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On the Likelihood-Based Approach to Modulation Classification Fahed Hameed, Octavia A. Dobre, Senior Member, IEEE, and Dimitrie C. Popescu, Senior Member, IEEE
Abstract—In this paper, likelihood-based algorithms are explored for linear digital modulation classification. Hybrid Likelihood Ratio Test (HLRT)- and Quasi HLRT (QHLRT)- based algorithms are examined, with signal amplitude, phase, and noise power as unknown parameters. The algorithm complexity is first investigated, and findings show that the HLRT suffers from very high complexity, whereas the QHLRT provides a reasonable solution. An upper bound on the performance of QHLRT-based algorithms, which employ unbiased and normallydistributed non-data aided estimates of the unknown parameters, is proposed. This is referred to as the QHLRT-Upper Bound (QHLRT-UB). Classification of binary phase shift keying (BPSK) and quadrature phase shift keying (QPSK) signals is presented as a case study. The Cramer-Rao Lower Bounds (CRBs) of non-data aided joint estimates of signal amplitude and phase, and noise power are derived for BPSK and QPSK signals, and further employed to obtain the QHLRT-UB. An upper bound on classification performance of any likelihood-based algorithms is also introduced. Method-of-moments (MoM) estimates of the unknown parameters are investigated and used to develop the QHLRT-based algorithm. Classification performance of this algorithm is compared with the upper bounds, as well as with the quasi Log-Likelihood Ratio (qLLR) and fourth-order cumulant based algorithms. Index Terms—Cramer-Rao lower bounds, joint parameter estimation, likelihood ratio test, modulation classification.
TABLE I L IST OF A BBREVIATIONS AND A CRONYMS ALRT ALRT-UB ASK BPSK CRB GLRT HLRT LB LRT MC ML MLE MoM NDA PDF PR PSK QAM QHLRT QHLRT-UB qLLR QPSK SNR w.r.t
Average Likelihood Ratio Test ALRT-Upper Bound Amplitude-Shift-Keying Binary Phase-Shift-Keying Cramer-Rao Lower Bound Generalized Likelihood Ratio Test Hybrid Likelihood Ratio Test Likelihood-based Likelihood Ratio Test Modulation Classification Maximum Likelihood Maximum Likelihood Estimates Methods of Moments Non Data Aided Probability Density Function Pattern Recognition Phase-Shift-Keying Quadrature Amplitude Modulation Quasi HLRT QHLRT-Upper Bound quasi Log-Likelihood Ratio Quadrature Phase-Shift-Keying Signal-to-Noise Ratio With respect to
I. I NTRODUCTION
I
N a world of rapid growth of commercial wireless services, accommodating the explosive demand for spectrum access, efficiency and reliability becomes increasingly technically challenging. Furthermore, implementation of advanced information services for military applications in a crowded electromagnetic spectrum is a challenging task for communication engineers. A solution is provided by flexible intelligent radios, capable of sensing and adapting to the environment. In such radios, modulation classification (MC) is an important task. A modulation classifier essentially involves two steps: signal preprocessing and application of a classification algorithm. Signal preprocessing tasks may include estimation of signal Manuscript received July 2, 2008; revised January 4, 2009 and June 17, 2009; accepted July 28, 2009. The associate editor coordinating the review of this paper and approving it for publication was L. Yang. F. Hameed and O. A. Dobre are with the Faculty of Engineering and Applied Science, Memorial University of Newfoundland, 300 Prince Phillip Dr., St. John’s, NL, A1B 3X5, Canada (e-mail:
[email protected],
[email protected]). D. C. Popescu is with the Department of Electrical and Computer Engineering, Old Dominion University, 231 Kaufman Hall, Norfolk, VA 23529 (e-mail:
[email protected]). This work has been supported in part by the National Sciences and Engineering Research Council of Canada and was presented in part at the 2006 IEEE CCECE [1] and the 2007 IEEE Sarnoff Symposium [2]. Digital Object Identifier 10.1109/TWC.2009.12.080883
amplitude and phase, and noise power, symbol timing and waveform recovery, etc. Generally, likelihood-based (LB) and statistical pattern recognition (PR) approaches are used to tackle the MC problem. A likelihood ratio test (LRT) is used for decisionmaking with the former [1]–[11], whereas features extracted from the received signal with the latter [3], [12]–[15]. The LB method formulates the MC problem as a multiple composite hypothesis testing problem, whose solution depends on the modeling of the unknown quantities1 . The number of hypotheses equals the number of modulations to classify. Under the hypothesis 𝐻𝑖 , that modulation 𝑖 is received, the probability density function (PDF) of the received signal can be computed either by averaging over the unknown quantities or using their estimates. This can lead to the Average Likelihood Ratio Test (ALRT) [3]–[10], Generalized Likelihood Ratio Test (GLRT) [3], [10], or Hybrid Likelihood Ratio Test (HLRT) [1]–[3], [10], [11]. The ALRT treats the unknown quantities as random variables, and the PDF of the received signal under each hypothesis is computed by averaging over them. A distribution has to be a priori assigned to each unknown. 1 The unknown quantities refer to the unknown signal constellation points and parameters, such as signal amplitude, phase, and noise power.
