous amplitude, instant phase, and instantaneous frequency from received RCB signals and converts them into images by the proposed signal rearrangement to complete the classification of radio modulation using existing ImageNet classification models.The aims of [47] are to resolve the issue of low identifying precision under single-feature parameters and to decrease the sampling system performance requisites by using a blind modulation classification method based on the compressed sensing with a high-order, cumulative, and cyclic spectrum associated with the decision-tab support vector machine classifier.


## 3. System Model

The system model and the proposed algorithm for the classification of M-QAM signals are discussed in detail in this section. Figure1 shows the proposed system model in which the received signal is first preprocessed and discrete Fourier transform (DFT) is applied. After extracting parameters and then selecting distinct features, random graphs are generated. The threshold decides the modulation scheme. The received signal in generalized form is given by
(1)r=Hs+η,where η is additive white Gaussian noise with zero mean and variance of σ2, H is a matrix of channel coefficients with order Nr×Nt. s is Nt×K vector of transmitted symbols, and r is Nr×K the received signal vector, where Nt and Nr are the number of transmitter and receiver antennas.Figure 1
Proposed system model.


## 4. Proposed Algorithm

The proposed algorithm for M-QAM signal classification based on sparse DFT (SDFT) of the received signal with powerp has the following steps.Step 1.
Calculate theSDFTrp, where p=2,4,8.Step 2.
To generate the random graphs, the detailed process is as follows:(1)
Normalization of sparse samples is given as(2)χnormz=χz−minχzmaxχz−minχz,where χnorm is the normalized discrete sample of the received signal and 0≤χnormz≤1(2)
Rayleigh cumulative distribution function (RCDF) is employed to obtain the uniform distribution(3)
Uniformly quantized the distribution for predefined quantization levelsQa(4)
Count the numbers of peaks and set the numbers of nodesn equal to Qa(5)
Calculate the probability by using equation(3)P=1−1−2Qaκ−1,where κ is the number of observed samples
(6)
To construct the adjacency matrix, Bernoulli trials were performed with a biased coinStep 3.
Calculate the maximum numbers of edges using the relationship:(4)Ne=QaQa−12.
Calculate the estimated number of edges:(5)N~e=numbersofonesinadjacencymatrix2.
Hence, ifNe=N~e, then the graphs are fully connected; otherwise, graphs are disconnected. However, that is a strict decision for graph connectivity. The threshold level for deciding the connectivity of the graph is defined as follows:
(6)λ=floor0.9×Ne.
Now to check the connectivity of a random graph, compareN~e with λ instead of Ne. The decision function for the connectivity of RGs is defined as follows:
(7)GM,p=1,N~e≥λ,0,ow,where M=8,32,64,128 and p∈2,4,8. Figure 2(a) depicts the fully connected graph for 32-QAM, 64-QAM, and 128-QAM, and Figure 2(b) shows the disconnected graph for 8-QAM when SDFT is taken with a power p=2 at SNR of 10dB. Similarly, RGs for p=4 are also generated which shows fully connected for 32-QAM while disconnected graph for 64-QAM and 128-QAM. For, p=8, the graph is fully connected for 128-QAM and disconnected for 64-QAM. The pseudocode of the proposed algorithm is shown in Algorithm 1.Figure 2
Fully connected and disconnected graphQa=10, NT=1, NR=1, and Rayleigh channel. (a) 32-QAM, 64-QAM, and 128-QAM and (b) 8-QAM.
(a)(b)Algorithm 1: Algorithm of the proposed problem.
1 initialization;2 whileM-QAM≠classifieddo3  evaluate⟶ SFT (r) (where r is received signal);4  apply⟶ Normalization (transformed r);5  evaluate⟶ Rayleigh CDF (normalized signal);6  for j=1:K do7    for k=1:K-1 do8      connect each ith edge to jth edge;9    end10  end11 ifgraph ==fully connectedthen12  class 1 is separated;13 else14  take next power and continue classification;15 end16 end


