COGNITIVE RADIO (CR) represents a promising solution

Transcription

1 26 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 1, FEBRUARY 2012 Second-Order Cyclostationarity of Mobile WiMAX LTE OFDM Signals Application to Spectrum Awareness in Cognitive Radio Systems Ala a Al-Habashna, Octavia A. Dobre, Senior Member, IEEE, Ramachran Venkatesan, Senior Member, IEEE, Dimitrie C. Popescu, Senior Member, IEEE Abstract Spectrum sensing awareness are challenging requirements in cognitive radio (CR). To adequately adapt to the changing radio environment, it is necessary for the CR to detect the presence classify the on-the-air signals. The wireless industry has shown great interest in orthogonal frequency division multiplexing (OFDM) technology. Hence, classification of OFDM signals has been intensively researched recently. Generic signals have been mainly considered, there is a need to investigate OFDM stard signals, their specific discriminating features for classification. In this paper, realistic comprehensive mathematical models of the OFDM-based mobile Worldwide Interoperability for Microwave Access (WiMAX) third-generation Partnership Project Long Term Evolution (3GPP LTE) signals are developed, their second-order cyclostationarity is studied. Closed-from expressions for the cyclic autocorrelation function (CAF) cycle frequencies (CFs) of both signal types are derived, based on which an algorithm is proposed for their classification. The proposed algorithm does not require carrier, waveform, symbol timing recovery, is immune to phase, frequency, timing offsets. The classification performance of the algorithm is investigated versus signal-to-noise ratio (SNR), for diverse observation intervals channel conditions. In addition, the computational complexity is explored versus the signal type. Simulation results show the efficiency of the algorithm is terms of classification performance, the complexity study proves the real time applicability of the algorithm. Index Terms Long-term evolution (LTE), mobile Worldwide Interoperability for Microwave Access (WiMAX), orthogonal frequency division multiplexing (OFDM), signal classification, signal cyclostationarity. I. INTRODUCTION COGNITIVE RADIO (CR) represents a promising solution to the spectrum scarcity problem. There are many potential commercial military applications for CR. Using Manuscript received March 30, 2011; revised August 08, 2011 September 25, 2011; accepted October 14, Date of publication December 21, 2011; date of current version January 18, This work was supported in part by the Natural Sciences Engineering Research Council of Canada, was presented in part at the 2010 IEEE Global Communications Conference (Globecom). The associate editor coordinating the review of this manuscript approving it for publication was Prof. Philippe Ciblat. A. Al-Habashna was with the Faculty of Engineering Applied Science, Memorial University of Newfoundl, St. John s, NL A1B 3X5, Canada. He is now with Stratos Global, St. John s, NL A1C 2G, Canada ( Ala a.al- Habashna@stratosglobal.com). O. A. Dobre, R. Venkatesan are with the Faculty of Engineering Applied Science, Memorial University of Newfoundl, St. John s, NL A1B 3X5, Canada ( odobre@mun.ca; venky@mun.ca). D. C. Popescu is with the Department of Electrical Computer Engineering, Old Dominion University, Norfolk, VA USA ( dpopescu@odu.edu). Digital Object Identifier /JSTSP CR, the spectrum utilization will be significantly improved by opening the licensed bs to be exploited by other users, without interfering with the licensed users. CR dynamically accesses the available spectrum, adapts its transmission parameters according to the changes in the radio environment. In order to acquire information on the surrounding, CR needs to detect classify the on-the-air signals [1] [3]. As orthogonal frequency division multiplexing (OFDM) has been chosen for the physical layer of many wireless stards, intensive research has been done recently on the detection, classification, parameter estimation of the OFDM signals [4] [18]. Most of the classification methods are developed for generic signals rely on cyclostationarity, with some of them employing the detection of the cyclic prefix (CP)-induced peaks in the cyclic autocorrelation function (CAF) [5] [9]. In these methods, the CAF magnitude is either searched over a large delay range to find the peaks [5] [8], which introduces computational complexity, or the location of the peaks is assumed a priori known [9]. Although this location is known for stard signals, it can be unreliable for classification, as the cognitive users sharing the spectrum may also employ the OFDM modulation with close useful symbol duration. Another method is presented in [10] [11], uses the cyclostationarity signatures intentionally embedded in the OFDM signals. The drawback of this method is the extra overhead that results from embedding such signatures. Furthermore, pilot-induced cyclostationarity is exploited in [12] under the assumption that the pilot pattern repeats in time frequency with some distributions, there are identical pilot symbols in each such pattern group, having application to WiFi, Digital Video Broadcasting- Terrestrial, fixed Worldwide Interoperability for Microwave Access (WiMAX) signals. In [13], the authors exploit the second-order cyclostationarity to classify diverse IEEE stard signals. A theoretical analysis of the second-order cyclostationarity induced by pilots, with application to the IEEE a signals is carried out in [14]. A non-cyclostationarity approach is presented in [17], [18], based on the kurtosis of the decoded symbols. The useful symbol duration, CP duration, carrier frequency offset, time delay are jointly estimated by performing a search to minimize this kurtosis, the OFDM signal type is identified based on the useful symbol duration estimate. As it was mentioned for the cyclostationarity-based approach, although the useful symbol duration is specific to diverse stards, this information can be unreliable for classification /$ IEEE