c 2009 IEEE 1536-1276/09$25.00 ⃝
HAMEED et al.: ON THE LIKELIHOOD-BASED APPROACH TO MODULATION CLASSIFICATION
If the assumed and actual distributions of the unknowns coincide, then the method provides maximum probability of classification. However, with an increased number of unknown parameters, ALRT suffers from high computational complexity and even mathematical intractability [4]–[6]. With phase as the unknown parameter, an approximation of the PDF of the received signal is employed in [4], [5], yielding a feature-based algorithm. With the GLRT, the PDF of the received signal is computed by employing maximum likelihood estimates (MLE) of the unknown quantities. The disadvantage of GLRT is its failure to identify nested signal constellations, such as Quadrature Amplitude Modulation (QAM) with 16 and 64 points in the signal constellation [10]. This is overcome with the HLRT (hybrid of ALRT and GLRT), by averaging over the signal constellation points in calculating the PDF of the received signal [10]. In [10], classification of linear digital modulations is investigated, with signal amplitude and phase as the unknown parameters. MLE of the unknown parameters are obtained by performing an exhaustive search over a likelihood function. Non-MLE of the signal amplitude and phase are employed in [6], leading to the so called QuasiHLRT (QHLRT)-based algorithm. In this paper, we consider classification of linear digital modulations under the assumption of unknown signal amplitude, phase, and noise power. The analytical closed form expressions for the ML estimators of the unknown parameters are derived, and used to develop the HLRT. Findings show that the HLRT suffers from very high complexity, whereas QHLRT provides a reasonable solution, and, hence, the QHLRT is further explored. An upper bound on the classification performance of the QHLRT-based algorithm is proposed for the case when unbiased and normally distributed non-data aided (NDA) estimates of unknown parameters are available. We refer to it as to the QHLRT-Upper Bound (QHLRT-UB). A case study is presented for binary phase-shift-keying (BPSK) and quadrature phase-shift-keying (QPSK) classification, with the signal amplitude, phase, and noise power as unknown parameters. The Cramer-Rao lower bounds (CRBs) of NDA joint estimates of these unknown parameters are derived for BPSK and QPSK signals, and further used to obtain the QHLRT-UB. Furthermore, an upper bound on the classification performance of any LB algorithm is derived, which is referred to as the ALRT-Upper Bound (ALRT-UB). The QHLRT which employs method of moments (MoM) estimates of the unknown parameters (QHLRT-MoM) is investigated, and its classification performance compared to QHLRT-UB, ALRT-UB, quasi Log-Likelihood Ratio (qLLR) [4], [5] and fourth-order cumulant based algorithms [12], [13]. The rest of the paper is organized as follows. The signal model is presented in Section II. The HLRT- and QHLRTbased algorithms and the ALRT-UB are introduced in Section III. The QHLRT-UB is proposed in Section IV. Furthermore, theoretical developments for the CRBs of NDA joint estimates of unknown signal amplitude, phase, and noise power for BPSK and QPSK signals are reported, and used to develop the QHLRT-UB for the BPSK and QPSK case study. Numerical results for the CRBs and QHLRT-UB are shown in Section V. In addition, MoM estimates of signal amplitude, phase, and noise power are studied, and classification
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performance of the QHLRT-MoM compared to QHLRT-UB, ALRT-UB, qLLR and fourth-order cumulant based algoithms. Conclusions are finally drawn in Section VI. Key steps in the derivation of the CRBs are presented in the Appendix. II. S IGNAL M ODEL Let the received baseband signal sequence at the output of the matched filter be r = 𝛼𝑒𝑗𝜑 s(𝑖) + n,
(1)
where r = [𝑟1 ⋅ ⋅ ⋅ 𝑟𝐾 ]† is the vector of samples at the output of receive matched filter, taken at the symbol rate, with 𝐾 as the number of processed symbols at the receive-side and † as the (𝑖) (𝑖) transpose, s(𝑖) = [𝑠1 ⋅ ⋅ ⋅ 𝑠𝐾 ]† is the sequence of transmitted symbols corresponding to a modulation 𝑖, n = [𝑛1 ⋅ ⋅ ⋅ 𝑛𝐾 ]† is the noise vector, √ 𝛼 is the signal amplitude, 𝜑 is the signal phase, and 𝑗 = −1. The noise components {𝑛𝑘 }𝐾 𝑘=1 are independent circular complex zero-mean Gaussian random variables, with real and imaginary parts of variance 𝑁/2. The (𝑖) sequence {𝑠𝑘 }𝐾 𝑘=1 is independent and identically distributed, with values drawn from an alphabet specific to the modulation 𝑖. Without loss of generality, we consider unit variance constellations and define the signal-to-noise ratio (SNR) as 𝛾 = 𝛼2 /𝑁 . III. LB M ODULATION C LASSIFICATION A LGORITHMS A. The LB Approach to Modulation Classification Under the assumption of statistically independent received symbols, the likelihood function is given by (𝑖)
𝑓 (r∣{𝑠𝑘 }𝐾 𝑘=1 , Θ) = 𝐾 } { ∏ (𝑖) = (𝜋𝑁 )−1 exp −𝑁 −1 ∣𝑟𝑘 − 𝛼𝑒𝑗𝜑 𝑠𝑘 ∣2 𝑘=1 } { = (𝜋𝑁 )−𝐾 exp −𝑁 −1 ∣∣r − 𝛼𝑒𝑗𝜑 s(𝑖) ∣∣2 ,
(2)
where Θ = [𝛼 𝑁 𝜑]† is the vector of the unkown parameters and ∥ ⋅ ∥ is the vector norm. Under the ALRT approach, the PDF of r under 𝐻𝑖 is obtained by averaging (2) over the nuisance parameters, i.e., the unknown symbols and the parameter vector Θ [3], [6], [8], [9]. Under the HLRT and QHLRT approaches, the averaging is over the unknown symbols only, and the dependence with Θ is dealt with by replacing it with ˆ [6], [10], [11]. Thus, with the former, the suitable estimate Θ PDF of r under 𝐻𝑖 can be expressed as (𝑖)
(𝑖)
𝑓ALRT (r) = 𝐸{𝑠(𝑖) }𝐾 𝐸Θ [𝑓 (r∣{𝑠𝑘 }𝐾 𝑘=1 , Θ)], 𝑘
𝑘=1
(3)
while with the latter as (𝑖) ˆ (𝑖) ∣{𝑠(𝑖) }𝐾 𝑓(Q)HLRT (r) = 𝐸{𝑠(𝑖) }𝐾 [𝑓 (r, Θ 𝑘 𝑘=1 )], 𝑘
𝑘=1
(4)
where 𝐸{𝑠(𝑖) }𝐾 [⋅] and 𝐸Θ [⋅] are the expectations with respect 𝑘=1 𝑘 to (w.r.t) the unknown signal constellation points (averaging is performed over all possible sequences of 𝐾 symbols corresponding to modulation 𝑖) and vector of unknown parameters, ˆ (𝑖) is the vector of the unknown parameter respectively, and Θ estimates under hypothesis 𝐻𝑖 . With the PDFs calculated under all hypotheses, the decision is made by applying an LRT [16, Ch. 2].
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{
B. The ALRT-UB The ALRT with perfect knowledge of the parameter vector Θ is of interest, as it provides an upper bound on the performance of any MC algorithm, including LB algorithms discussed in this paper [8]. We refer to it as to the ALRT-UB. In such case, it is straightforward to express the PDF of r under 𝐻𝑖 as [8], [9]2 (𝑖)
𝑓ALRT-UB (r) =
𝐾 ∏
𝑀𝑖−1 ×
𝑘=1
(5)
𝑀𝑖 [ }] { ∑ (𝑖) (𝜋𝑁 )−1 exp −𝑁 −1 ∣𝑟𝑘 − 𝛼𝑒𝑗𝜑 𝑠𝑘,𝑚 ∣2 , × 𝑚=1
where 𝑀𝑖 is the number of points in the signal constellation (𝑖) corresponding to modulation 𝑖, and 𝑠𝑘,𝑚 , 𝑚 = 1, ⋅ ⋅ ⋅ , 𝑀𝑖 , represents the symbol over the 𝑘-th period, drawn from this signal constellation. The criterion used to make a decision on the modulation is [8], [9] ˆ𝑖 = arg max ln{𝑓 (𝑖) ALRT-UB (r)},
where ˆ𝑖 is the estimate of the modulation, and ln{⋅} is the natural logarithm. For the BPSK and QPSK case study, one can easily find the analytical closed-form expressions for the (BPSK) (QPSK) PDFs 𝑓ALRT-UB (r) and 𝑓ALRT-UB (r) respectively as 𝐾 ∏ { } (BPSK) 𝑓ALRT-UB (r) = (𝜋𝑁 )−1exp −𝑁 −1 (∣𝑟𝑘 ∣2 + 𝛼2 ) ×
× cosh(2𝛼𝑁
−1
Re{𝑟𝑘 𝑒
−𝑗𝜑
{ − ln
(7)
}),
and 𝐾 ∏ { } (QPSK) 𝑓ALRT-UB (r) = (𝜋𝑁 )−1exp −𝑁 −1 (∣𝑟𝑘 ∣2 + 𝛼2 ) × 𝑘=1 √ × cosh(√2𝛼𝑁 −1 Re{𝑟𝑘 𝑒−𝑗𝜑 })× × cosh( 2𝛼𝑁 −1 Im{𝑟𝑘 𝑒−𝑗𝜑 }),
C. The HLRT-based MC Algorithm with Signal Amplitude, Phase, and Noise Power as Unknown Parameters With the HLRT approach, MLE of the unknown parameters are used to calculate the PDF of r in (4). By using the joint parameter estimation method [17, Ch. 7], one can find the analytical closed form expressions for the ML estimators of the unknown signal amplitude, noise power, and phase respectively as {√ } 𝛼 ˆ (𝑖) = Re (s(𝑖)𝐻 r)(r𝐻 s(𝑖) ) /∥s(𝑖) ∥2 , (9) ˆ (𝑖) = 𝐾 −1 (∥r∥2 − ∣r𝐻 s(𝑖) ∣2 )/∥s(𝑖) ∥2 ), 𝑁 ( (𝑖)𝐻 ) 𝑗 r s 𝜑ˆ(𝑖) = − ln 𝐻 (𝑖) , 2 r s
(
𝑓HLRT (r) = (8)
𝑖=BPSK,QPSK
modulation 𝑖 conditioned on the event that modulation 𝑗 is actually received, is approximately given by ⎞ ⎡ ⎛√ (BPSK) 𝐾𝜇 1⎣ BPSK,QPSK ⎠ 𝑃𝑐𝑐 ≈ 1−𝑄⎝ √ Λ 2 (BPSK) 𝜐Λ ⎛√ ⎞⎤ (QPSK) 𝐾𝜇 ⎠⎦ , + 𝑄⎝ √ Λ (QPSK) 𝜐Λ (𝑖)
where 𝑄(⋅) represents the Q-function, and 𝜇Λ and 𝜐Λ are the mean and variance of the variable 2 In this case, the average over all possible sequences of 𝐾 symbols reduces to the average over all possible values of each symbol.