## 5. Simulation and Results

The simulation parameters for the M-QAM signal classification are presented in Table1. Various types of systems are considered for simulation such as single-input single-output (SISO), single-input multiple-output (SIMO), multiple-input single-output (MISO), and multiple-input multiple-output (MIMO). To validate the results, MATLAB built-in functions and communication system toolbox are used. M-QAM signal classification is also carried out under different fading channel conditions, i.e., Rayleigh, Rician, and Pedestrian channels. Spatial multiplexing is employed for multiple antennas at the transmitter and receiver sides. The figure of merit is the average percentage of correct classification (APCC).Table 1
Simulation parameters.
ParameterValuesNumbers of iterations50ChannelPedestrianNumbers of samples (K)8000Quantization levels10Channel length4Path delays[0, 110, 190, 410] nsAverage path gains[0, -0.97, -12.9, -22.8] dBMaximum Doppler shift8 HzSample rate12.5 MHzRayleigh CDF parameter0.25Figure3 represents the decision tree for the M-QAM classification, where p represents the order of the DFT. The G=0 and 1 represents that the random graphs are disconnected and fully connected, respectively. The higher SNR is considered in Figure 3 for the optimum decision-making. At lower SNRs, the decision-making process is suboptimum, and there are more chances of misclassification. M-QAM signal classification using RGs is achieved into three steps at SNR of 10 dB with 10 quantization levels:
(1)
In the first step, ifp=2, then there are two possibilities: if a graph is fully connected, then the received signal may be modulated using by using 32-QAM, 64-QAM, and 128-QAM. If a graph is disconnected, then the received signal is modulated using 8-QAM(2)
In 2nd stepp=4. If a graph is fully connected, then the received signal is identified to be 32-QAM and the MC process stops here; otherwise, the signals may be modulated using 64-QAM or 128-QAM(3)
In last step,p=8. If a graph is fully connected, then the received signal is modulated using the 128-QAM scheme otherwise 64-QAMFigure 3
Decision tree for M-QAM signal classification.The APCC for M-QAM signals using RGs on the Rayleigh and Rician fading channels with AWGN is simulated in Figure4. The parameters for the Rayleigh and Rician channels such as path delays and path gains are 0,110 nanosecond and 0,−9.7 dB. K-factor for the Rician channel is selected 6,6 dB with a maximum Doppler shift of 8 Hz. As shown from Figure 4, APCC for the AWGN channel is 100% because random graph is fully connected for 64-QAM constellations, while for the case of Rayleigh channel model, the APCC is 100% at 8 dB of SNR and for Rician channel the APCC is 100% at 4 dB of SNR. The fully connected random graphs show 100% accuracy at lower SNRs, while disconnected random graphs show lower APCC for lower SNRs.Figure 4
APCC for AWGN, Rayleigh, and Rician channels (64-QAM).The proposed algorithm is also validated on various types of systems such as SISO, SIMO, MISO, and MIMO. Therefore simulation scenarios are divided into four cases, and in all cases, the ITU-R Pedestrian fading channel is taken for the classification of M-QAM signals using RGs. For the simplicity of the results, the 8-QAM candidate constellation is selected to check the validity of our proposed cases.Case 1.
In the first case, the number of transmitter and receiver antenna is 1, i.e., SISO system. The APCC for Case1 is approaching 100% at 6 dB of SNR. Figure 5 shows the APCC is more than 95% at an SNR of 0dB.Figure 5
APCC for Case1 (SISO).Case 2.
In the second case, there is only one transmitter antenna and multiple receive antennas, i.e., 2, 3, and 4. The APCC is approaching 100% at 6 dB of SNR. At lower SNRs and forNr=2; APCC is better as compared to Nr=3 and Nr=4. The APCC for Case 2 (SIMO) is given in Figure 6. At higher SNRs, larger number of receive antennas gives better APCC as compared to small number of receive antennas. The performance of proposed algorithm is worst on lower SNRs, i.e., below -4 dB. As the signal is dominant over noise, the APCC for Nr=4, which is better as compared to the Nr=2 and 3. For example, at SNR=−2 dB, the APCC is 97.5% at Nr=4, while at Nr=2, the APCC is approximately 96%.Figure 6
APCC for Case2 (SIMO).Case 3.
Multiple transmitter antennas such as 2, 3, and 4 are considered with a single receiver antenna. The APCC is shown in Figure7, and it is clear from Figure 7 that the accuracy approaches 100% at lower SNRs. The APCC is an approximate approach to 99% at 0 dB of SNR for Nt=2, 3, and 4.Figure 7
APCC for Case3 (MISO).Case 4.
In Case4, multiple transmit and receive antennas are considered, and APCC is shown in Figure 8. At 2 dB of SNR, the APCC is 99.6% for the 2×2 system and 100% for 3×3 and 3×4. From Figure 8, it is clear that the overall APCC of the MIMO system is much better at lower SNRs as compared to SISO, SIMO, and MISO.Figure 8
