arXiv:1709.05460v1 [cs.NI] 16 Sep 2017 ehd olso detection. collision method, u lsicto fTM,ODA n DAi outto exi robust outperforms that is contention CDMA show of methods. detection and Simulations while OFDMA, blind SNR. load contention- TDMA, is network received of detect method the classification proposed to our of Our t cumulant independent methods. using same and access detector the collision channel novel of based CDMA. a variance and proposes sample modulati OFDMA, stage TDMA, second cumulant-based between The fourth-order distinguish to existing sta classifier first classify The an network. to primary algorithm a extends by stage used two method a access channel propose we holes, temporal hswr a oewieMhradPuowr h tdnsa U at students PhD were Paulo danijel and [email protected], Mihir An [email protected], while CA Los E-mail: done California, Clara, was of Santa University work the Semiconductor, This with Marvell i.e. is Cabric with standards, Danijela is of Urriza class Paulo the which identify category, on, second focusing The be family. will 802.11 we the be of distinguish to members times inter-arrival and lengths learni frame supervised the uses packet [13] de- as format. on preamble such and based characteristics structure used PHY/MAC standard of specific knowledge the tailed determine [11 fall as to such standard aim work works [12] First, particular standards. 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NTRODUCTION tdn ebr IEEE, Member, Student ontv ai Applications Radio Cognitive [email protected]. ee,C 90095. CA geles, 55,and 95054, al ria n aieaCabric, Danijela and Urriza, Paulo CLA. tween gof ng and ] sting 2121, or s cess ious the on he ge re s- s l , eevn inl rmantokof network a from signals receiving oe h inltasitdb the by transmitted signal the model Let themselves. amongst oprsnwt xsigwr spoie nScinIV. Section V. Section in and in provided concluded results is is Simulation paper work the III. Finally, proposed existing Section Our with in II. comparison Section described in is described method is notation and model methods. and access cumulant channel same detecting contention-based the for detect of method variance thus, sample novel the a using propose collusins we and CDMA, Second, TDMA, OFDMA. between distinguish to [15] modulat classifier cumulant-based type fourth-order existing an extend we Contributions B. OFDMA [14]. and in CDMA Further, addressed data. not supervised labeled are of systems a lack channe the fading such to unknown in due However, limitations has networks. approach learning ALOHA multiple pure (SVM) ALOHA, sense slotted and Machine (CSMA/CA), carrier The avoidance Vector TDMA, collision [14]. with Support between is access a classify category to this employ in approach [14] of only of know The authors we CDMA2000. that for work CDMA existing and 802.11n for OFDMA where inltasitdb h U fatv.Bt h activity the Both active. if PU, the by transmitted signal ucrir n ylcpexo length of prefix cyclic a and subcarriers ucrir,subset subcarriers, d,w suethat assume we ods, network. PU the by n h signal the and rnmtiga time at transmitting ae cee ont i.e., not, do schemes based t code its eoyls ouain uha A.TM enforces TDMA QAM. i.e., time, as in such orthogonality modulation, memory-less ecnie ytmcnitn fasnl esn node sensing single a of consisting system a consider We h eto h ae sognzda olw.Tesystem The follows. as organized is paper the of rest The First, follows. as are work this of contributions key two The o DA the CDMA, For o ohTM n otninbsdcanlacs meth- access channel contention-based and TDMA both For o OFDMA, For n c i stetm index, time the is ( n ) flength of s i ( s n s i i ) ( ( S n i n eedo h hne cesmto used method access channel the on depend hP’ aastream data PU’s th I S II. ) i ) s n i sa FMmdltdsga with signal modulated OFDM an is x = r sindt the to assigned are ( n tews,and otherwise, 0 and i n ( L c ) n c YSTEM i U ) sasnl are inlwt linear with signal carrier single a is : ( Í n = Í eirMme,IEEE Member, Senior eteidxsto h U.We PUs. the of set index the be i N = i a N mod total = 1 a i total 1 ( i n ( M a n ) a i s ) L ( i ODEL i N n ( i ( c hP as PU th n n total ≤ ) = ) ∈ ) ) d i d ( 1 1 U communicating PUs n i i N hl contention- while , hPU. th ( ) N n fthe if . { ∪ ) p s fthese Of . i ssra using spread is ( 0 n } ∈ ) . i hP is PU th C sthe is a i N N ion ( (1) (2) n sc sc ls 1 ) 2

