Likelihood-Based Modulation Classification for Multiple-Antenna ...

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 9, SEPTEMBER 2013

Likelihood-Based Modulation Classification for Multiple-Antenna Receiver Ali Ramezani-Kebrya, Student Member, IEEE, Il-Min Kim, Senior Member, IEEE, Dong In Kim, Senior Member, IEEE, François Chan, and Robert Inkol, Senior Member, IEEE

Abstract—Likelihood-based algorithms for the classification of linear digital modulations are systematically investigated for a multiple receive antennas configuration. Existing modulation classification (MC) algorithms are first extended to the case of multiple receive antennas and then a critical problem is identified that the overall performance of the multiple antenna systems is dominated by the worst channel estimate of a particular antenna. To address the performance degradation issue, we propose a new MC algorithm by optimally combining the log likelihood functions (LLFs). Furthermore, to analyze the upper-bound performance of the existing and the proposed MC algorithms, the exact Cramér-Rao Lower Bound (CRLB) expressions of nondata-aided joint estimates of amplitude, phase, and noise variance are derived for general rectangular quadrature amplitude modulation (QAM). Numerical results demonstrate the accuracy of the CRLB expressions and verify that the results reported in the literature for quadrature phase-shift keying (QPSK) and 16-QAM are special cases of our derived expressions. Also, it is demonstrated that the probability of correct classification of the new algorithm approaches the theoretical bounds and a substantial performance improvement is achieved compared to the existing MC algorithm. Index Terms—Cramér-Rao lower bound, modulation classification, multiple antennas, non-data-aided estimation, rectangular QAM.

I. I NTRODUCTION UTOMATIC (blind) modulation classification (MC) is to determine the unknown modulation type of the received signal from a set of modulation candidates, e.g., general rectangular quadrature amplitude modulation (QAM). For various civilian and military applications, MC is an important task. As

A

Manuscript received December 31, 2012; revised May 21 and July 13, 2013. The editor coordinating the review of this paper and approving it for publication was R. C. M. da Silva. This work was supported in part by the Defense Research and Development Canada (DRDC), and in part by the Ministry of Knowledge Economy (MKE), Korea, under the Information Technology Research Center (ITRC) support program supervised by the National IT Industry Promotion Agency (NIPA) (NIPA-2012-(H0301-12-1005)) and the NRF grant funded by the Korea government (MEST) (No. 2012-047720). A. Ramezani-Kebrya is with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, Ontario, M5S 3G4, Canada (email: [email protected]). I.-M. Kim is with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, Ontario, K7L 3N6, Canada (e-mail: [email protected]). D. I. Kim is with the School of Information and Communication Engineering, Sungkyunkwan University (SKKU), Suwon, Korea (e-mail: [email protected]). F. Chan is with the Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, Ontario, K7K 7B4, Canada (email: [email protected]). R. Inkol is with Royal Military College (RMC), Kingston, ON, Canada. Digital Object Identifier 10.1109/TCOMM.2013.073113.121001

an example, when cognitive radios, which are able to sense and adapt to the environment, change the constellation size, MC might need to be used at the receiver side. In the case of fading environments, MC becomes more challenging as there are multiple unknown parameters (e.g., phase, amplitude, and noise variance) that should be estimated before the modulation scheme can be recognized. From a system viewpoint, MC algorithms work in two stages. The first stage includes estimation of signal carrier frequency, phase, amplitude, noise power, and the symbol period. The specific preprocessing tasks needed depend on the a priori knowledge of signal parameters available at the receiver, such as their estimates or distributions. The second stage is the actual modulation classification. MC can be classified into two classes [1]–[4]: likelihoodbased and statistical pattern recognition-based known as feature-based. The former is based on the likelihood function tests of the received signal [1]–[10], while the latter is based on the features of the received signal [1], [11]–[13]. Likelihoodbased algorithms can be optimal in Bayesian sense and they provide probabilistic sense. Optimum decision making typically suffers from high computational complexity, which might be challenging in practical systems. On the other hand, featurebased algorithms are not optimal in Bayesian sense. Using feature-based algorithms, computational complexity can be very low and corresponding algorithms are easy to implement, which might be particularly useful for certain applications. The features are usually selected heuristically in order to improve the probability of correct classification. In this work, we focus on likelihood-based algorithms rather than feature-based algorithms (which are typically heuristically derived), because the probability of false classification would be minimized if true values of parameters were available. Recently, there is ongoing research on providing better and better non-data aided parameter estimates. Once accurate estimates of unknown parameters are available, likelihood-based algorithm might achieve high probability of correct classification within a very short observation time. Many likelihood-based algorithms have been investigated in the literature such as Average Likelihood Ratio Test (ALRT) [1]–[3], [5]–[8], Generalized Likelihood Ratio Test (GLRT) [1], [2], [4], and Hybrid Likelihood Ratio Test (HLRT) [1]–[4], [7], [8]. However, these algorithms have various limitations. The ALRT algorithm requires a priori knowledge about the actual distributions of the unknown parameters, i.e., channel amplitude, phase, and noise variance [2], [3], [6]–[8]. With GLRT, the probability density function (PDF) of the received

c 2013 IEEE 0090-6778/13$31.00

RAMEZANI-KEBRYA et al.: LIKELIHOOD-BASED MODULATION CLASSIFICATION FOR MULTIPLE-ANTENNA RECEIVER

signal is computed by employing maximum likelihood estimates (MLE) of unknown quantities. However, GLRT fails to identify the nested signal constellations correctly, such as 16-QAM and 64-QAM, as this approach can lead to the same value of the likelihood function (LF) [10]. To address the nested constellations problem, HLRT can be used where the MLE of the unknown parameters are obtained in lieu of taking expectation. Unfortunately, HLRT and ALRT involve high computational complexity due to exhaustive search over the LFs. With Quasi Hybrid Likelihood Ratio Test (QHLRT), non-data-aided non-MLEs of the unknown parameters are used to calculate the LFs, and thus, the computational complexity reduces significantly [2], [3], [7], [8]. Likelihoodbased algorithms were proposed for slow fading channels [4], for wireless sensor networks [9], and for identifying M-ary phase-shift keying (MPSK) modulations [13]. In [7], an MC algorithm was proposed for a multiple antenna configuration with unknown phase and amplitude. To the best of our knowledge, however, likelihood-based MC for the case of multiple antennas with unknown amplitude, phase, and noise variance has not been studied in the literature. In this paper, we propose a new likelihood-based MC algorithm, namely a weighted-sum algorithm, assuming amplitude, phase, and noise variance to be unknown for a multiple receive antennas scenario. Furthermore, the performance upperbounds of the proposed and QHRLT-based algorithms are studied. It is well-known that the Cramér-Rao Lower Bounds (CRLBs) give the lower bounds on the variances of joint estimates of unknown parameters. In [3], [14], [15], the CRLBs of non-data-aided joint estimates of the unknown parameters were derived for binary phase-shift keying (BPSK), quadrature phase-shift keying (QPSK), and square QAM. In this paper, we derive the CRLBs for the most general rectangular I ×J-QAM signals and show the derivations in the literature are special cases of our derived results. Numerical results demonstrate the merit of using our proposed MC algorithm and the correctness of our CRLB analysis. The rest of this paper is organized as follows: The system model is described in Section II-A; the problem is formulated in II-B; the extension of the existing algorithms to multiple antennas is discussed in Section II-C; and state-of-the-art estimators are discussed in Section II-C5. Our new MC algorithm is proposed in Section III. In Section IV, we derive the CRLBs of the unknown parameters for rectangular I×J-QAM signals. Numerical results for the CRLBs are presented in Section V. Conclusions are drawn in Section VI. Notation: We use A := B to denote that A, by definition, equals B. For a complex number C, |C| and C denote the norm and phase of C, respectively. The transpose and Hermitian transpose of vector ν, are represented by ν † and ν H , respectively. We use E[·] to represent the expectation operation and IK to represent a K × K identity matrix. Also, z1 ∼ N (η1 , ω1 ) denotes that z1 is a real Gaussian random variable with mean η1 and variance ω1 . Finally, z2 ∼ CN (η2 , Ω2 ) denotes that z2 is a circularly symmetric complex Gaussian (CSCG) random vector with mean vector η2 and covariance matrix Ω2 .

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II. S YSTEM M ODEL AND MC FOR M ULTIPLE A NTENNAS This section begins by describing the system model and formulating the MC problem. Then MC algorithms are developed for multiple receive antennas by modifying the existing algorithms. Finally, the enhanced estimators for multiple receive antennas configuration are discussed. A. System Model Suppose that signals are sent from a transmitter. Our aim is to determine the unknown modulation type of the signals using (m) (m) a receiver with N antennas. Let s(m) := [s1 , · · · , sK ]† denote the column vector of the transmitted symbols corresponding to the constellation Sm with |Sm | = Mm for m = 1, · · · , Nmod , where Nmod denotes the number of modulation candidates. We use K to denote the number of samples (m) of the received signals. It is assumed that {sk }K k=1 are independently and uniformly distributed random variables over the signal constellation Sm . Let hi denote the complex channel coefficient from the transmitter to the i-th receive antenna. We assume channels are independent in space and are constant over the K observation times, i.e, block fading environment. For analysis in this paper, no assumption needs to be made on the distribution of hi ; but, for simulation purpose only in Section V, |hi | will be assumed to be Rayleigh. When the mth modulation is used for the transmitted signal, the equivalent baseband received signal at the i-th receive antenna is given by for i = 1, · · · , N, (1) ri = hi s(m) + ni , where ri = [ri,1 , · · · , ri,K ]† denotes the received signal sequence at the i-th receive antenna and ni = [ni,1 , · · · , ni,K ]† is the additive white Gaussian noise (AWGN) with ni ∼ CN (0, σi2 IK ). We assume that the AWGN is spatially white, i.e., E[n†i nl ] = 0 for l = i. We suppose that the channel 2 N coefficients {hi }N i=1 and {σi }i=1 are unknown at the receiver side, since providing the training symbols to measure the instantaneous channel state information (CSI) perfectly for an unknown transmitter, i.e., data-aided approach, is impossible or it causes a too high burden to the destination. Note that this is a common assumption for MC [1], [3], [4], [7], [8]. Let ui := [αi , ϕi , σi2 ]† denote the vector of the unknown parameters, where αi and ϕi represent unknown channel amplitude and phase of the i-th channel, respectively. B. Problem Formulation: MC for Multiple Antennas The MC problem can be modeled as a multiple hypotheses testing problem. Modulation candidate m corresponds to the hypothesis Hm . In the maximum likelihood (ML)-based MC, modulation m is selected if the LF of the received signal given Hm is larger than those of all other hypotheses. Conditional upon the transmitted symbols s(m) and the unknown parameter vectors, the conditional LF under the m-th hypothesis is given by (m) f ({ri }N , {ui }N i=1 |s i=1 , Hm ) =

N K k=1 i=1

(m)

Ri,k ,

(2)

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where

−1 |ri,k − (m) Ri,k = πσi2 exp −

(m) αi ejϕi sk |2 σi2

.