2 AL-HABASHNA et al.: SECOND-ORDER CYCLOSTATIONARITY OF MOBILE WIMAX AND LTE OFDM SIGNALS 27 to the second-order cyclostationarity of mobile WiMAX LTE OFDM signals, respectively, Appendix D shows that the test statistics are independent of the phase, frequency, timing offsets. II. MOBILE WiMAX OFDM SIGNAL MODEL AND ITS SECOND-ORDER CYCLOSTATIONARITY A. Mobile WiMAX OFDM Signal Model Fig. 1. TDD frame structure for mobile WiMAX. Fig. 2. OFDM frequency description [21]. In this paper, we provide realistic comprehensive mathematical models of the mobile WiMAX OFDM Long Term Evolution (LTE) OFDM signals, which take into account preambles, pilots, reference signals (RS). We also study the preamble-, CP-, RS-induced second-order cyclostationarity of these signals, based on the findings, we propose an algorithm for signal classification in common frequency bs. We show that the phase offset has no effect on CAF, while the frequency timing offsets affect only the CAF phase. The cycle frequencies (CFs) are not affected by such signal impairments. Furthermore, the CAF-based statistics used for signal classification are independent of phase, frequency, timing offsets. The performance of the proposed classification algorithm is studied versus signal-to-noise ratio (SNR), for different observation intervals (number of samples) channel conditions. Analysis of the computational complexity is performed versus the considered signals the number of samples. Apparently, there is a tradeoff between performance complexity. Results show a good performance, as well as real-time applicability of the proposed algorithm. In addition, the proposed algorithm does not require carrier, waveform, symbol timing recovery, is immune to phase, frequency, timing offsets. The rest of this paper is organized as follows. In Sections II III, we introduce the developed signal model findings on the second-order cyclostationarity of mobile WiMAX LTE OFDM signals, respectively. In Section IV we present the proposed cyclostationarity-based signal classification algorithm. We investigate the performance computational complexity of the proposed algorithm in Section V, followed by conclusions final remarks in Section VI. Appendix A presents fundamental concepts on the second-order signal cyclostationarity, Appendices B C discuss proofs related Fig. 1 presents the IEEE e time-division duplex (TDD) frame structure, as per the current mobility certification profiles [19] [21]. The stard frame duration can range from 2 ms to 20 ms; however, all WiMAX equipments support only a 5-ms frame [22]. The frame is divided into two subframes, one for the downlink (DL) another one for the uplink (UL). The DL-to-UL subframe ratio is variable, to support different traffic profiles. Transition gaps separate the adjacent DL UL subframes. In Fig. 1, TTG represents the DL-UL gap is referred to as the transmit/receive transition gap, while RTG represents the UL-DL gap is referred to as the receive/transmit transition gap. Note that the terminology used here is according to the IEEE e stard [20], [21]. The DL subframe starts with a preamble as the first symbol, which is used for time frequency synchronization uniquely identifies a serving base-station. Therefore, a cognitive user within the coverage area of a base-station will periodically receive the same preamble. The OFDM frequency-domain description is presented in Fig. 2. One can note four types of subcarriers: data subcarriers to transmit information, pilot subcarriers for estimation purposes, null subcarriers for guard bs, direct current (DC) subcarrier [21]. The first two types of subcarriers are called the used subcarriers. The pilot symbol on subcarrier is generated as where is a value taken from a pseudorom binary sequence that is different for each OFDM symbol [21]. The distribution of the pilot subcarriers might differ from one OFDM symbol to another in the frame, while this repeats every frame, i.e., it is the same for each th OFDM symbol of a frame [19] [21]. Note that for the preamble symbol, for the DL OFDM symbols (excluding the preamble), for the UL OFDM symbols, with as the number of OFDM symbols in the DL (excluding the preamble) in the frame, respectively. Note that more than one pilot distribution might be employed in the DL or UL subframes; each pilot distribution is used in a certain set of OFDM symbols in the DL or UL subframes [19] [21]. The pilot symbols are usually transmitted with boosted power over the data symbols. The preamble contains only null subcarriers subcarriers used for transmitting preamble data. According to the stard, the preamble data symbols are transmitted every third subcarrier out of the set of subcarriers, starting from the subcarrier up to where is the number of the preamble data symbols [19] [21]. Fig. 3 shows the frequency domain description of the preamble when.

