CLASSIFICATION IN CHANNELS USING ARRAYS

Ali Abdi1, Octavia A. Dobre1, Rahul Choudhry1, Yeheskel Bar-Ness1, and Wei Su2

1CCSPR, Dept. of ECE, New Jersey Institute of Technology, Newark, NJ 07102, USA 2CECOM, Fort Monmouth, NJ 07703, USA

ABSTRACT ϕ0 is the channel phase (which also includes the carrier ()i N phase offset), {}skk=1 represents the N complex Blind modulation classification (MC) is an intermediate transmitted data symbols, taken from the i-th finite- step between signal detection and demodulation, and plays a alphabet modulation format, and † is the transpose key role in various civilian and military applications In this operator. Depending on the classification approach, in paper, first we provide an overview of decision-theoretic Section IV we consider α0 and ϕ0 as random variables, MC approaches. Then we derive the average likelihood ratio denoted by α and ϕ , respectively, and average over (ALR) based classifier for linear and nonlinear them, whereas in Section V, α and ϕ are considered as , in noisy channels with unknown carrier phase 0 0 unknown deterministic quantities, for which αˆ0 and ϕˆ0 offset and also in channels. Since these are provided as their estimates. In any case, we treat the ALR-based classifiers are complex to implement, we then ()i N data symbols {}skk=1 as independent and identically develop a quasi hybrid likelihood ratio (QHLR) based distributed (iid) randoms, and average over them. With classifier, where the unknown parameters are estimated symbol period T0 , rectangular unit-amplitude pulse shape using low-complexity techniques. This QHLR-based ut() of length T , and j 2 =−1 , st(;u ) is given by T0 0 i,0 classifier is much simpler to implement and is also N jϕ0 ()i applicable to any fading distribution, including Rayleigh and st(;uikT,0 )=−−≤≤α 0 e s () tu ( t ( k 1) T 0 ),0 t NT 0 . (1) Rice. Afterwards, we propose a generic multi-antenna ∑k =1 0 classifier for linear and nonlinear modulations, using an Eq. (1) is applicable to M-ary ASK, PSK, QAM, and at the receiver. This classifier has the FSK modulations. In MASK we have potential to improve the performance of traditional single- sM(MASK) ∈ {±± 1, 3,..., ± ( − 1)} , where M is even. The antenna classifiers, including the proposed QHLR-based symbols of MPSK are given by se(MPSK) = jθm , in which algorithm, via spatial diversity. Simulation results are θm = 2π mM , m=− 0, 1, ..., M 1 , with M as a power of provided to show the performance enhancement offered by two. For square and cross QAM symbols, see [1]. In the new QHLR-based multi-antenna classifier, in a variety MFSK, the symbols are given by ste(MFSK)()= jft 2π m , such of channel and fading conditions. that fmd= ±±ff00, 3 d , ..., ± ( Mf − 1) d 0, with fd 0 as the frequency spacing between any two adjacent constellation I. INTRODUCTION points and M as a power of two. As mentioned in the Modulation identification of a received signal is of footnote of this page, for linear modulations, the data importance in a variety of military and commercial symbols do not depend on t. This property simplifies the applications. Some examples include spectrum monitoring general classifier derived in Section IV. and management, surveillance and control of Let us define the variance of the zero-mean i-th activities, adaptive transmission schemes, and electronic ()i ()i 2 constellation as ϑ = Es[|k | ] , iN= 1,2,..., mod , where E[.] warfare. Our comprehensive literature survey [3] shows denotes the mathematical expectation. Then the signal that although this topic has been extensively studied, less power, defined by attention has been paid to modulation classification in NT0 fading channels, and also the utilization of the spatial ()i −12 SNTEstdt00= () [|(;)|]ui ,0, (2) diversity, provided by antenna arrays, in such channels. In ∫0 ()i 2()i this paper, we study these two topics in a systematic way. can be shown to be S00= α ϑ , after substituting (1) into (2). For equiprobable Mi constellation points of the i-th II. SIGNAL, , AND CHANNEL MODELS ()ii− 1Mi ()2 modulation, obviously one has ϑ = Msim∑m=1||. Note Let st(;u ) represent the noise-free baseband complex ()i 2 i,0 that for PSK and FSK ||1sm = , mM= 1,2,..., i , whereas envelope of the received signal, coming from the i-th ()i 2 in ASK and QAM, ||sm takes different values. modulation format, iN= 1,2,..., , where N is the mod mod In the presence of noise, for the baseband received number of candidates modulations that we are looking at. complex envelope we have The vector ui,0 , which corresponds to the i-th modulation format, denotes the vector of unknown quantities at the rt()= st (;ui,0 )+≤≤ nt (),0 t NT 0 , (3) receiver. In this paper we consider a frequency-flat slowly- where nt() is the complex additive white Gaussian noise varying multipath fading channel with (AWGN) with two-sided power N ()i N † 1 0 * uikk,0= [αϕ 0 0 {}]s = 1 , where α0 is the channel amplitude, (W/Hz), and the correlation Entn[() ( t+=τ )] N0δτ (), such ______that * is the complex conjugate and δ(.) is Dirac delta. 1 As we discuss in the first paragraph after eq. (1), only for frequency shift keying (FSK), the data symbols depend on t. For others such as amplitude shift keying (ASK), phase shift keying not depend upon t. To simplify the notation, we most often drop ()i (PSK), and quadrature (QAM), sk ’s do this t-dependence, unless otherwise mentioned.