(3)

Because both s(m) and {ui }N i=1 are unknown, it is not possible to directly use the conditional LF (m) f ({ri }N , {ui }N i=1 |s i=1 , Hm ) for MC. Therefore, by taking expectation (if the distributions are known) or by finding their estimates, the unconditional LF or unconditional log LF (LLF) must be determined, which is denoted by g({ri }N i=1 |Hm ). Then the MC problem is formulated by

4) QHLRT: To resolve the complexity issue of HLRT, one can use non-MLEs of the unknown parameters, which do not depend on the transmitted signals, leading to blind MC. In [3], [7], [8], an algorithm was used to estimate the unknown channel amplitude and noise variance. For multiple antennas, an MC scheme was proposed in [7] with unknown channel amplitude and phase, and with known noise variance. For the case of unknown noise variance, it is not difficult to extend the results of [7] as follows: M K N m 1 ˆ (m) N R log , (6) gQHLRT({ri }i=1 |Hm ) = Mm i=1 i,k,p p=1 k=1

m ˆ = arg

max

m∈{1,··· ,Nmod }

g({ri }N i=1 |Hm ),

(4)

where m ˆ is the estimate of the modulation. In the following subsections, g(·) is determined.

where ˆ (m) R i,k,p

⎞ ⎛ (m) (m) 2 (m) î ej ϕî sk,p | |ri,k − α 1 ⎠. = exp ⎝−

2 (m)

2 (m) πσ σ i

C. Extension of the Existing Algorithms and Enhanced Estimators For a single receive antenna configuration, likelihood-based MC was studied with unknown amplitude, noise variance, and phase in [3]; and for a multiple receive antennas configuration, it was studied with unknown amplitude and phase, and with known noise variance, in [7]. To the best of our knowledge, likelihood-based MC for the multiple receive antennas configuration has not been studied for the case with unknown amplitude, phase, and noise variance. In the following, extending the results of [3], [7] to such a case is discussed. 1) Ideal MC algorithm: When perfect knowledge of the instantaneous CSI and noise variances of all branches, {ui }N i=1 , is available at the receiver side, it is easy to show that the LLF under the m-th hypothesis is given by gIdeal ({ri }N i=1 |Hm ) =

K k=1

log

Mm N 1 (πσi2 )−1 Mm p=1 i=1

(5)

(m) |ri,k − αi ejϕi sk,p |2 , · exp − σi2 (m)

where sk,p denotes the p-th constellation point drawn from Sm in the k-th time slot. This MC algorithm provides an ultimate upper-bound on the performance of any MC methods and it was referred to as ALRT-Upper Bound (ALRT-UB) in [3]. In practice, however, it is not realistic to know {ui }N i=1 perfectly. 2) ALRT: When the distributions of s(m) and {ui }N i=1 are available, it is possible to use ALRT by taking expectation of (2) and by taking log(·). However, obtaining the distributions of {ui }N i=1 requires a very high signaling overhead or could be completely impossible. 3) HLRT: When the distributions of {ui }N i=1 are unknown, the MLEs of the unknown parameter vectors can be used to be substituted into LLFs [3]. Since the MLEs employed under the HLRT algorithm are functions of s(m) , it is necessary to take into account all symbol sequences of length K. This leads K to computational complexity in the order of O(N Mm ) given Hm , meaning that it grows exponentially with the number of observation time K [3].

i

(m) (m)

2 (m) are given The explicit expressions of α ˆ i , ϕî , and σ i in [7]. The computational complexity of QHLRT is in the order of O(KMm ), which is much lower than that of HLRT. Unfortunately, the estimators [3], [7], [8] used in the existing QHLRT do not perform well in the multiple receive antennas systems since the spatial diversity has not been taken into account. This performance issue of the estimators is addressed in the following subsection. 5) Best Estimators for QHLRT: To address the poor performance issue of the estimators of [3], [7], [8], in this paper we adopt a state-of-the-art estimator proposed in [16], which was actually proposed for SNR estimation of multiple receive antennas systems, rather than MC. The main idea behind [16] is to take advantage of the correlation of the received signal between different antenna pairs. The non-dataaided channel phase estimates depend on the type of the hypothesized modulation scheme. For QAM, the state-of-theart non-data-aided phase estimate proposed in [17] is used. To estimate the phase for PSK signals, [3, eq. (15)] is used. When these enhanced estimates are substituted into QHLRT of (6), the performance is much improved over that of the existing QHLRT methods [3], [7], [8]. However, we will see the performance of QHLRT is still bad for multiple receive antennas because the overall performance is dominated by the worst channel estimates in the multiple receive antennas configuration. This performance issue will be addressed in the next section by proposing a new MC algorithm.

III. P ROPOSED MC A LGORITHM For the multiple antenna configuration, QHLRT of (6) has a structural problem. Specifically, if at least one channel corresponding to a particular receive antenna falls in a low signal-to-noise-ratio (SNR) range, the QHLRT of (6) suffers from a severe performance degradation. Mathematically N ˆ (m) speaking, the product i=1 R i,k,p in (6) approaches zero if ˆ (m) , becomes very small: at least a single LF, i.e., a single R i,k,p N ˆ (m) (m) ˆ ˆ (m) R → 0, when min { R i i,k,p i,k,p } → 0. Note that Ri,k,p i=1 becomes small when the estimates of the corresponding ui are not accurate, which is highly likely in fading environments. Consequently, even if the estimates for most receive antennas


ˆ (m) for most receive are accurate, giving large LF values of R i,k,p antennas, the smallest LF for a particular receive antenna dominates the overall performance of (6). In the following, a new MC algorithm is proposed to address the problem. Let Li (ri |Hm ) denote the LLF of the received signal from the i-th receive antenna with the estimates in Section II-C5: Mm K 1 1 log Li (ri |Hm ) =

2 M Mm p=1 σ k=1 4 (m) (m) 2 |ri,k − α ˆ i,M4 ej ϕî sk,p | · exp − , (7)

2 M σ 4

for i = 1, · · · , N. In order to eliminate the problem discussed above, we propose a new structure where a weighted summation of the LLFs of N receive antennas is used to perform MC. To this end, we define the vector L(m) of LLFs and the vector λ of the weighting coefficients: L(m)

=

[L1 (r1 |Hm ), L2 (r2 |Hm ), · · · , LN (rN |Hm )]†

λ

=

[λ1 , λ2 , · · · , λN ]† .

Then we propose that the LLF corresponding to the m-th hypothesis is given by † (m) = gProposed ({ri }N i=1 |Hm ) = λ L

N

λi Li (ri |Hm ), (8)

i=1

where the vector norm of the weighting vector is unity: λ = 1. In this paper, we optimize the weight vector λ such that gProposed ({ri }N i=1 |Hm ) is maximized. For this, we use the Cauchy-Schwarz inequality: λ† L(m) ≤ λL(m)

(9)

where the equality holds if and only if λ = ζL(m) for a constant ζ. Satisfying the constraint of λ = 1, the optimal coefficients {λopt i }, for the proposed MC algorithm based on the weighted sum structure, are obtained as follows: λopt i

Li (ri |Hm ) = N 2 i=1 Li (ri |Hm )

(10)

for i = 1, · · · , N. It can be seen that, in the proposed MC algorithm of (8), small LLF values due to inaccurate estimates do not dominate opt the entire term, i.e., N i=1 λi Li (ri |Hm ) will not approach zero even if mini {Li (ri |Hm )} → 0. When some Li (ri |Hm ) opt values are small, their corresponding optimal coefficients λi N become small, and thus, their impact on gProposed ({ri }i=1 |Hm ) is small. This means that the proposed structure is robust to the non-ideal (noisy) estimates of the unknown parameters. In our proposed scheme, even if most of the channels are poor, the overall performance is still good as long as a single estimate is good. The only case where our proposed scheme fails is when all channels are bad at the same time.1 Note that λopt are optimal only in the sense that they i N maximize the weighted-sum of LLFs, i=1 λi Li (ri |Hm ). 1 In

order to address this problem, one may think that some useful information for MC might be sent from the transmitter to the receiver. However, such signaling cannot be used in MC problems because no cooperation between the transmitter and receiver is assumed in MC.