(10) (11)
where the superscript 𝐻 denotes Hermitian transpose. It is noteworthy that the ML estimators depend on the 𝐾 symbol sequence s(𝑖) . As such, these cannot be directly estimated in an MC problem, in which the symbols are unknown. However, by using (2), (4) and (9)-(11), one can obtain the expression for the PDF of r under 𝐻𝑖 as (𝑖)
where cosh denotes the hyperbolic cosine function, and Re{⋅} and Im{⋅} are the real and imaginary parts, respectively. Furthermore, under the assumption of equally likely hypotheses and by following [8], one can show that the average BPSK,QPSK = probability defined as 𝑃𝑐𝑐 ∑of correct classification, (𝑖∣𝑗) −1 (𝑖∣𝑖) 𝑃𝑐 , with 𝑃𝑐 the probability of deciding 2
(𝑖)
{ }]} [ (QPSK) 4 ∣𝑟𝑘 − 𝛼𝑒𝑗𝜑 𝑠𝑘,𝑚 ∣2 1 1 ∑ exp − 4 𝑚=1 𝜋𝑁 𝑁
given the hypothesis 𝐻𝑖 is true, 𝑖 = BPSK,QPSK.
(6)
𝑖
𝑘=1
Λ(𝑟𝑘 ) = ln
{ }]} [ (BPSK) 2 ∣𝑟𝑘 − 𝛼𝑒𝑗𝜑 𝑠𝑘,𝑚 ∣2 1 1 ∑ exp − 2 𝑚=1 𝜋𝑁 𝑁
(𝑖)
𝐾 𝜋𝑒𝑀𝑖 ∣∣r∣∣2 (𝑖)
𝐾 )𝐾 𝑀 𝑖 ∑
𝑚=1
1 (𝑖)2
(1 − 𝜌𝑚 )𝐾
,
(12)
(𝑖)
where 𝜌𝑚 = ∣r𝐻 s𝑚 ∣/(∥r∥ ⋅ ∥s𝑚 ∥), 𝑚 = 1, ⋅ ⋅ ⋅ 𝑀𝑖𝐾 , is (𝑖) (𝑖) the correlation coefficient between r and s𝑚 , with s𝑚 as a sequence of 𝐾 symbols drawn from the signal constellation of modulation 𝑖. With (12) calculated under all hypotheses, the modulation is identified by correspondingly using (6). As one can easily notice, the complexity to compute the PDF of r under 𝐻𝑖 is of the order of 𝒪(𝑀𝑖𝐾 ), and increases with the number of symbols, 𝐾 (exponential increment) and modulation order, 𝑀𝑖 . D. QHLRT-based MC Algorithm with Signal Amplitude, Phase, and Noise Power as Unknown Parameters With the QHLRT, NDA non-MLE of the unknown parameters are used to calculate the PDF of r in (4). NDA estimators of the unknown parameters are of particular interest for MC, as they do not rely on the presence of pilot symbols. As the likelihood function is multimodal [18], [19], this cannot provide a unique solution, and one resorts to statistic-based estimators. When MoM estimates are employed, we refer to the QHLRT-based algorithm as to QHLRT-MoM. The MoM estimators of signal amplitude, 𝛼, and noise power, 𝑁 for PSK signals are respectively given by [6], [20] )1/4 ( ˆ2 −𝑀 ˆ 𝑟,42 𝛼 ˆ (𝑖) = 2𝑀 , (13) 𝑟,21
HAMEED et al.: ON THE LIKELIHOOD-BASED APPROACH TO MODULATION CLASSIFICATION
and ˆ (𝑖)
𝑁
ˆ 𝑟,21 − =𝑀
(
ˆ2 2𝑀 𝑟,21
ˆ 𝑟,42 −𝑀
)1/2
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1
,
(14)
g2(γ)
0.9
g1(γ)
0.8
𝑘=1
and depends on the hypothesis 𝐻𝑖 (through the modulation order, 𝑀𝑖 ). By using (2) and (4) with the MoM estimates, the PDF of r under 𝐻𝑖 can be easily expressed as 𝐾 ∏
𝑀𝑖 ∑
(𝑖) ˆ (𝑖) )−1 × 𝑀𝑖−1 (𝜋 𝑁 𝑓QHLRT-MoM (r) = 𝑚=1 𝑘=1 { } (𝑖) ˆ (𝑖) )−1 ∣𝑟𝑘 − 𝛼 × exp −(𝑁 ˆ (𝑖) 𝑒𝑗 𝜑ˆ 𝑠𝑘,𝑚 ∣2 .