3 28 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 1, FEBRUARY 2012 TABLE I OFDM PARAMETERS FOR THE MOBILE WiMAX Signal [22] Fig. 3. OFDM frequency description of the preamble symbol (S =0). According to the above description, we express the discretetime mobile WiMAX OFDM signal affected by noise as 1 where are the signal components corresponding to the preamble, the DL (excluding the preamble) UL subframes, respectively, is the additive zero-mean Gaussian noise. Furthermore, are given respectively as 1 (1) (1.a) (1.b) (1.c) where are the number of used subcarriers in the preamble, DL, UL symbols, respectively, are the amplitude factors equal to respectively, is the OFDM symbol period 2 equal to the useful OFDM symbol duration, plus the CP duration, is th preamble data symbol transmitted in the th subcarrier of the preamble, with as the position of the first subcarrier used to transmit preamble data are the symbols (data pilot) transmitted on the th subcarrier within 1 Note that the effect of phase, frequency, timing offsets are not included in the signal expressions to simplify streamline presentation. However, they are taken into account their effect on the second-order signal cyclostationarity is discussed in Section II-C. 2 Note that durations are expressed in number of samples. For durations expressed in time periods, we use the symbol T instead of D, e.g., T;T ; T ; instead of D; D ; D, respectively. the th 3 OFDM symbol which belongs to the DL UL subframes, respectively (note that the distribution of pilot subcarriers could be different for different groups of OFDM symbols), is the number of OFDM symbols in the UL subframe, is the impulse response of the transmit the receive filters in cascade, is the total duration of the transition gaps within each frame, with as the TTG RTG transition gaps, respectively. The data symbols are taken either from a quadrature amplitude modulation (QAM) or phase shift keying (PSK) signal constellation, are assumed to be zero-mean independent identically distributed (i.i.d.) rom variables. The fast Fourier transform (FFT) size for generating OFDM symbols is equal to the total number of subcarriers (used guard b subcarriers), equals. To ease the understing of the expressions for the signal model, here we provide an explanation for the in (1.a). The inner summation on the right-h side of (1.a) is over the number of data subcarriers, while the outer summation is over the OFDM symbol (preamble) index. Specific to (1.a) is the position of the preamble, which appears at the beginning of each frame. As such, the preamble index,, is an integer multiple of the number of OFDM symbols in a frame. Furthermore, for the position of the preamble, we need to take into account the total duration of the transition gaps within each frame, which yields the shift of the exponential with where provides the frame index. It is noteworthy that the data symbol on each subcarrier is the same for all preambles ( does not depend on the preamble index ). The OFDM parameters for mobile WiMAX signals are presented in Table I. As one can notice, the FFT size is scalable with the bwidth: when the available bwidth increases, the FFT size also increases, such that the useful symbol duration (equal to the reciprocal of the subcarrier frequency spacing, ) is fixed. This in turn leads to a constant useful OFDM symbol duration. B. CAF Set of CFs 4 for the Mobile WiMAX OFDM Signals According to derivations in Appendix B, the CAF set of CFs for the mobile WiMAX OFDM signals are as follows. 3 Note that the OFDM symbol index, l, can be further expressed in terms of the frame index, bln c; with b1c as the floor function, OFDM symbol index within a frame, ; with = l mod N. As such, l = bln cn + ; = 0;...;N For the definition of CAF, R (; ); set of CFs, fg; the reader is referred to Appendix A.

4 AL-HABASHNA et al.: SECOND-ORDER CYCLOSTATIONARITY OF MOBILE WIMAX AND LTE OFDM SIGNALS 29 Fig. 4. CAF magnitude for the mobile WiMAX signal at = D = 512 versus CFs. Case (1): For delays equal to zero (due to the correlation of the signal with itself) (CP-induced cyclostationarity) Fig. 5. CAF magnitude for the mobile WiMAX signal at = d512=3e =171 versus CFs. Case (2): For delays integer (structure repetition of the preamble induced-cyclostationarity), with denoting the ceiling function (4) where with 5 as the number of data pilot subcarriers, as the variance of data pilot symbols in the th DL OFDM symbol (excluding the preamble), respectively, as their UL counterparts. Noteworthy to mention that the CFs are integer multiples of the reciprocal of the frame duration, not of the OFDM symbol duration, as one can expect based on the analysis of generic OFDM signals [4] [6]. This is due to the existence of the transition gaps in the WiMAX frame. Nevertheless, we expect that the CAF magnitude at CFs close to integer multiples of predominates, as the transition gaps are small when compared with the frame duration. Fig. 4 shows the CAF magnitude at delay versus CFs, with the signal parameters set as in Section V. One can see that, indeed, although the CFs are integer multiples of the reciprocal of the frame duration, the CAF at integer multiples of the symbol duration predominates. 5 Note that the number of pilot subcarriers, the variance of the data symbols, the variance of the pilot symbols are the same in all the OFDM symbols belonging to the same group. For the convenience of notation, we provide here a more general case where these parameters are different for each OFDM symbol,. (2) (3) the set of CFs is given by (3). As the preamble repeats every frame, it is expected that the CFs are integer multiples of the frame duration. The CAF magnitude due to the preamble, at delays is plotted versus CFs in Fig. 5. Case (3): For delays with as the difference between two consecutive pilot subcarriers in DL (pilot induced-cyclostationarity) where the CFs are given in (3), with as the variances of the pilot data symbols, respectively, represents the set of pilot subcarriers in DL. Similar expressions as in (5) can be also obtained for CAF at delays the same CFs. In the expressions for CAF at are, respectively, replaced by their UL counterparts,. (5)