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However, the proposed weighted-sum structure (8) itself is not necessarily optimal in the Bayesian sense; this issue particularly matters if very accurate estimates of the unknown parameters are available. However, the non-data-aided estimates, especially the phase estimates, are far from the true values. Since only the non-ideal non-data-aided estimates are available in practical scenarios, the proposed MC method of (8) performs much better than the QHLRT approach in practice. The actual performance of the proposed scheme will be presented in Section V. From the weighted-sum structure, it is possible to consider two special cases: equal gain LLF combining (this will be referred to as equal gain combining (EGC)) and selecting the best LLF (this will be referred to as selection combining (SC)). Mathematically, they are given by gEGC ({ri }N i=1 |Hm ) =

N

Li (ri |Hm ),

(11)

i=1

gSC ({ri }N i=1 |Hm ) =

max

i∈{1,··· ,N }

Li (ri |Hm )

(12)

where the EGC of (11) is obtained from the proposed algorithm of (8) by setting λopt = 1 and the SC of (12) is i obtained from (8) by selecting the best LLF. On the other hand, it is not straightforward to derive the EGC or SC from N ˆ (m) QHLRT of (6), because the product i=1 R i,k,p exists inside the log function in (6). Regarding the complexity, all three algorithms including SC, EGC, and the proposed one have the polynomial worst-case complexity; specifically, their optimum code implementations involve the complexity with O(N ). In general, with the actual estimates of the unknown parameters, the mathematical performance analysis of QHLRT and the proposed algorithm is an extremely difficult task. Hence, we study the performance upper-bounds for both algorithms in the next section. IV. CRLB FOR P ERFORMANCE U PPER -B OUND A NALYSIS The performance upper-bounds for the QHLRT methods such as the algorithms of [3], [7], [8] and the proposed algorithm are studied in this section. For unbiased joint estimates of the unknown parameters, using the CRLBs as the variances of the estimates provides an upper-bound on the MC performance [3]. Thus, we consider the QHLRT-based and proposed MC methods using optimal estimates achieving the CRLBs, which will be referred to as QHLRT-UB and Proposed-UB, respectively. Assuming the parameter estimates are normally distributed,2 the LLFs under the m-th hypothesis are expressed as gQHLRT-UB({ri }N i=1 |Hm ) = gProposed-UB ({ri }N i=1 |Hm ) =

K k=1 N

log g˜({ri }N i=1 ) , (13) õpt L ˜ i (ri |Hm ), λ i

(14)

i=1 2 The actual estimates are not necessarily normally distributed; however, it is a very typical approach to use normally distributed estimates for CRLBs as variances to study performance upper-bound [3]. When K is large, they are accurately modeled by normal distributions according to the central limit theorem.

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Mm N 2 (m) −1 1 where g˜({ri }N ) = i=1 p=1 i=1 π σi Mm (m) (m) j ϕ (m) 2 ˜ |ri,k −α ˜ i e i sk,p | exp − . In the above equations, 2 (m) σi

2 (m) are optimum estimates, which are and σ i (m) (m) (m) ∼ N αi , CRLB(ˆ αi ) , ϕ˜i ∼ distributed as α ˜i (m)

2 (m) (m) 2 2 N ϕi , CRLB(ϕî ) , and σi ∼ N σi , CRLB(σi ) , where CRLB(·) represents the CRLB of an unknown param˜ opt are obtained by substituting ˜ i (ri |Hm ) and λ eter. Also, L i the optimum estimates into (7) and (10), respectively. Hence, to evaluate the performance of QHLRT-UB and ProposedUB, CRLB(·) must be derived. In the following, we derive (m) (m)

2 (m) ) for the genCRLB(ˆ αi ), CRLB(ϕî ), and CRLB(σ i eral rectangular QAM constellations. Note that, for modulation classification, it is important to consider a broader set of constellations such as rectangular QAM (rather than square QAM only). For example, using rectangular QAM, the constellation size M = 2k can be changed by incrementing k to k + 1 or decrementing k to k − 1. For square QAM, however, k should be incremented to k + 2 or decremented to k − 2. Compared to the other types of irregular QAM, rectangular QAM is the most frequently used constellation for practical scenarios [19], [20], because rectangular QAM can be implemented very easily at the transmitter using two independent PAM signals, i.e., in-phase and quadrature components, and easily demodulated at the receiver side. Rectangular QAM has been used and considered for many different practical applications [21]– [23]. Overall, studying rectangular QAM is very important not only for MC but also for many current and emerging practical systems. Therefore, the reference value of the analysis given in this section is to provide the most general CRLB expressions for such important rectangular QAM, which can be used for many other applications as well as MC. as the unknown Considering [u†1 , u†2 , · · · , u†N ]† parameter vector, we derive the actual joint CRLBs for the multiple antenna configuration. Since directly taking expectation with respect to (m) Mm N K |ri,k −αi ejϕi sk,p |2 1 2 −1 exp − p=1 k=1 Mm i=1 (πσi ) σ2 (m) α ˜i ,

(m) ϕ˜i ,

i

is analytically intractable, we resort to the approach based on the exact CRLBs for a single antenna configuration. In [14] and [15], the CRLBs of square QAM were derived for unknown SNR and phase, respectively. However, those results cannot be directly extended to the general rectangular QAM, because the corresponding LLFs, designed specifically for square QAM, not applicable to rectangular QAM. A specific difference is the step to simplify the expression of the LLF even when the constellation points have not full symmetry across I and Q axes which happens for square QAM. To the best of our knowledge, the CRLBs for rectangular I × J-QAM with amplitude, phase, and noise variance as the unknown parameters have not been derived in the literature. For notational simplicity, we drop the subscript i and superscript (m) in the following equations. We first derive a simplified and closed-form expression for the LLF conditional upon that the received signal has rectangular I × J-QAM modulation scheme. Then, using the simplified LLF, we obtain the CRLBs. We let 2dI,J denote the minimum distance

between two adjacent constellation points of the rectangular I ×J-QAM constellation with I = 2q and J = 2t where q and t are arbitrary positive integers. Then the constellation points can be written as S = {±(2l − 1)dI,J ± j(2p − 1)dI,J }, for l = 1, 2, · · · , I/2, p = 1, 2, · · · , J/2. In order to have a unit variancerectangular I × J-QAM constellation, we assume dI,J = I 2 +J3 2 −2 . We use the notation γ := α2 /σ 2 to represent the received SNR. Denoting {sk,p }IJ p=1 as the possible transmitted symbols drawn from the constellation in the k ∈ [1, K]-th time slot and averaging over all constellation points, the conditional LF given the unknown parameter vector u is obtained by IJ K 1 |rk − αejϕ sk,p |2 exp − . (15) f (r|u) = πIJσ 2 p=1 σ2 k=1

The computational complexity of (15) is in the order of O(KIJ). Our goal is to derive the CRLBs. To this end, we need a simpler form of f (r|u) in (15). This simpler form is presented in the following lemma, which can be shown by combining four exponential terms corresponding to each pair of (l, p), i.e., ±(2l − 1)dI,J ± j(2p − 1)dI,J . Lemma 1: The conditional LLF corresponding to (15) can be simplified to I/2 K LLF(r|u) = C + gl cosh(ul {rk e−jϕ }) log k=1

+ log

J/2

l=1

gp cosh(up {rk e−jϕ }) ,

(16)

p=1

where gl = exp (−α2 d2I,J [(2l − 1)2 − 1]/σ 2 ), ul = 2α(2l − 1)dI,J /σ 2 for l = 1, 2, · · · , I/2, and C = 4 2 2 2 2 2 K log( πIJσ 2 ) − 2Kα dI,J /σ − r /σ . Note that the complexity to compute LLF(r|u) in (16) reduces to the order of O(K(I + J)). In order to find the CRLBs of the unknown parameters, one should first derive the Fisher information matrix (FIM). The FIM is denoted by I(u) := [Ib,c ], for b, c =1, 2, and 3: ∂2 Ib,c = −E LLF(r|u) , (17) ∂ub ∂uc where ub is the b-th unknown parameter. This is derived in the following lemma. Lemma 2: The FIM for the unknown parameter vector assuming a rectangular I × J-QAM signal is given by

⎞ ⎛ 2 σ − σ 2 GI,J (γ) 0 αHI,J (γ) 2K ⎝ 2 2 ⎠, 0 α σ 1 − FI,J (γ) 0 I(u) = 4 σ 1 α2 0 − K (γ) αHI,J (γ) I,J 2 σ2 (18)

where

⎧ F (γ) ⎪ ⎪ ⎨ I,J GI,J (γ) ⎪ ⎪ HI,J (γ) ⎩ KI,J (γ)

= = = =

2

2

J −1 1 I −1 ( 21 + γ I 2 +J 2 −2 )FI (γ) + ( 2 + γ I 2 +J 2 −2 )FJ (γ), 1/2GI (γ) + GJ (γ) , 1/2HI (γ) + HJ (γ) , 1/2 KI (γ) + KJ (γ) , (19)

and FI (γ), GI (γ), HI (γ) and KI (γ) are given in the Appendix by (A.6), (A.12), (A.16), and (A.20), respectively.


Proof: See the Appendix A.

QPSK and BPSK modulations are given by

Note that the steps to derive the FIM is different (and more difficult) compared to those of [14], [15] due to less symmetry of rectangular QAM compared to square QAM. In the following theorems, we now derive the CRLBs of the unknown parameters for rectangular I × J-QAM and M PAM, i.e., for one- and two-dimensional amplitude modulation schemes. To the best of our knowledge, these results have not been reported in the literature. Theorem 1: The CRLBs of unbiased and joint estimates of α, ϕ, and σ 2 , for rectangular I × J-QAM with even I and J, are given by

α) = CRLBQPSK (ˆ CRLBQPSK (ϕ) ˆ =

2 ) = CRLBQPSK (σ α) = CRLBBPSK (ˆ CRLBBPSK (ϕ) ˆ =

2 ) = CRLBBPSK (σ

α2 1 − 2γKI,J (γ) , 2Kγ DI,J (γ) 1 1 , CRLBI×J-QAM (ϕ) ˆ = 2Kγ 1 − FI,J (γ) 4

2 ) = σ 1 − GI,J (γ) , CRLBI×J-QAM (σ K DI,J (γ) CRLBI×J-QAM (ˆ α) =

(20) (21) (22)

where DI,J (γ) = 1 − GI,J (γ) − 2γKI,J (γ) + 2 (γ) . 2γ GI,J (γ)KI,J (γ) − HI,J Proof: Taking the inverse of the FIM (18), (20)–(22) are obtained. The above theorem is valid only for even I and J, which is not applicable to an M -PAM case. The result for an M -PAM is given in the following. Theorem 2: The CRLBs, given the modulation scheme is M -PAM with even M, are given by

α2 1 − 2γKM-PAM (γ) , 2Kγ DM-PAM (γ) 1 1 CRLBM-PAM (ϕ) , ˆ = 2Kγ 1 − FM-PAM (γ) 4