(16)
With (16) calculated under all hypotheses, the modulation is identified by correspondingly using (6). One can easily notice that the complexity to compute (16) is of order of 𝒪(𝐾𝑀𝑖 ), and, apparently, much lower when compared to HLRT. Note that NDA estimators of the unknown parameters for QAM signals (including MoM estimators) can be used with the QHLRT algorithm. For such estimators one can see, e.g., [6], [18, Ch. 5], [20], [21]. IV. A N U PPER B OUND ON THE P ERFORMANCE OF QHLRT-BASED MC A LGORITHMS A. Proposed Upper Bound on the Performance of QHLRTbased MC Algorithms Results in [3] and [7] show that an improved classification performance is attained with more accurate parameter estimates. For unbiased estimates, minimum variance estimates which reach the CRB provide the best accuracy, and, accordingly, lead to the best classification performance. An upper bound on the performance of QHLRT-based algorithms which employ unbiased NDA joint estimates of the unknown parameters is thus achieved for variances equal to the CRBs, and calculation of the CRBs is crucial for the performance bound. When considering estimates which are also normally distributed, we refer to this bound as the QHLRT-UB. With signal amplitude, phase, and noise power as unknown parameters, the QHLRT-UB is obtained by using 𝛼 ˆ (𝑖) ∈ ℵ(𝛼, CRB𝑖 (ˆ 𝛼)), (𝑖) (𝑖) ˆ ˆ 𝑁 ∈ ℵ(𝛼, CRB𝑖 (𝑁 )), and, 𝜑ˆ ∈ ℵ(𝜑, CRB𝑖 (𝜑)) ˆ in (16), with ℵ(𝑚, 𝜎 2 ) denoting a normal distribution with mean 𝑚 ˆ ), and CRB𝑖 (𝜑) and variance 𝜎 2 , and CRB𝑖 (ˆ 𝛼), CRB𝑖 (𝑁 ˆ as (𝑖) ˆ (𝑖) (𝑖) the CRB of 𝛼 ˆ , 𝑁 , 𝜑ˆ , respectively. The modulation is identified by correspondingly applying (6).
0.7 0.6
2
g (γ), g (γ)
0.5
1
ˆ 𝑟,21 and 𝑀 ˆ 𝑟,42 are the estimates of the second-order/ where 𝑀 one-conjugate and fourth-order/ two-conjugate moments of ˆ 𝑟,21 = the PSK received signal, given respectively by 𝑀 ∑𝐾 ∑𝐾 −1 2 −1 4 ˆ 𝐾 𝑘=1 ∣𝑟𝑘 ∣ and 𝑀𝑟,42 = 𝐾 𝑘=1 ∣𝑟𝑘 ∣ . Note that (13)-(14) hold regardless the PSK modulation order, and the signal amplitude and noise power MoM estimators do not actually depend on the hypothesis 𝐻𝑖 when recognizing PSK signals of different modulation orders. The MoM phase estimator for PSK signals is given by [18, Ch. 5] ) (𝐾 ∑ 𝑀𝑖 −1 (𝑖) , (15) 𝑟𝑘 𝜑ˆ = 𝑀𝑖 arg
0.4 0.3 0.2 0.1 0 −30
−20
−10
0 SNR, γ (dB)
10
20
30
Fig. 1. Numerical evaluation of the functions 𝑔1 (𝛾) and 𝑔2 (𝛾) in (18) and (19), respectively, versus SNR.
B. The QHLRT-UB for BPSK and QPSK Modulation Classification As already mentioned, the QHLRT-UB requires calculation of the CRBs of NDA joint estimates of the unknown parameters. This is presented below for the case study of BPSK and QPSK modulations (𝑖 =BPSK,QPSK), with unknown signal amplitude, phase, and noise power. We refer interested readers to [17, Ch. 3] for complete procedure details. The Fisher information matrix for BPSK signals is given by (see Appendix for derivations) ⎛ 𝑁 − 𝑁 𝑔1 (𝛾) 2𝐾 ⎝ 𝛼𝑔1 (𝛾) JBPSK = 2 𝑁 0 ⎞ 0 𝛼𝑔1 (𝛾) ⎟ 1 𝛼2 ⎠, − 𝑔1 (𝛾) 0 2 𝑁 2 2 0 𝛼 𝑁 − 𝛼 𝑁 𝑔2 (𝛾) where exp{−𝛾} 𝑔1 (𝛾) = √ 2𝜋 and
∫
exp{−𝛾} 𝑔2 (𝛾) = √ 2𝜋
∞
−∞
∫
𝑢2 exp{−𝑢2 /2} √ 𝑑𝑢, cosh(𝑢 2𝛾)
∞
−∞
exp{−𝑢2 /2} √ 𝑑𝑢. cosh(𝑢 2𝛾)
(17)
(18)
(19)
The functions 𝑔1 (𝛾) and 𝑔2 (𝛾) are calculated through numerical integration and plotted versus the SNR, 𝛾, in Fig. 1. They both decrease monotonically from one towards zero as the SNR increases. By taking the inverse of the FIM matrix in (17), one obtains the expressions for the CRBs of joint estimates of 𝛼, 𝑁 , and 𝜑, respectively as 𝛼2 1 − 2𝛾𝑔1 (𝛾) , 2𝛾𝐾 1 − (2𝛾 + 1)𝑔1 (𝛾)
(20)
2 1 − 𝑔1 (𝛾) ˆ) = 𝑁 , CRBBPSK (𝑁 𝐾 1 − (2𝛾 + 1)𝑔1 (𝛾)
(21)
CRBBPSK (ˆ 𝛼) = 3
3 It can be shown that the CRB for the estimate of the noise power equals 𝛼4 /4 times the CRB for the fourth-order/ two-conjugate normalized cumulant used for MC in [13] (which assumes perfect knowledge of 𝑁 ), taking into account that the noise is complex-valued.
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4
10
1 1 . CRBBPSK (𝜑) ˆ = 2𝛾𝐾 1 − 𝑔2 (𝛾)
(22) 10
As one can easily notice, the CRBs depends on 𝛾 and 𝐾. Furthermore, the CRBs of the estimates of 𝛼 and 𝑁 also depend on the actual value of the parameter, whereas the CRB of the estimate of 𝜑 does not. It should be noted that the phase CRB is actually decoupled from those of amplitude and noise power. For QPSK signals, one can similarly show that ⎛ 𝑁 − 𝑁 𝑔1 (𝛾/2) 2𝐾 𝛼𝑔1 (𝛾/2) JQPSK (Θ) = 2 ⎝ 𝑁 0 ⎞
0 𝛼𝑔1 (𝛾/2) ⎟ 1 𝛼2 ⎠. − 𝑔1 (𝛾/2) 0 2 𝑁 0 𝛼2 𝑁 − 𝛼2 𝑁 (1 + 𝛾)𝑔2 (𝛾/2)
2 KCRBi (α)/α ˆ
2
10
1
10
0
10
−1
10
−2
10 −10
(23) Fig. 2.
(25)
1 1 . 2𝛾𝐾 1 − (𝛾 + 1)𝑔2 (𝛾/2)
(26)
and CRBQPSK (𝜑) ˆ =
10
i =QPSK i =BPSK
1
10
0
10 −10
The above remarks on the CRBs in case of BPSK hold for QPSK, as well. V. N UMERICAL R ESULTS
5
2
(24)
2 1 − 𝑔1 (𝛾/2) ˆ) = 𝑁 CRBQPSK (𝑁 , 𝐾 1 − (2𝛾 + 1)𝑔1 (𝛾/2)
0 SNR, γ (dB)
10
ˆ )/N 2 KCRBi (N
𝛼2 1 − 2𝛾𝑔1 (𝛾/2) , 2𝛾𝐾 1 − (2𝛾 + 1)𝑔1 (𝛾/2)
−5
𝐾CRB𝑖 (ˆ 𝛼)/𝛼2 versus SNR, for 𝑖=BPSK,QPSK.
