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An OFDM Signal Identification Method forWireless Communications Systems

Ali Gorcin∗, Huseyin Arslan∗†,∗Department of Electrical Engineering, University of SouthFlorida, 4202 E. Fowler Ave., ENB-118, Tampa, FL,

33620, USA† School of Engineering and Natural Sciences, Istanbul Medipol University, Beykoz, Istanbul, 34810, Turkey

Emails: agorcin@mail.usf.edu, arslan@usf.edu

1 Abstract—Distinction of OFDM signals from single carriersignals is highly important for adaptive receiver algorithms andsignal identification applications. OFDM signals exhibit Gaussiancharacteristics in time domain and fourth order cumulants ofGaussian distributed signals vanish in contrary to the cumulantsof other signals. Thus fourth order cumulants can be utilized forOFDM signal identification. In this paper, first, formulatio ns ofthe estimates of the fourth order cumulants for OFDM signalsare provided. Then it is shown these estimates are affectedsignificantly from the wireless channel impairments, frequencyoffset, phase offset and sampling mismatch. To overcome theseproblems, a general chi-square constant false alarm rate Gaus-sianity test which employs estimates of cumulants and theircovariances is adapted to the specific case of wireless OFDMsignals. Estimation of the covariance matrix of the fourth ordercumulants are greatly simplified peculiar to the OFDM signals.A measurement setup is developed to analyze the performanceof the identification method and for comparison purposes. Aparametric measurement analysis is provided depending onmodulation order, signal to noise ratio, number of symbols,anddegree of freedom of the underlying test. The proposed methodoutperforms statistical tests which are based on fixed thresholdsor empirical values, while a priori information requiremen t andcomplexity of the proposed method are lower than the coherentidentification techniques.

I. Introduction

Modulation identification has always been a part of commu-nications surveillance systems and electronic warfare, howevergained more importance with the adaptive communicationsystems. Seamless and efficient communications should beachieved between a wide variety of communications systemswhich may employ different modulation techniques. Therefore,simple, robust, and fast modulation identification methodsare required at the implementation of adaptive and intelli-gent receivers. Blind receiver design algorithms and signaltransmission procedures can also benefit from the modulationrecognition information.

Improving the efficiency of the wireless communicationssystems, on the other hand, has substantial importance due tothe increasing demand for higher capacity. Ascending numberof users, networks, and deployment of a wide variety of

1Accepted for publication in the IEEE Transactions on Vehicular Technol-ogy. Copyright (c) 2013 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

systems led the wireless communications towards heteroge-neous networks, small cell deployments and implementationof dynamic spectrum access techniques. Therefore emergingtechnologies require as much information as possible aboutthe communications medium. Wireless communications mea-surement systems should also adapt themselves to the newwireless communications paradigm by providing more infor-mation such as modulation type, carrier number, bandwidth,and other waveform parameters automatically. To this end,radiocommunication sector of International Telecommunica-tion Union (ITU-R) initialized an incentive, and publishedthe“Technical identification of digital signals” document as themain guideline for the wireless communications measurementsystems to identify unexpected or unlicensed emissions, tode-tect interference sources, and to provide waveform parametersvia signal identification [1].

A. Related Work

When the contemporary communications systems are con-sidered, orthogonal frequency-division multiplexing (OFDM)is a multi-carrier (MC) multiplexing scheme which maintainsadaptive communications features by employing sub-carriersin a flexible way. Deployments of OFDM based systems areincreasing rapidly. Distinction of OFDM signals from thesingle carrier (SC) signals is an important issue from theadaptive transceiver design, signal identification and spectrummonitoring perspectives. Initial research on this field indi-cates that modulation identification or recognition methodsfor OFDM systems can be classified into two groups; co-herent techniques which include maximum likelihood (ML)modulation classifiers along with other methods and non-coherent statistical methods which do not require a prioriinformation, in general. Coherent techniques are based on theassumption that certain characteristics of the OFDM signalsare known to the receiver. Statistical methods mostly dependon high order statistics (HOS) of cumulant or moment esti-mates of the received signals. The background of cumulantbased modulation classification laid down by [2] for fourthand sixth order cumulants. Later, fourth order cumulants areadapted to OFDM modulation identification [3]-[5]. Moreover,fourth order moments of OFDM signals are combined withthe amplitude estimation to improve signal detection in [6].On the other hand, support vector machines in combination

2

with specific fourth order cumulants are used for the samepurpose in [7]. Tabular values of high order moments whichwere implemented for SC modulation identification in [8] areadapted to a minimum mean-square error (MMSE) estimatorfor OFDM signals in [9]. A low complexity single carriermodulation identification algorithm based on fourth ordercumulants for multiple-input and multiple-output systemsisalso proposed in [10].

The ML modulation classification algorithms are knownwith better estimation results when compared to statisticalmethods, however they suffer from computational complexityand require channel estimation [11]. On the other hand, boot-strap technique is introduced as a pattern recognition basedmethod in [12]. This method conducts SC modulation separa-tion along with the SC/MC distinction. A wavelet transformalgorithm is proposed to extract transient characteristics ofOFDM signals in [13]. Moreover, time-frequency representa-tion of the wireless signals is employed to distinguish OFDMsignals from SC frequency-shift keying (FSK) and phase-shiftkeying (PSK) signals in [14]. OFDM signals have uniquesecond order cyclostationary features and a feature extractionalgorithm based on the magnitude of the cyclic cumulantsof OFDM signals is implemented in [15]. Furthermore, sec-ond order cyclostationary features of block transmitted-singlecarrier linearly digitally (BT-SCLD) modulated signals areutilized for distinction from OFDM signals and for parameterestimation in [16]. Another second order cyclostationary algo-rithm to detect and classify both worldwide interoperability formicrowave access (WiMAX) and long term evolution (LTE)signals is developed in the context of cognitive radio systemsand spectrum awareness concept in [17]. An analysis andcomparison of second order cyclostationary, matched filter,MMSE, and normalized kurtosis methods can be found in [18]and for a general overview of coherent techniques, [19] canbe consulted.