2 ) = σ 1 − GM-PAM (γ) , CRLBM-PAM (σ K DM-PAM (γ) α) = CRLBM-PAM (ˆ

3821

(23) (24) (25)

where DM-PAM (γ) = 1 − GM-PAM (γ) − 2γKM-PAM (γ) + 2 (γ) . Also, GM-PAM (γ), 2γ GM-PAM (γ)KM-PAM (γ) − HM-PAM FM-PAM (γ), HM-PAM (γ), and KM-PAM (γ) are given by (B.3). Proof: See the Appendix B. As a special case of Theorem 1, we can obtain √ the CRLBs of square M -QAM by substituting I = J = M /2 into (A.6), (A.12), (A.16), and (A.20). The obtained results are essentially the same as the results reported in [14], [15], although the final mathematical expressions are not equal. In Section V, we numerically demonstrate that they provide exactly the same results. Furthermore, as special cases of Theorems 1 and 2, we present the CRLBs for QPSK and BPSK. Corollary 1: The CRLBs of the unknown parameters for

where ⎧ ⎨ GQPSK (γ) = ⎩ FQPSK (γ) = and ⎧ ⎨ GBPSK (γ) = ⎩ FBPSK (γ) =

1 − 2γGQPSK (γ) α2 , 2Kγ 1 − (2γ + 1)GQPSK (γ) 1 1 , 2Kγ 1 − (1 + γ)FQPSK (γ) σ4 1 − GQPSK (γ) , K 1 − (2γ + 1)GQPSK (γ) 1 − 2γGBPSK (γ) α2 , 2Kγ 1 − (2γ + 1)GBPSK (γ) 1 1 , 2Kγ 1 − FBPSK (γ) σ4 1 − GBPSK (γ) , K 1 − (2γ + 1)GBPSK (γ)

2 exp(−γ/2) √ 2π 2 exp(−γ/2) √ 2π

2 exp(−γ) √ 2π 2 exp(−γ) √ 2π

(26) (27) (28) (29) (30) (31)

+∞

exp(−x2 /2)x2 √ cosh( γx) dx, 0 +∞ exp(−x2 /2) √ 0 cosh( γx) dx,

(32)

+∞

exp(−x2 /2)x2 √ dx, 0 cosh( 2γx) +∞ exp(−x 2 /2) √ dx. 0 cosh( 2γx)

(33)

Proof: For QPSK, using Theorem 1, we can show that GQPSK (γ) = HQPSK (γ) = KQPSK (γ). Furthermore, the expression of FI (γ) in (A.6) can be simplified to FQPSK (γ). By substituting GQPSK (γ) into (20) and (22), one can derive (26) and (28). Finally, (27) is obtained by substituting FQPSK (γ) into (21). Similarly, (29)–(31) can be derived for BPSK. The expressions of the CRLBs are identical to those reported in [3, eqs. (20)–(26)]. In this sense the results of Theorems 1 and 2 are generalizations of [3]. Another important issue is to estimate the unknown SNR, which is of interest in digital communication systems for adaptive modulation and power control. In the following lemma, the CRLB of the SNR estimate is derived. Since SNR is typ2 ically given in the logarithmic dB scale, η(uγ ) = 10 log10 α σ2 is considered in the following, where uγ := [α, σ 2 ]† denotes the partial unknown parameter vector corresponding to SNR estimation. Lemma 3: The CRLB of the SNR estimate for rectangular I × J-QAM signal is given by CRLBSNR (ˆ γ) 100 2 + γ 1 − GI,J (γ) − 4KI,J (γ) + 4HI,J (γ) = . K ln2 (10)γDI,J (γ) (34) Proof: The partial FIM for rectangular I × J-QAM is given by 2 αHI,J (γ) σ − σ 2 GI,J (γ) 2K 2 . I(uγ ) = 4 1 α σ αHI,J (γ) 2 − σ2 KI,J (γ) (35) The CRLB for the unknown SNR estimate is obtain by [18], CRLB(ˆ γ) =

∂η(uγ ) ∂η(uγ )† −1 I (uγ ) , ∂uγ ∂uγ

(36)

3822


5

4

10

10

16-QAM (by Theorem 1) QPSK (by Theorem 1) 3

10

4-PAM (by Theorem 2)

−0.4

3

10

2

10

10

BPSK (by Theorem 2)

2

10

10

8-QAM (by Theorem 1)

10

−0.3

4

CRLB Asym (by Lemma 4)

−0.5

K-CRLB(ϕ) ˆ

K-CRLB(ˆ α)

10 1

10

0

10

0

1

2

16-QAM (by Theorem 1) QPSK (by Theorem 1) 8-QAM (by Theorem 1) 4-PAM (by Theorem 2) BPSK (by Theorem 2) 16-QAM (by [15]) QPSK (by [15]) CRLB Asym (by Lemma 4)

1

10

0

10 −1

10

−1

10

−2

−2

10

10

−3

10 −10

−3

10 −10

−5

0

5 10 SNR, γ [dB]

15

20

25

Fig. 1. K-CRLB(ˆ α) versus SNR, γ, for BPSK, 4-PAM, 8-QAM, QPSK, and 16-QAM.

20 −10 † where ∂uγγ = [ α ln(10) σ2 ln(10) ] denotes the derivative of the parameter transformation with respect to the partial parameter vector. Substituting the FIM of (35) into (36), and after some algebraic manipulations, (34) is obtained. Finally, the asymptotic behaviors of the CRLBs obtained in Theorem 1 and Lemma 3 are derived in the following lemma. Lemma 4: When γ tends to infinity, the CRLBs of the unknown amplitude, phase, noise variance, and SNR estimates for rectangular I × J-QAM are given by

α2 , 2Kγ 1 CRLBAsym (ϕ) , ˆ = 2Kγ 4

2 ) = σ , CRLBAsym (σ K 100 CRLBAsym (ˆ γ) = . K ln2 (10)

0

5

10 SNR, γ [dB]

15

20

25

Fig. 2. K-CRLB(ϕ) ˆ versus SNR, γ, for BPSK, 4-PAM, 8-QAM, QPSK, and 16-QAM.

5

∂η(u )

CRLBAsym (ˆ α) =

−5

10

16-QAM (by Theorem 1) QPSK (by Theorem 1) 8-QAM (by Theorem 1) 4-PAM (by Theorem 2) BPSK (by Theorem 2)

2 ) K-CRLB(σ


0

10

(37) −5

(38) (39)

10 −10

−5

0

5 10 SNR, γ [dB]

15

20

25

2 ) versus SNR, γ, for BPSK, 4-PAM, 8-QAM, QPSK, Fig. 3. K-CRLB(σ and 16-QAM.

(40)

Proof: As γ tends to infinity, FI,J (γ), GI,J (γ), HI,J (γ), and KI,J (γ) approach zero. In fact, these functions are all multiples of exp(−γd2 ). Also, from (A.6), (A.12), (A.16), and (A.20), it is possible to show that the integrands approach zero when γ increases as well. This is because in the numerators of the integrands, there is multiplication of gl gp in which both gl and gp approach zero exponentially, while there is only gp in the denominator. In both the numerator and denominator, these terms asymptotically dominate the cosh(·) and sinh(·) terms √ which are functions of γ. Hence, by substituting FI,J (γ) = GI,J (γ) = HI,J (γ) = KI,J (γ) = 0 into (20), (21), (22), and (34), we obtain (37)–(40). The asymptotic behaviors of the CRLBs of amplitude, phase, noise variance, and SNR estimates are verified by numerical integrations in Section V. V. N UMERICAL R ESULTS In this section, numerical results for the CRLBs of channel amplitude, phase, noise variance, and SNR estimates are first presented. Then the probabilities of correct classification are evaluated for the QHLRT of (6) and the proposed algorithm of (8); and they are compared to the upper-bounds of the ideal

MC of (5), the QHLRT-UB of (13), and the proposed-UB of (14). We assume a block Rayleigh fading environment. We set hi ∼ CN (0, 1) for i = 1, · · · , N. Unless otherwise mentioned, K is set to 100. We set σi2 = σ 2 for i = 1, · · · , N and change σ 2 to adjust the SNR values. The number of Monte Carlo trials to find the probability of correct classifications given Hm is set to 104 . We use the average probability, PrCC , of correct classification to compare the performance of MC algorithms. In order to be consistent with the estimates in Section II-C5, we take the average over the estimated noise variances of receive antennas and consider only nonnegative estimated amplitudes. For fair performance comparison of the proposed algorithm and QHLRT, the enhanced estimators in Section II-C5 are used for both schemes. Using (20)–(22), (23)–(25), (26)–(31), and substituting α = 1, Figs. 1, 2, and 3 show the numerical results of CRLB(ˆ α),

2 ), respectively for BPSK, 4-PAM, 8CRLB(ϕ), ˆ and CRLB(σ QAM, QPSK, and 16-QAM. One can notice that all curves decrease as SNR increases. As expected, for low SNR values, the CRLBs corresponding to BPSK and 4-PAM tend to converge and the same behavior is observed for QPSK and


3823

6

10

16-QAM (by Lemma 3) QPSK (by Lemma 3) 8-QAM (by Lemma 3) 4-PAM (by Lemma 3) BPSK (by Lemma 3) 16-QAM (by [14]) QPSK (by [14])

5

10

4

CRLB(ˆ γ ) [dB 2 ]

10

1 1 0.95 0.98 0.9

0.96


3

10

0.94

0.85 2

10

0.92

0.8

1

PrCC

10

0

10

0.9 0.75 0.88

1

2

3

0.7

−1

10 −15

−10

−5

0

5 SNR, γ [dB]

10

15

20

25

0.65

Fig. 4. CRLB(ˆ γ ) versus SNR, γ, for BPSK, 4-PAM, 8-QAM, QPSK, and 16-QAM for K = 100.

Ideal MC by (5) QHLRT-UB by (13) using CRLBs

0.6

Proposed-UB by (14) using CRLBs 0.55

Proposed algorithm by (8) QHLRT by (6)

0.5 1

0

5

10

15

SNR, γ [dB]

1 0.95

Fig. 6. The probability PrCC of correct classification of the QHLRT and proposed scheme along with Ideal MC, QHLRT-UB, and Proposed-UB when recognizing a modulation scheme out of BPSK, 4-PAM, and 16-QAM with K = 100 for N = 2 (dotted lines) and N = 4 (solid lines).