1 of 7 III. DECISION THEORY AND LIKELIHOOD search, to find the ML estimates [18], for each hypothesis. To design the modulation classifier, in order to Moreover, for nested constellations such as BPSK/QPSK, and 16-QAM/64-QAM, the likelihood function in GLRT determine what modulation has been received, out of N mod equally likely candidates, we take the decision-theoretic can take the same numerical values, which in turn leads to approach, which is optimal under certain conditions [2]. incorrect classification [18] [26]. This approach, which works based upon the maximum HLRT is a combination of ALRT and GLRT, in an likelihood (ML) principle, requires the likelihood function attempt to avoid the disadvantages of both techniques, of rt() over the interval 0 ≤≤tNT0 . Using the complex while employing their useful properties. In HLRT, the Gaussian distribution of nt() and for the i-th hypothesis unknown data symbols are considered as random variables

Hi (the i-th modulation format), the conditional likelihood and are averaged out, whereas the unknown parameters function of rt(), conditioned on the unknown vector ui,0 , are treated as deterministic unknowns, eventually replaced can be shown to be [2] [5] by their ML estimates. To write the likelihood function, we Ξ=[()|rtu , H ] break ui,0 to two vectors w0 and si such that ii,0 ††† uwsii,0= [] 0 . The vector w0 contains the unknown NT NT † 21002 parameters, whereas s = [...]ss s represents the exp Rer ( t ) s* ( t ; ) dt s ( t ; ) dt , iN12 uuii,0− ,0 (4) unknown data symbols. With this notation, the likelihood NN∫∫00 00 function, conditioned on w , is where Re[.] gives the real part. To derive the likelihood 0 ()i ()i function of rt() from (4), Ξ [()]rt , three techniques are Ξ=Ξ[()|rtwwsss00 ] [()| rt ,ii , H ]( p i| H ii ) d . (9) proposed in the literature, that we discuss in the sequel. ∫ ()i With the ML estimate of w0 for each candidate Once Ξ [()]rt is calculated for all the possible N mod candidate modulations, based on the observed rt() over modulation

0 ≤≤tNT0 , one can make the decision according to ML() i wwˆ i,0 = arg maxΞ [rt ( ) | ] , (10) Choosei as the received modulation w the HLRT likelihood function is eventually given by ifirt=Ξ arg max()i [ ( )]. (5) ()i ()iML 1≤≤iN mod Ξ=ΞHi[()]rt [()| rt wˆ ,0 ]. (11) Obviously our modulation classification is a multiple Averaging over the data symbols in HLRT removes the composite hypothesis testing problem, due to the unknown nested constellations problem of GLRT. However, finding ()i data symbols {}sk , as well as the unknown parameters, the ML estimates of the unknown parameters in HLRT which are α0 and ϕ0 in this paper. Based on our still entails an exhaustive search, which makes its comprehensive literature survey [3] [4], three methods are implementation complex. proposed so far, to handle the unknown quantities: average In this paper, first we derive, in Section IV, closed-form likelihood ratio test (ALRT) [6]-[18], generalized likelihood expressions for the ALRT-based likelihood function in ratio test (GLRT) [19], [20], and hybrid likelihood ratio AWGN and Rayleigh fading channels, for any kind of test (HLRT) [19], [21]-[25]. memoryless modulation, including ASK, PSK, QAM, FSK, The unknown quantities in ALRT are considered as etc. Due to the exponential complexity of these ALRT- random variables, with a certain joint probability density based classifiers, as well as not being applicable to other function (PDF), pH(|uii ), and the likelihood function is types of fading such as Rice, Weibull, Nakagami [1], we derived by averaging the conditional likelihood function then propose a quasi HLRT (QHLRT) approach in Section with respect to it V. In QHLRT, we use non-ML parameter estimators, which are simple yet accurate enough to provide a good Ξ=Ξ()i [()]rt [()| rtuu|u , H ]( p H ) d . (6) Aiiiii∫ classification performance, with much less computational complexity, also applicable to any fading distribution. If the chosen pH(|uii ) is the same as the true PDF, then ALRT is the optimal classifier, i.e., maximizes the Then in Section VI we introduce a multi-antenna classifier, probability of correct classification. Otherwise, the which significantly improves the performance of classifiers, optimality is not guaranteed. Furthermore, the including the new QHLRT of Section V, with a moderate multivariate integration in (6) is mathematically tractable increase in the hardware complexity. only for few cases, and normally one has to resort to IV. AVERAGE LIKELIHOOD APPROACH approximations [8]-[12]. By inserting (1) into (4), the conditional likelihood In GLRT, on the other hand, the unknown quantities function can be written as are treated as unknown deterministics, and the likelihood 2 2αα()iiT () function is obtained by replacing the unknown quantities 000− jϕ0 Ξ=[()|rtuii,0 , H ] exp Re e R N −η N , (12) in the conditional likelihood function, with their ML NN00 estimates where ()i ML N Ξ=ΞGii[()]rt [()| rtuˆ ,0 , H ], (7) ()ii () RNk= R , (13) where the ML estimate of u is obtained by ∑k =1 i,0 such that ML uuˆiii,0 =Ξarg max [rt ( ) | , H ] . (8) kT0 ()ii ()* ui RrtstdtkNiNkk===( ) ( ) , 1,..., , 1,..., mod , (14) ∫(1)kT− GLRT is a reasonable alternative to ALRT. However, the 0 N ML estimator is (6) usually does not have a closed from, ()ii () 2 ηNk= ||s . (15) and one needs to carry out a multidimensional exhaustive ∑k =1