For both OFDMA and CDMA, we assume that all PUs TABLE I SAMPLEMEANANDVARIANCEOF C ESTIMATED FROM J SAMPLES FOR are transmitting simultaneously, i.e., ai(n) = 1. Though this 42 UNIT POWER NOISE-LESSSIGNALS appears to be a strong assumption, squelching the received 1 signal and normalization of the test statistic ensure that it does Constellation C42 J var(Cˆ42) from [15] J var(Cˆ42) not impact the performance of our algorithm. BPSK -2.0000 36.00 0.00 We denote the tuple of channel access method and modula- PAM(4) -1.3600 34.72 10.24 tion type used by M. Non-contention based channel access PAM(8) -1.2381 32.27 9.98 PAM(16) -1.2094 31.67 9.90 methods covered by this work are listed in Table II and PAM(32) -1.2024 31.52 9.88 collected in the set M. PAM(64) -1.2006 31.49 9.88 The packet arrival rate at the PUs determines the rate of PAM(∞) -1.2000 31.47 9.87 collisions in a contention-based scheme. For simplicity, we PSK(≥4) -1.0000 12.00 0.00 assume the following conditions: 1) all packets have the same V32 -0.6900 9.70 1.42 length, 2) a packet is generated from each PU according to V29 -0.5816 8.75 1.77 QAM(4,4) -0.6800 9.54 1.38 a Poisson process with rate λi, 3) packets that collide are QAM(8,8) -0.6191 8.82 1.39 not retransmitted. As a result, the aggregate messages to the QAM(16,16) -0.6047 8.65 1.39 channel will also be a Poisson arrival process with parameter QAM(32,32) -0.6012 8.61 1.39 N QAM(∞ ) -0.6000 8.59 1.39 G = = λi. This is referred to as offered load. i 1 BPSK-OFDM 0 – ∼8 TheÍ downconverted signal received at the sensing node is QPSK-OFDM 0 – ∼4