0.98 0.9 0.96 0.85

PrCC

0.94 0.8 0.92 −1

0

1

0.75

Ideal MC by (5)

0.7

QHLRT-UB by (13) using CRLBs Proposed-UB by (14) using CRLBs

0.65

Proposed algorithm by (8) QHLRT by (6) −2

0

2

4 SNR, γ [dB]

6

8

10

Fig. 5. The probability PrCC of correct classification of the QHLRT and proposed scheme along with Ideal MC, QHLRT-UB, and Proposed-UB when recognizing a modulation scheme out of BPSK and QPSK, with K = 100 for N = 2 (dotted lines) and N = 4 (solid lines).

16-QAM. In addition, the gap between all curves tends to zero when SNR increases, which is consistent with the asymptotic CRLBs in (37)–(39). According to Fig. 2, the CRLB(ϕ) ˆ curves obtained by Theorems 1 and 2 perfectly overlap with the curves obtained by [15, eq. (44)] for 16-QAM and QPSK.

2 ) is presented; for this, however, there is In Fig. 3, CRLB(σ no explicit existing result to compare with in the literature. The asymptotic CRLBs are consistent with the derivations by Lemma 4 in Figs. 1–4. Fig. 4 shows the CRLB of the SNR estimate for BPSK, 4-PAM, 8-QAM, QPSK, and 16-QAM substituting K = 100 into (34). One can see that the curves obtained by Lemma 3 exactly overlap with the curves obtained by [14, eqs. (11) and (50)] for QPSK and 16-QAM.

In Figs. 5 and 6, the correct classification probability, PrCC , is presented for the QHLRT of (6) and proposed algorithm of (8) with N = 2 and 4 receive antennas. Also, their performance is compared to the upper-bounds of Ideal MC of (5), QHLRT-UB of (13), and Proposed-UB of (14). Particularly, using the CRLB analysis in Section IV, QHLRTUB and Proposed-UB are obtained. Specifically, for Proposed˜ opt are obtained by substituting the ˜ i (ri |Hm ) and λ UB, L i optimum CRLB estimates into (7) and (10), respectively. Then gProposed-UB ({ri }N i=1 |Hm ) of (14) is obtained. In Fig. 5, a modulation scheme is determined out of BPSK and QPSK; and in Fig. 6, a modulation scheme is identified out of BPSK, 4-PAM, and 16-QAM. As expected, the proposed scheme significantly outperforms the QHLRT. It can be seen that the probabilities of correct classification of Ideal MC, QHLRTUB, proposed-UB, and the proposed scheme approach one when SNR increases. Increasing the number of receive antennas improves the classification performance of all schemes except the QHLRT. QHLRT-UB performs very well and its performance is similar to that of Ideal MC algorithm having no error in estimation. However, the performance of QHLRT degrades as the number of antennas increases. The reason for the odd performance of the QHLRT scheme is due to the inaccurate non-data-aided estimates, especially the phase estimate. Specifically, as discussed in Section III, the worst LF value dominates the overall performance in QHLRT N ˆ (m) ˆ (m) of (6): i=1 R i,k,p → 0, when mini {Ri,k,p } → 0. With more ˆ (m) } → 0 get higher receive antennas, the chances of mini {R i,k,p for QHLRT. It can be demonstrated that the performance of QHLRT improves with the number of antennas, if the non ideal phase estimates are replaced with the optimal estimates

3824


1

1

0.37

0.9

N =4

K=100 0.36

0.95

0.35

0.9

0.97

0.8

0.96 0.95 N =2

PrCC

0.34 −4

−3

PrCC

0.7 K=10

−2

0.94

0.85

−2.5

0.6 0.8

−2

−1.5

Proposed algorithm by (8)

0.5 Selection combining by (12)

0.75

0.4

−5

QHLRT-UB by (13) using CRLBs, N = 2 QHLRT-UB by (13) using CRLBs, N = 3 QHLRT-UB by (13) using CRLBs, N = 4 −4

−3

−2

−1

0 SNR, γ [dB]

1

2

3

4

Equal gain combining by (11) 0.7 −6

5

Fig. 7. The probability PrCC of correct classification of QHLRT-UB when recognizing a modulation scheme out of QPSK, 8-QAM, and 16-QAM with K = 10 and K = 100 for N = 2, N = 3, and N = 4.

−4

−2

0

2 SNR, γ [dB]

4

6

8

10

Fig. 9. The probability PrCC of correct classification of the proposed scheme along with selection combining, and equal gain combining when recognizing a modulation scheme out of BPSK and QPSK, with K = 100 for N = 2 (dotted lines) and N = 4 (solid lines).

1

1 0.95

0.9

0.9 0.8

Ideal MC by (5)

0.85

PrCC

PrCC

Proposed algorithm by (8) 0.7

0.6

0.8 QHLRT by (6) 0.75 0.7

Ideal MC by (5) QHLRT-UB by (13) using CRLBs

0.65

Proposed-UB by (14) using CRLBs

0.5

Proposed algorithm by (8)

0.6

QHLRT by (6) 0.4 20

30

40

50

60

70 K

80

90

100

110

120

0.55

2

3

4

5 N

6

7

8

Fig. 8. The probability PrCC of correct classification of the QHLRT and proposed scheme along with Ideal MC, QHLRT-UB, and Proposed-UB when recognizing a modulation scheme out of BPSK, 4-PAM, and 16-QAM with SNR = 2 dB for N = 2 (dotted lines) and N = 4 (solid lines).

Fig. 10. The probability PrCC of correct classification of the QHLRT and proposed scheme along with Ideal MC when recognizing a modulation scheme out of BPSK and QPSK, with K = 100 for SNR = −2dB (dotted lines) and SNR = 1dB (solid lines).

using CRLBs, keeping the non ideal amplitude and noise variance estimates. In Fig. 6, the performance of QHLRT is still bad even in high SNR. This is due to the poor (or limited) performance of the non-ideal (but, state-of-the-art practical) phase estimator. In particular, the impact of non-ideal phase estimator on the performance of QHLRT is more significant for higher modulation schemes such as 16-QAM having shorter minimum Euclidean distance among the constellation points. The impact is adversely more enhanced in QHLRT with multiple antennas, because the overall performance of QHLRT is dominated by the worst phase estimates of all antennas. In Fig. 7, the correct classification probability of QHLRTUB is plotted when recognizing a modulation scheme out of QPSK, 8-QAM, and 16-QAM for N = 2, 3, 4 with K = 10 and 100. One can see that increasing the number of antennas may result in performance improvement or performance degradation depending on the SNR regime. In the low SNR region where the estimates are not accurate (although they are optimal), increasing the number of receiving antennas does not lead to performance improvement; instead, the performance

degrades. This is because if at least one channel corresponding to a particular receive antenna is bad, the overall performance of QHLRT-UB degrades substantially. In high SNR, as one can naturally expect, the performance of QHLRT-UB improves with the number of receive antennas. Recall that when the non-optimal, non-data-aided estimates are used as in QHLRT, the performance degrades as the number of receive antennas grows even up to (very) high SNR range. In Fig. 8, the classification performance is presented over different values of K. One can see that increasing K improves PrCC and the proposed scheme outperforms the QHLRT scheme for all values of K. The performance of QHLRT is bad even for large K, which is due to the poor performance of the phase estimator, as discussed for Fig. 6. In Fig. 9, the probability PrCC of correct classification of the proposed scheme along with SC, and EGC when recognizing BPSK and QPSK, with K = 100 for N = 2, 4 is shown. One can see that the proposed algorithm outperforms the other two algorithms slightly, and the gap increases with the number of antennas. In Fig. 10, PrCC is presented for the QHLRT, proposed


algorithm, and Ideal MC with N = 2, · · · , 8 and SNR = −2, 1dB when classifying BPSK and QPSK. As expected, the proposed scheme significantly outperforms the QHLRT, and moreover, the performance of the proposed scheme is very close to that of the Ideal MC. Remark 1: In each of the simulation settings, we have tested two or three different modulation schemes at the same time. If many modulation schemes are tested at the same time, the performance of all schemes including the proposed one and QHLRT becomes very poor. In fact, this is not only the problem of our own proposed scheme; but it is a limitation of all likelihood-based modulation classification schemes because the performance of likelihood-based algorithms heavily depends on the quality of phase estimation. As discussed before, no existing phase estimators works well enough to ensure good performance of MC with many modulation schemes. Remark 2: It is an interesting topic to consider MC for other type of modulations such as frequency shift keying (FSK) or orthogonal frequency division multiplexing (OFDM). For MC of FSK, the carrier frequency offset, phase jitter, and time offset must be estimated in addition to the amplitude, phase, and noise variance. However, this is a very difficult task and it seems almost impractical to develop such likelihood-based MC methods that perform reasonably well. For essentially the same reason, MC for multi-carrier systems such as OFDM is very difficult. In fact, the state-of-the-art of MC for multicarrier systems is to distinguish between single carrier linearly digitally modulated signals (SCLD) and OFDM, which is only a binary decision [24]. Even this binary deicsion is never simple. To classify those more complex modulations, featurebased algorithms might be possibly a better approach, rather than likelihood-based algorithms. For example, feature-based algorithms based on cyclostationarity might be an efficient approach to distinguish among SCLD, cyclic-prefix (CP)SCLD, and OFDM signals [24]. VI. C ONCLUSIONS In this paper, we have explored likelihood-based algorithms for linear digital modulation classification. We have considered a multiple receive antennas configuration with channel amplitude, phase, and noise variance as the unknown parameters. We have first extended the existing MC algorithms to the multiple receive antennas configuration and then identified a critical problem where the overall performance of MC is dominated by the worst channel estimates. To address this performance issue, we have proposed a new scheme, namely the weighted-sum MC algorithm. Furthermore, the CRLBs for general rectangular I × J-QAM and M -PAM modulations have been derived to be used for the performance upperbounds of QHLRT and proposed algorithms. Numerical results demonstrated the accuracy of these CRLB expressions and confirmed that our derived CRLB expressions generalize existing results in the literature. It was shown that the performance of the proposed MC algorithm approached the theoretical upper-bounds and significantly exceeded that of the existing QHLRT algorithm. Because the performance of likelihood-based algorithms depends heavily on the quality of phase estimation, if there