APCC for Case4 (MIMO).To evaluate the effect of phase offset on classifier performance, the APCC is shown in Table2. Degradation and variation in the classifier performance can be observed at various values of SNR from -10 to 10 dB. From Table 3, it does not mean that bigger offset leads to better classification accuracy. For example, at 5 dB of SNR and π/3 offset, the APCC is 96.59% and at π/12 the APCC is 97.33%. At lower SNRs (i.e., -5 dB), the noise is dominant and the behavior of the classifier is random, e.g., at π/4, the APCC is 81% while at π/12 the APCC is 81.78%.Table 2
Effect of phase offset on APCC.
SNR (dB)Phase offsetπ/12π/6π/4π/3-1066.70%60%70%76.67%-581.78%77.78%81%85.11%089.07%86.44%88%90.63%597.33%96.3%96.31%96.59%10100%100%100%100%Table 3
Comparison with state of art existing techniques.
SNR (dB)Complexity analysis-10-8-6-4-20246810[24]28%33%46%57%68%76%81%90.5%96%96.5%98.5%OKD2[31]80.2%84.8%88.6%95.7%99%100%100%100%100%100%100%OKND2ON2[48]47%61%72%83%86%87%90%91%92.5%93%94%OKND2ON2[49]81%81.5%81.5%83%86%91%93%95%96.5%97%97.5%OND2Proposed classification schemeSISO60%70.1%77.45%84.34%89.42%93.23%95.9%97.69%98.8%100%100%ON2SIMO80%86.37%90.49%94.22%96.67%98.4%99%99.75%99.96%100%100%ON2MISO93.3%96%97.47%99.6%99.87%100%100%100%100%100%100%ON2MIMO96.67%98.3%98.8%99.64%99.9%100%100%100%100%100%100%ON2APCC is also affected via increasing the number of samples as shown in Figure9. A various number of samples are taken, i.e., K=2000 to K=10000. Increasing the numbers of samples, the APCC increased and vice versa. In Figure 9, the SISO system with the Pedestrian channel and 10 quantization levels is employed.Figure 9
Effect ofK on the classification performance 64-QAM signal.The performance of proposed algorithm is also compared with the state of art existing techniques as shown in Figure10. From Figure 10, it is clear that the proposed algorithm performs much better as compared with the existing techniques for any number of transmitter and receiver antennas.Figure 10
Performance compassion of proposed M-QAM classifier with state of art existing techniques.The number of samples is 8000 withITR=50 and APCC is shown in Figure 11, which shows the approximately same results as for DFT and SFFT. The time complexity of DFT is a little bit more as compared to SFFT of the 3rd level.Figure 11
APCC for DFT and multilevel SFFT (M-QAM).The comparison of APCC and their complexity analysis with state of art existing techniques is shown in Table4. The proposed algorithm performs much better as compared to the existing techniques. The proposed classifier achieves approximately 100% classification accuracy at lower SNRs. Furthermore, authors in [28] considered an AWGN channel model as compare to the proposed simulations the APCC is 100% for lower SNRs as shown in Figure 4. In Table 4, ITU Pedestrian channel model is considered for all the cases.Table 4
Execution time of DFT and SFFT.
Features extraction techniqueTime (sec)DFT11.18 s1st step SFFT10.64 s2nd step SFFT10.16 s3rd step SFFT9.98 sIn this research paper, SFFT is employed to reduce the proposed algorithm time complexity. Table4 shows the multilevel SFFT with the time in seconds.Table4 shows the time complexity comparison of using DFT and sparse FFT in seconds. The feature extraction from the DFT and 3rd step SFFT time complexity is 11.18 seconds and 9.98 seconds, respectively. The time complexity reduces at 3rd step SFFT which will cause the lower classification accuracy. The time saving affects the classification accuracy as shown in Figure 11.


## 6. Conclusions

The purpose of this research is to improve the classification accuracy of M-QAM signals under low SNR or unsatisfactory channel conditions. In this research, the undirected random graphs are constructed with DFT and sparse transform-based features to classify the M-QAM signals with various system models such as SISO, SIMO, MISO, and MIMO. Sparse Fourier transform is taken for 2nd, 4th, and 8th power of the received signal with the different fading channel scenarios to confirm the validity of the results. The novel classifier proposed in this article has an advantage that it requires neither channel estimates nor the timing/frequency offsets. Additionally, the proposed classifier is also compared with the state-of-the-art existing techniques, and from the simulation results, it is evident that all the considered M-QAM signals are classified accurately even at lower SNRs with less time complexity.The further classification accuracy may be enhanced using different statistical features and for the application area of massive MIMO.



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