Abstract
Abstract-Non-cooperative identification of unknown communication signals is a popular research area with widespread civilian and military applications. Multiple input multiple output (MIMO) systems employing multi-antenna transmission pose new challenges to signal identification systems, such as the classification of the employed space time block code (STBC) and modulation in the presence of the self-interference inherent to the multi-antenna transmission. In the existing literature, these two classification problems have been handled separately, despite the fact that they are interrelated. This letter presents a novel approach to MIMO signal identification by considering the modulation type and the STBC classification tasks as a joint classification problem.
I. INTRODUCTION
N ON-COOPERATIVE identification of unknown communication signals finds application in both military and commercial contexts, such as in electronic warfare, radio surveillance, spectrum monitoring and cognitive radio. Signal identification for multiple input multiple output (MIMO) systems presents new challenges, which do not exist for conventional single input-single-output (SISO) systems, such as the classification of the employed space time block code (STBC). Furthermore, modulation classification (MC) algorithms designed for SISO systems assuming the presence of a single modulated signal at the receiver, are unable to classify MIMO signals, where multiple signals, one from each transmit antenna, are present at each antenna at the receiver [1] .
I. INTRODUCTION
In the existing literature, STBC and modulation classification for MIMO systems are handled as separate problems, despite the fact that these two tasks are fundamentally interrelated. Likelihood based (LB) methods for MC in the literature decide for the modulation that maximizes the likelihood function of the received signal, requiring a-priori knowledge on the employed STBC. In [2] such algorithms are proposed for MIMO systems with spatial multiplexing (SM). In [3] Similarly, LB STBC classification methods in the literature assume the presence of perfect knowledge on the employed modulation, the SNR and code block timing [7] . Their FB counterparts usually exploit the cyclostationary behavior of the received signal induced by the coding operation, which has the benefit of not requiring any a-priori information on the modulation type, however, they are sub optimal [8] - [10] .
I. INTRODUCTION
The interdependence between the MC and STBC classification tasks is due to the fact that the likelihood function of the received signal in a MIMO system depends jointly on both the modulation and the STBC. Hence, an optimal classification in a realistic non-cooperative scenario where both are unknown can only be achieved by considering these two problems jointly. Recognizing this interdependence, this work proposes, for the first time in the literature, a novel LB approach to the MIMO signal identification problem, where the modulation type and the STBC employed in a MIMO signal are jointly classified. In addition to providing an optimal approach to the MIMO signal identification problem as a whole by formulating it as a joint classification problem, the proposed method, for the first time in the literature, enables modulation classification in absence of a-priori information on the STBC employed in the transmission.
I. INTRODUCTION
First, the optimal joint classifier is presented, where the channel matrix, noise variance and the block timing are assumed to be known a priori. Subsequently, classifiers for more realistic scenarios are introduced, where some or all of these parameters are unknown. The proposed joint classification algorithms are evaluated via simulations, and compared with a more conventional sequential classification approach, where the STBC classification is performed prior to the modulation classification, providing the modulation classifier with the required information on the employed STBC.
II. SYSTEM MODEL AND ASSUMPTIONS
We consider a MIMO transmitter with n t transmit antennas where the k'th transmit signal block of size n t ×L can be writ- 
II. SYSTEM MODEL AND ASSUMPTIONS
where W[k] is the circular complex additive white Gaussian noise matrix with variance σ 2 , and H is the n r × n t channel matrix whose elements are modeled as independent zeromean circular complex Gaussian random variables with unit variance. We assume a flat block fading channel over the observation interval. The aim of this work is to perform the joint classification of the modulation type and the STBC employed in a MIMO system using the received signal block 
II. SYSTEM MODEL AND ASSUMPTIONS
where K M j is the number of discrete states in the modulation type M j , L i is the length and r c i is the code rate of C i and ||.|| F represents the Froebenius norm. The proposed optimal joint classifier for the modulation type and STBC is obtained by jointly maximizing the natural logarithm of the average likelihood function wrt the modulation type and the STBC i.e.,
II. SYSTEM MODEL AND ASSUMPTIONS
This classifier, which we refer to as the joint average likelihood ratio (J-ALRT) classifier, can be considered as optimal in the Bayesian sense, given that the channel matrix H, the noise variance σ 2 and the code block timing τ , are known; hence, its performance can be regarded as an upper performance bound for the joint classification problem considered in this work. In many practical scenarios involving signal classification, no cooperation between the transmitter and receiver is possible, thus, these parameters are usually unknown to the receiver.