3825

are many unknown modulation candidates, feature-based algorithms might be considered as a future work. Another interesting topic as a future work is to study MC algorithms for FSK or OFDM signals. A PPENDIX A P ROOF OF L EMMA 2 In order to obtain the FIM, we derive each element of I in the following. Let us start by deriving I2,2 . Using (16) and I/2 −jϕ denoting LLFI,k := log }) , we l=1 gl cosh(ul {rk e can obtain (A.1), where

⎧ δ1,ϕ ({rk e−jϕ }) ⎪ ⎪ ⎪ ⎪ ⎨ δ2,ϕ ({rk e−jϕ }) ⎪ −jϕ ⎪ }) ⎪ δ3,ϕ ({rk e ⎪ ⎩ δI ({rk e−jϕ })

ul {rk e−jϕ } sinh(ul {rk e−jϕ }) cosh(up {rk e−jϕ }), u2l cosh(ul {rk e−jϕ }) cosh(up {rk e−jϕ }), ul up sinh(ul {rk e−jϕ }) sinh(up {rk e−jϕ }), I/2 −jϕ }). p=1 gp cosh(up {rk e (A.2) αejϕ sk,q + n, where n denotes AWGN. and imaginary parts of rk e−jϕ are given = · = = =

Assume rk = Then the real by α{sk,q } + v and α {sk,q } + v , respectively, where ne−jϕ = v + jv . Denoting F I ({rk e−jϕ }) := I/2 I/2 ˜ I ({rk e−jϕ }) := − p=1 l=1 gl gpδ1,ϕ ({rk e−jϕ }) and F I/2 I/2 −jϕ −jϕ δ g g ({r e }) − δ ({r e }) , l p 2,ϕ k 3,ϕ k l=1 p=1 conditioned on that the known constellation point sk,q is transmitted, E[LLFI,k |sk,q ] is given by (A.3). After changing the variables α{sk,q } + v = x and α {sk,q } + v = y, it can be observed that sk,q vanishes from (A.3) and we have E

where J1,I =

∂ 2 LLFI,k ∂ϕ2

α2 d 2

α2 d 2 2 exp − σ2I,J √ = −J1,I + J2,J I πσ2 2 x +∞ exp(− σ2 )F˜ I (x) × −∞ dx, δI (x)

I,J 2 exp − σ2 √ 2 I πσ

+∞ −∞

α2 d 2 I,J

2 exp − σ2 √ and J2,J = J πσ2 J/2 stituting {gp , up }p=1 into

2

exp(− σx 2 )x

+∞ −∞

I/2

l=1

(A.4)

ul gl sinh(ul x)dx

2

exp(− σy 2 ) y 2 δJ (y)dy. Sub-

δJ (y), we have (y−(2p−1)αdI,J )2 +∞ J/2 J2,J = J √1πσ2 −∞y 2 p=1 exp − σ2 (y+(2p−1)αdI,J )2 dy + exp − σ2 J/2 2 2 2 2 2 = J . p=1 σ /2 + α dI,J (2p − 1) (A.5) J/2 2 2 Using (2p − 1) = J(J − 1)/6, we obtain J = 2,J p=1 σ2 2 J 2 −1 + α . Similarly, it can be shown that J 1,I = 2 I 2 +J 2 −2 2α2 I 2 −1 K σ2 I 2 +J 2 −2 . Since {rk }k=1 are independent random vari∂ 2 LLF

2

I (r|u) ables, we have E[ ∂ LLF ] = KE[ ∂ϕ2I,k ]. Furthermore, ∂ϕ2 ∂2 C it is not difficult to show that E[ ∂ϕ 2 ] = 0. Substituting J1,I and J2,J into (A.4) and after some manipulations, we 2 1 obtain I2,2 = 2Kγ[1 − ( 12 + γ I 2J+J−1 2 −2 )FI (γ) − ( 2 + 2 γ I 2I+J−1 2 −2 )FJ (γ)], where

FI (γ) =

2 exp(−γd2I,J ) √ I 2π

+∞ 0

ˆ I (x) exp(−x2 /2)F √ dx, 2γ(2p − 1)xdI,J p=1 gp cosh (A.6)

I/2

I/2 I/2 ˆ I (x) F = gl gp 4(2l − p=1 √ l=1 √ 2 2 1) dI,J cosh 2γ(2l − 1)xdI,J cosh 2γ(2p − 1)xdI,J −

and

3826


2

I/2 I/2

∂ LLFI,k = ∂ϕ2

p=1

l=1 gl gp

2α2 d2 4 exp − σ2I,J

− δ1,ϕ ({rk e−jϕ }) + 2 {rk e−jϕ } δ2,ϕ ({rk e−jϕ }) − δ3,ϕ ({rk e−jϕ }) . 2 δI ({rk e−jϕ })

+∞ +∞

˜ I (α{sk,q } + v ) F I (α{sk,q } + v ) + (α {sk,q } + v )2 F

πIJσ 2 δI (α{sk,q } + v ) −∞ −∞ 2 2 (α{sk,q } + v ) + (α {sk,q } + v ) × exp − δJ (α {sk,q } + v )dv dv . σ2 √ √ 2 4(2l−1)(2p−1)d 2γ(2l−1)xdI,J sinh 2γ(2p− I,J sinh 1)xdI,J . Taking partial derivatives of LLFI,k with respect to α, ϕ, we have (A.7), where ⎧ 2αd2I,J ul ⎪ β1 ({rk e−jϕ }) = [(2p − 1)2 − (2l − 1)2 ] + α1 ⎪ 2 ⎪ σ ⎪ ⎪ · sinh(ul {rk e−jϕ }) cosh(up {rk e−jϕ }), ⎪ ⎪ ⎨ u2 β2 ({rk e−jϕ }) = αl {rk e−jϕ } cosh(ul {rk e−jϕ }) ⎪ · cosh(up {rk e−jϕ }), ⎪ ⎪ ⎪ u u −jϕ ⎪ }) = lα p {rk e−jϕ } sinh(ul {rk e−jϕ }) ⎪ ⎪ β3 ({rk e ⎩ · sinh(up {rk e−jϕ }). (A.8) Taking expectation and changing variables similar to the proof ∂ 2 LLF of (A.4), we have E[ ∂ϕ∂αI,k ] = 0 since δJ (y) is an even α2 d 2 2 2 exp − σ2I,J +∞ √ y exp(− σy 2 )δJ (y)dy = function of y and −∞ J πσ2

GI (γ) =


(A.1)

× (A.3)

ˆ I (x) exp(−x2 /2)G dx, √ 2γ(2p − 1)xdI,J (A.12) I/2 I/2 2 2 2 0.5(2l − 1) x + p=1 l=1 4dI,J gl gp √ 2 2 +∞

I/2

0

p=1 gp cosh

Î (x) = and G 2 2 − 1][(2l − 1) − (2p 1) ] cosh 2γ(2l − γdI,J [(2l − 1)√ − √ 1)xdI,J cosh 2γ(2p− − 1)xd 2γxd − 1)[(2l − I,J I,J (2l √ √ 1)2 − (2p − 1)2 ] sinh 2γ(2l − 1)xdI,J cosh 2γ(2p − √ − (2l−1)(2p−1) x2 sinh 2γ(2l − 1)xdI,J 1)xdI,J 2 √ sinh 2γ8(2p − 1)xdI,J . In order to derive I1,3 , we take partial derivatives of LLFI,k , which is given by (A.13), where ⎧ −jϕ −jϕ ⎪ μ ({r e }) = {r e } 3ul dI,J α[(2l − 1)2 ⎪ 1 k k ⎪ ⎪ ⎪ ⎪ 2 ⎪ −(2p − 1) ] − 2(2l − 1) ⎪ ⎪ ⎪ ⎪ −jϕ ⎪ }) cosh(up {rk e−jϕ }), · sinh(u ⎪ l {rk e 2 ⎪ ∂ C d2 α2 0. In addition, it is easy to show that E[ ∂ϕ∂α ] = 0 and hence ⎪ ⎨ μ2 ({rk e−jϕ }) = 2dI,J α[(2l − 1)2 − 1] I,J 1)2 σ2 [(2p − I1,2 = 0. Similarly, it can be shown that I2,1 = I3,2 = I2,3 = 2 2 ⎪ ul σ 2 −jϕ ⎪ 0. ⎪ −(2l − 1)2 ] + 1 − dI,J } ⎪ α {rk e ⎪ ⎪ In order to find I1,1 , we take the second derivative of ⎪ −jϕ −jϕ ⎪ }) cosh(up {rk e }), · cosh(ul {rk e ⎪ ⎪ LLFI,k with respect to α, which is given by (A.9), where ⎪ ul up σ2 2 −jϕ −jϕ −jϕ ⎪ ⎪ }) = dI,J α {rk e } sinh(ul {rk e }) μ3 ({rk e ⎪ ⎧ ⎪ ⎩ −jϕ 2 2 −jϕ −jϕ δ (2p − 1) {r ({r e }) = 4u − (2l − 1) e } · sinh(u {r e }). 1,α k l k ⎪ p k ⎪ ⎪ ⎪ (A.14) · sinh(ul {rk e−jϕ }) cosh(up {rk e−jϕ }), ⎪ ⎪ ⎪ 2d2I,J α2 ⎪ Taking expectation and changing variables, we obtain −jϕ 2 2 ⎪ }) = 2[(2l − 1) − 1]( σ2 [(2l − 1) ⎨δ2,α ({rk e α2 d 2 4(2l−1)2 2 2 −jϕ 2 2 exp − σ2I,J −(2p − 1) ] − 1) + {r e } 2 k σ ⎪ ∂ LLFI,k ⎪ √ = J4,I − J5,I + × E ∂σ2 ∂α ⎪ · cosh(ul {rk e−jϕ }) cosh(up {rk e−jϕ }), ⎪ I πσ2 (A.15) ⎪ ⎪ 4(2l−1)(2p−1) 2 x2 ˜ ⎪ −jϕ −jϕ exp(− σ2 )HI (x) +∞ ⎪ δ ({r e }) = {r e } 3,α k 2 k ⎪ σ × −∞ dx, ⎩ δI (x) · sinh(ul {rk e−jϕ }) sinh(up {rk e−jϕ }). I/2 I/2 gl gp dI,J (A.10) where H ˜ I (x) = μ1 (x) + μ2 (x) + l=1 p=1 σ4 After taking expectation and changing variables, we have 2 μ (x) − 2d α[(2l − 1) − 1] cosh(u 3 I,J l x) cosh(up x) α2 d 2 I,J 2 2 exp − σ2 +2(2l − 1)x sinh(ul x) cosh(up x) , J4,I = ∂ LLFI,k √ = −J3,I + E × α2 d2I,J ∂α2 I πσ2 (A.11) 4d2 α exp − +∞ 2 I/2 I,J σ2 +∞ exp(− σx22 )G˜I (x) √ exp(− σx2 ) l=1 [(2l − −∞ σ4 I πσ2 × −∞ dx, δI (x) − 1]gl cosh(ul x)dx, and J5,I = 1)2 α2 d2I,J 2 g d g I/2 l p I,J I/2 I/2 +∞ ˜I (x) = x2 where G δ1,α (x) + δ2,α (x) − 4dI,J exp √− σ2 − l=1 p=1 σ2 l=1 (2l −∞ exp(− σ2 ) σ4 I πσ2 2α I 2 −4 2 and 1)gl x sinh(ul x)dx. We can show that J4,I = σ4 I 2 +J 2 −2 and δ3,α (x) + 2[(2l − 1) − 1] cosh(ul x) cosh(up x) α2 d2I,J I 2 −1 ∂2C 4Kα 3 J5,I = 2α +∞ 2 I/2 4d2I,J exp − σ2 σ4 I 2 +J 2 −2 . Also, we have E[ ∂σ2 ∂α ] = σ4 I 2 +J 2 −2 . x 2 √ J3,I = exp(− ) [(2l − 1) − Using (A.15) and after some manipulations, we have l=1 σ2 −∞ σ2 I πσ2 2 1 1 I1,3 = I3,1 = 2Kα 1]gl cosh(ul x)dx. We can show that J3,I = 2 2(I2 −4) . σ4 [ 2 HI (γ) + 2 HJ (γ)], where 2 2