5 30 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 1, FEBRUARY 2012 Fig. 8. FDD DL frame structure in the LTE OFDM-based systems [24]. TABLE II LTE OPERATION MODES AND ASSOCIATED SIGNAL PARAMETERS Fig. 6. CAF magnitude for the mobile WiMAX signal at = d =14e =110versus CFs. C. Effect of Phase, Frequency, Timing Offsets on CAF CF Fig. 7. CAF magnitude for the mobile WiMAX signal at = D = versus CFs. Fig. 6 displays the CAF magnitude at versus CFs. As the distribution of the pilot symbols repeats every two DL OFDM symbols, one may expect that the CFs are integer multiples of. Nevertheless, this periodicity does not appear in the UL OFDM symbols. Hence, the CAF has CFs at integer. However, the CAF magnitude at CFs close to integer multiples of predominates; this can be explained based on the relatively large number of DL OFDM symbols in the frame. Case (4): For delays equal to integer multiples of (repetition of the preamble induced-cyclostationarity) the set of CFs is given in (3). Fig. 7 shows the CAF magnitude due to the preamble at delay versus CFs. We should note that all previously mentioned results for the estimated CAF agree with theoretical results; these are not presented here due to space consideration. (6) For a received signal affected by phase, frequency, timing offsets,, respectively, one can find the effect of these impairments on CAF CFs by following the derivation steps provided in Appendix B, as: the phase offset has no effect on CAF; the frequency offset affects the phase of CAF at delay CF by ; the timing offset affects the phase of CAF at delay CF by ; the CFs are not affected by phase, frequency, timing offsets. III. LTE OFDM SIGNAL MODEL AND ITS SECOND-ORDER A. LTE OFDM Signal Model CYCLOSTATIONARITY Fig. 8 shows the frequency division duplex (FDD) DL frame structure used in the LTE systems [24]. The frame time duration is 10 ms, each frame is divided into 20 slots, with the slot duration equal to 0.5 ms. Each slot contains OFDM symbols, where depends on the CP length useful symbol duration (equal to the reciprocal of the subcarrier frequency spacing) parameters of the OFDM signal. The LTE stard allows multimedia broadcast multicast services be performed either in a single cell mode or in a multi-cell mode. For the latter, transmissions from different cells are synchronized to form a multicast broadcast single frequency network (MBSFN) [24]. Here, we consider the case where a single operational mode is employed in each cell, i.e., either MBSFN or non-mbsfn [24]. The LTE operation modes, along with the values of their parameters, i.e., 2, are summarized in Table II.

6 AL-HABASHNA et al.: SECOND-ORDER CYCLOSTATIONARITY OF MOBILE WIMAX AND LTE OFDM SIGNALS 31 Fig. 9. Slot structure resource grid in the FDD DL frame [24]. The slot structure associated resource grid used in the FDD DL frame are illustrated in Fig. 9. The slot can be represented as a two dimensional resource grid consisting of OFDM symbols in time domain subcarriers in frequency domain, with as the number of resource blocks as the number of subcarriers in a resource block. Note that represents the number of subcarriers in an OFDM symbol. A resource block is defined as consecutive OFDM symbols in time domain consecutive subcarriers in frequency domain. equals for the LTE signals with 15 khz 7.5 khz subcarrier spacing, respectively. then depends on the signal bwidth; for possible values of this parameter the reader is referred to [25]. The smallest entity of the resource grid is called resource element; a resource block consists of resource elements. Reference signals (RSs) are embedded in the resource blocks of the transmission frame for channel estimation cell search/ acquisition purposes [24]. An RS is assigned to each cell of the network acts as a cell identifier. Therefore, the RS repeats each DL frame. Here we study two types of RSs: the cell-specific RS associated with the non-mbsfn mode the MBSFN RS associated with the MBSFN mode. Note that the terminology used here is according to [24]. The RSs are interspersed over the resource elements, usually transmitted on some of the subcarriers of one or two non-consecutive symbols in each slot. Fig. 10 shows the distribution of the cell-specific RS for short CP over one resource block two consecutive slots ( OFDM symbols per slot subcarriers per resource block): the cell-specific RS is transmitted on the first seventh subcarriers of the first OFDM symbol on the fourth tenth subcarriers of the fifth OFDM symbol in each slot. For the distribution of other RSs, one is referred to [24]. Following this description, we express the received LTE OFDM signal with short CP corresponding RS distribution, which is affected by additive Gaussian noise as 1 Fig. 10. Resource element mapping of cell-specific RS in LTE signal with non- MBSFN mode short CP [24]. where is the amplitude factor equal to is the repetition period for the RS distribution (in number of OFDM symbols), which equals 7 in this case corresponds to the number of OFDM symbols in a slot, are the sets of subcarriers on which the RS is transmitted in corresponding OFDM symbols, is the transmit pulse shape window (associated with the first symbol in the slot) the impulse response of the receive filter in cascade, is the transmit pulse shape window (associated with remaining symbols in the slot) the impulse response of the receive filter in cascade, are the duration of the first OFDM symbol in the slot, the duration of the remaining OFDM symbols in the slot, the duration of the slot, respectively, are the data RS symbols transmitted on the th subcarrier within the th OFDM symbol, respectively, is the additive zero-mean Gaussian noise. Note that the data symbols are taken either from a QAM or PSK signal constellation, while the RS symbols are taken from the BPSK signal constellation. Data RS symbols are zero-mean i.i.d. rom variables. To ease the understing of the expressions for the signal model in (7), here we provide an explanation for that. (7)