By taking the inverse of the FIM matrix in (23), the expressions for the CRBs of joint estimates of 𝛼, 𝑁 , and 𝜑 are respectively obtained as 𝛼) = CRBQPSK (ˆ
i =QPSK i =BPSK
3
Fig. 3.
−5
0 SNR, γ (dB)
5
10
ˆ )/𝑁 2 versus SNR, for 𝑖=BPSK,QPSK. 𝐾CRB𝑖 (𝑁
Numerical results for the CRBs of NDA joint estimates of signal amplitude, phase, and noise power are subsequently reported, along with simulation results for MoM estimates of these parameters. Classification results for QHLRT-MoM, QHLRT-UB, and ALRT-UB are presented, and a link between QHLRT-MoM and QHLRT-UB is explained. A comparison of the classification performance of the QHLRT-MoM, qLLR [4], [5] and fourth-order cumulant based algorithms [12], [13] is performed. The applicability of the QHLRT-MoM to MC in Ricean block fading channel is additionally shown.
By using (20)-(22) and (24)-(26), results for ˆ )/𝑁 2 , 𝛼)/𝛼2 , 𝐾CRB𝑖 (𝑁 and 𝐾CRB𝑖 (𝜑), ˆ 𝐾CRB𝑖 (ˆ 𝑖 = BPSK, QPSK, are plotted versus SNR in Figs. 2, 3, and 4, respectively. As expected, all curves decrease as the SNR increases, and results for BPSK are below or close to those for QPSK. In addition, the gap between BPSK and QPSK increases with a decrease in SNR.
A. Simulation Setup
C. MoM Estimates of the Unknown Parameters
Unless otherwise mentioned, the signal amplitude, 𝛼, is set to 1, and the phase, 𝜑, is uniformly distributed over the [−𝜋/4, 𝜋/4) range and fixed over the duration of the data sequence. The number of symbols processed at the receiveside is set to 𝐾 = 100. The number of Monte Carlo trials used to compute the variance of MoM estimates of the unknown parameters and the probability to correctly decide on the (𝑖∣𝑖) modulation 𝑖, 𝑃𝑐 , 𝑖 = BPSK, QPSK, 16-QAM, V.29, 4 equals 10 . The average probability of correct classification is employed as performance measure, being calculated as the arithmetic mean of the probabilities of correct classification for corresponding modulations.
The variance of MoM estimates of 𝛼 and 𝑁 for the QPSK signal, along with corresponding CRBs, is plotted versus SNR in Fig. 5. The variance is lower bounded by the CRB above a certain SNR, when MoM estimates are unbiased; such a range is shown in Fig. 5. As expected, the variance decreases as the SNR, 𝛾, increases; this is relatively close to corresponding CRB, especially for signal amplitude. Although results for MoM estimates are presented here only for 𝛼 and 𝑁 for the QPSK signal, we should mention that a similar behavior has been observed for 𝜑. Moreover, this remains valid for BPSK, with the remark that the SNR above which the estimates are unbiased is much lower when compared with
B. CRBs of Non-Data Aided Joint Estimates of the Unknown Parameters
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i =QPSK i =BPSK
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ALRT−UB QHLRT−UB QHLRT−MoM
0.82 −2
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Fig. 4.
−5
0 SNR, γ (dB)
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𝐾CRB𝑖 (𝜑) ˆ versus SNR, for 𝑖=BPSK,QPSK.
ˆ VarQPSK and CRBQPSK for α ˆ and N
Var and CRB for Nˆ
−6
−4
ˆ CRBQPSK (N)
VarQPSK CRBQPSK
VarQPSK (ˆα)
10
0 2 SNR, γ (dB)
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6
8
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TABLE II VARIANCE AND CRB OF THE PARAMETER ESTIMATES AT 3.5 D B AND 2 D B SNR, RESPECTIVELY, FOR QPSK AND 𝐾 = 100.
ˆ VarQPSK (N)
−2
−2
Fig. 6. Performance of QHLRT-MoM, and QHLRT-UB and ALRT-UB, when recognizing BPSK and QPSK with 𝐾 = 100.
−1
10
0.8 −8
CRBQPSK (ˆα)
𝛼 ˆ 0.0041 0.0044
𝜑 ˆ 0.0111 0.01442
ˆ 𝑁 0.009 0.009
Var and CRB for αˆ −3
10
−4
10
0
2
4
SNR, γ (dB)
6
8
10
Fig. 5. Variance of MoM estimates of the signal amplitude and noise power, and the corresponding CRBs versus SNR, for QPSK and 𝐾 = 100.
QPSK. Simulations have been also carried out to study the distribution of MoM estimates of the unknown parameters for BPSK and QPSK signals. A normal distribution occurs at higher SNR, whereas this distribution does not hold at lower SNR. For example, MoM estimates of the phase approach a uniform distribution as the SNR decreases. This is in agreement with results obtained for the variance of MoM estimates of the phase, which approaches 𝜋 2 /48 as the SNR decrease. This represents the variance of a uniform distributed random variable, with values in the [−𝜋/4, 𝜋/4) range. D. QHLRT-MoM, QHLRT-UB, and ALRT-UB Simulation results for the QHLRT-MoM algorithm, as well as for the QHLRT-UB and ALRT-UB are presented in Fig. 6, when recognizing BPSK and QPSK signals. The average probBPSK,QPSK , is plotted versus ability of correct classification, 𝑃𝑐𝑐 SNR, for a range of values of practical interest (above 0.8). Note that classification results for the QHLRT-MoM reflect the behavior of MoM estimates of all unknown parameters BPSK,QPSK range for both BPSK and QPSK signals. For the 𝑃𝑐𝑐 shown in Fig. 6, both QHLRT-UB and ALRT-UB provide upper bounds on the classification performance of QHLRT-
MoM. As expected, QHLRT-UB yields a tighter bound than ALRT-UB. Although the latter is a very optimistic upper bound, it is still of interest, as providing the best performance that can be achieved with any MC algorithm, including LB algorithms. A link between the performance of the QHLRT-MoM and QHLRT-UB is shown, as follows. According to Fig. 6, an average probability of correct classification of almost one, BPSK,QPSK ≈ 1, is achieved with QHLRT-MoM at 3.5 𝑃𝑐𝑐 BPSK,QPSK value at dB SNR. QHLRT-UB provides such a 𝑃𝑐𝑐 around 2 dB SNR. Apparently, achieving a probability of cor(QPSK∣QPSK) rect classification for QPSK of almost 1, 𝑃𝑐 ≈ 1, requires higher SNR when compared to correctly identifying BPSK with the same accuracy and number of symbols. The performance is limited for QPSK, in which case 3.5 dB SNR (QPSK∣QPSK) ≈ 1 with QHLRT-MoM, is required to attain 𝑃𝑐 whereas only 2 dB is needed with QHLRT-UB. The variances and CRBs corresponding to the MoM estimates of the signal ˆ , for which this amplitude, 𝛼 ˆ , phase, 𝜑, ˆ and noise power, 𝑁 performance is achieved, are given in the first and second row of Table II. One can easily notice the similarity of the values of the variance and CRB for each parameter, which explains similar results for the QHLRT-MoM and QHLRT-UB at the aforementioned SNRs.