2 of 7 ()i Notice that for linear modulations, since sk is independent 2 ()ii ()* ()iiiN 2αα000 () T () of t, (14) simplifies to Rsrkkk= , such that Ξ=[()|{rt skk }=10 , H i ] I  |R N | exp −η N , (20) kT0 NN00 rrtdtk = () . (16) where I (.) is the zero order modified Bessel function of ∫(1)kT− 0 0 ()i the first kind. Further averaging of (20) with respect to Both Rk and rk can be regarded as the output of some {}s()i N yields the required unconditional likelihood function matched filters at the receiver, sampled at kk=1 ()i  2αα 2T  tkTk==0, 1, ..., N. Also note that ηN in (15), when ()iii000 () () Ξ=[()]rt E()i I |R | exp −η . (21) ()i ACPO− {}s N  0  N N  divided by N, represents an estimate of ϑ , the k k =1  NN00 constellation variance of the i-th modulation format. For the i-th modulation, note that the number of terms Now we write the conditional likelihood function of (12) required to calculate E [.] in (21) is M N . Therefore, {}s()i N i in a convenient product form k k =1 N one may consider the exponential complexity of OM()i for N 2 (21). Obviously the classifier which implements (21) is 2αα000()iiT () Ξ=[rt ( ) |u , H ] exp Re[ e− jϕ0 R ] − | s |2 .(17) ii,0 ∏  k k much more complex than (18). NN00 k =1 C. Rayleigh Fading In the sequel, we consider three cases and derive the In this more realistic case, both and in (1) are associated likelihood function, in compact forms, by α0 ϕ0 averaging (17) with respect to the random quantities. In all unknown. Since in the ALRT approach, unknown the situations, data symbols will be treated as iid random quantities are visualized as random variables, we replace variables and will be averaged out. both with α and ϕ , respectively, to reflect their randomness. In Rayleigh fading channels, αe jϕ is a A. AWGN with no Unknown Parameters complex Gaussian variable and α and ϕ are independent. The Rayleigh PDF of α is given by This is an optimistic case where there is no unknown 2 −α / Ω0 2 quantity in (1), except for the data symbols, and AWGN pe()αα=Ω (2/0 ) , α ≥ 0 , with Ω=0 E[]α as the in (3) is the only source of uncertainty in modulation average fading power, whereas for ϕ we have classification. Hence, it can serve as a benchmark, i.e., the p()ϕ = 1/(2)π over [,)−π π . Now we derive a new compact best performance that one can expect. form for the likelihood function, applicable to both linear and nonlinear modulations. With both α0 and ϕ0 known in (1), obviously ()i N † For the vector of unknown parameters and data uuiikk,0==[{s }= 1 ] . Due to the independence of the data ()i N symbols, now we have uu==[{}]αϕ s()i N † . Notice symbols, averaging of (17) with respect to {}skk=1 results ii,0 kk= 1 in the following unconditional likelihood function that the average fading power Ω0 is not included in the Ξ=()i [()]rt vector, as it is assumed to be known. Otherwise, we have AAWGN− to assign a pdf to it, for example, lognormal or gamma N 2αα2T [28], and average over it. This makes it very difficult, if not 000−jϕ0 ()ii () 2 EeRs()i exp Re[ ]− | | . s kk(18) ∏ k =1 k impossible, to derive an expression for the likelihood NN00 function. Using the following integral identity Note that E ()i [.] in (18) is nothing but a finite summation sk ∞ 2 over all the M possible alphabets of the i-th modulation, 2 1 c2 i xcxIcxdxexp()()−= exp , (22) divided by M , for the k-th interval. So, the 102  i ∫0 24cc11 implementation complexity of (18) for the i-th modulation where c and c are positive constants, averaging of (20) may be considered as OMN(), where O(.) denotes the 1 2 i first with respect to α , the random representative of α , order. When compared to the other two classifiers derived 0 and then over the data symbols results in the subsequently in this section, implementation of the unconditional likelihood function of interest classifier built upon (18) is less complex. −1 Ω T η()i B. AWGN with Unknown Carrier Phase Offset ()i 00 N Ξ=+[()]rt E ()i  1 A− Rayleigh {}s N  k k =1  Now we consider a less optimistic case, where α in (1)  N 0 0  is known but ϕ0 , regarded as the carrier phase offset ()i −1 (CPO), is not known. We change the notation from ϕ to  ()i 2  0 Ω00T ηN Ω ||R  ϕ , to emphasize its random nature in the ALRT approach. ×+exp 10 N  . (23) 2 Assuming a uniform distribution for over [,), NN00 ϕ −π π   different approximations to the likelihood function are  proposed in the literature, for a variety of modulation Note that (23) for Rayleigh fading is simpler than (21) for schemes. In what follows, we derive a new closed-form AWGN with the unknown CPO, as it does not have the expression for the likelihood function, which applies to Bessel function. However, still it suffers from the same exponential implementation complexity, due to the ASK, PSK, QAM, and FSK. ()i N ()i N † ()i ()i averaging over the data symbols {}skk=1 . Clearly uuii,0==[{ϕ s kk }]= 1 . Let RkI, / RkQ, be the ()i real/imaginary parts of Rk , respectively. Also let V. QUASI HYBRID LIKELIHOOD APPROACH ()ii () () i RRNNINQ=+,,j R. Now we rewrite (12) as As we discussed in Section III, HLRT is a reasonable Ξ=[()|rtuii , H ] alternative to ALRT and GLRT, yet still has a high computational complexity, to calculate the ML estimates of 2 2αα000()ii ()T () iunknown parameters. Here we propose the QHLRT expRR cosϕϕη+− sin . (19) NI,, NQ N approach, in which we average over the data symbols, NN00 Using eq. 3.338-4 [27], averaging of (19) with respect to ϕ yields whereas for the unknown parameters, we use simple non-