7 32 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 1, FEBRUARY 2012 TABLE III OFDM PARAMETERS FOR THE LTE SIGNALS [24] Case (1): For delays equal to zero (due to the correlation of the signal with itself) (CP-induced cyclostationarity) (8) (9) In all terms on the right h-side of (7), the inner summation is over the number of subcarriers, while the outer summation is over the OFDM symbol index. The first two terms model the first OFDM symbol in the slot, which includes data RS symbols, respectively. The first term represents the subcarriers where data symbols are transmitted, while the second term represents those where RS symbols are transmitted. Furthermore, for the position of the first symbol in the slot, we need to take into account the duration of the preceding slots, which yields the shift of by with providing the slot index. Similarly, the third fourth terms model the fifth OFDM symbol in the slot, which includes data RS symbols. Furthermore, for the position of the fifth symbol in the slot, we need to take into account the duration of the preceding slots, as well as the duration of the preceding OFDM symbols in the same slot, which are the first OFDM symbol in the slot, (which is longer than the other symbols) along with other three OFDM symbols,. This yields the shift of by. The fifth term expresses the remaining OFDM symbols where only data symbols are transmitted. Here, for the position of the OFDM symbol in the slot, we need to take into account the duration of the preceding slots, as well as the duration of the preceding OFDM symbols in the same slot, which are the first OFDM symbol in the slot, along with other OFDM symbols,. This yields the shift of by. Similar signal models can be written for long CP with different operation modes frequency separations. In this case, all symbols have the same duration, the signal models can be easily obtained. The OFDM parameters for LTE signals are presented in Table III. As one can notice, the FFT size is scalable with the bwidth: when the available bwidth increases, the FFT size also increases such that the useful symbol duration the subcarrier spacing are fixed. As previously mentioned, two different values are used for the subcarrier spacing with the LTE signal ( khz 15 khz), this results in two different values for the useful OFDM symbol duration ( s s, respectively). B. CAF Set of CFs for LTE OFDM Signals According to derivations in Appendix C, the CAF set of CFs for the LTE OFDM signals with short CP are as follows. where represents the variance of RS data symbols. Case (2): For delays equal to integer multiples of (RSinduced cyclostationarity) (10) at the CFs in (9). Here represents the number of subcarriers on which RS are transmitted in each OFDM symbol which includes RS; as noted in Section II.B, this is equal for. Similarly, one can show that CAF set of CFs for LTE OFDM signals with long CP are as follows. Case (1): For delays equal to zero (due to the correlation of the signal with itself) (CP-induced cyclostationarity) (11) integer (12) (RS- Case (2): For delays equal to integer multiples of induced cyclostationarity) (13) at the CFs in (9). Here are the sets of the OFDM symbols in which the RSs are transmitted, which is different than for the LTE OFDM signals with short CP [24]. It is worth noting that the CFs are integer multiples of for short CP in all cases long CP only at delays integer multiples of while the CFs are integer multiples of for long CP at delays equal to zero due to the same duration of all OFDM symbols in the frame. The theoretical CAF magnitudes for the LTE signal with non-mbsfn mode short CP non-mbsfn mode long CP at [Case (1)] are presented versus CFs in Figs , respectively. The signal parameters are set as in Section V. The range corresponding to the first four positive negative CFs is shown. As one can notice, the CAF magnitude is lower for the LTE signal with the non-mbsfn

8 AL-HABASHNA et al.: SECOND-ORDER CYCLOSTATIONARITY OF MOBILE WIMAX AND LTE OFDM SIGNALS 33 Fig. 11. CAF magnitude for the LTE signal with non-mbsfn mode short CP at = D =512versus CFs. Fig. 13. CAF magnitude for the LTE signal with non-mbsfn mode short CP at = D =76800versus CFs. Fig. 14. CAF magnitude for the LTE signal with non-mbsfn mode long CP at = D = versus CFs. Fig. 12. CAF magnitude for the LTE signal with non-mbsfn mode long CP at = D = 512 versus CFs. mode short CP; this is due to a lower in such a case. Additionally, the CAF magnitude at CFs close to integer multiples of predominates, as the duration of the first symbol in the slot is only slightly different when compared with the duration of the remaining symbols. The theoretical CAF magnitude for the LTE signal with non-mbsfn mode short CP, with non-mbsfn mode long CP, with MBSFN mode long CP ( khz) at is presented versus CFs in Figs , respectively. As one can see, the CAF magnitude is higher in case of MBSFN mode. This can be easily explained, as the MBSFN RS is induced on more subcarriers when compared with the cell-specific RS for the non-mbsn mode. Furthermore, due to diverse RS distributions for different transmission modes, the CF values depend on the mode. We would also like to note that results for estimated CAF agree with theoretical results; these are not presented here due to space consideration. Furthermore, the effect of phase, Fig. 15. CAF magnitude for the LTE signal with MBSFN mode 1f =15 khz at = D = versus CFs. frequency, timing offsets on CAF CFs is as presented in Section II-C for mobile WiMAX OFDM signals.