OFDM signals also exhibit time-domain Gaussianity. This isdue to fact that the assignment of random data over orthogonalsub-carriers in a simultaneous way can be assumed as a com-position of large number of independent, identically distributed(i.i.d.) random variables and central limit theorem impliesGaussian distribution for large enough data sets. A Gaussianitytest based on empirical distribution function is introduced forOFDM systems in [20]. In addition to that, a general chi-squared constant false alarm rate (CFAR) Gaussianity testbased on the estimates of the third and fourth order cumulants[21] is adapted to OFDM signals without considering thechannel effects in [22].

B. Proposed Method

When the coherent methods are taken into account, thereare two main drawbacks when compared to the statisticalmethods: first, the dependence on the specific signal parame-ters such as symbol duration, cyclic-prefix size, active carriernumbers for the identification of each signal type and standard.Second, the algorithm complexity can increase the softwareand/or hardware complexity thus complicate the feasibility ofimplementation of the method. Moreover, complexity issues

can lead to extension of identification duration. On the otherhand, it is known that statistical methods are simpler thanthe coherent methods however they are highly susceptible tothe changes in the wireless channel conditions along withother impairments such as frequency offset, phase offsetdue to sampling mismatch and other factors [23]. Moreover,the statistical tests in the literature depend on either fixedthresholds or empirical values of various HOS. However HOScan significantly fluctuate under wireless channel conditionsand the identification performances of these tests degradesignificantly.

It is known that considering all combinations of possiblelags is a computationally consuming process for cumulantbased statistical classification methods. However peculiar tothe OFDM signals, inclusion of certain lags in the analysis issufficient to retain Gaussianity. Therefore in this study, firstthe fourth order cumulant equations for the OFDM signalsare provided for limited number and combination of lags andlag distances. Second, in [23] a general analysis of the effectsof wireless impairments over the fourth order cumulants ofOFDM signals were conducted. Furthermore in this paper, thefourth order cumulant estimates including the wireless effectsare provided. These equations imply fundamental impact ofimpairments over the cumulant values. Third, an OFDM signalidentification method which is based on a modified chi-squaredGaussianity test is proposed to distinguish MC OFDM signalsfrom SC signals. The method employs the estimates of thefourth order cumulants and their covariance matrix in thedecision process. Therefore, a certain level of protectionisachieved against the effects of multipath fading channels, whencompared to the statistical methods which are based on onlythe estimates of cumulants or moments of wireless signals.Moreover as detailed in Section III, instead of fixed numericalvalues, the proposed identification method utilizes the degreeof freedom of the underlying chi-squared test which providesrelative isolation from the effects of wireless impairments.

Fourth, the proposed method includes the computation ofthe estimate of the covariance matrix of fourth order cumu-lants. In general, this is a complex and detailed operationbecause the combinations of products of the second andfourth order moments are included in the estimation process.However, in this paper the estimation process is reduced downto three particular equations peculiar to the OFDM signals.The only parameter required is the symbol duration whichdetermine the degree of freedom of the modified chi-squaredtest. The numerical values that satisfy the confidence levelsfor correct detection can be found iteratively thus the testcanbecome fully blind. Fifth, a measurement setup is developedand the proposed method is implemented to achieve parametricperformance analysis in a realistic way. A signal generationsoftware is developed and integrated to the signal generatorto enable parametric flexibility during the signal transmissionprocess. Such an approach led to detailed parametric analysisof the proposed method based on modulation order, signalto noise ratio (SNR), number of symbols, and degree offreedom of the underlying test against FSK, PSK, and quadra-ture amplitude modulation (QAM) SC signals. The detectionperformance of the proposed method is also compared with

3

three statistical methods which are based on HOS and acoherent identification method which employs second ordercyclostationary features of OFDM signals for identification.

The OFDM signal model which is used at the imple-mentation of the proposed method is defined in Section II.The fourth order cumulant expressions for OFDM signals,discussion on the effects of wireless impairments on thecumulants, the modified chi-squared test which is employedby the proposed method, the decision mechanism, and the flowof the identification method are provided in Section III. Theestimates of the covariances of the cumulants are derived inthe Appendix. The measurement setup is detailed in SectionIV. Parametric performance analysis of the proposed signalidentification method based on the measurements is given inSection V. Finally, conclusion constituted the Section VI.

II. Signal Model

Inverse discrete Fourier transform (IDFT) is employed byOFDM systems for transmission. The baseband continuous-time OFDM signal is given by

xn(t) =K

∑

k=1

S n(k)e j 2πα(k)tTD − TG ≤ t < TD, (1)

where j at the exponential term is the imaginary unit,α isthe set of K subcarriers in total,S n(k) is the nth OFDM datasymbol which is transmitted at thekth subcarrier from theset of α(k), TD is the useful data duration,TG is the cyclicprefix (CP) duration, and the total OFDM symbol duration isTS = TD + TG. The transmitted total signal can be written as

x(t) =∞∑

n=−∞xn(t − nTS ). (2)

The signal is modulated after digital analog (D/A) conver-sion and passed through mobile radio channel. The channelcan be modeled as a time-variant linear filter

h(t) =L

∑

m=1

hm(t)δ(t − τm), (3)

where L is the number of taps andτi is the excess delayfor each tap. It is assumed that the taps are sample spacedand the channel is constant for a symbol but time-varyingacross multiple OFDM symbols. Therefore, baseband modelof the received signal after down conversion also consideringthe channel model given in eqn. (3) can be given by

y(t) = e jθte j2πξtL

∑

m=1

x(t − τm)hm(t) + n(t), (4)

where ξ is the carrier frequency offset due to inaccuratefrequency synchronization,θt is the phase offset, andn(t)corresponds to additive white Gaussian noise (AWGN) samplewith zero mean and variance ofσ2

n. The received signal issampled with sampling time of∆t at the analog to digitalconverter (ADC) and discrete-time received signal can berepresented by

y(i) = e j∆θie j2π∆ f iL

∑

m=1

x(i∆t − τm)hm(i∆t) + n(i), (5)

where i = 1, . . . ,N, ∆θi and ∆ f are normalized phase andcarrier frequency offsets. Models of FSK, PSK, and QAMmodulated SC signals are not listed here, but can be foundin [3] and references therein. Finally, at the initial stageofthe identification process, it is assumed that the receiver isfocused on a single channel which may be utilized by aSC or a MC OFDM signal comprised of sub-carriers, thebandwidth and carrier frequency are estimated, the signal isfiltered, down-converted and sampled beforehand. It is alsoassumed that in case of signals that are overlapping on wirelessspectrum, before the initialization of the identification process,the proposed method can benefit from the signal separationprocesses introduced in [24].