Ntotal r (n) = h (n) x (n) + ν (n) (3) i i ˜ Õi=1 Since the C42 is normalized by signal power, its mean value is independent of any attenuation due to flat-fading [15]. 2 where ν(n)∼ CN(0, σν ) is white Gaussian noise with variance The additivity of the unnormalized C [16, Theorem 2 42 σν and hi(n) includes the PU’s transmit power, the fading 2.3.1(vi)] can be used to express the mean and variance of the channel from the PU to the sensing node, and path loss with estimated C˜42 when U PUs are transmitting simultaneously: exponent γ. Let y(n) be the squelched received signal. 2 u∈U C , C42,u(M) E C˜ ( f ) M = 21 u (5) 42,U Í 2 III.IDENTIFYING CHANNEL ACCESS METHOD   u∈U C21,u 4 ˜ TDMA, OFDMA, and CDMA have distinguishable features u∈UÍC , var
C42,u( f ) M var C˜ ( f ) M = 21 u due to their signal structure viz., modulation used in TDMA, 42,U Í  4  C , the effect of the inverse fourier transform in OFDMA, and the   u∈U 21 u 8 σν varÍC˜42,ν ( f )
pseudonoise sequence used by users in CDMA. Contention- + . (6)  4  based methods are distinguished by the probability of colli- u∈U C21,u sions between users. We now show that these properties can be Í
˜ distinguished using the sample mean and sample variance of Expressions to compute C42,u(M) and var C42,u( f ) M are the normalized fourth-order cumulant of the squelched signal. provided in the Appendix while values for noise-less signal s are listed in Table I as computed from (5) and (6). In the interest of readability, the left hand side of (5) and (6) does A. Normalized 4th-Order Cumulant C42 and its Properties not explicitly mention the conditional dependence of the mean Cumulants of multiple random variables are particular poly- and variance on the signal powers and noise variance. nomial combinations of their joint higher order moments [16]. We define kth-order cumulants Cki, for i ∈ {0,..., k}, of B. Proposed Method the zero mean complex signal y(n) as the cumulant of k − i ∗ Since collisions would modify test statistics that depend on repeated copies of y(n) and i repeated copies of y (n). Also, the signal structure, we use a 2 stage algorithm that classifies , k−i ∗ i let Mki E[y (n)(y (n)) ]. between TDMA, CDMA, and OFDMA first and then detect Our proposed test statistic requires estimating the cumu- whether there are collisions in the second stage. The first stage lants of frames of J samples each. The unbiased maximum is a multihypothesis test using the sample mean of C˜42( f ) to likelihood estimators for the cumulants of the f th frame are classify as TDMA, CDMA, or OFDMA. This is an extension ˆ denoted by Cki( f ) and are computed by formulae derived in of the modulation classification method proposed in [15]. The [15]. Similar to [15], the cumulant estimates are normalized 2 second stage proposes a novel binary hypothesis test to detect by the signal power Cˆ21( f )− σν : collisions by thresholding the sample variance of C˜42( f ). −2 We divide the squelched received signal {y(n)} ˜ = ˆ ˆ 2 n∈{1,...,JF } C42( f ) C42( f ) C21( f )− σν . (4) into F frames of J samples each. As proposed in [15], The   sample mean W of the normalized C42 can be used to identify Note that C21 of a signal is the signal power and does not depend on the modulation type. We denote the u’th user’s the modulation type of the received signal. We now extend it signal’s C by C , . If the value depends on the modulation 1 ki ki u Derivation of var[Cˆ42] in [15] missed an O(1/J) term. Details provided type or channel access method, it is denoted by Cki,u(M). in the appendix. 3