σ (I +J −2)

∂ C 12K Furthermore, we can show that E[ ∂α 2 ] = − σ 2 (I 2 +J 2 −2) . Using (A.11) and after some manipulations, we have I1,1 = 2K 1 1 σ2 [1 − 2 GI (γ) − 2 GJ (γ)], where

HI (γ) =


+∞ 0

ˆ I (x) exp(−x2 /2)H √ dx, 2γ(2p − 1)xdI,J p=1 gp cosh (A.16)

I/2


I/2 I/2

2

∂ LLFI,k = ∂ϕ∂α

l=1 gl gp {rk e

p=1

∂ 2 LLFI,k = ∂α2

I/2 I/2 p=1

l=1

gl gp d2I,J σ2

I/2 I/2

2

∂ LLFI,k = ∂σ 2 ∂α

p=1

−jϕ

} β1 ({rk e−jϕ }) + β2 ({rk e−jϕ }) − β3 ({rk e−jϕ }) . 2 δI ({rk e−jϕ })

δ1,α ({rk e−jϕ }) + δ2,α ({rk e−jϕ }) − δ3,α ({rk e−jϕ }) . 2 δI ({rk e−jϕ })

gl gp dI,J l=1 σ4

μ1 ({rk e−jϕ }) + μ2 ({rk e−jϕ }) + μ3 ({rk e−jϕ }) . 2 δI ({rk e−jϕ })

I/2 I/2 2 ˆ I (x) and H = (2l − l=1 2dI,J gl gp p=1 2 2 2 2 2 1) x + γdI,J [(2l − 1) − 1][(2l √ − 1) − (2p − √ 1)2√ ] cosh 2γ(2l − 1)xdI,J cosh 2γ(2p − 1)xd √ I,J − 2 2 1.5 2γxd (2l − 1)[(2l − 1) − (2p − 1) ] sinh 2γ(2l − I,J √ 1)xdI,J cosh 2γ(2p − 1)xdI,J − (2l − 1)(2p − √ √ 1)x2 sinh 2γ(2l − 1)xdI,J sinh 2γ(2p − 1)xdI,J . In order to find I3,3 , we take the second-order derivative of LLFI,k with respect to σ 2 , which is given by (A.17), where ⎧ δ1,σ2 ({rk e−jϕ }) = {rk e−jϕ } − 2ul dI,Jα[(2l − 1)2 ⎪ ⎪ ⎪ ⎪ −(2p − 1)2 ] + 4(2l − 1) ⎪ ⎪ ⎪ −jϕ ⎪ · sinh(u }) cosh(up {rk e−jϕ }), ⎪ l {rk e ⎪ ⎪ 2 2 ⎪ ⎪δ 2 ({r e−jϕ }) = d α[(2l − 1)2 − 1] dI,J α [(2l − 1)2 ⎨ k I,J 2,σ σ2 u2 σ2 2 2 −jϕ l ⎪ −(2p − 1) ] − 2 + {r e } k ⎪ dI,J α ⎪ ⎪ ⎪ ⎪ · cosh(ul {rk e−jϕ }) cosh(up {rk e−jϕ }), ⎪ ⎪ ⎪ u u σ2 ⎪ ⎪ δ3,σ2 ({rk e−jϕ }) = dlI,Jp α 2 {rk e−jϕ } sinh(ul {rk e−jϕ }) ⎪ ⎪ ⎩ · sinh(up {rk e−jϕ }). (A.18) Taking expectation and changing variables, we have E

2

∂ LLFI,k ∂σ4

=

α σ2 (−J4,I

×

+∞

+ 2J5,I ) +

α2 d 2 2 exp − σ2I,J √ × I πσ2

x2 ˜ exp(− σ 2 )KI (x) dx, δI (x) −∞

(A.19) ˜ I (x) = I/2 I/2 gl gp d6I,J α δ1,σ2 (x) + δ2,σ2 (x) − where K l=1 p=1 σ δ3,σ2 (x)+ 2dI,J α[(2l − 1) 2 − 1] cosh(ul x) cosh(up x) −4(2l − ∂2 C 1)x sinh(ul x) cosh(up x) . In addition, we have E[ ∂σ 4] = K 2kα2 3 − σ4 + σ6 ( I 2 +J 2 −2 + 1) . Using (A.19) and after some 2 manipulations, we obtain I3,3 = σK4 − Kα σ6 [KI (γ) + KJ (γ)], where 4 exp(−γd2I,J ) √ KI (γ) = I 2π

+∞ 0

ˆ I (x) exp(−x2 /2)K √ dx, g cosh 2γ(2p − 1)xdI,J p=1 p (A.20)

I/2

2 ˆ I (x) = and K 2(2l − 1)2 x2 + p=1 l=1 dI,J gl gp √ 2 2 − 1][(2l − 1)2 − (2p γd2I,J [(2l − 1)√ −√1) ] cosh 2γ(2l − 1)xdI,J cosh 2γ(2p−1)xd I,J −2 2γxd I,J (2l−1)[(2l− √ √ 2 cosh√ 2γ(2p − − 1) ] sinh 2γ(2l − 1)xd 1)2 − (2p I,J − 2(2l − 1)(2p − 1)x2 sinh 2γ(2l − 1)xdI,J √ 1)xdI,J sinh 2γ(2p − 1)xdI,J . Substituting Ib,c for b, c = 1, 2, and 3 into (17), (18) is obtained and the proof is completed. I/2 I/2

3827

(A.7)

(A.9)

(A.13)

A PPENDIX B P ROOF OF T HEOREM 2 Employing the approach used in Lemma 1, the conditional LLF given M -PAM signal is obtained by ⎛ ⎞ M/2 K LLF(r|u) = CM-PAM + log ⎝ gl cosh(ul {rk e−jϕ })⎠ , k=1

l=1

(B.1) 2 2 2 2 2 2 where CM-PAM = K log( πMσ 2 ) − Kα d /σ − r /σ . To derive the FIM, following the steps in Appendix A, we obtain (B.2), where ⎧ 2 exp(−γd2M -PAM ) √ ⎪ GM-PAM (γ) = ⎪ M 2π ⎪ ⎪ ˆ M -PAM (x) +∞ ⎪ exp(−x2 /2)G ⎪ · 0 dx, M/2 √ ⎪ ⎪ g cosh( 2γ(2p−1)xd M -PAM ) ⎪ p=1 p ⎪ 2 ⎪ 2 exp(−γdM -PAM ) ⎪ ⎪ √ FM-PAM (γ) = ⎪ ⎪ M 2π ⎪ +∞ ˆ M -PAM (x) ⎪ exp(−x2 /2)F ⎪ ⎨ · 0 dx, M/2 √ p=1 2

gp cosh( 2γ(2p−1)xdM -PAM )

2 exp(−γdM -PAM ) ⎪ √ HM-PAM (γ) = ⎪ ⎪ ⎪ +∞M 2π exp(−x2 /2)H ⎪ ˆ M -PAM (x) ⎪ ⎪ · 0 dx, M/2 √ ⎪ ⎪ p=1 gp cosh( 2γ(2p−1)xdM -PAM ) ⎪ ⎪ 2 ⎪ 2 exp(−γdM -PAM ) ⎪ √ ⎪ KM-PAM (γ) = ⎪ ⎪ ⎪ +∞M 2π exp(−x2 /2)K ˆ M -PAM (x) ⎪ ⎩ · 0 dx. M/2 √ p=1

gp cosh( 2γ(2p−1)xdM -PAM )

(B.3) ˆ M-PAM (x), H ˆ M-PAM (x), ˆM-PAM (x), F In the equations above, G ˆ M-PAM (x) are obtained by substituting I = M and and K 3 dM-PAM = M 2 −1 into the corresponding functions in (A.12), (A.6), (A.16), and (A.20), respectively. Taking the inverse of the FIM (B.2), (23)–(25) are derived. This completes the proof. ACKNOWLEDGMENT The authors would like to thank Professor Octavia A. Dobre and the anonymous reviewers for their constructive comments, which have improved the presentation of the paper. R EFERENCES [1] O. A. Dobre, A. Abdi, Y. Bar-Ness, and W. Su, “Survey of automatic modulation classification techniques: classical approaches and new trends,” IET Commun., vol. 1, pp. 137–156, Apr. 2007. [2] J. L. Xu, W. Su, and M. Zhou, “Likelihood-ratio approaches to automatic modulation classification,” IEEE Trans. Systems, Man, and Cybernetics—Part C, vol. 41, pp. 455–469, July 2011.