II. SYSTEM MODEL AND ASSUMPTIONS
For such practically more relevant cases, we propose a joint hybrid likelihood ratio test (J-HLRT) which approximates, for each hypothesis, the average likelihood function in (3) by substituting the actual values of the channel matrix and the noise variance with their blind estimates, i.e.,
II. SYSTEM MODEL AND ASSUMPTIONS
andσ 2 (i,j) are the blind estimates of H and σ 2 respectively, generated with the assumption that the modulation M j and the STBC C i have been transmitted.
1) Estimation of the Channel Matrix:
The blind channel estimation strategy employed in this work consists of two steps. First, the higher order statistics (HOS) based MIMO blind channel estimation algorithm proposed in [11] is used, which can be employed for a large set of codes, including SM, orthogonal, quasi-orthogonal and non-orthogonal STBCs, to form a pre-estimate of the channel matrix. Subsequently, the channel ambiguities that may have remained are removed by using a blind phase estimation algorithm if necessary. For a detailed derivation and analysis of the algorithm, the reader is referred to [11] . It should be noted that, while independent of the modulation type, the estimation requires the knowledge of the STBC used in the signal, thus, the pre-estimateH (i) is generated for each hypotesized STBC C i ∈ C individually. Furthermore, the pre-estimate contains STBC-specific ambiguities, which, for a large class of codes, have been derived and listed in [11] . Amongst the STBCs considered in this work, which are given in the Appendix, only SM and the Alamouti code lead to phase and permutation ambiguities. It is straightforward to show that the permutation ambiguities are not relevant to the classification, leaving only the phase ambiguities to be resolved. For the remaining codes, there solely exist a sign ambiguity, which does not have any effect on the classification due to the symmetry of the employed modulation types, thus, for these STBCs, the pre-estimate of the channel matrix can be used directly, i.e.,Ĥ (i,j) =H (i) . For SM and the Alamouti code, the generation ofĤ
1) Estimation of the Channel Matrix:
requires the estimation and correction of the phase offset terms in the corresponding phase ambiguity matrices.
2) Blind Phase Recovery for SM and the Alamouti Code:
For the case of SM, eq. (2) reduces to
2) Blind Phase Recovery for SM and the Alamouti Code:
where y[k] is the k'th received signal vector. We employ a phase offset estimation strategy similar to those in [5] . We first recover an estimate of the transmit signal vector s[k] using the pre-estimate of the channel matrixH (i) , i.e.,
2) Blind Phase Recovery for SM and the Alamouti Code:
Due to the phase ambguities in the channel estimation, the components ofs[k] are noisy and phase-rotated versions of the components of the actual transmit signal s[k]:
2) Blind Phase Recovery for SM and the Alamouti Code:
where v l [k] is a noise term and ϕ l is a random phase offset, which can now be estimated with the assumption that the modulation M j has been transmitted. As in [2] and [5] , we employ the blind phase recovery in algorithm in [12] exploiting the higher order moments of the signal. With the assumption that M j has a 2π V j rotationally symmetric constellation, the phase offset estimate is given aŝ
2) Blind Phase Recovery for SM and the Alamouti Code:
where
2) Blind Phase Recovery for SM and the Alamouti Code:
th order moment of a unit power signal using the modulation M j . After estimating the phase offset for each component ofs [k] , the phase corrected channel estimate for the modulation M j can be expressed aŝ
2) Blind Phase Recovery for SM and the Alamouti Code:
l . The phase ambiguity matrix for the case of the Alamouti code is similar to that of SM, except the fact that the two phase offsets are related to each other [11] . Thus, a similar phase recovery procedure can be employed for Alamouti coded signals.
3) Estimation of the Noise Variance:
The blind channel estimation algorithm in [11] is based on the assumption of unit power transmit signal components and employs this as a constraint in solving the cost minimization problem. Thus, it is straightforward to show that, for all the modulation types and the STBCs considered in this work, a method-of-moments estimator for the noise variance can be given aŝ
3) Estimation of the Noise Variance:
is the sample covariance matrix of the transmit signal recovered with the channel estimateĤ (i,j) .
4) Classification in Absence of the Code Block Timing
Information: For the most general case, where the code block timing τ is also unknown, a decision theoretic approach is formulated by realizing that for C i ∈ C , it can only take on a finite number of discrete values within the set
4) Classification in Absence of the Code Block Timing
Thus, the J-HLRT algorithm is extended for this case by maximizing (4), for each hypothesis C i , with respect to the unknown block timing parameter τ ∈ T i :
4) Classification in Absence of the Code Block Timing
where Y (τ ) is the received signal block which has been parsed with the assumption of the code block timing τ .