to identify TDMA, OFDMA, and CDMA. In particular, we TABLE II classify the received signal as being one of the classes listed LISTOFCHANNELACCESSMETHODSANDMODULATIONSIN M IN (12). ˜ in Table II. Let C42,U ( f ) be the estimated normalized C42 if Class Label Channel Access Method Modulation Type & Levels U ⊆ U( f ) indexed PUs were transmitting in frame f . Let M1 BPSK M2 4/8/16/32/64-PSK U( f ) denote the set of PUs transmitting simultaneously in TDMA ˜ M3-M7 4/8/16/32/64-PAM frame f . Then, the measured C42( f ) can be written as: M8-M10 16/64/256-QAM M11 4-QAM ˜ = ˜ OFDMA C42( f ) 1{U=U(f )}C42,U ( f ). (7) M12 16-QAM UÕ⊆U M13 BPSK ˜ M14 CDMA 4-QAM Since C42( f ) has a Gaussian distribution due to the central M15 16-QAM limit theorem, (7) implies that C˜42( f ) has a Gaussian mixture distribution with mean and variance given by If ςˆ2 < τ then we declare the inference Mˆ from (12) to be E C˜42( f ) M = P (U = U( f )) E C˜42,U ( f ) M (8) correct. If not, then we infer that the channel access method   UÕ⊆U   is contention-based. var C˜42( f ) M = P (U = U( f )) var C˜42,U ( f ) M   UÕ⊆U    2 IV. RESULTS AND COMPARISONS + E C˜42,U ( f ) M − E C˜42( f ) M . (9) The theoretical distributions of both the proposed test    
o Further, assuming that the same number, say K, of PUs statistics have been described in the previous section. Since transmit at any time, then (8) and (9) can be simplified to our proposed algorithm consists of a multi-hypothesis test, it is not possible to derive a closed form expression for E C˜42( f ) M = C42,U′ (M) and (10) the classification accuracy. Hence, in this section, we use  ˜  = = ˜ var C42( f ) M P(U U( f )) var C42,U ( f ) M simulations to study the performance of our proposed channel   UÕ∈U   access method classification algorithm. |U |=K (11) where U′ ⊆ U and |U′| = K. For TDMA, K = 1, while A. Simulation System K = Ntotal for CDMA and OFDMA. Since this simplification Consider N PUs communicating by a TDMA, CDMA, is not possible for contention-based channel access methods, OFDMA, or contention-based channel access method. The sig- we separate the contention detection to the second stage. nals transmitted by these users are as described in Section II. The first stage of our algorithm uses (10), (11), and the We assume a Rayleigh flat fading channel between the PUs ˆ sample mean W of C˜42( f ) to find the most likely class M and our sensor node. The received SNR of each individual amongst those listed in Table II. user is exponentially distributed [18] and we vary the average ′ SNR. The offered load by the system is varied from 0.1 to 1. Mˆ = arg max P (C42 = W|M ) (12) M′ ∈M Our metric is the probability of correctly classifying channel For the second stage of our algorithm, detection of con- access methods. We have chosen the parameter PC|T as 0.05. tention, we use (9) to note that varying number of simultane- We present results averaged over the classes listed in Table II. ously transmitting PUs increases the sample variance: We also implemented the SVM-based classifier proposed by Hu et al. in [14] to distinguish between TDMA and var C˜42( f ) M < M > var C˜42( f ) M ∈ M . slotted ALOHA. The SVM is trained using features of received     energy, idle time, and busy time of the channel. In order to We use this fact to estimate our confidence in the inference ˆ simulate a blind scenario, we trained the SVM with about M. We define PC |T ∈ (0, 1) as the maximum probability of detecting collisions for non-contention based medium access 60,000 realizations consisting of 50 realizations from each modulation type, SNR, number of users, and traffic load. control protocols. We compute the second moment of C˜42( f ) around the theoretically expected C42(M)ˆ : B. Effect of Traffic Load 1 F 2 ςˆ2 = C˜ ( f )− C Mˆ . (13) F 42 42 The traffic load does not affect the classification of TDMA, Õf =1    CDMA, and OFDMA because we squelch the input signal. If the correct class has been detected, then ςˆ2 is the sample Fig. 1 shows this classification accuracy as a function of variance of C˜42( f ) and by Cochran’s Theorem [17], the sample the normalized load. However, the number of collisions in variance is distributed as a contention-based method increase with load and causes the classification accuracy of contention-based channel access 2 −1 ˜ 2 ςˆ ∼ F var C42( f ) M χF (14) method to increase with normalized load. 2 2   SNR affects the variance of C˜42( f ) for all classes but its where χF is a χ distribution with F degrees. We choose a 2 threshold τ such that PC |T > P ς > τ M < HC : mean is affected only for contention-based channel access methods. Therefore, low SNR affects the classification of = −1 ˜ ˆ −2
τ F var C42( f ) M χF 1 − PC |T . (15) contention-based methods more than that of the other classes. h i

4

1 1 1

0.8 0.8 0.8

SNR Proposed Hu et al. 0.6 0dB 0.6 0.6 2dB 4dB 6dB 0.4 0.4 0.4 SNR CDMA OFDMA 8dB 0dB 10dB 2dB 0.2 0.2 0.2 4dB Probability of Detecting TDMA Probability of correct classification Probability of Detecting Contention 6dB 8dB 0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Normalized Load Normalized Load Normalized Load (a) 4 user TDMA (b) 4 user contention-based. Legend same as (a). (c) 64 subcarrier OFDMA and 16 user CDMA Fig. 1. Probability of correctly identifying the channel access method for TDMA, OFDMA, and CDMA and comparison with [14] for detection of TDMA.

