9.1 Introduction

In previous chapters we have taken a theoretical view of modulation classification. The listed classifiers have mostly been developed based on general assumptions. The performance comparison in Chapter 8 gives a general impression of how each classifier performs in different scenarios.

In this chapter we visit some of the civilian communication systems with special AMC requirements that have not been covered previously. First, we investigate systems where higher-order modulations are deployed. Secondly, the application of modulation classifiers in link adaptation is revised with specific classifiers designed to exploit some of the properties of the system. Lastly, the case of blind modulation classification in MIMO systems is discussed.

9.2 Modulation Classification for High-order Modulations

In real world communication systems there are many instances where high-order modulations are employed for high data rate transmission. These systems mostly rely on wired transmission links to achieve ultrahigh spectrum efficiency. Among them, 64-QAM and 256-QAM are used for digital terrestrial television and its high-definition version. Broadband over power line (BPL) uses 1024-QAM and 4096-QAM modulations. To design a classifier for high-order modulation over wired channel, we need to understand the following characteristics of these systems.

First, the wired channel is relatively stable compared with wireless channels. Therefore, the classifier does not need to adapt to the change of channel conditions. In other words, the channel estimation can be implemented at the initialization of the system. The estimated channel parameters can be used in the remainder of the transmission. Secondly, the high-order modulations are normally more difficult to classify, as concluded in Chapter 8. Therefore, to guarantee successful classification, a classifier of high classification accuracy is needed. Among all the modulation classifiers, the ML classifier, distribution test classifier and the higher-order cumulant features are reported to have superior performance for high-order modulations. Thirdly, because of the large number of symbol states, in a single piece of signal frame, it cannot be guaranteed that the symbol assignment will be equally probable. In certain cases some of the symbol states will not be used in the transmission. This condition has a significant impact on the distribution test-based classifiers. For a KS test classifier, when some of the modulation symbol is not observed in the received signal, the empirical distribution function will exhibit big differences at the missing symbols, since in the theoretical CDF these symbol positions normally possess higher probability values. For higher-order cumulant-based classifiers, it is obvious that the signal length is of great importance. From Figure 8.17 we can see that among three of the QAM modulations considered in the simulation, the higher-order QAM modulations make a higher impact when limited signal length is available for analysis. When the modulation is increased to 1024-QAM or 4096-QAM, it is clear that the higher-order cumulants will not be able to provide the demanded classification accuracy. On the other hand, the ML classifier has suggested superior performance for both high-order modulation as well as limited signal length. Therefore, the ML classifier is the ideal option for the classification of high-order modulations in wired communication systems.

With the above analysis, we construct the AMC solution in a wired communication system using high-order modulations. The overall process is illustrated in Figure 9.1. The heaver dashed line indicates the transmission of pilot samples to the channel estimator in system initialization. Thanks to the pilot samples, many state-of-the-art channel estimators can be used to acquire the CSI rather easily. The thinner dashed line indicates the exchange of CSI and modulation candidate pool from the channel estimator to the ML classifier. The CSI and information of the modulation candidate pool are then saved in the ML classifier to assist modulation classification. They are not updated until the system is reconfigured. During normal transmission, the signals will be transmitted through the wired channel indicated by the heavy solid line. The ML classifier will analyze each signal and evaluate the corresponding likelihood for each modulation candidate in the pool using equation (3.3). When the likelihood values for each modulation hypothesis are obtained, the classification decision can be easily reached using the maximum likelihood criteria given in equation (3.13). The demodulator is then informed of the modulation type of the received signal from the ML classifier. If the classification is correction, the transmission is successfully completed.

9.3 Modulation Classification for Link-adaptation Systems

Link adaptation, also known as adaptive modulation and coding, adaptively selects the modulation method depending on the channel conditions. The mechanism of the LA system with AMC has been briefly discussed in Chapter 1. The system configuration is illustrated in Figure 9.2.

Since the selection of the modulation scheme depends on the channel conditions, the completed posteriori probability of the signal being received with a modulation type can be expressed as a combination of the prior probability of symbol s_m being transmitted and the likelihood of the received signals belonging to the transmitted symbol, as shown in equation (9.1).

(9.1)

The prior probability can be automatically achieved before the next batch of signals is transmitted. The incorporation of the prior probability in a modulation classifier is most natural for a likelihood-based classifier. Depending on the implementation, the link adaptation could depend on the estimated channel or the acknowledgement and negative acknowledgement protocol. However, this is outside the scope of this book. We assume that the prior probability with regard to the channel information is known to the classifier. Instead of the maximum likelihood criterion one can adopt the maximum a posteriori (MAP) criterion to accommodate the added information of modulation selection prior probability. The application of MAP in modulation classification was first suggested by Haring, Chen and Czylwik (2010). The log MAP likelihood could be calculated using equation (9.2).