3 of 7 ML estimators, which are accurate enough. Method-of- where similar to (1) moment (MOM) estimators are attractive candidates due N ()i to their simplicity, whereas there is a possibility of getting st(;u )=−−≤≤α ejϕ0,A s () tut ( ( k 1) T ),0 t NT.(32) AAikT,0 0,∑ 0 0 0 near-ML performance, depending on the estimation k =1 problem at hand [29]. In the rest of this section, we Furthermore, for A = 1,2,...,L , ntA()’s, are independent concentrate on linear modulations, and QAM in particular, complex AWGNs, with the same two-sided power spectral and consider MOM estimators for the amplitude and † density N 0 . Let r()trtrtrt= [12 () ()...L ()]. Then phase. conditioned on the unknown vector u = [{αϕ }LLN { } {s()i } ]† , the conditional likelihood To estimate α0 in (1), for linear modulations we use ikk,0 0,AA=== 1 0, AA 1 1 function of r()t , based on (4), can be written as the output of the matched filter rk in (16). After substituting (1) into (3) and then the result into (16), we Ξ=[()|r tHuii,0 , ] get L NT NT ()i 002 jϕ0  21*  resTnkkkk=+=α00, 1,2,..., N, (24) exp Rertst ( ) ( ; ) dt st ( ; ) dt .  AAuuii,0− A ,0  ∏ NN∫∫00 where A=1 00

kT0 (33) nntdtk = () . (25) ∫(1)kT− Now let us define (13) and (14) for the A -th branch, i.e. 0 N Based on (24) and the independence of signal and noise, ()ii () RNk,,AA==Ri, 1,..., N mod, (34) one can show that ∑k =1