1 D. Computational Complexity Computing cumulant estimates from F frames of J samples 0.8 each requires O(FJ) operations. Computing the statistics for each class in M requires O(|M|) operations. Hence, the first 0.6 stage of the algorithm requires O(FJ) operations. By reusing 4 user TDMA values computed in the first stage, computing ςˆ2 and τ for the 16 user CDMA

. 64 subcarrier OFDMA second stage requires O(F) operations. Thus, our algorithm Probability of Correct0 Classification 4 4 user Contention based requires O(FJ) computations dominated by the first stage. 0 100 200 300 Number of Frames V. CONCLUSION Fig. 2. Probability of correctly identifying channel access methods as number of frames are increased. System: J = 500, 5dB SNR at 0.5 normalized load. In this article, we have presented a new algorithm to identify the channel access method utilized by a primary network. Our methods are not restricted to any specific standards. Furthermore, in case a few “hidden” PUs have very low SNRs We extended a cumulant-based modulation type classification due to, say, distance or deep fading channels, the classification technique to differentiate between OFDMA, CDMA, and accuracy will reduce for all classes and the detection of TDMA. We proposed a novel collision detection method using contention will suffer the most. However, if the remaining the sample variance of the cumulant estimator and, thus, PUs have higher SNR, then the classification accuracy will identify contention-based channel access methods such as increase. CSMA. These test statistics were chosen to make our methods Further, Figs. 1(a) and 1(b) compares with the SVM based robust to channel fading and size of the PU network. classifier proposed in [14] which considers only TDMA and contention-based schemes. Since [14] separates only two REFERENCES classes, note that our proposed algorithm has significantly higher combined classification accuracy. The SVM proposed [1] S. Haykin, “Cognitive radio: Brain-empowered wireless communica- tions,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. in [14] tends to classify all signals as contention-based at 2005. low loads possibly because channel idle and busy times [2] Y. Chen, Q. Zhao, and A. Swami, “Joint Design and Separation Principle depend more on the load than contention for the channel. for Opportunistic Spectrum Access in the Presence of Sensing Errors,” IEEE Trans. Inf. Theory, vol. 54, no. 5, pp. 2053–2071, May 2008. However, further study is required to improve our algorithm’s [3] W. Wellisch and A. Barreto, “Spectrum sensing of TETRA systems classification of contention-based schemes at low SNRs. through timefrequency analysis,” in IEEE LATINCOM, Nov. 2012, pp. 1–6. [4] S.-Y. Tu, K.-C. Chen, and R. Prasad, “Spectrum Sensing of OFDMA Systems for Cognitive Radio Networks,” IEEE Trans. Veh. Technol., vol. 58, no. 7, pp. 3410–3425, Sep. 2009. C. Number of Frames [5] F. Bernardo, R. Agusti, J. Perez-Romero, and O. Sallent, “A Self- Organized Spectrum Assignment Strategy in Next Generation OFDMA Networks Providing Secondary Spectrum Access,” in IEEE ICC, Jun. Fig. 2 shows the classification accuracy as the number 2009, pp. 1–5. of frames F is increased. Increasing F reduces the sample [6] M. Li, N. B. Stella, A. P. Dimitris, T. Melodia, J. M. Michael, and variance of C˜ which increases the probability of correctly D. M. John, “Cognitive code-division channelization with blind primary- 42 system identification,” in IEEE MILCOM, Oct. 2010, pp. 1460–1465. classifying TDMA, CDMA, and OFDMA. Increasing F also [7] M. G. Khoshkholgh, K. Navaie, and H. Yanikomeroglu, “Adaptive increases the probability of observing a collision in contention- Multiple Time-Scale Power Allocation for Spectrum Sharing in DS- based methods. So, the probability of correctly classifying CDMA Networks,” in IEEE ICC Workshops, May 2008, pp. 466–470. [8] Y. Liu, N. Kundargi, and A. Tewfik, “Channel Idle Time Statistics Based contention-based methods increases. For comparison, note that Spectrum Accessing Strategies With CSMA Based Primary Networks,” Fig. 1 shows results for F = 200. IEEE Trans. Signal Process., vol. 62, no. 3, pp. 572–582, Feb. 2014. 5