(9.2)

The classification decision can be given by finding the modulation candidate with the highest a posteriori likelihood, as determined by equation (9.3).

(9.3)

9.4 Modulation Classification for MIMO Systems

Multiple-input and multiple-output (MIMO) systems have been the key technology in many recent communications innovations. A MIMO system employs spatially differing transmission and receiving antenna arrays. MIMO systems enable the transmission of multiple signal streams through spatially different signal paths, which is known as spatial multiplexing (SM). The diversity of the signal paths also provides the possibility of improved link reliability. When space-time coding (STC) is used, multiple versions of the signal symbol are transmitted simultaneously. With accurate channel knowledge, the received signals can be recovered more accurately.

MIMO systems differ from most modulation classifiers due to the multiple paths between the transmission antenna arrays and the receiving antenna arrays. Since each receiver provides a mixture of single symbols from all transmitters, the modulation classification approach in single-input and single-out system cannot be used any more. The existing modulation classifiers must be modified to meet the new requirements as shown in Figure 9.3.

The MIMO systems are composed of N_i transmitting antennas and N_r receiving antennas. A Rayleigh fading channel with time-invariant path gains is considered. The resulting channel matrix H is given by an N_r × N_i complex matrix with the element h_j,i representing the path gain between ith transmitting antenna and jth receiving antenna. Assuming perfect synchronization, the nth received MIMO-SM signal sample vector r_n = [r_n(1), r_n(2), …, r_n(N_r)] ^T in a total observation of N samples is expressed as given by equation (9.4),

(9.4)

where s_n = [s_n(1), s_n(2), …, s_n(N_t)] ^T is the nth transmitted signal symbol vector and ω_n = [ω_n(1), ω_n(2), …, ω_n(N_r)] ^T is the additive noise observed at the nth signal sample. The transmitted symbol vector is assumed to be independent and identically distributed with each symbol assigned from the modulation alphabet with equal probability. The additive noise is assumed to be white Gaussian with zero mean and variance σ², which gives , where I_Nr is the identity matrix of size N_r × N_r.

From an ML classifier, Choqueuse et al. developed the MIMO version of the likelihood-based classifier (Choqueuse et al., 2009), with an updated version of the likelihood function, given in equation (9.5),

(9.5)

where is the Frobenius norm. The possible transmitted symbol set S = [S₁, S₂ … S_M] gathers all the combinations of transmitted symbols from N_i number of antennas. Given a modulation with L number of states, there exist number of transmitted symbol vectors and a transmitted symbol set of size . The channel matrix and noise variance are assumed to be known. The corresponding log-likelihood function can be derived as equation (9.6).

(9.6)

The subsequent classification decision can be made by finding the modulation candidate which provides the highest likelihood, as calculated using equation (9.7).

(9.7)

The method provides high classification accuracy, but with a rather high computational complexity. Alternative classifiers with reduced complexity have been proposed in several publications (Kanterakis and Su, 2013; Mühlhaus et al., 2013). Kanterakis and Su suggested treating independent component analysis (ICA) recovered signal components as independent processes. The recovered source signal components are acquired by using equation (9.8),

(9.8)

where is the inverse of the estimated channel matrix. For each independent transmitted signal stream, the updated likelihood function is given by equation (9.9).

(9.9)

In addition to the likelihood-based classifier, Choqueuse et al. also suggested the use of independent component analysis for the estimation of the channel matrix when it is not readily known. Cardoso and Souloumiac’s JADE algorithm was suggested (Cardoso and Souloumiac, 1993). The resulting estimation of channel matrix has a phase shift and needs phase correction. The ICA estimation provides channel matrix estimation but the estimation for noise variance is still lacking. In addition, there are limits to the MIMO arrangement that can be considered. When the number of transmitting antennas exceeds the number of receiving antennas, the ICA estimation cannot be implemented. To overcome these issues related to an ICA-aided modulation classifier for the MIMO system, we suggest an expectation maximization estimator for more practical implementation of the likelihood-based classifier for MIMO systems.