III. Proposed Identification Method

The proposed method benefits from the high order cumu-lants of the received signals for signal identification. Thecumulants of a process are defined as the generalization ofthe autocorrelation functionEy(i)y(i + i1). Thus, the generalformulation of third and fourth order cumulants for the sta-tionary random processes with zero mean is given by

c3y(i1, i2) , Ey(i)y(i + i1)y(i + i2) (6)

and

c4y(i1, i2, i3) , Ey(i)y(i + i1)y(i + i2)y(i + i3)− c2y(i1)c2y(i2 − i3) − c2y(i2)c2y(i3 − i1)

− c2y(i3)c2y(i2 − i1), (7)

wherec2y(i1) , Ey(i)y(i+ i1). The third and fourth order cu-mulants have different characteristics; thekth order cumulantswill vanish for k > 3 if y(i) is Gaussian [25]. However, thirdorder cumulants can converge to zero althoughy(i) is non-Gaussian but symmetrically distributed [26]. Therefore, thirdorder cumulants will be ignored and fourth order cumulantswill be analyzed for OFDM signal identification. On the otherhand, when the calculation of the estimates of the cumulantsare considered, the estimated cumulants, ˆcky, should be abso-lutely summable [27] and the signal model for wireless OFDMsignals which is given in eqn. (5) satisfies this condition.Under these conditions, the estimate autocorrelation becomesc2y(i1) = 1

N

∑N−i1i=1 y(i)y(i+ i1), and 4th-order cumulant estimate

is given by

c4y(i1, i2, i3) ,1N

N−i1∑

i=1

y(i)y(i + i1)y(i + i2)y(i + i3)

− c2y(i1)c2y(i2 − i3) − c2y(i2)c2y(i3 − i1)

− c2y(i3)c2y(i2 − i1), (i1, i2, i3) ∈ IN4 . (8)

The region of all combinations of all lags is given byI∞k , 0 ≤ ik−1 ≤ · · · ≤ i1 ≤ ∞. However, it is shown in [28]that cumulant lags that comprise distribution characterizationof the series are finite and it’s not required to estimate thecumulant values for the majority of the possible lag combina-tions. Therefore, the region of the lags that should be takeninto consideration for the analysis is given by

IN4 = 0 ≤ i3 ≤ i2 ≤ i1 ≤ N, (9)

4

and the sample estimates in eqn. (8) should only be calculatedover IN

4 as defined in eqn. (9). The Gaussianity test will uselags of c4y that can be collected into anNc × 1 vector. Thisvector constitutes a 3 dimensional triangular region and itslength is given byNc = N(N + 1)(N + 2)/6.

A. Fourth Order Cumulants for OFDM Signals

When the wireless signals are considered, the length of thedata can be orders of thousands or tens of thousands of sam-ples depending on the sampling rate and the recording time.Defining theIN

4 lag region and computing the components ofvector c4y can become impractical in many cases.

The data set can be divided into smaller sections andprocessed in parallel or the data set can be shortened. However,even though the choice ofN and consequentlyNc is appli-cation dependent, in practice when a strong non-Gaussianity(e.g., due to the multipath fading channel) present in shortrecords i.e, N < 200, this will weaken the Gaussianityassumption and will lead to unreliability for the signal identi-fication methods. Moreover, even for the shortest data sets,the size of the ˆc4y vector can become very large and thecomputation of the cumulants can be impractical. However,[29] and [22] showed that, first, shrinking the lag regioninto IM

4 = i3 = 0 ≤ i2 = i1 ≤ M does not lead toloss of significant distribution information for communicationssignals. The terms that are left over have very small influenceon the features of the modulation type of the signal whencompared to the set in the hand. Second, the lag distancecan also be limited withM ≈ 1.5T s whereT s is the symbolduration for the signal under test. Therefore the cumulantsshould be evaluated only for the set of lags of (λ, λ, 0) wherei1 = i2 = λ ∈ [0, 1.5T s]. Under these conditions eqn. (8) canbe written as

c4y(λ, λ, 0) ,1N

N−λ∑

i=1

y(i)y(i + λ)y(i + λ)y(i)

− c2y(λ)c2y(λ) − c2y(λ)c2y(−λ)− c2y(0)c2y(0), (i1, i2, i3) ∈ IM

4 , (10)

and the open form becomes

c4y(λ, λ, 0) =

1N

N−λ∑

i=1

y2(i)y2(i + λ) −[

1N

N−λ∑

i=1

y(i)y(i + λ)

]2

(11.1)

−[

1N

N−λ∑

i=1

y(i)y(i + λ)

][

1N

N∑

i=λ+1

y(i)y(i − λ)]

(11.2)

−[

1N

N∑

i=1

y2(i)

]2

. (11.3)

(11)

When the signal model in the eqn. (5) is replaced into theeqn. (11.1), the statements that are given in the eqn. (16) isachieved. Instead of expanding whole eqn. (5), eqn. (16) issufficient for the analysis at the next step. The results canthen be generalized for the rest of the eqn. (5). Note thatthe noise componentn(i) is independent from the rest of the

terms that constitute the received signal and vanishes dueto its Gaussianity. Therefore any term that included noisecomponent is eliminated from the expressions given in eqn.(16). When further mathematical operations on eqn. (16)is sought, the HOS theory [26] indicates that ifηu, whereu = 1, . . . , k − 1, are constants then

cky(η1i1, . . . , ηk−1ik−1) =

( k−1∏

u=1

ηu

)

cky(i1, . . . , ik−1), (12)

and based on this property of cumulants, for the linear andtime-invariant systems it can be written that

y(i) =∞∑

j=−∞x(i − j)h( j) + n(i),

∞∑

i=−∞|h(i)| < ∞ (13)

andk-th-order cumulants becomes [27]

cky(i1, i2, i3, . . . , ik−1) =

ckx(0, . . . , 0)∞∑

i=−∞h(i)h(i + i1) . . . h(i + ik−1). (14)