[9] T. Suzuki, T. Fujii, and O. Takyu, “An intelligent secondary MAC shown in the following derivation. Using the definition protocol using shared spectrum with IEEE802.11 wireless LAN,” in 2 α , M42,y − M , Proc. ICUFN, Jun. 2011, pp. 358–362. 21,y [10] Y. C. Cheng, E. H. Wu, and G. H. Chen, “A Decentralized MAC Protocol for Unfairness Problems in Coexistent Heterogeneous Cognitive Radio (J − 1)(J − 2)(J − 3) Networks Scenarios With Collision-Based Primary Users,” IEEE Syst. var[Mˆ 2 ] = M4 J., vol. 10, no. 1, pp. 346–357, Mar. 2016. 21,y J3 21,y [11] M. Di Benedetto, S. Boldrini, C. Martin, and J. Diaz, “Automatic 6(J − 1)(J − 2) network recognition by feature extraction: A case study in the ISM + 2 2 M21,y M42,y band,” in CROWNCOM 2010, Jun. 2010, pp. 1 –5. J3 [12] R. Hachemani, J. Palicot, and C. Moy, “A new standard recognition α 2 − M2 + + O(1/J2) sensor for cognitive radio terminals,” in Proc. EUSIPCO, Sep. 2007, 21,y J pp. 856–860.   [13] S. A. Rajab, W. Balid, M. O. A. Kalaa, and H. H. Refai, “Energy 6 4 6 2 = 1 − M + M M42,y detection and machine learning for the identification of wireless MAC  J  21,y J 21,y technologies,” in Proc. IWCMC, Aug. 2015, pp. 1440–1446. α 2 [14] S. Hu, Y.-D. Yao, and Z. Yang, “MAC protocol identification using 2 + + 2 − M21,y O(1/J ) support vector machines for cognitive radio networks,” IEEE Wirel.  J  Commun., vol. 21, no. 1, pp. 52–60, Feb. 2014. 4 2 2 ≈ M (M , − M ) (19) [15] A. Swami and B. Sadler, “Hierarchical digital modulation classification J 21,y 42 y 21,y using cumulants,” IEEE Trans. Commun., vol. 48, no. 3, pp. 416–429, Mar. 2000. [16] D. R. Brillinger, Time Series. Society for Industrial and Applied − 6 M4 Mathematics, 2001. The error is in failing to take into account the J 21,y part [17] W. G. Cochran, “The distribution of quadratic forms in a normal system, of the first term. A similar derivation also gives the corrected with applications to the analysis of covariance,” Math. Proc. Camb. expression Philos. Soc., vol. 30, no. 02, pp. 178–191, Apr. 1934. [18] J. Proakis and M. Salehi, Digital Communications, 5th ed., ser. McGraw- Hill higher education. McGraw-Hill Education, 2007. ˆ ˆ 2 = 2 [19] M. Shi, A. Laufer, Y. Bar-Ness, and W. Su, “Fourth order cumulants in cov[M42,y, M21,y ] M21,y(M63,y − M42,y M21,y ) (20) distinguishing single carrier from OFDM signals,” in IEEE MILCOM, J Nov. 2008, pp. 1–6. Following similar derivations we can find the rest of the terms in (17) as APPENDIX STATISTICS OF SAMPLE ESTIMATES OF CUMULANTS 2 2 2 4 4 var[|Mˆ , | ] = M , |M , | − |M , | 20 y J 42 y 20 y J 20 y ( ) 2 ∗2 Consider J samples of a complex signal r n as given by (3). + Re{M40,y M } (21) = 1 J J 20,y Define y(n) r(n)− J n=1 r(n) so as to obtain a zero-mean 2 2 ∗ form of the received signal.Í We wish to derive the mean and cov[Mˆ , , |Mˆ , | ] = Re{M , M } 42 y 20 y J 62 y 20,y variance of the estimator C˜42,y where the subscript y indicates 2 2 that the cumulant is computed from the signal y(n). − M , |M , | (22) J 42 y 20 y We begin by studying the unbiased estimator for the (un- 2 2 4 ∗ normalized) Cˆ cumulant: cov[|Mˆ ,y | , Mˆ ] = M ,y Re{M ,y M } 42,y 20 21,y J 21 41 20,y 4 2 2 ˆ = ˆ 2 ˆ2 − M21,y |M20,y | (23) C42,y M42,y −|C20,y | − 2C21,y . (16) J