kT0 M =+α 2()2ϑ i TNT, (26) ()ii ()* 200 00 RrtstdtkNiNkk,AA===( ) ( ) , 1,..., , 1,..., mod . (35) ∫(1)kT− M =++ααϑ4442()322Es[|()i | ]T 4i NT 2 NT, (27) 0 40k 0 0 0000 Therefore, (33) can be written as where M = Er[| |2 ] and M = Er[| |4 ] are the second and 2 k 4 k Ξ[()|r tHuii,0 , ]= fourth absolute moment of rk . By calculating NT00 from (26) and substituting it into (27), eventually we get a  2 LLT  −jϕ0,A ()ii0 2 () result similar to [30], which provides the basis for exp Reααη0,AAe RNN ,− 0, A . (36) NN∑∑AA==11() estimating α 00 0 Clearly, (36) reduces to (12) for L = 1 . We also rewrite the 4()4ii2 2()i 4 ()2 − 1 αϑ00TEs=−(2MM 24 - )[2 ( [|k | ]/ ϑ )] . (28) above conditional likelihood function in a convenient For 16QAM, 32QAM, 64QAM, and 128QAM, one can product form N show that Es[|()i |4() ]/ϑ i 2 = 1.32, 1.31, 1.38, and 1.35 , k Ξ=[()|r tHu , ] respectively. Since these are close enough, we use the ii,0 ∏ ()i 4()i 2 k =1 approximation Es[|k | ]/ϑ ≈ 1.35 for all of them. This simplifies the estimator for the right-hand side of (28) to  2 LLT  −jϕ0,A ()ii0 22 () (2MMˆˆ2 - )/ 0.65 , which is independent of the modulation exp Reαα0,AAeRkk ,− 0, A | s | , (37) 24 NN∑∑AA==11() type. Note that Mˆ = Nr−12∑N || and Mˆ = Nr−14∑N ||. 00 2 k =1 k 4 k =1 k which is the natural generalization of (17) for antenna To estimate ϕ in (1) for QAM, we employ this 0 diversity. property of QAM at the output of the matched filter [5] When everything is known, expect for the data 44j ϕ0 Er[k ]== Ce , k 1,2,..., N, (29) symbols, the optimal array classifier is the one that relies where C is a positive constant, independent of the number on ALRT, and its likelihood function can be derived by of constellation points. This suggests the following estimator averaging (37) with respect to the data symbols N N 1 4 ()i  ϕˆ = angle r . (30) Ξ=[()]r tE()i 0 k ArrayALRTnoUnknown s 4 (∑k =1 ) ∏ k  k =1  Simulation results for the proposed QHLRT QAM LL  2 ()iiT ()  classifier are provided in Section VII. − jϕ0,A 0 22 exp Reαα0,AAeRkk ,− 0, A | s | . (38) NN∑∑AA==11() VI. MULTI-ANTENNA CLASSIFIER AND 00 SPATIAL DIVERSITY On the other hand, when in addition to the unknown data symbols, there are some unknown parameters, i.e., The spatial diversity offered by an antenna arrays is an LL effective tool in communications [1]. In this section {}α0,AA= 1 and {}ϕ 0, AA= 1 , integration over the unknown we develop a generic decision-theoretic multi-antenna parameters becomes more difficult for the array classifier L − jϕ0,A ()i L 2 modulation classifier, by deriving the associated likelihood (notice the summations ∑A=1α0,AAeRk , and ∑A=1α0,A in function, which is applicable to both linear and nonlinear (37), which used to be single terms in (17), the single- modulations of any order, in multipath fading channels. In antenna classifier of Section IV). HLRT does not seem to [24], the utility of an antenna array for classification of be a good solution also, as finding the ML estimates of BPSK versus QPSK in an AWGN channel is studied. several parameters, depending on the number of antennas, can be very time consuming. Therefore, we propose the Consider an L branch receive antenna array, such that QHLRT approach for the array classifier, where the model given in (3) holds for each branch LL {}α0,AA= 1 and {}ϕ 0, AA= 1 will be estimated using the low- complexity estimators that rely on (28) and (30), rtAA( )=+≤≤= st ( ;ui,0 ) nt A ( ), 0 t NT 0 ,A 1,2,..., L, (31)