9 34 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 1, FEBRUARY 2012 IV. PROPOSED CLASSIFICATION ALGORITHM BASED ON CP-, PREAMBLE-, AND RS-INDUCED SECOND-ORDER CYCLOSTATIONARITY As previously discussed, both mobile WiMAX LTE OFDM signals exhibit CP-induced cyclostationarity. However, this information can be unreliable for identifying primary WiMAX LTE signals, as the cognitive users sharing the spectrum may employ OFDM signals with close useful symbol durations. Hence, the distinctive preamble-induced cyclostationarity of the mobile WiMAX OFDM-based signals (Cases (2) (4) in Section II) the distinctive RS-induced cyclostationarity of the LTE OFDM-based signals (Case (2) in Section III) are used in addition to the CP-induced cyclostationarity for signal classification. We propose a classification algorithm for these signals, which makes use of their CP-, preamble-, RS-induced cyclostationarity. The distinctive features of the mobile WiMAX signals used with this algorithm are the non-zero CAF at: CF delay (Case (1)); CF delay [Case (2)]; CF delay [Case (4)]. Note that only is used at as other CFs depend on the CP length, which is unknown. Also note that the pilot-induced cyclostationarity is not used here, as different pilot distributions result in different locations for the peaks, the distribution is not a priori known. On the other h, the distinctive features of the LTE signals used with the proposed algorithm are the non-zero CAF at: CF delay [Case (1)]; CF delay [Case (2)]. Note that only is used at for the same reason as for the WiMAX signals. Also note that only is used at as other CFs depend on the RS distribution, which is not a priori known. This characteristic can be employed to further identify the operation mode; this topic is beyond the scope of the paper. A. Description of the Proposed Algorithm The proposed classification algorithm is a binary tree classifier, as depicted in Fig. 16. At Node 1, we decide whether or not the received signal is WiMAX based on the previously mentioned distinctive features. The cyclostationarity test proposed in [26] is applied to check that is a CF at (with s). 2 In addition, the test in [27], which represents an extension of the one in [26] to multiple CFs, is applied to check that a frequency is a CF at, respectively (with s ms). 2 Both tests make a decision on the existence of a CF at a certain delay by comparing a CAF-based statistic with a threshold; details on the tests are provided in next section. If the tested s are decided to be CFs at the corresponding delays, the decision is mobile WiMAX. Otherwise, the classification algorithm performs further testing at the next node. At Node 2, we decide whether or not the received signal is LTE with khz. For that we use the previously mentioned distinctive features of the LTE signals ( khz). The cyclostationarity test in [26] is employed to check whether or not is a CF for delays equal to respectively (with s ms). 2 The decision that Fig. 16. Flowchart of the proposed cyclostationarity-based signal classification algorithm. the LTE signals with khz (MBSFN or non-mbsfn) is present is made if is found to be a CF at both delays. Otherwise, further testing is carried out at the next node. Finally, at Node 3, we decide whether or not the received signal is an LTE signal with khz. The cyclostationarity test in [26] is employed to check whether or not is a CF for delays equal to, respectively (with s ms) 2. The decision that the LTE signal with this subcarrier spacing is present is made if is found to be a CF at both delays. Otherwise, the decision that other signals are present in the environment is made. For the classification of other signals, such as single carrier linearly digitally modulated signals, cyclically prefix linearly digitally modulated signals, other OFDM signal types, further nodes can be added [27]. We note that the CP- induced cyclostationarity based features are stronger than the preamble- RS-induced cyclostationarity-based features (see Figs. 4, 5, 7, 11 14). In addition, preamble- RS-induced cyclostationarity-based features (Case (4) for WiMAX Case (2) for LTE) require at least a two frame observation interval for estimation. When the statistics corresponding to the latter features do not pass the cyclostationarity tests, e.g., because the observation interval is too short 6, the classification algorithm can still provide an indication based on the CP-induced cyclostationarity. We would like to reiterate that such a decision is not fully reliable (only one out of the three two CAF-based statistics pass the test at Nodes 1, Nodes 2 3, respectively), as OFDM with close useful symbol duration can be employed by CR users, as well. B. Cyclostationarity Tests for Decision-Making The cyclostationarity tests in [26] [27], which are used for decision-making with the proposed algorithm, are described below. With the test in [26], the presence of a CF is formulated 6 One can go even further, if the CP-induced cyclostationarity-based feature fails the test, the kurtosis-based algorithm in [17], [18] can be triggered, which can provide the partial information with a very short observation time, at the price of increased computational complexity.