Since these terms are correlated, we have Substituting (18)–(23) into (17) we find the general expression for the asymptotic variance of the C42,y estimate as follows: ˆ = ˆ + ˆ 2 + ˆ 2 var[C42,y] var[M42,y] var[|M20,y | ] 4var[M21,y] ˆ ˆ 2 ˆ ˆ 2 ˆ 2 − 2 cov[M42,y, |M20,y | ]− 4 cov[M42,y, M21,y ] J var[C42,y]≈ M84,y − M42,y 2 2 + 4 cov[|Mˆ 20,y | , Mˆ ]. (17) + 2 2 21,y 8M21,y 2M21,y (M42,y − M21,y −|M20,y | ) h + ∗ + From the Appendix in [15] we have 2 Re{M41,y M20,y }− M63,y M21,y M42,y i + ∗ ∗ 2 Re{M20,y (M40,y M20,y − 2M62,y)} 1 2 + 2 2 var[Mˆ , ] = (M , − M ) (18) 2|M20,y | (3M42,y − 2|M20,y | ). (24) 42 y J 84 y 42,y

ˆ 2 The asymptotic analysis of both var[M21,y ] and Now, y(n) is a noisy signal. We will rewrite (24) in terms ˆ ˆ 2 cov[M42,y, M21,y ] as derived in [15] are incorrect since of cumulants so that we can quantify the effect of noise using they fail to take into account some O(1/J) terms as the additive property of cumulants. The following moment– 6 cumulant equivalence relations are easy to derive: Similarly, for signals modulated by constellations having four- fold symmetry, such as QAM, we use C20 = 0 = C21 to get 2 M84,y =C84,y + 16C63,yC21,y + 12 Re{C64,yC20,y } J var[C˜42,y] = C84(M) + |C40(M)| + 2 + 2 + 2 2 72C , C42,y 18C , 16|C41,y | C21,y C21,y 21 y 42 y + 8 C (M) + 20 C (M) + 2 + ∗ 2 2 63 2 42 |C40,y | 6 Re{C40,yC20,y }  C21,y − σν  C21,y − σν  ∗ 2 4 + 96 Re{C C }C + 36|C | C C21,y 41,y 20,y 21,y 20,y 42,y + 4 + 17C (M)2 (29) 2 42 + 2 2 + 4 + 4  C21,y − σν  72|C20,y | C21,y 24C21,y 9|C20,y | (25) = + + 2 + 3 This is the corrected form of [15, Eqns. 13]. M63,y C63,y 6 Re[C20,yC43,y] 9|C20,y | C21,y 6C21,y Table I lists the statistics of the C estimation for unit power + C C 42 9 21,y 42,y (26) noise-less signals having different modulation types. = + 2 + 2 M42,y C42,y |C20,y | 2C21,y = + 2 M40,y C40,y 3C20,y B. Statistics for OFDMA Signals M21,y =C21,y . (27) The use of fourth order cumulants for distinguishing OFDM signals from single carrier signals is proposed and analyzed in [19]. The moments for OFDM are found to be Gaussian noise has all the relevant cumulants zero except for 2 C21,ν = σν . By slight abuse of notation, let Cki,x be the Cki M84 ≈ 24, M63 ≈ 6, M40 ≈ 0, M42 ≈ 2, M21 = 1. cumulant of the noiseless signal component of y(n). Then, Using these moments in the corrected expression in (24) gives except for C21,y, we can rewrite all the relevant cumulants as 2 that J var[Cˆ42] ≈ 4 which coincidentally also matches the Cki,y = Cki,x . C21,y can be rewritten as C21,y = C21,x + σν . Using these relations and (25)-(27), we can rewrite (24) in result derived in [19] from the incorrect expression of [15]. terms of cumulants. Note however that this result only applies for OFDM with subcarrier modulations that satisfy the fourfold symmetry such as QPSK. Using these moments the variance of the fourth order cumulant for a noisy OFDM signal can be found using (29). A. Single User Signals