Assuming that thex(i) is i.i.d., in case of non-Gaussian sig-nals, zero-lag fourth order cumulants will converge to the finitemoments of ˆc4x(0, 0, 0), 0 and in case of Gaussian distributedsignals ˆc4x(0, 0, 0) = 0. Thus ˆc4y(0, 0, 0) will also converge tozero theoretically. However, this is not the case for wirelesssignals because, the signal model given in Section II impliesthat wireless channel is time-variant across multiple symbols.Moreover, phase and frequency offset can not be assumedconstant over the period ofN symbols which can extend toa couple of hundreds as indicated in Section III-A. Thereforeit is not possible to shift the channel coefficients, phase andfrequency offset components out of the cumulant equations.The i.i.d. property will not perfectly hold because of theseadditional components introduced to each sample of the signalwhen compared to eqn.(1). Hence the high order cumulants ofwireless signals will not only be determined by the transmittedsignal, but also will depend on the communications mediumcharacteristics. Furthermore, the probabilistic convergence ofestimated cumulants ˆcky to the cky can be possible wheny(i)samples are independent and well separated in time. This iscalled mixing conditions. However, the autocorrelation ˆc2y(i1)is estimated using sample averaging because instead of theseriesy(t), the samples of the original seriesy(i) are employedin the process. Moreover, there is no synchronization processinvolved in the sampling stage. Therefore an irreducible erroris introduced to the estimation of the cumulants. Under theseconditions ˆc4y will not converge perfectly to zero and can takedifferent numerical values based on the channel conditions andcommunications medium. On the other hand, when the fourthorder cumulants are written in the form of

c4y(i1, i2, i3) = Ey(i)y(i + i1)y(i + i2)y(i + i3)− Eg(i)g(i + i1)g(i + i2)g(i + i3), (15)

whereg(i) is a Gaussian random process, it can be inferredfrom the eqn. (15) that fourth order cumulants provide ameasure of distance of the distribution of the signal samplesform Gaussianity. Considering the fact that the transmitted SC

5

1N

N−λ∑

i=1

y2(i)y2(i + λ) −[

1N

N−λ∑

i=1

y(i)y(i + λ)

]2

=1N

N−λ∑

i=1

e2 j(∆θi+∆θi+λ)e j2π∆ f i

( L∑

m=1

x(i∆t − τm)hm(i∆t)

)2( L∑

m=1

x((i + λ)∆t − τm)hm((i + λ)∆t)

)2

−[

1N

N−λ∑

i=1

e2 j(∆θi+∆θi+λ)e j2π∆ f i(i+λ)L

∑

m=1

x(i∆t − τm)hm(i∆t)L

∑

m=1

x((i + λ)∆t − τm)hm((i + λ)∆t)

]2

(16)

signals will also be affected from same propagation character-istics and consequently their estimates of HOS will change,it is still plausible to assume that estimates of the fourthorder cumulants of OFDM signals will converge toclosernumerical values to zero when compared to SC carrier signals.However, under wireless channel conditions and propagationcharacteristics, the statistical identification methods whichare based on fixed thresholds or empirical numerical valuessuch as [4], [6], [8], [9] will fail to identify OFDM signalsconsistently. To this end, a new approach is required to theidentification of OFDM signals for wireless communicationssystems.

B. The Gaussianity Test and Decision Mechanism

It is known from [30] that asymptotic Gaussianity of thesummable cumulants leads to

√N(c4y − c4y)

dist.∼N→∞

Dr(0,Σc) (17)

where Dr denotes real Gaussian distribution,c4y is the 4th-order cumulant vector which holds the theoretical cumulantvaluesc4y , limN→∞ Ec4y, andΣc is the asymptotic covari-ance matrix ofc4y which has the close form of

Σc , limN→∞

NE(c4y − c4y)(c4y − c4y)′. (18)

where′ denotes the transpose operation. Therefore, the time-domain Gaussianity test can be formulated as a binary hypoth-esis problem of

H0 : c4y ∽ Dr(0,N−1Σc)

vs.

H1 : c4y ∽ Dr(c4y,N−1Σc),∽ c4y , 0. (19)

To be able to solve the problem, in this study, thedG,4 testwhich was proposed in [21] is adopted to the peculiar case ofOFDM signal identification under multipath wireless channels.Therefore the binary hypothesis problem which is given in eqn.(19) can be addressed by a chi-square test called asdG,4 whichis defined as

dG,4 , Nc4yΣ−1c c4y, (20)

and from [21] it can be seen that

dG,4 = Nc′

4yΣ−1c c4y

dist.∽

N→∞χ2

Nc. (21)

The pseudo-inverse ofΣ−1 should replace inverse in theequations below ifΣ rank-deficient. Therefore, for anα levelsignificance the test in eqn. (19) becomes a chi-squared testwhich can be defined by

dG,4H1

≷H0

tG = χ2Nc

(α). (22)

tG is χ2 disribution table value with degree of freedom ofα.The probability of the false alarm is given by

PF , α ≤ P[dG,4 ≥ χ2Nc|H0] (23)

UnderH1, the distribution ofdG,4 is estimated from eqn. (19)as

dG,4 ∼ Dr[Nc′

4yΣ−1c c4y, 4Nc

′

4yΣ−1c c4y] (24)

and the probability of detection is given by

PD , α ≤ P[dG,4 ≥ tG|H1]. (25)

The fourth order cumulant and covariance matrix estima-tion processes derived herein is utilized by the introducedasymptotic chi-squared CFAR test. Thus the effects of fadingchannels which can lead to stronger non-Gaussian compo-nents are minimized by the involvement of covariances ofthe cumulants. Selection of the threshold is conducted byemploying theχ2 test tables based on the degree of freedomof the distribution. On the other hand, fourth order cumulantcovariance matrix estimate computation is a long and complexprocedure which requires involvement of products of the crossmoment terms. Moreover, computation of theΣc can becomeimpractical when ˆc4y vector becomes very large. However,peculiar to the OFDM signals, under the conditions defined inSection III-A, covariance matrix estimate calculation becomesa simpler process. The general form of covariance estimatesof the cumulants for OFDM signals is derived based on thecomputations in [27] (eqn. 2.3.8) and [21] and reduced to threeequations which are consist of several moments of the receivedsignal. The computation process can be found in the Appendix.Furthermore, the test is originally applied to real processes andin this paper the same approach is maintained because thedistributional characteristics of the complex signals arealsoretained by the imaginary and real parts. Besides, operating inreal or imaginary domain reduces the computation time for thetest output when compared to complex domain. Therefore, theanalysis for wireless OFDM signals can be given over eitherof these domains and the analysis can be extended to com-plex domain [22]. The proposed OFDM signal identificationmethod can be executed as follows:

• calculate the estimates of the fourth order cumulants ofthe received signal based on given degree of freedom, i.e.,M, using equation (11),

• calculate theΣ−1c of the fourth order cumulants based on

the equations given in Appendix,• calculate the chi-squared test result from the equation

(20), using the estimates computed in the previous twosteps, and

6

Parameter ValueCarrier frequency 2140 MHzTx/Rx distance 18.24 feetModulation type SC-PSK, SC-FSK, SC-QAM, OFDM-QAMModulation order 2,4,8,16,32,64,128,256Number of symbols 64,128,256,512,1024,2048SNR level -4 to 15 dBSymbol rate 250 kHzFilter type Root raised cosine,α = 0,3Frequency Offset (∆ f ) 10 Hz (max.)

TABLE IMeasurement setup: the values of the used parameters

• compare the test result with the tabular value of the testwith the given degree of freedom and make the decisionbased on equation (22).

The calculation of the estimates of the cumulants at thefirst step of the proposed method is achieved by a set of mul-tiplication, shifting, and summation operations for a limitednumber of lags. Moreover in the Appendix, the estimationof the covariance matrix in the second step of the proposedmethod is reduced down to calculation of a set of high-order moments which does not include any shifting operations.These mathematical operations can be implemented over mi-croprocessors or micro-controllers of the measurement devicesby the utilization of smart mathematical algorithms whichavoid calculation duplications. Therefore, the implementationof the proposed method can be greatly simplified. Furthermore,the construction of the decision process at the last two stepsconsist of very straightforward multiplication and comparisonoperations which do not introduce an additional complexityto implementation of the proposed method over the wirelesscommunications systems.

IV. M easurement Setup

A measurement setup is developed to investigate the identifi-cation performance of the proposed method. The measurementsystem consists of an Agilent ESG E4438C signal generator(Tx), an Agilent E4440A PSA series spectrum analyzer (Rx)and accompanying vector signal analyzer (VSA) software torecord the signals, a tuple of omni-directional antennas tocover the wireless communications bands of interest, a laptopcomputer, cables, and connectors. All the measurement pro-cess is controlled by the laptop computer. The signal generatorand the spectrum analyzer are connected to the network routerand the laptop computer commands both devices throughthe transmission control protocol and internet protocol. Thesignal transmission and the reception starts and ends withthe commands that are sent by the laptop computer. Thesetup is depicted in Fig. 1(a). Moreover, the signal generatorsupports the generation and transmission of waveforms thatare designed separate from the generator. Therefore, a signalgeneration software is developed in MATLAB to be able toobtain the full control of the data generation and transmissionprocedures.

The developed software generates the SC-PSK, SC-FSK,SC-QAM, and OFDM modulated signals with various mod-ulation orders based on the signal model in Section II. The

OFDM signals employ QAM as the underlaying modulationtechnique over 64 active carriers with 250 kHz spacing.TG

is also selected as 1/4TD. The signal generation softwaregenerates the designed waveforms at the laptop and uses ven-dor provided library functions to access the signal generator.As the software accesses the generator, it uploads generatedwaveforms to the device and commands the signal generatorto transmit the predefined number of symbols of the waveformat the defined frequency band, with the predefined signalpower. Moreover, signal parameters such as modulation typeand order can also be adjusted by the software. Thereforethe parametric analysis which is based on the measurementparameters such as SNR, modulation type, data length, andmodulation order can be done methodologically. On the otherhand, the VSA software provided by the vendor is employedfor signal acquisition and recording purposes. SNR of thereceived signal is also determined by this software. Therefore,all the signal transmission loop is controlled and completedon the laptop computer while the transmitter and the receiveris allocated to different locations in the laboratory. The list ofthe all parameters used in the measurement process is givenin TABLE I. The measurements are repeated for 200 timesfor each changing parameter by the aid of signal generationsoftware.

The measurement environment also has an important effectover the performance of the proposed method. TheTx/Rx

devices are fixed to certain allocations in the measurementlaboratory which has scattering surfaces such as metal panels,other measurement and calibration equipment, tables, chairsetc., as shown in Fig. 1(b). Therefore, the measurement envi-ronment is a highly reflective indoor environment. It shouldbenoted that the performance of any signal identification methodwill be affected under these conditions, because the wirelesschannel is defined by the heavily reflective nature of the indoorenvironment. These circumstances can be assumed to be thebottom line performance conditions and it can also be assumedthat the identification performance of the proposed method willincrease as the system is deployed outdoors with relativelylessreflections.

V. Measurement Results

A CFAR detector is implemented based on the proposedmethod and the equations (20)-(25) in Section III-B forPF of0.001, 0.01, 0.1 and 200 measurements. On the other hand, afrequency shift of∆ f = 10 Hz is added to the signal generationprocess to imitate the Doppler shift and other impairments thatcause the shift in the frequency. In Fig. 2, 3, and 4 thePD

results for OFDM vs. SC signals are given for 1024 symbols,modulation order of 32,M = 1.5TS , and various SNR levels.The results indicate that best separation is achieved againstSC-FSK, SC-PSK and SC-QAM consecutively. Especially forPF of 0.1, and 0.01, a certain confidence level is achieved at2 dB of SNR for SC-FSK signals, at 6 dB for SC-PSK, andat 9 dB for SC-QAM.

The detection performance of the proposed method fallssharply in a range of couple of dBs at the end of positiveSNR region for all signals and identification becomes impos-sible when the SNR is negative. The measurement results

7

(a) Measurement setup: transmitter, receiver, laptop com-puter, router and their connections.

(b) A general overview of the measurement environment.

Fig. 1. The measurement setup and environment.

−4 −3 −2 −1 0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR [dB]

PD

PF = 0.1

PF = 0.01

PF = 0.001

Fig. 2. Probability of detection for SC-FSK v.s. OFDM:N = 1024,modulation order= 32, M = 1.5TS .