E. Comparison of the Performance of QHLRT-MoM, qLLR, and Fourth-order Cumulant Based Algorithms Classification results achieved with the QHLRT-MoM are compared with those for the qLLR and fourth-order cumulant based algorithms proposed in [4], [5] and [12], [13], respectively. Classification of BPSK versus QPSK and 16-QAM versus V.29 is considered. The qLLR is derived as
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1
P BPSK,QPSK cc
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Residual channel effect, 2 σ =0.2
0.9 No residual channel effect
h
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0.75 −2
−1
0
1 2 SNR, γ (dB)
3
4
5
Fig. 7. Performance comparison between QHLRT-MoM, qLLR, and fourthorder cumulant based algorithms when recognizing BPSK and QPSK, with 𝐾 = 100 and no residual channel effect (solid line), residual channel effect 2 = 0.1 (dashed-dot line), and 𝜎 2 = 0.2 (dashed line). with 𝜎ℎ ℎ
a low-SNR large-sample approximation of the ALRT-based algorithm, with phase as the unknown parameter [4], [5]. Test signal statistics are obtained based on this approximation, and a near-optimal decision threshold is set by using approximate expressions for the PDFs of the statistic under hypotheses corresponding to different modulations [4], [5]. For 𝑀 -ary PSK modulation classification, the algorithm is a ∑𝐾 binary decision tree, where the statistic 𝑞𝑀𝑖 = ∣ 𝑘=1 𝑟𝑘𝑀𝑖 ∣ 4 is compared against a threshold to make a decision whether the modulation is 𝑀𝑖 -PSK or 𝑀𝑗 -PSK (𝑀𝑗 > 𝑀𝑖 ) at each node [4]. For QAM classification, the qLLR is used for binary hypothesis testing problems, and applied to distinguishing 16QAM and V.29 [5]. Proposed statistics ∑𝐾 with 𝐾 = 100∑symbols 𝐾 are 𝑞4 = ∣ 𝑘=1 𝑟𝑘4 ∣, 𝑝4 = 𝑘=1 ∣𝑟𝑘4 ∣ (see footnote 4 ), and 𝑡4 = 0.0135𝑞4 − 0.0246𝑝4. Thresholds for decision making are set similarly to the PSK case. In [12], [13], fourth-order cumulant based features are proposed to classify linear digital modulations. In [12], a hierarchical classifier is developed to identify a large pool of modulations, such as 𝑀 -ary PSK, QAM, and ASK, V.29, V.32, and V.29c. The magnitude of the fourth-order/ zero-conjugate normalized cumulant, ∣˜ 𝑐𝑠,40 ∣, is employed for the classification of 16-QAM versus V.29, as well as BPSK versus QPSK. In [13], the fourth-order/ two-conjugate normalized cumulant, 𝑐˜𝑠,42 , is used to classify a reduced pool of modulations when compared to [12]; the algorithm is not applicable to 𝑀 -ary PSK, with 𝑀 > 4 5 . In [12], [13], normalization of the fourth-order cumulants is performed to the second power of the second-order/ oneconjugate cumulant of the signal component. The test statistics are estimated and compared against thresholds for decision making. The thresholds are set as the arithmetic average of the test statistic means under hypotheses considered in the ascending order of these means. Results for BPSK versus QPSK achieved with the QHLRTMoM and algorithms proposed in [4] and [12], [13] are 4 One can easily notice that these statistics represent magnitudes of signal moments multiplied by the number of symbols, 𝐾. 5 An extension to the time dispersive channel is performed in [13], while only the additive white Gaussian noise channel is considered in [4], [5], [12], and this work. Nevertheless, the effect of a residual channel on the classification performance of the investigated algorithms is shown here.
presented here, with signal amplitude, phase, and noise power as unknown parameters. Neither qLLR nor fourth-order cumulant based algorithms require phase estimation; both require estimation of signal amplitude. In addition to the aforementioned unknown parameters, the effect of a residual channel is investigated. A three-tap channel [1 ℎ1 ℎ2 ] is considered, with ℎ1 and ℎ2 as zero-mean independent Gaussian random variables, each with variance 𝜎ℎ2 . This variance is set to 0.1 and 0.2, respectively. The average probability of correct BPSK,QPSK , achieved with each of the four classification, 𝑃𝑐𝑐 previously mentioned algorithms is plotted versus SNR in Fig. 7, when no residual channel effect is considered (solid line), 𝜎ℎ2 = 0.1 (dashed-dot line), and 𝜎ℎ2 = 0.2 (dashed line). With no residual channel effect, QHLRT-MoM and qLLR perform similarly, slightly better than the fourth-order/ twoconjugate cumulant based algorithm, and much better than the fourth-order/ zero-conjugate cumulant based algorithm. On the other hand, the fourth-order/ two-conjugate cumulant based algorithm is the most robust to residual channel effects, followed by the QHLRT-MoM and qLLR. The fourth-order/ zero-conjugate cumulant based algorithm is outperformed by the three other algorithms under considered scenarios. Classification results achieved by using the same algorithms are also reported for 16-QAM versus V.29, with the phase as unknown parameter. The average probability of correct 16-QAM,V.29 , is plotted versus SNR in Fig. 8. classification, 𝑃𝑐𝑐 Note that results obtained with the qLLR 𝑝4 , 𝑞4 , and 𝑡4 , and normalized fourth-order/ zero- and two-conjugate based algorithms are reproduced from [5] and [12], respectively. One can notice that the average probability of correct classification saturates for qLLR 𝑝4 , 𝑞4 and 𝑡4 , and normalized fourth-order/ zero- and two-conjugate cumulant based algorithms, whereas it does not for the QHLRT-MoM. The saturation values are as follows: 0.675 and 0.71 for qLLR 𝑝4 and 𝑞4 , respectively, around 0.78 for the normalized fourth-order/ zero- and twoconjugate cumulant based algorithms, and 0.875 for the qLLR 𝑡4 . With the QHLRT-MoM, the average probability of correct classification approaches 1 as the SNR exceeds 20 dB. In addition to not exhibiting saturation, QHLRT-MoM outperforms both qLLR and fourth-order cumulant based algorithms for average probabilities of correct classification of interest (above 0.8). F. QHLRT-MoM Under Fading Conditions Performance of the QHLRT-MoM under fading conditions is presented in Fig. 9. The average probability of correctly BPSK,QPSK , is plotted versus classifing BPSK and QPSK, 𝑃𝑐𝑐 the Rice factor, 𝑅. The values 𝑅 = 0 and 𝑅 = ∞ correspond to the cases of Rayleigh fading and no fading, respectively. Simulation results show that QHLRT-MoM yields a reasonable performance for the entire range of 𝑅 values. VI. C ONCLUSION In this paper we investigate likelihood-based algorithms for linear digital modulation classification. Our results indicate that the HLRT is very computationally demanding, while the QHLRT algorithm offers a reasonable alternative. An upper bound on classification performance of QHLRT-based
HAMEED et al.: ON THE LIKELIHOOD-BASED APPROACH TO MODULATION CLASSIFICATION
a promising candidate algorithm. Hence, we intend to further explore the QHLRT algorithm in future work, by extending the case study presented here to a larger pool of modulations, more unknown parameters, and more complex propagation environments.