4 of 7 respectively. Based on (37), the associated likelihood ALRT in AWGN gets some improvement from antenna function can be written as diversity, only at low SNRs. The four curves of Fig. 1 N discussed here are obtained according to (38). ()i  Ξ=[()]r tE()i ArrayQHLRT s  Clearly it is not possible in practice to find perfect ∏ k  k =1 estimates of the signal power and channel phase. The 2 LLT  classification results of the proposed QHLRT, together − jϕˆ0,A ()ii0 22 () exp Reααˆˆ0,AAeRkk ,− 0, A | s | . (39) with the previously described MOM estimators, is shown in ∑∑AA==11() NN00  Fig. 1 for Rayleigh fading. As expected, we get some Interestingly, the simulation results of the HQLRT performance degradation, when compared to the array classifier, given in the next section, show that by hypothetical perfect estimates. However, by adding only a adding even one antenna to the standard single-antenna second antenna, we again see the performance classifier, one gets a significant performance improvement. enhancement. For example, at the 10 dB SNR, Pcc in Rayleigh fading with imperfect estimates jumps from 0.82 VII. SIMULATIONS AND DISCUSSION to 0.94. Of course this will be further improved by using A. Simulation Setup and Parameters more antennas. Note that with QHLRT and L = 2 in Rayleigh fading, Pcc does not increase with SNR. This is In simulations, we generate normalized constellations, mainly because of the simple phase estimator we have ()i ()i 2 i.e., ϑ ==Es[|k | ] 1 for 16QAM, 32QAM, and 64QAM, ()i used, for which the performance does not improve at high which makes the power of the signal S0 in (2) SNR [31] and becomes less accurate for large QAMs [5]. independent of the modulation type. Furthermore, instead Here we have used it just to demonstrate the utility of the of working with α0 , the channel amplitude in (1), it is QHLRT concept and of course one can consider more more convenient to work with the signal power for which efficient but complex estimators, to get better performance. 2 we have S00= α . Then, as discussed in Section V and also The two curves of Fig. 1 with MOM estimates and based on (28), we estimate the signal power using QHLRT are obtained via (39). ˆ −22ˆˆ 12 ST00= [(2MM 24 - )/ 0.65] . We also define the received signal-to-noise ratio (SNR) per symbol as γ 0000= ST N . In C. Effect of the Number of Antennas all the simulations we set T0 = 1 , and by keeping S0 = 1 , In Fig. 2 we have plotted, versus the number of SNR is changed by varying N 0 . When simulating the antennas L, the P of ALRT in AWGN and Rayleigh fading, the average fading power is set to 1, i.e., cc 2 fading, with no unknown parameters, as well as the new Ω=0 E[]1α = . QHLRT with MOM estimates in Rayleigh fading. All the To estimate ϕ0 , the channel phase in (1), we have used curves are generated according to (38) and (39). As eq. (30), which has a four-fold phase ambiguity [5], since expected, the performance enhances with L in all the QAM has a π 2 symmetry. However, as expected, our situations. The performance improvement is more simulation results have confirmed that the probability of significant at low SNR. Furthermore, the largest jump in correct classification, defined below, has a π 2 symmetry Pcc versus L comes by adding the second antenna. for QAM as well. So, the phase ambiguity of the estimator Interestingly, as we observed in Fig. 1, classification in has no effect on the classifier performance. When phase is AWGN does not get that much improvement from the assumed to be known to the classifier, we set it to ϕ0 = 0 . antenna array. For SNR= 10 dB , Pcc in Fig. 2 is almost 1, To evaluate the performance of classifiers, we have used whereas with SNR= 5 dB , Pcc in AWGN falls below the the average probability of correct classification, defined by Pcc in Rayleigh fading for L = 4, and 5 . To explain this, −1(|)Nmod ii one can argue that in AWGN with no unknown PNcc= mod ∑i=1 P c for equiprobable modulations, in (|ii′ ) parameters, the performance is affected only by SNR, which Pc is the conditional classification probability. (|ii′ ) which is increased by an antenna array. However, in fading More specifically, Pc is the probability to declare that the i-th modulation is received, where indeed the i′ -th channels, the antenna array not only increases SNR, but also combats the deep fades, which are major sources of modulation has been originally transmitted. Here N mod = 3 (|)ii performance loss. and Pc for each modulation has been estimated via 1000 Monte Carlo trials. The number of symbols is N = 500 . D. Effect of the Correlation among the Antennas B. Single & Multi-Antenna Classifiers in AWGN & Fading In the derivation of the array classifier in Section VI According to Fig. 1, for a single-antenna classifier when and the simulations so far, we have assumed the antenna we have perfect (error free) estimates of all the parameters elements are far apart, and hence the branches are (so, there is no unknown parameter), ALRT with uncorrelated. To see the impact of correlation on the averaging over the unknown data symbols in AWGN performance, in Fig. 3 we have considered Rayleigh fading shows the best performance. However, the same ALRT and these two-branch classifiers: ALRT with perfect classifier in Rayleigh fading, again with perfect estimates of estimates in (38) and QHLRT in (39). The correlation * the unknown parameters and averaging over the unknown between the two antennas is defined by ρ12= Ezz[] 1 2 , jϕ1 jϕ2 data symbols, exhibits a significant performance loss. This where ze11= α and ze22= α are two zero-mean motivates the application of antenna array, to take complex Gaussian variables. A non-complex correlation advantage of spatial diversity. By adding only a second coefficient is simulated, such that 01≤ ρ12 ≤ , which antenna, we obtain a huge performance improvement, corresponds to fading channels with the common model of when compared to the single antenna ALRT in Rayleigh isotropic scattering [32]. As expected, Pcc decreases when fading. For example, at Pcc = 0.9 we get a 5 dB in ρ12 increases. The performance degradation seems to be SNR. By adding a second antenna, ALRT in Rayleigh less at high SNRs. Overall, the array classifiers which fading also converges to ALRT in AWGN. Interestingly, assume uncorrelated branches, appear to be reasonably