−4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR [dB]

PD

PF = 0.1

PF = 0.01

PF = 0.001

Fig. 3. Probability of detection for SC-PSK v.s. OFDM:N = 1024,modulation order= 32, M = 1.5TS .

indicate that the noise dominates the test statistics in thisregion, therefore identification performance deteriorates. SC-QAM follows other modulation types in terms of performancein all measurements because the underlying modulation that

−4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR [dB]

PD

PF = 0.1

PF = 0.01

PF = 0.001

Fig. 4. Probability of detection for SC-QAM v.s. OFDM:N = 1024,modulation order= 32, M = 1.5TS .

is employed by the OFDM signal is also QAM. For themodulation order of 32, 1024 symbols, andPF = 0.1, detectionperformance above 0.9 is achieved at 0 dB of SNR for OFDMvs. SC-FSK, at 2 dB for OFDM vs. SC-PSK, and at 3 dB forOFDM vs. SC-QAM.

The measurement results are compared with the simulationresults which are based on [23] for each signal type andPF = 0.01 in Fig. 5, 6, and 7. Except for a set of negative SNRvalues, in general the comparison plots indicate that simulationresults are slightly better and in some cases -especially inthe initial positive SNR region, an increase of 0.1 at thePD

values can be observed for all signals. This is the indicationof the significant effect of measurement environment depictedin Section IV over the wireless channel and consequentlyon the performance of the proposed method in contrary tothe four taps simulated channel implemented in [23]. On theother hand, transmitter and the receiver are directly connectedwith a cable and the measurement process is synchronized bythe available software libraries in both of the devices. Theseconditions imply only the effect of AWGN is introduced tothe received signal during the transmission, thus the plotsthat are given in the same figures with the label of AWGN

8

−4 −3 −2 −1 0 1 2 3 4 5 6 7 8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

PD

measurementsimulationAWGN

Fig. 5. PD performance comparison for SC-FSK v.s. OFDM: AWGN channel,simulations, measurements. (PF = 0.01), N = 1024, modulation order= 32,M = 1.5TS

−4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

PD

measurementsimulationAWGN

Fig. 6. PD performance comparison for SC-PSK v.s. OFDM: AWGN channel,simulations, measurements. (PF = 0.01), N = 1024, modulation order= 32,M = 1.5TS

can be considered as the upper bound performance limits forthe proposed signal identification method. These measurementindicate PD above 0.9 for 0 dB of SNR in case of FSK vs.OFDM, 1 dB in case of PSK vs. OFDM, and 5 dB in case ofQAM vs. OFDM.

The PD performance of the proposed method is also in-vestigated by employing QPSK modulated SC wideband codedivision multiple access (WCDMA) and OFDM based LTEsignals. The transmission bandwidth for WCDMA is 5 MHzand LTE system is also configured to 5 MHz transmissionbandwidth with 15 KHz spaced carriers and fast Fourier trans-form size of 512. The signals are generated from the wirelessstandards based libraries available in the Agilent ESG E4438Csignal generator. The rest of the measurement parameters arekept same as given in TABLE I. The measurement results showsimilarity with that of generic SC-PSK measurements however,PD leads to certain identification at higher SNR values whencompared to the generic case.

The effect of the change of modulation order on the test

−4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

PD

measurementsimulationAWGN

Fig. 7. PD performance comparison for SC-QAM v.s. OFDM: AWGNchannel, simulations, measurements. (PF = 0.01), N = 1024, modulationorder= 32, M = 1.5TS

−4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR [dB]

PD

PF = 0.1

PF = 0.01

PF = 0.001

Fig. 8. Probability of detection for WCDMA v.s. LTE:M = 1.5TS .

statistics is investigated in Fig. 9 when the SNR is 7 dB,PF = 0.1, and 1024 symbols are recorded. While SC-FSKseparation barely depends on the modulation order, as theorder increase above 8, the proposed method performs betterfor SC-PSK steadily, andPD reaches to the levels above 0.95.In case of SC-QAM signals, the best performance is achievedas the modulation order reaches to 32 withPD > 0.96.The measurement results indicate no significant effect ofmodulation order over identification performance.

Data length or number of symbols used is also anotherimportant parameter which affects the performance of thestatistical tests. Fig. 10 displays the detection performance ofthe proposed method with the changing number of symbolsfor 7 dB SNR,PF = 0.1, and modulation order of 32 for eachsignal type. Below 128 symbols the proposed method performspoorly due to the problems discussed in Section III. Whilethe test performs well for FSK signals over 256 symbols withPD > 0.95, for all SC signals the test performance is optimizedas the number of symbols reaches 512. The degradation in thetest performance after 1024 symbols is due to central limit

9

2 4 8 16 32 64 128 2560.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Modulation Order

PD

OFDM vs. QAMOFDM vs. PSKOFDM vs. FSK

Fig. 9. Effect of modulation order on the identification: (PF = 0.1), SNR=7 dB, N = 1024, M = 1.5TS .

64 128 256 512 1024 20480.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Number of Symbols

PD

OFDM vs. QAMOFDM vs. PSKOFDM vs. FSK

Fig. 10. Effect of number of symbols on the identification: (PF = 0.1), SNR= 7 dB, modulation order= 32. M = 1.5TS .

theorem.The lag distance which also determine the degree of free-

dom of the test has an important effect on the identificationperformance of the proposed method. More information aboutthe distribution of the received signal is introduced to thedecision process as the lag distance increases. However, thecomputation time required to calculate the estimates of thecumulants will also increase as the lag distance increases.The detection improvements for increased lag distance interms of symbol duration is given in Fig. 11 forN = 1024,SNR = 7 dB, modulation order of 32 andPF = 0.01. Theoversampling rate is double the Nyquist rate, therefore, forinstance,M = 0.5TS corresponds to 2 samples. A significantgain of 0.4 in the detection performance can be noticed incase of QAM signals as the lag distance increase from 0.5to 2.5, however computation time requirements also increasesignificantly. For other signal types thePD performances aresteadyi.e., above 0.9.