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A PPENDIX D ERIVATION OF (17)
0.7 0.65 0.6 0.55 0.5
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SNR, γ (dB)
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Fig. 8. Performance comparison between QHLRT-MoM, qLLR, and fourth-order cumulant based algorithms when recognizing 16-QAM and V.29, with K=100.
The derivation of the first element of the Fisher Information Matrix for the BPSK case, [JBPSK (Θ)]1,1 = 𝐸[−∂ 2 ln 𝑓BPSK (r∣Θ)/∂𝛼2 ], is presented here. Other elements of the matrix can be similarly derived. The likelihood function 𝑓BPSK (r∣Θ) is computed by averaging }𝐾 𝑓 (r∣{𝑠(BPSK) 𝑘=1 , Θ) over the constellation points correpond𝑘 ing to the BPSK modulation; this is given by (7). By using (7), one can easily show that ∂ 2 ln 𝑓BPSK (r∣Θ) = −2𝐾𝑁 −1 ∂𝛼2 𝐾 ∑ +4𝑁 −2 (𝑟𝑘,𝐼 cos 𝜑 + 𝑟𝑘,𝑄 sin 𝜑)2 ×
1
0.95
(27)
𝑘=1
P cc
BPSK,QPSK
×sech2 (2𝛼𝑁 −1 (𝑟𝑘,𝐼 cos 𝜑 + 𝑟𝑘,𝑄 sin 𝜑)), where 𝑟𝑘,𝐼 and 𝑟𝑘,𝑄 are the in-phase and quadrature components of the signal samples 𝑟𝑘 , 𝑘 = 1, ⋅ ⋅ ⋅ , 𝐾, respectively. Futher, by applying expectation w.r.t 𝑓BPSK (r∣Θ) to (27), one can write
0.9
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∫
0.8 K=100, SNR=5 dB K=100, SNR=3 dB 0.75 −infinity
0
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[JBPSK (Θ)]1,1 20
R (dB)
Fig. 9. Performance of QHLRT-MoM when discriminating BPSK and QPSK in Ricean fading, with 𝐾 = 100.
algorithms is proposed (QHLRT-UB), for the case when unbiased and normally distributed non-data aided estimates of the unknown parameters are available. This requires calculation of the CRBs of non-data aided parameter estimates. As a case study we investigate classification of BPSK and QPSK modulations with signal amplitude, phase, and noise power as unknown parameters. The CRBs of joint non-data aided parameter estimates are derived for this case, and further employed to develop the QHLRT-UB. MoM estimates of the unknown parameters are explored and used with the QHLRT, leading to the QHLRT-MoM. According to simulation results, when the MoM estimates are unbiased, their variance is relatively close to corresponding CRBs, and the QHLRT-MoM provides a reasonable performance. A comparison between the QHLRT-MoM and qLLR algorithms reveals that although both provide a similar performance when discriminating PSK signals with no residual channel effects, the QHLRT-MoM is more robust to this model mismatch. On the other hand, the fourth-order/ two-conjugate cumulant based algorithm is the most robust under such conditions. When compared with qLLR and fourth-order cumulant based algorithms, QHLRT can be successfully applied to distinguish QAM modulations. In addition, QHLRT provides an acceptable performance under fading conditions. We note that QHLRT displays good performance with reasonable complexity, and appears to be
∂ 2 ln 𝑓BPSK (r∣Θ) 𝑓BPSK (r∣Θ)𝑑r ∂𝛼2 (28) = 𝔈1 + 𝔈2 , = −
where 𝔈1 = 𝐸[2𝐾𝑁 −1 ] = 2𝐾𝑁 −1
𝐾 ∑ and 𝔈2 = −𝐸[ 𝔏(𝑟𝑘 )] 𝑘=1
with
𝔏(𝑟𝑘 ) = 4𝑁 −2 (𝑟𝑘,𝐼 cos 𝜑 + 𝑟𝑘,𝑄 sin 𝜑)2 × ×sech2 (2𝛼𝑁 −1 (𝑟𝑘,𝐼 cos 𝜑 + 𝑟𝑘,𝑄 sin 𝜑)).
By using that {𝑟𝑘 }𝐾 𝑘=1 are independent and identically distributed random variables, and the expectation is linear, one can drop the dependency on 𝑘, and write 𝔈2 as 𝔈2 = −𝐾𝐸[𝔏(𝑟)].
(29)
The expectation in (29) is w.r.t 𝑓BPSK (𝑟∣Θ), in which 𝐾 = 1 and the dependency on 𝑘 is dropped. By using that 𝑟𝑘 𝑒−𝑖𝜑 = (𝑖) 𝛼𝑠𝑘 + 𝑤𝑘,𝐼 + 𝑗𝑤𝑘,𝑄 , with 𝑤𝑘,𝐼 and 𝑤𝑘,𝑄 as the in-phase and quadrature components of 𝑛𝑘 𝑒−𝑗𝜑 , respectively, and by dropping the dependency on 𝑘 and changing variables, one can show that (29) can be written as 𝔈2
=
−4𝐾𝜋 −1 𝑁 −3 exp{−𝑁 −1 𝛼2 } ∫ ∞ exp{−𝑁 −1 (𝑤𝑄 )2 }𝑑𝑤𝑄 ℌ(𝑠(𝑖) , 𝛼, 𝑁 ), (30) −∞
(𝑖)
where ℌ(𝑠 , 𝛼, 𝑁 ) = ∫ ∞ (𝛼𝑠(𝑖) + 𝑤𝐼 )2 exp{−𝑁 −1 (𝛼𝑠(𝑖) + 𝑤𝐼 )2 } 𝑑𝑤𝐼 . = cosh(2𝛼𝑁 −1 (𝛼𝑠(𝑖) + 𝑤𝐼 )) −∞
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By using that the integrand in the expression of ℌ(𝑠(𝑖) , 𝛼, 𝑁 ) is an even function of 𝛼𝑠(𝑖) + 𝑤𝐼 , one can easily show that ℌ(𝑠(𝑖) , 𝛼, 𝑁 ) is invariant under 𝑠(𝑖) = −1 and 𝑠(𝑖) = 1. As such, the result for 𝔈2 does not change if one averages (30) w.r.t the points in the BPSK signal constellation. By applying this average, and after some manipulations, one can show that ∫ 2𝐾 exp{−𝛾} ∞ 𝑢2 exp{−𝑢2 /2} √ √ 𝔈2 = − 𝑑𝑢. (31) 𝑁 2𝜋 −∞ cosh(𝑢 2𝛾) Finally, by replacing results for 𝔈1 and 𝔈2 in (28), it is straightforward to obtain the expression for [JBPSK (Θ)]1,1 as in (17). Derivation of the FIM matrix for QPSK can be similarly carried out. Note that results obtained here for [J𝑖 ]𝑝,𝑞 , 𝑝, 𝑞 = 1, 2 are consistent with those reported in [22], where the CRB of NDA estimates of SNR is derived. In addition, numerical results achieved for [J𝑖 ]3,3 by using the analytical expression derived here are consistent with those reported in [23]. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their constructive comments on the paper. R EFERENCES [1] O. A. Dobre and F. Hameed, “Likelihood-based algorithms for linear digital modulation classification in fading channels," in Proc. 19th IEEE Canadian Conf. Electrical Computer Engineering – CCECE 2006, Ottawa, Ontario, Canada, May 2006, pp. 1347-1350. [2] ——, “On performance bounds for parameter estimation and modulation classification," in Proc. IEEE Sarnoff Symposium, 2007, pp. 1-5. [3] O. A. Dobre, A. Abdi, Y. Bar-Ness, and W. Su, “A survey of automatic modulation classification techniques: classical approaches and new trends," IET Commun., vol. 1, pp. 137-156, 2007. [4] C. Y. Huang and A. Polydoros, “Likelihood methods for MPSK modulation classification," IEEE Trans. Commun., vol. 43, pp. 1493-1504, 1995. [5] C. Long, K. Chugg, and A. Polydoros, “Further results in likelihood classification of QAM signals," in Proc. IEEE MILCOM, 1994, pp. 