IV. CLASSIFICATION RESULTS
In this section, the proposed joint classification algorithms are evaluated using simulations. We consider the set of possible modulation types M = {BPSK, QPSK, 8PSK, 16QAM}.
IV. CLASSIFICATION RESULTS
Two sets of possible STBCs is considered. For n t = 2, we consider the set C (2) = {SM (2) , C AL } and for n t = 3, C (3) = {SM (3) , C 1 , C 2 }, where SM (n t ) represents SM with n t transmit antennas. (See the Appendix for the code matrices of C AL , C 1 and C 2 ). For n t = 2, the classification is performed within the set (2) = M × C (2) , with 8 hypotheses, whereas for n t = 3, the set (3) = M × C (3) with 12 hypotheses is considered. For each hypothesis, 1000 Monte Carlo trials have been performed. We use the average probability of correct joint classification P cc as a performance measure, which, (4) and the sequential approach, for (2) , n r = 4, 6. with the assumption of equiprobable hypotheses, is given as
IV. CLASSIFICATION RESULTS
] the probability of correct joint classification of the modulation M j and STBC C i . Figures 1 and 2 display the performance of the proposed J-ALRT and J-HLRT classifiers given in eqs. (3) and (4), for the sets of hypotheses (2) with n r = 4, 6, and for (3) with n r = 6 respectively. The classification has been performed for N = 500 and 750. Clearly, the use of the blind estimates of the channel matrix and noise variance in the J-HLRT instead of their true values leads to a considerable decrease in the classification performance compared to the ideal J-ALRT. As expected, the performance increases as the number of observed vector samples N and the number of receive antennas n r increase. Both for (2) and (3) , the proposed joint classifiers exhibit a high classification performance even in the low SNR regime. Furthermore, in both figures, the performance of the proposed joint classification approach is compared with a conventional sequential method, where the STBC classification is performed prior to the modulation classification, providing the modulation classifier with the information on the employed STBC. Since this approach can only be used with a sub optimal STBC classification method that does not require the Fig. 3 . Effect of the unknown code block timing on the performance of the J-HLRT classifier for (2) , n r =4, 6. modulation type of the transmit signal, we use the cyclostationarity based classification approach in [10] for the set of codes C (2) and C (3) respectively. The subsequent modulation classification is performed by maximizing the likelihood function in (4) only wrt. the modulation type, using the STBC information from the previous step. While this sequential method exhibits a considerably lower computational complexity, the results show that the proposed joint approach outperforms the sequential method in all cases, and the performance gap increases with increasing n t and decreasing N. Figure 3 illustrates the practically most relevant scenario where, in addition to H and σ 2 , the code block timing τ is also unknown to the receiver. The classification is performed using eq. (10) for n t = 2, n r = 4, 6 for the set of hypotheses (2) , and the results are compared with the case of known τ (4) . Clearly the proposed approach in eq. (10) is able to handle the lack of the block timing information without any essential decrease in the performance. It should be noted, however, that this approach leads to a considerable increase in the computational complexity, since the maximization is to be performed for each possible value of τ for each C i ∈ C . V. CONCLUSION This letter presents a novel approach to the signal identification problem for MIMO systems by considering the classification of the modulation type and the STBC as a joint classification problem, in contrast to the existing literature, where these two problems are considered separately. This approach, for the first time in the literature, enables MIMO modulation classification in absence of the a-priori knowledge of the employed STBC. First, an optimal LB joint classifier is presented, which considers the ideal but unrealistic scenario, where the channel matrix, the noise variance and the code block timing are known to the receiver, whose performance can be considered as an upper bound to the joint classification problem. Subsequently, suboptimal J-HLRT classifiers are proposed for practically more relevant cases, where some or all of these parameters are unknown. The numerical results show that the proposed algorithms exhibit good classification performance for relatively low values of SNR, and outperform a more conventional sequential approach. Future work will include the extension of this method to fast fading channels.
APPENDIX THE STBCS CONSIDERED FOR CLASSIFICATION
The joint classifiers proposed in this letter can be used for any linear modulation and any linear STBC. In the simulations, we have limited ourselves to cases with n t = 2 and n t = 3. For n t = 2, the set of considered STBCs is C (2) = {SM (2) , C AL }, where SM (n t ) represents SM with n t transmit antennas and C AL is the well known Alamouti code given in [13] . The set of STBCs considered for n t = 3 is C (3) = {SM (3) , C 1 , C 2 }, where the code matrices of C 1 [14] and C 1 [15] are given as:

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