Finally the identification performance of the proposedmethod is compared with three of the non-coherent statisticalmethods and a coherent method that are available in the

0.5 1 1.5 2 2.5

0.4

0.5

0.6

0.7

0.8

0.9

1

Lag Distance (M) [Symbol duration (TS)]

PD

OFDM vs. QAMOFDM vs. PSKOFDM vs. FSK

Fig. 11. Effect of lag distance (degree of freedom) on the identification:(PF = 0.01), N = 1024, SNR= 7 dB, modulation order= 32.

literature. One of these methods, which is introduced in [8]depends on the fourth and the sixth orders of moments ofthe OFDM signals and introduces an MMSE algorithm basedon these moment functions (MMSE-M). Secondly in [6], amethod based on the combination of fourth order momentsand the amplitude estimation is introduced (AE-M) and third,sums of fourth order cumulants are used for OFDM v.s. SCclassification in [4] (SUM-C). The coherent method employsa two stage identification procedure to distinguish SLCD,BT-SLCD, and OFDM signals from each other based onthe unique second order cyclostationary features of thesesignals. First the SLCD signals are distinguished from theBT-SLCD/OFDM pair, then these two signals are identifiedin the second stage [16] (BT-SLCD), however the first stageof the signal classification algorithm is investigated for thecomparison purposes assuming that the there are two possiblesignal types. The performance comparison with these methodscan be found in Fig. 12 for the SC-PSK case. Modulation orderis selected as 32,PF = 0.01, andN = 1204.

The proposed method performs better than the non-coherentmethods because of the underlying test structure which doesnot depend on fixed thresholds and utilization of the estimatesof the covariances of the cumulants in the identification pro-cess. At 1 dB SNR region, the proposed method outperformsother methods with thePD margin of 0.2. For the 5 dB ofSNR, the margin is 0.15 with SUM-C, 0.08 with AE-M, and0.05 with MMSE-M. As the SNR reaches to 10 dB, thetests converge to closerPD rates. On the other hand, whilethe coherent BT-SLCD algorithm outperforms the proposedmethod on the low-SNR region, the identification performanceof the proposed method converges to the BT-SLCD aroundthe 8 dB SNR region. These results are expected when itis considered that in general coherent techniques have betterperformance than the statistical test, trading a certain level ofincrease in the algorithm complexity and availability of certaininformation to the receiver beforehand with improvements interms identification performance. The performance compari-son for SC-FSK and SC-QAM signals are not included, butsimilar results are observed.

10

0 1 2 3 4 5 6 7 8 9 10

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR [dB]

PD

SUM−CAE−MMMSE−M

Proposed (χ2)BT−SCLD

Fig. 12. Performance comparison with other algorithms: SC-PSK v.s. OFDM,(PF = 0.01), N = 1024, modulation order= 32, M = 1.5TS .

VI. Conclusion

Signal identification is a vital component of modern wirelesscommunications systems. In this paper first, fourth ordercumulant estimates of wireless OFDM signals are given. Sec-ondly, it is shown that wireless impairments have significantimpacts on the distribution characteristics of wireless signals.Then, an OFDM signal identification method which employsestimates of fourth order cumulants and their covariances forlimited number of lags is proposed. The cumulant and the co-variance matrix calculations are greatly simplified particularlyfor OFDM signals. The method utilizes a modifiedχ2 test atthe decision process and symbol duration information is usedto determine the number of cumulant legs or the degree offreedom of the underlying Gaussianity test. In practice, degreeof freedom which isM > 10 can satisfy certain confidencelevels, thus the requirement for the symbol duration can alsobe waived. A measurement setup is developed to validatethe performance of the proposed method realistically. Theidentification performance of the method is provided utilizingparameters such as modulation order and type, data length,and degree of freedom of theχ2 test. A CFAR detector isused in the performance analysis. The method performs betterthan other statistical identification methods under the wirelessmultipath fading channel conditions and it is robust againstthe changing modulation order and data length. However theidentification performance is affected under low-SNR, becauseof the time-domain Gaussianity of the OFDM signals.PD ofmore than 0.9 is achieved for all signal types whenPF = 0.1.

Appendix

The asymptotic covariances of sample cumulants can beestimated based on the straightforward asymptotic covarianceexpression for the cumulant orders up tok = 3 [28]. However,the covariance estimation becomes a complex procedure fork > 3 due to the combinations of products of the secondand fourth order moments. The estimate of the 4-th order

cumulants can be written in terms of moments as [21]

c4y(i1, i2, i3) = m4y(i1, i2, i3) − m2y(i1)m2y(i2 − i3)

− m2y(i2)m2y(i3 − i1) − m2y(i3)m2y(i1 − i2)(A.1)

where

mky(ik) ,1N

T−1∑

t=0

y(t)y(t + i1)y(t + i2) . . . y(t + ik−1), (A.2)

and ik , (i1, i2, . . . , ik−1). The computation of each entrycovc4y(i1, i2, i3), c4y( j1, j2, j3) of the Σc becomes a compu-tationally consuming process underIN

4 . However the compu-tation of the diagonal entries defined byIM

4 is sufficient forOFDM signals. Ifλ = i1 = i2, β = j1 = j2, i3 = j3 = 0 and0 ≤ β ≤ M, the open for of covariance of ˆc4y can be given inthree parts and the first part is consists of covariances of twofourth order moments as

c1 = covm4y(λ, λ, 0), m4y(β, β, 0), (A.3a)

and the second part will be consist of various combinations ofcovariances of fourth order and second order moments. Theopen expression of these terms are given by

c2 = covm4y(λ, λ, 0), m2y(β)m2y(β) (A.3b1)

+ covm4y(λ, λ, 0), m2y(β)m2y(−β) (A.3b2)

+ covm4y(λ, λ, 0), m2y(0)m2y(0) (A.3b3)

+ covm4y(β, β, 0), m2y(λ)m2y(λ) (A.3b4)

+ covm4y(β, β, 0), m2y(λ)m2y(−λ) (A.3b5)

+ covm4y(β, β, 0), m2y(0)m2y(0), (A.3b6)

and finally the third term will be constituted by the covari-ances of all possible second order moments of the estimatedcumulants as

c3 = covm2y(λ)m2y(λ), m2y(β)m2y(β) (A.3c1)

+ covm2y(λ)m2y(λ), m2y(β)m2y(−β) (A.3c2)

+ covm2y(λ)m2y(λ), m2y(0)m2y(0) (A.3c3)