To evaluate likelihood for the ML classifier, the complex channel matrix H and noise variance σ² must be estimated beforehand. Since the modulation is unknown to the receiver, many data-aided approaches using pilot symbols are not suitable. Expectation maximization has been employed for joint channel estimation through an iterative implementation of maximum likelihood estimation (Wautelet et al., 2007; Das and Rao, 2012). In MIMO systems, we consider the received signal R = [r₁, r₂ … r_N] as the observed data. Meanwhile, the membership Z of the observed samples is considered as the latent variable. Z is an M × N matrix with the (m,n)th element being the membership of the nth signal sample r_n , given the transmitted symbol vector S_m. The possible transmitted symbol set S = [S₁, S₂ … S_M] gathers all the combinations of transmitted symbols from N_t number of antennas. Given a modulation with L number of states, there exist number of transmitted symbol vectors and a transmitted symbol set of size . With Θ = {H, σ²} representing the channel parameters, the complete likelihood is given by equation (9.10),

(9.10)

where p(R|S, Θ) is the probability of the received signal being observed given transmitted symbols vector S and channel parameter Θ. Since the additive noise is assumed to have a complex Gaussian distribution, p(R|S, Θ) can be calculated as given in equation (9.11).

(9.11)

Meanwhile, p(S|R, Θ) represents the probability of S being transmitted given the observed signal R and the channel parameter Θ. In Wautelet et al. (2007), this probability is acquired by an a posteriori probability computation which is not presented. In this book, we replace the a posteriori probability with a soft membership z_nm representing the likelihood of the nth transmitted symbol vector being S_m, with . Since assignment of the transmitted symbol is independent of the channel parameter, p(S|Θ) is a constant, 1/M, when equal probability is assumed. The estimation of Θ is achieved by iterative steps of expectation evaluation and maximization.

The evaluation step (E-step) provides the expected log-likelihood under the current estimate of Θ^t at the tth iteration. The expectation is then subsequently maximized for the updated estimation of Θ. From equation (9.11) the expected value of the complete log-likelihood is derived as shown in equation (9.12),

(9.12)

where is the probability of the nth received signal vector being observed given the current estimation of channel matrix H_t and noise variance . The soft membership z_nm is evaluated equation (9.13).

(9.13)

The update of the parameter estimation is achieved through the maximization of the current expected log-likelihood (M-step). To derive the close form update function for the channel matrix and noise variance, we first find the derivatives of Q(R,S|Θ^t ) with respect to H and σ² separately. Given that equation (9.14) holds, then

(9.14)

the derivative of Q(R,S|Θ^t ) with respect to the individual element h_j,i of the channel matrix is given by equation (9.15).

(9.15)

In the same way, the derivative of Q(R,S|Θ^t ) with respect to the noise variance σ² is found as shown in equation (9.16).

(9.16)

When the derivatives are set to zero, the update functions of h_j,i and σ² can be derived from equations (9.15) and (9.16). However, it is obvious that different channel parameters are coupled. To simplify the maximization process, the coupled channel parameters are estimated in turn. The path gain h_j,i is estimated with the rest of the channel matrix known and is represented with the latest estimate for each path gain. The path gains are updated in ascending order with respect to j and i. The resulting update function for h_j,i is given by equation (9.17),

(9.17)

where h_k,i is the latest estimate of path gain. At the tth iteration, if it has not been updated, or if it has been updated. After the channel matrix is completely updated, H_i+1 is used to acquire the noise variance estimation, given in equation (9.18).

(9.18)

The EM algorithm with such a maximization process is known as expectation condition maximization. ECM shares the convergence property of EM (Meng and Rubin, 1993) and can be constructed to converge at a similar rate as the EM algorithm (Sexton, 2000). The ECM joint estimation of channel parameters has previously been successfully applied in BMC for SISO systems (Chavali and Da Silva, 2011, 2013; Soltanmohammadi and Naraghi-Pour, 2013).

The final estimation of channel matrix H and noise variance σ² is achieved when the iterative process is terminated by one of two conditions. The first condition terminates the process when the estimation reaches convergence. The condition is represented numerically with the difference between the expected likelihoods of the current iteration and the previous iteration along with a predefined threshold. In the second condition, termination is triggered when the predefined number of iterations has been reached. The estimate at termination is used to replace the known channel matrix and noise variance in equation (9.6) and classification decision to be made using equation (9.7). An illustration of the system is given in Figure 9.4.

9.5 Conclusion

In this chapter, three civilian communication systems, which pose their unique requirement on modulation classification, are considered for the practical design of some modulation classifiers. The high-order modulation classification challenge is addressed for wired communication systems with associated channel effect. The knowledge of system configuration in a link adaptation system is exploited for improved modulation classification performance through a maximum a posteriori classifier. The update issue of modulation classification for MIMO systems is solved by an EM-ML classifier. Updated likelihood evaluation and channel estimation are both presented for MIMO systems.

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