5 of 7 robust to some possible correlations that may exist the BPSK versus QPSK case,” in Proc. IEEE MILCOM, between the branches. 1988, pp. 431-436. [9] A. Polydoros and K. Kim, “On the detection and E. Effect of K Factor in Rice Fading with Antenna Array classification of quadrature digital modulations in broad-band noise,” IEEE Trans. Commun., vol. 38, pp. 1199-1211, 1990. As discussed in Section IV, one of the disadvantages of [10] C. Y. Huang and A. Polydoros, “Likelihood methods for ALRT in fading channels with unknown parameters is that MPSK modulation classification,” IEEE Trans. Commun., the classifier is going to be designed for a certain fading vol. 43, pp. 1493-1504, 1995. distribution, e.g., eq. (23) for Rayleigh fading, via [11] P. C. Sapiano and J. D. Martin, ”Maximum likelihood PSK averaging over the channel amplitude and phase. However, classifier,” in Proc. IEEE ICASSP, 1996, pp. 1010-1014. when the fading distribution is changed, then the classifier [12] C. Long, K. Chugg and A. Polydoros, “Further results in likelihood classification of QAM signals,” in Proc. IEEE should be re-designed, which most likely does not have a MILCOM, 1994, pp. 57-61. closed form, due to the complicated distributions of the [13] L. Hong and K. C. Ho, “Classification of BPSK and QPSK channel amplitude and phase. For example, look at [1] for signals with unknown signal level using the Bayes technique,” the complicated amplitude and phase distributions in Rice in Proc. IEEE ISCAS, 2003, pp. IV.1-IV.4. fading. On the other hand, The QHLRT proposed in [14] B. F. Beidas and C. L. Weber, “Higher-order correlation- Section V does not depend on the fading model. based approach to modulation classification of digitally frequency-modulated signals,” IEEE J. Select. Areas To observe the utility of the QHLRT approach, see Fig. Commun., vol. 13, pp. 89-101, 1995. 4 where Pcc is plotted versus the Rice K factor, for [15] B. F. Beidas and C. L. Weber, “Asynchronous classification different number of antennas. The reasonable performance of MFSK signals using the higher order correlation domain,” of QHLRT in (39) for all K’s is noteworthy for L = 1 , IEEE J. Select. Areas Commun., vol. 46, pp. 480-493, 1998. [16] B. F. Beidas and C. L. Weber, “Higher-order correlation- which is significantly enhanced by adding only one extra based classification of asynchronous MFSK signal,” in Proc. antenna, i.e., L = 2 . The results of ALRT in Rice fading IEEE MILCOM, 1996, pp. 1003-1009. with error free estimates of the channel amplitudes and [17] A. E. El-Mahdy and N. M. Namazi, “Classification of multiple phases in (38) are also included as a reference, to which M-ary frequency-shift keying signals over a Rayleigh fading QHLRT becomes quite close when L ≥ 2 . channel,” IEEE Trans. Commun., vol. 50, pp. 967-974, 2002. [18] C. Y. Hwang and A. Polydoros, “Advanced methods for Note that for K =∞0and , Rice fading reduces to digital quadrature and offset modulation classification,” in Rayleigh fading and no fading, respectively [1]. This is why Proc. IEEE MILCOM, 1991, pp. 841-845. all the Pcc ’s in Fig. 4 increase with K. [19] P. Panagiotou, A. Anastasoupoulos, and A. Polydoros, ”Likelihood ratio tests for modulation classification,” in Proc. VIII. CONCLUSION IEEE MILCOM, 2000, pp. 670- 674. In this paper, after an overview of decision-theoretic [20] N. Lay and A. Polydoros, “Modulation classification of signals in unknown ISI environments,” in Proc. IEEE approaches to blind modulation recognition, we have MILCOM, 1995, pp. 170-174. focused on linear and nonlinear modulations in fading [21] K. M. Chugg, C. S. Long, and A. Polydoros, “Combined channels, via a unified notation. After formulating the likelihood power estimation and multiple hypothesis problem in a systematic way, we have derived closed-form modulation classification,” in Proc. ASILOMAR, 1996, pp. expressions for the average likelihood function, required to 1137-1141. implement the classifier, under different channel conditions. [22] L. Hong and K. C. Ho, “BPSK and QPSK modulation classification with unknown signal level,” in Proc. IEEE Due to the complexity of these classifiers, a new low- MILCOM , 2000, pp. 976-980. complexity quasi-hybrid classifier is introduced. Then the [23] L. Hong and K. C. Ho, “Modulation classification of BPSK classification problem using an antenna array is formulated and QPSK signals using a two element antenna array and solved. From the simulation results one can conclude receiver,” in Proc. IEEE MILCOM, 2001, pp. 118-122. the usefulness of the proposed quasi hybrid technique, as [24] L. Hong and K. C. Ho, “An antenna array likelihood well as the significant performance enhancement offered by modulation classifier for BPSK and QPSK signals,” in Proc. IEEE MILCOM, 2002, pp. 647-651. multiple antennas. Interestingly, by adding even a second [25] O. A. Dobre, J. Zarzoso, Y. Bar-Ness, and W. Su, “On the antenna to a single-antenna classifier, one can get a classification of linearly modulated signals in fading channel,” reasonable improvement in the classification accuracy. in Proc. CISS, 2004. [26] S. M. Kay, Fundamental of Statistical Signal Processing: REFERENCES Detection Theory. Englewood Cliffs, NJ: PTR Prentice-Hall, 1998. [1] G. L. Stuber, Principles of Mobile Communication, 2nd ed., [27] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, Boston, MA: Kluwer, 2001. and Products, 5th ed., A. Jeffrey, Ed., San Diego, CA: [2] H. L. Van Trees, Detection, Estimation and Modulation Academic, 1994. Theory: Part I. New York: Wiley, 1968. [28] A. Abdi, H. A. Barger, and M. 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6 of 7 16QAM, 32QAM, 64QAM, N=500 symbols, 1000 trials for each modulation 16QAM, 32QAM, 64QAM, N=500 symbols, 1000 trials for each modulation, L=2 1 1 AWGN (L=1,2) 0.98 0.95 SNR=15 dB