10 AL-HABASHNA et al.: SECOND-ORDER CYCLOSTATIONARITY OF MOBILE WIMAX AND LTE OFDM SIGNALS 35 as a binary hypothesis-testing problem, i.e., under hypothesis the tested frequency is not a CF at delay under hypothesis the tested frequency is a CF at delay. The test consists of the following three steps: The CAF of the received signal, is estimated (from samples) at tested frequency delay a vector is formed as with as the real imaginary parts, respectively, as the estimate of. A statistic is computed for the tested frequency delay as with the superscripts denoting the matrix inverse transpose, respectively. is an estimate of the covariance matrix of (14) where the covariances are given, for zero-mean process, respectively by [26] (15) (16) with as the cumulant operator as the second-order lag product. The estimators for the covariances are given, respectively, by [26] (17) (18) where is a spectral window of length. The test statistic is compared against a threshold, for decision making. If, we decide that the tested frequency is a CF at delay. The threshold is set for a given probability of false alarm, which is defined as the probability to decide that the tested frequency is a CF at delay when this is actually not. This can be expressed as. By using that the statistics has an asymptotic chi-square distribution with two degrees of freedom under the hypothesis [26], the threshold is obtained from the tables of the chi-squared distribution for a given value of this probability. The cyclostationarity test proposed in [27] represents an extension of the one presented previously to multiple CFs. In this case, the test statistic is with as the set of tested frequencies at each tested frequency calculated as previously explained. The threshold is similarly set, based on the asymptotic distribution of the test statistic under the hypothesis. As shown in [27], this is a chi-square distribution with degrees of freedom, where represents the number of tested frequencies. Apparently, for this test reduces to the one proposed in [26]. Effect of Phase, Frequency, Timing Offsets on the Test Statistics: By considering the effect of these signal impairments on CAF (see Sections II-C III-B), along with (14), (17), (18), after tedious trigonometric calculations, one can show that thus are independent of these impairments (see Appendix D for proof). Comment on the Asymptotic Algorithm Performance: According to Section IV-A, three statistics are tested at Node 1, while two statistics are tested at Nodes 2 3; the thresholds are set according to a certain as stated above. Hence, asymptotically, the probability to decide that a signal is WiMAX when it is not (Node 1) equals while the probability to decide that a signal is LTE with khz khz when it is not (Nodes 2 3) equals, respectively. V. CLASSIFICATION PERFORMANCE AND COMPUTATIONAL COMPLEXITY OF THE PROPOSED ALGORITHM A. Simulation Setup The signals are simulated with 5 MHz double-sided bwidth. For WiMAX, the number of subcarriers is 512, while for LTE this is 512 with khz 1024 with khz. For the mobile WiMAX signal equals 1/8, while for the LTE signal both long ) short ( for the first symbol in the slot for the remaining symbols) CPs are employed. For both WiMAX LTE signals, QAM with 16 points unit variance of the signal constellation is used to modulate the data subcarriers. The pilot subcarriers in mobile WiMAX are modulated according to the IEEE e stard [21]. A raised root cosine pulse shape window is used at the transmitter with a roll-off factor of For the WiMAX signal, the number of symbols in the UL DL subframes equals 35 12, respectively, the RTG duration is 60 s whereas the TTG duration equals s [19]. The sampling frequency at the receiver is set to 8.4 MHz, the signal is affected by a phase offset uniformly distributed in a timing offset uniformly distributed in a carrier frequency offset. The AWGN ITU-R pedestrian vehicular A fading channels are considered. The maximum delay spreads for the ITU-R pedestrian vehicular A fading channels are 410 ns 2.51 s, respectively. For details on the power-delay profiles of the fading channels one is referred to [29]. The maximum Doppler frequency equals 7.28 Hz Hz for the ITU-R pedestrian vehicular A fading channels, respectively. At the receive-side, a filter is used to remove the out-of-b

11 36 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 1, FEBRUARY 2012 Probability of correct classification versus SNR for the signals of in- Fig. 17. terest. noise, the SNR is set at the output of this filter. The probability of correct classification is used as performance measure, with mobile WiMAX signals, LTE signals with khz (non-mbsfn mode long CP, non-mbsfn mode short CP, MBSFN mode) LTE signals with khz (MBSFN mode). Note that the LTE signal with non-mbsfn mode long CP is referred to LTE in figures, for the convenience of notation. This probability is estimated based on 1000 Monte Carlo simulation trials. The Kaiser window with parameter 10 is used in (17) (18), with. Unless otherwise mentioned, the observation interval is 30 ms (equivalent to six WiMAX frames three LTE frames), the thresholds used with