5761. [6] A. Abdi, O. A. Dobre, R. Choudhry, Y. Bar-Ness, and W. Su, “Modulation classification in fading channels using antenna arrays," in Proc. IEEE MILCOM, 2004, pp. 211-217. [7] O. A. Dobre, A. Abdi, Y. Bar-Ness, and W. Su, “Blind modulation classification: a concept whose time has come," in Proc. IEEE Sarnoff Symposium, 2005, pp. 223-228. [8] W. Wei and J. Mendel, “Maximum-likelihood classification for digital amplitude-phase modulations," IEEE Trans. Commun., vol. 48, pp. 189193, 2000. [9] J. A. Sills, “Maximum-likelihood modulation classification for PSK/QAM," in Proc. IEEE MILCOM, 1999, pp. 217-220. [10] P. Panagiotou, A. Anastasopoulos, and A. Polydoros, “Likelihood ratio tests for modulation classification," in Proc. IEEE MILCOM, 2000, pp. 670-674. [11] O. A. Dobre, J. Zarzoso, Y. Bar-Ness, and W. Su, “On the classification of linearly modulated signals in fading channels," in CD, Conf. Inform. Sciences Systems (CISS), Princeton University, 2004. [12] A. Swami and B. M. Sadler, “Hierarchical digital modulation classification using cumulants," IEEE Trans. Commun., vol. 48, pp. 416-429, 2000. [13] H.-C. Wu, M. Saquib, and Z. Yun, “Novel automatic modulation classification using cumulant features for communications via multipath channels," IEEE Trans. Wireless Commun., vol. 7, pp. 3098-3105, Aug. 2008. [14] C. M. Spooner, “On the utility of sixth-order cyclic cumulants for RF signal classification," in Proc. IEEE ASILOMAR, 2001, pp. 890-897. [15] D. Boudreau, C. Dubuc, F. Patenaude, M. Dufour, J. Lodge, and R. Inkol, “A fast automatic modulation recognition algorithm and its implementation in a spectrum monitoring application," in Proc. IEEE MILCOM, 2000, pp. 732-736.
[16] H. L. Van Trees, Detection, Estimation and Modulation Theory. New York: Wiley, 2001. [17] S. M. Kay, Fundamentals of Statistical Signal Processing, Estimation Theory, Vol. 1. Englewood Cliffs, NJ: Prentice Hall, 1993. [18] U. Mengali and A. N. D’Andrea, Synchornization Techniques for Digital Receivers. New York: Plenum, 1997. [19] T. R. Benedict, “The joint estimation of signal and noise from sum envelope," IEEE Trans. Inf. Theory, vol. IT-13, pp. 447-454, 1967. [20] D. R. Pauluzzi and N. C. Beaulieu, “A comparison of SNR estimation techniques for the AWGN channel," IEEE Trans. Commun., vol. 48, pp. 1681-1691, 2000. [21] H. Xu, Z. Li, and H. Zheng, “A non-data aided SNR estimation algorithm for QAM signals," in Proc. ICASSP, 2004, pp. 999-1003. [22] N. S. Alagha, “Cramer-Rao bounds of SNR estimates for BPSK and QPSK modulated signals," IEEE Commun. Lett., vol. 5, pp. 10-12, 2001. [23] W. G. Cowley, “Phase and frequency estimation for PSK packets: bounds and algorithms," IEEE Trans. Commun., vol. 44, pp. 26-28, 1996. Fahed Hameed received his Bachelor’s Degree in Electrical Engineering from National University of Sciences and Technology, Pakistan in 2003 and Master’s Degree in Electrical Engineering from Memorial University of Newfoundland, Canada, in 2006. Currently, he is working in the Telecoms O&M department of Qatar Electric Supply Company (Kahramaa). His research interests include blind parameter estimation, modulation classification, and multihop cellular networks.
Octavia A. Dobre received the Engineering Diploma and Ph. D. degrees in Electrical Engineering from Politehnica University of Bucharest (formerly the Polytechnic Institute of Bucharest), Romania, in 1991 and 2000, respectively. In 2001 she joined the Wireless Information Systems Engineering Laboratory at Stevens Institute of Technology in Hoboken, NJ, as a Fulbright fellow. Between 2002 and 2005, she was with the Department of Electrical and Computer Engineering at New Jersey Institute of Technology (NJIT) in Newark, NJ, as a Research Associate. Since 2005 she has been an Assistant Professor with the Faculty of Engineering and Applied Science at Memorial University of Newfoundland, Canada. Her current research interests include blind modulation classification and parameter estimation, cognitive radio, multiple antenna systems, multicarrier modulation techniques, cyclostationarity applications in communications and signal processing, and resource allocation in emerging wireless networks. Dr. Dobre is an Associate Editor for the IEEE C OMMUNI CATIONS L ETTERS , and has served as the Chair of the Signal Processing and Multimedia Symposium of the 2009 IEEE Canadian Conference on Electrical and Computer Engineering (CCECE). Dimitrie C. Popescu received the Engineering Diploma and M.S. degrees in 1991 from the Polytechnic Institute of Bucharest, Romania, and the Ph.D. degree from Rutgers University in 2002, all in Electrical Engineering. His research interests are in the areas of wireless communications, digital signal processing, and control theory. He is currently an Assistant Professor in the Department of Electrical and Computer Engineering, Old Dominion University. Between 2002 and 2006 he was with the Department of Electrical and Computer Engineering, the University of Texas at San Antonio. He has also worked for AT&T Labs in Florham Park, New Jersey, on signal processing algorithms for speech enhancement, and for Telcordia Technologies in Red Bank, New Jersey, on wideband CDMA systems. He has served as technical program chair for the vehicular communications track of the IEEE VTC 2009 Fall, finance chair for the IEEE MSC 2008, and technical program committee member for the IEEE GLOBECOM, WCNC 2006, and VTC conferences. His work on interference avoidance and dispersive channels was awarded second prize in the AT&T Student Research Symposium in 1999. He is the co-author of a monograph book on interference avoidance methods for wireless systems published in 2004 by Kluwer Academic Publishers.