If it is known that the received signal r(n) consists of a C. Statistics for CDMA Signals single user’s signal, i.e., no collisions have occurred and it A CDMA signal which uses BPSK chips can be viewed as is not a CDMA or OFDMA signal, then we can use the the sum of Ntotal BPSK signals given as ˜ modulation type M of the signal to describe var C42,y . We N 2 total   2 = do this by assuming that the normalizing factor Cˆ21,y − σν x(n) si (n) Õi=1 is perfectly estimated. Then, after normalization, Cki,x are replaced by Cki(M) where Cki(M) is the Cki cumulant of a unit where si (n) is a BPSK formed by spreading the data to be power signal having modulation M. After normalization, C21,y transmitted with the particular code assigned to that user and 2 would be replaced by C21,y /(C21,y − σν ). Using these relations is given by (2). Thus we can find the mean of the normal- and (25)-(27), we can rewrite (24) in terms of cumulants for ized fourth-order cumulant of a noiseless CDMA signal as signals modulated by real constellations as: E[Cˆ42] = −2/Ntotal where we invoke the additivity property. In effect, the mean normalized fourth order cumulant approaches 0 as the number of users increases. C21,y J var[C˜42,y] = C84(M) + 4C63(M) As for the variance of Cˆ we can use the general expression  C − σ2  42 21,y ν in (24) once the moments are found. Due to the blind nature + ∗ + 2 12 Re{C62(M)C20(M)} 17C42(M) of our classification problem, we do not have knowledge 2 C21,y of the true spreading code used by each user. As a result, − 8 C (M) + 34|C (M)|2C (M) 2 42 20 42 the correlations from one chip to another within the same  C , − σ  21 y ν symbol period cannot be known. However, unlike the single C21,y + 16|C (M)|2 + 24 Re{C (M)∗C (M)} carrier signals presented in this appendix, these correlations 41 41 20 2  C21,y − σν  are clearly non-zero and are dependent on the codes used. To 2 ∗ 2 + |C40(M)| + 6 Re{C40(M) C20(M) } circumvent this issue we will assume that such correlations 4 are negligible. Note that with this assumption we are treating C21,y + 24 (28) CDMA signals to be similar to a sum of BPSK signals in 2 C21,y − σν  which symbols from different symbol periods and different users are regarded as i.i.d. This is clearly an approximation, where we use the fact that real constellations have all real but we have found through simulations that the discrepancy is moments, i.e., M20 = M21, M40 = M41 = M42, and M62 = M63. negligible in practice. 7

With this assumption we can proceed to derive the moments of a sum of Ntotal BPSK signals to be 4 N 1 1 M = total + 42 2 2 2 N    Ntotal total 6 4 N 1 6 N 2 M = total + total 63 2 2 3 3 4 2 3     Ntotal    Ntotal + 1 2 Ntotal 8 6 4 N 1 8 4 N 3 M = total + total 84 2 2 2 4 4 4 2 3 J4      Ntotal     + 8 + 1 8 Ntotal 2 + 1 6 2 4 2 4 3       Ntotal Ntotal

The variance of C˜42 of the noisy signals can then be found through (27).