+ covm2y(λ)m2y(−λ), m2y(β)m2y(β) (A.3c4)

+ covm2y(λ)m2y(−λ), m2y(β)m2y(−β) (A.3c5)

+ covm2y(λ)m2y(−λ), m2y(0)m2y(0) (A.3c6)

+ covm2y(0)m2y(0), m2y(β)m2y(β) (A.3c7)

+ covm2y(0)m2y(0), m2y(β)m2y(−β) (A.3c8)

+ covm2y(0)m2y(0), m2y(0)m2y(0). (A.3c9)

Therefore the covariance of ˆc4y is given by

covc4y(λ, λ, 0), c4y(β, β, 0) = c1 + c2 + c3. (A.3d)

The derivation of these three groups can be based onthe computable sample approximations of the asymptoticcovariances established in [31], [32] for the mixing discrete-time stationary processes such as defined in eqn. (14). Thecovariance of the moments is defined as

limN→∞

Ncovmky(ik), mky( jl) =∞∑

−∞covy(0)y(i1) . . . y(ik−1),

y(τ)y(τ + j1) . . . y(τ + jl−1). (A.4)

11

Following covab = Eab − EaEb based on eqn. (A.4)and replacing the moments with their estimates as defined ineqn. (A.2), (A.3a) can be written as

c1 = covm4y(λ, λ, 0), m4y(β, β, 0)

≈ 1N

KN∑

τ=−KN

[

m8y(λ, λ, 0, τ, τ + β, τ + β, τ)

− m4y(λ, λ, 0)m4y(β, β, 0)]

(A.5)

Limits of the sum in the eqn. (A.5) should satisfyKN → ∞asN → ∞ andKN/N → 0 asN → ∞. In practice, for OFDMsystemsKN can be selected asKN > τ wherec2xm(τ) ≈ 0 [21].The second group of sample estimates include the covariancesof the multiplications of the moment terms and theorem 2.3.2in [27] expresses such terms in the form of

cova, bc = covbcova, c+ covccova, b (A.6)

and eqn. (A.3a) can be written as

covm4y(λ, λ, 0), m2y(β)m2y(β)= Em2y(β)covm4y(λ, λ, 0), m2y(β)

+Em2y(β)covm4y(λ, λ, 0), m2y(β)= 2

[

Em2y(β)covm4y(λ, λ, 0), m2y(β)]

.

(A.7)

Depending on the estimator given in eqn. (A.2) and covariancedefinition in eqn. (A.4), similar to eqn. (A.5), the covariancesin eqn. (A.7) can be expressed in terms of moments as

covm4y(λ, λ, 0), m2y(β)m2y(β)

= 2[

m2y(β)KN∑

τ=−KN

[m6y(λ, λ, 0, τ, τ + β) − m4y(λ, λ, 0)m2y(β)]]

.

(A.8)

When the same derivations are applied to the rest of theequations and after some simplifications (A.3a - A.3b6) canbe written as

c2 = 2(

Ω(

2m2y(β) + m2y(−β))

+ Φm2y(β) + 2Ψm2y(0))

, (A.9)

where

Ω =1N

KN∑

τ=−KN

[m6y(λ, λ, 0, τ, τ+β)−m4y(λ, λ, 0)m2y(β)], (A.10)

Φ =1N

KN∑

τ=−KN

[m6y(λ, λ, 0, τ, τ − β) − m4y(λ, λ, 0)m2y(−β)],

(A.11)and

Ψ =1N

KN∑

τ=−KN

[m6y(λ, λ, 0, τ, τ) − m4y(λ, λ, 0)m2y(0)]. (A.12)

The IM4 leads to symmetricΣc therefore for eachΣc entry not

all six terms of second groups should be processed, estimationof the first half would be sufficient and the following terms willbe replicated as indicated in eqn. (A.9). Similar to the secondgroup, covariances of multiplications of the estimate momentswhich consist the third group are in the form ofcovab, cd.

According to eqn.(2.3.8) in [27] these terms can be expressedas

covab, cd = cova, ccovb, d+ cova, dcovb, c, (A.13)

then the eqn. (A.3b6) is given by

covm2y(λ)m2y(λ), m2y(β)m2y(β)= covm2y(λ), m2y(β)covm2y(λ), m2y(β)+ covm2y(λ), m2y(β)covm2y(λ), m2y(β). (A.14)

After applying the conversion defined in the eqn. (A.13) andrearranging all nine terms, (A.3b6 - A.3c9) can be written as

c3 =2Γ2 + 2ΓΛ + 2Θ2 + 2Γcovm2y(λ), m2y(−β)+ ∆(Γ + Λ) + 2Θcovm2y(−λ), m2y(0) + 2Υ2

+ 2Υcovm2y(0), m2y(−β) + 2[

covm2y(0), m2y(0)]2,(A.15)

where

Γ =1N

KN∑

τ=−KN

[m4y(λ, τ, τ + β) − m2y(λ)m2y(β)], (A.16)

Λ =1N

KN∑

τ=−KN

[m4y(λ, τ, τ − β) − m2y(λ)m2y(−β)], (A.17)

Θ =1N

KN∑

τ=−KN

[m4y(λ, τ, τ) − m2y(λ)m2y(0)], (A.18)

∆ =1N

KN∑

τ=−KN

[m4y(−λ, τ, τ − β) − m2y(−λ)m2y(−β)], (A.19)

Υ =1N

KN∑

τ=−KN

[m4y(λ, τ, τ + β) − m2y(λ)m2y(β)], (A.20)

and

covm2y(−λ), m2y(β)

=1N

KN∑

τ=−KN

[m4y(−λ, τ, τ + β) − m2y(−λ)m2y(β)], (A.21)

covm2y(−λ), m2y(0) = 1N

KN∑

τ=−KN

[m4y(−λ, τ, τ)−m2y(−λ)m2y(0)],

(A.22)

covm2y(0), m2y(−β) =1N

KN∑

τ=−KN

[m4y(0, τ, τ−β)−m2y(0)m2y(−β)],

(A.23)

covm2y(0), m2y(0) = 1N

KN∑

τ=−KN

[m4y(0, τ, τ) − m2y(0)m2y(0)].

(A.24)Each entry ofΣc now can be computed as the sum of equations(A.5), (A.9), and (A.15) underIM

4 .

12

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