Rayleigh fading 0.96 (L=2) 0.9 0.94

0.85 0.92 Rayleigh fading (L=1) 0.9

0.8 Pcc Pcc SNR=10 dB

0.88 0.75 AWGN(perfect estimate,L=1, ALRT) AWGN(perfect estimate,L=2, ALRT) 0.86 Rayleigh fading (perfect estimate, SNR = 15 dB, ALRT) Rayleigh fading(perfect estimate,L=1, ALRT) Rayleigh fading (perfect estimate, SNR = 10 dB, ALRT) 0.7 Rayleigh fading(perfect estimate,L=2, ALRT) Rayleigh fading (MOM estimate, SNR = 15 dB, QHLRT) Rayleigh fading(MOM estimate,L=1, QHLRT) 0.84 Rayleigh fading (MOM estimate, SNR = 10 dB, QHLRT) Rayleigh fading(MOM estimate,L=2, QHLRT) 0.65 0.82

0.8 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 SNR (dB) Correlation coefficient between two branches

Fig. 1. Performance comparison versus SNR: ALRT in AWGN Fig. 3. Performance comparison of a two-branch array classifier and Rayleigh fading, with no unknown parameters (perfect in Rayleigh fading, versus the correlation among the branches: estimates), as well as the proposed QHLRT in Rayleigh fading ALRT with no unknown parameters (perfect estimates), as well (MOM estimates). as the proposed QHLRT (MOM estimates).

16QAM, 32QAM, 64QAM, N=500 symbols, 1000 trials for each modulation 16QAM, 32QAM, 64QAM, N=500 symbols, 1000 trials for each modulation, SNR=10 dB 1 1

SNR=10 dB 0.98 0.95 L=4 L=2 0.96 0.9 0.94

0.85 0.92 L=1 SNR=5 dB 0.8 0.9 Pcc Pcc

0.88 0.75 Rice fading (perfect estimate, L=1, ALRT) AWGN(perfect estimate,SNR=10dB, ALRT) Rice fading (perfect estimate, L=2, ALRT) 0.86 Rayleigh fading(perfect estimate,SNR=10dB, ALRT) Rice fading (perfect estimate, L=4, ALRT) 0.7 Rayleigh fading(MOM estimate,SNR=10dB, QHLRT) Rice fading (MOM estimate, L=1, QHLRT) AWGN(perfect estimate,SNR=5dB, ALRT) Rice fading (MOM estimate, L=2, QHLRT) Rayleigh fading(perfect estimate,SNR=5dB, ALRT) 0.84 Rice fading (MOM estimate, L=4, QHLRT) Rayleigh fading(MOM estimate,SNR=5dB, QHLRT) 0.65 0.82

0.6 0.8 1 2 3 4 5 −infinity 5 10 15 Number of antennas L Rice K factor (dB)

Fig. 2. Performance comparison of a multi-antenna classifier Fig. 4. Performance comparison of a multi-antenna classifier in versus L: ALRT in AWGN and Rayleigh fading, with no Rice fading, versus the Rice K factor: ALRT with no unknown unknown parameters (perfect estimates), as well as the new parameters (perfect estimates), as well as the proposed QHLRT QHLRT in Rayleigh fading (MOM estimates). (MOM estimates).

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