3828


2

∂ LLFI,k = ∂σ 4

I/2 I/2 p=1

l=1

gl gp αdI,J σ6

δ1,σ2 ({rk e−jϕ }) + δ2,σ2 ({rk e−jϕ }) − δ3,σ2 ({rk e−jϕ }) . 2 δI ({rk e−jϕ })

⎞ ⎛ 2 0 σ − σ 2 GM -PAM (γ) αHM -PAM (γ) 2K ⎝ 2 2 ⎠. 0 α σ 1 − FM -PAM (γ) 0 IM -PAM (u) = 4 σ 1 α2 0 − K (γ) αHM -PAM (γ) M -PAM 2 2 σ

[3] F. Hameed, O. A. Dobre, and D. C. Popescu, “On the likelihoodbased approach to modulation classification,” IEEE Trans. Wireless Commun., vol. 8, pp. 5884–5892, Dec. 2009. [4] M. Derakhtian, A. A. Tadaion, and S. Gazor, “Modulation classification of linearly modulated signals in slow flat fading channels,” IET Signal Processing, vol. 5, pp. 443–450, Aug. 2011. [5] C. Y. Huang and A. Polydoros, “Advanced methods for digital quadrature and offset modulation classification,” in Proc. 1988 IEEE MILCOM, vol. 2, pp. 841–845. [6] C. Y. Hwang and A. Polydoros, “Likelihood methods for MPSK modulation classification,” IEEE Trans. Commun., vol. 43, pp. 1493– 1504, Apr. 1995. [7] A. Abdi, O. A. Dobre, R. Choudhry, Y. Bar-Ness, and W. Su, “Modulation classification in fading channels using antenna arrays,” in Proc. 2004 IEEE MILCOM, vol. 1, pp. 211–217. [8] O. A. Dobre and F. Hameed, “Likelihood–based algorithms for linear digital modulation classification in fading channels,” in Proc. 2006 IEEE CCECE, pp. 1347–1350. [9] J. L. Xu, W. Su, and M. Zhou, “Distributed automatic modulation classification with multiple sensors,” IEEE Sensors J., vol. 10, pp. 1779–1785, Nov. 2010. [10] P. Panagiotou, A. Anastasopoulos, and A. Polydoros, “Likelihood ratio tests for modulation classification,” in Proc. 2000 IEEE MILCOM, vol. 2, pp. 670–674. [11] H.-C. Wu, Y. Wu, J. C. Principe, and X. Wang, “Robust switching blind equalizer for wireless cognitive receivers,” IEEE Trans. Wireless Commun., vol. 7, pp. 1461–1465, May 2008. [12] O. A. Dobre, Y. Bar-Ness, and W. Su, “Cyclostationarity-based modulation classification of linear digital modulations in flat fading channels,” Springer Wireless Personal Commun. DOI: 10.1007/s11277009-9776-2, 2009. [13] A. Swami and B. M. Sadler, “Hierarchical digital modulation classification using cumulants,” IEEE Trans. Commun., vol. 48, pp. 416–429, Mar. 2000. [14] F. Bellili, A. Stéphenne, and S. Affes, “Cramér-Rao lower bounds for non-data-aided SNR estimates of square QAM modulated transmissions,” IEEE Trans. Commun., vol. 58, pp. 3211–3218, Nov. 2010. [15] F. Bellili, N. Atitallah, S. Affes, and A. Stéphenne, “Cramér-Rao lower bounds for frequency and phase non-data-aided estimation from arbitrary square QAM-modulated signals,” IEEE Trans. Signal Process., vol. 58, pp. 4517–4525, Sept. 2010. [16] A. Stéphenne, F. Bellili, and S. Affes, “Moment-based SNR estimation over linearly-modulated wireless SIMO channels,” IEEE Trans. Wireless Commun., vol. 9, pp. 714–722, Feb. 2010. [17] P. Campisi, G. Panci, S. Colonnese, and G. Scarano, “Blind phase recovery for QAM communication systems,” IEEE Trans. Signal Process., vol. 53, pp. 1348–1358, Apr. 2005. [18] S. M. Kay, Fundamentals of Statistical Signal Processing, Estimation Theory. Prentice Hall, 1993. [19] J. G. Proakis, Digital Communications, 4th ed. McGraw-Hill, 2001. [20] G. K. Karagiannidis, “On the symbol error probability of general order rectangular QAM in Nakagami-m fading,” IEEE Commun. Lett, vol. 10, pp. 745–747, Nov. 2006. [21] Standard and 8PSK (8QAM LDPC only) & 16QAM. Available: http: //www.all4sat.com/standard-and-8psk-8qam-ldpc-only-and16qam. html [22] B. Li, J. Huang, S. Zhou, K. Ball, M. Stojanovic, L. Freitag, and P. Willett, “MIMO-OFDM for high-rate underwater acoustic communications,” IEEE J. Ocean. Eng., vol. 34, pp. 634–644, Oct. 2009. [23] K. P. Peppas and C. K. Datsikas, “Average symbol error probability of general-order rectangular quadrature amplitude modulation of optical wireless communication systems over atmospheric turbulence channels,” IEEE J. Opt. Commun. Netw., vol. 2, pp. 102–110, Feb. 2010.

(A.17)

(B.2)

[24] A. Punchihewa, Q. Zhang, O. A. Dobre, C. Spooner, S. Rajan, and R. Inkol, “On the cyclostationarity of OFDM and single carrier linearly digitally modulated signals in time dispersive channels: theoretical developments and application,” IEEE Trans. Wireless Commun., vol. 9, pp. 2588–2599, Aug. 2010. Ali Ramezani-Kebrya (S’13) received the B.Sc. degree from the University of Tehran, Tehran, Iran, and the M.A.Sc degree from Queen’s University, Kingston, Canada, in 2010 and 2012, respectively, both in electrical engineering. In September 2012, he joined the Wireless Computing Lab (WCL) of the Department of Electrical and Computer Engineering, the University of Toronto, Toronto, Canada, where he is currently working towards the Ph.D. degree. His research interests include distributed signal processing techniques and optimization, detection and estimation theory, and statistical signal processing. Mr. Ramezani-Kebrya was the recipient of the IEEE Kingston Section M.Sc. Research Excellence Award. Il-Min Kim (SM’06) received the B.S. degree in electronics engineering from Yonsei University, Seoul, Korea, in 1996, and the M.S. and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Taejon, Korea, in 1998 and 2001, respectively. From October 2001 to August 2002 he was with the Dept. of Electrical Engineering and Computer Sciences at MIT, Cambridge, USA, and from September 2002 to June 2003 he was with the Dept. of Electrical Engineering at Harvard, Cambridge, USA, as a Postdoctoral Research Fellow. In July 2003, he joined the Dept. of Electrical and Computer Engineering at Queen’s University, Kingston, Canada, where he is currently an Associate professor. His research interests include cooperative diversity networks, physical layer security, compressive sensing, and cognitive radio. He was as an Editor for the IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS from 2005 to 2011. He is currently serving as an Editor for the IEEE W IRELESS C OMMUNICATIONS L ETTERS and as an Editor for the Journal of Communications and Networks (JCN). Dong In Kim (S’89-M’91-SM’02) received the Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, in 1990. He was a tenured Professor with the School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada. Since 2007, he has been with Sungkyunkwan University (SKKU), Suwon, Korea, where he is currently a Professor with the School of Information and Communication Engineering. His research interests include wireless cellular, relay networks, and cros-slayer design. Dr. Kim has served as an Editor and a Founding Area Editor of Cross-Layer Design and Optimization for the IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS from 2002 to 2011. From 2008 to 2011, he served as the Co-Editor-in-Chief for the Journal of Communications and Networks. He is currently the Founding Editor-in-Chief for the IEEE W IRELESS C OMMUNICATIONS L ETTERS and has been serving as an Editor of Spread Spectrum Transmission and Access for the IEEE T RANSACTIONS ON C OMMUNICATIONS since 2001.


François Chan received his B.Eng. degree in electrical engineering from McGill University, Montreal, Canada and his M.Sc.A. and Ph.D. degrees in electrical engineering from Ecole Polytechnique de Montréal, Canada. He is currently an associate professor in the Department of Electrical and Computer Engineering at Royal Military College of Canada, Kingston, Ontario, Canada. He was a visiting researcher at the University of California, Irvine in 2002 and 2005. His research interests include digital communications, wireless communications, and digital signal processing.

3829

Robert Inkol (M’73-SM’86) received the B.Sc. and M.A.Sc. degrees in Applied Physics and Electrical Engineering from the University of Waterloo in 1976 and 1978, respectively. From 1978 to 2012, he was a Defence Scientist with Defence Research and Development Canada (DRDC) where he was responsible for the technical leadership of various electronic warfare related projects and research programs. He was actively involved in the application of very large scale integrated circuit technology and digital signal processing techniques to electronic warfare systems. For these and other contributions, Mr. Inkol received the Queen Elizabeth II Diamond Jubilee medal. He is currently an Adjunct Professor with the Royal Military College of Canada and is a Senior Member of IEEE.