the cyclostationarity tests correspond to a probability of false alarm of, the channel is AWGN. B. Classification Performance of the Proposed Algorithm The performance of the proposed algorithm is investigated in terms of the probability of correct classification. Fig. 17 shows simulation results for this probability versus SNR for mobile WiMAX, LTE with khz (non-mbsfn mode long CP, non-mbsfn mode short CP, MBSFN mode) LTE signals khz (MBSFN mode). One can see that among the LTE signals, the best results are achieved for the LTE signals with the MBSFN mode, while lower performance is achieved for the LTE signals with the non-mbsfn mode long CP, the lowest performance is attained for the LTE signals with non-mbsfn mode short CP. Also, the performance for WiMAX signals is relatively close to that of the LTE signals with non-mbsfn mode short CP. Note that this is expected, being in agreement with CAF values used for discrimination, as per Sections II III. We should also mention that a probability of deciding that other signals, such as single carrier linearly digitally modulated (SCLD) signals, are present, when indeed this is the case, is almost 1 under the investigated conditions. This is expected, as per the discussion on the asymptotic performance analysis in Section IV-B. A confusion matrix is provided in Fig. 18, for db SNR. As expected based on the value, there are basically no cases in which the algorithm decides that a signal is of one of the types of interest (mobile Fig. 18. Confusion matrix for 07 db SNR 30 ms observation interval. WiMAX LTE), if it is not of that specific type; if the algorithm fails in classifying the signal of interest, the decision is practically Other signals. Figs show the simulation results for the probability of correct classification versus SNR for mobile WiMAX LTE with the non-mbsfn mode long CP (LTE ), respectively, with different observation intervals (minimum two WiMAX LTE frames, respectively), diverse thresholds used for decision-making. With respect to the thresholds used for decision making, corresponds to an asymptotic probability of false alarm of 0.01 (this equals 9.21 for 1 CF for 3 CFs), to an asymptotic probability of false alarm of (this equals for 1 CF for 3 CFs). As expected, a higher probability of correct classification is achieved with a lower threshold. Furthermore, an improved performance is achieved with increased observation interval (increased number of samples, L) for all signals. We note that at least two frames are needed to estimate CAF at (10 ms for WiMAX 20 ms for LTE signals); even with such a short observation interval, a probability of almost 1 is achieved at 3 db SNR for WiMAX db SNR for LTE signals, respectively. According to Section IV, the CP-induced cyclostationarity can provide a partial indication to the CR for lower SNRs, shorter observation intervals. Simulation results show that with 20 ms 30 ms observation intervals, the statistics associated to these features pass the cyclostationarity test with probability approaching 1 at SNRs above db db for the mobile WiMAX signal (Node 1), db db for the LTE signal with non-mbsfn mode long CP MBSFN mode with khz, db db for the LTE signal with non-mbsfn mode short CP (Node 2), db db for the LTE signal with MBSFN mode khz (Node 3). Moreover, for a shorter observation interval of 10 ms, these statistics pass the test at db SNR for the mobile WiMAX signal (Node 1), db for the LTE signal with non-mbsfn mode long CP MBSFN mode with khz, db for the LTE signal with non-mbsfn mode

12 AL-HABASHNA et al.: SECOND-ORDER CYCLOSTATIONARITY OF MOBILE WIMAX AND LTE OFDM SIGNALS 37 Fig. 19. Probability of correct classification of mobile WiMAX signals versus SNR with various observation intervals thresholds used for decision-making. Fig. 21. Probability of correct classification of mobile WiMAX signals versus SNR under different channel conditions. Fig. 20. Probability of correct classification of LTE signals (non-mbsfn mode with long CP) versus SNR with various observation intervals thresholds used for decision-making. short CP (Node 2), 9 db for the LTE signal with MBSFN mode khz (Node 3). The algorithm proposed in [17] [18] also provides partial information to the CR based on the useful symbol duration. To estimate this duration, the authors perform a minimization of the kurtosis of the decoded symbols over the useful CP durations, as well as frequency timing offsets. When compared to the CP-induced cyclostationarity approach, this method has the advantage that it does not depend on the CP duration. On the other h, this comes at the price of increased computational complexity, as it involves an exhaustive search over four parameters, including timing frequency offsets, whereas the CP-induced cyclsotationarity (zero CF) tests only the peak which corresponds to the useful symbol duration for a certain stard signal is immune to frequency timing offsets. When the observation interval CP duration are very short, the kurtosis-based method in [17], [18] provides better performance. For example, for a 5 OFDM symbol observation interval 4 db SNR, the partial information on mobile WiMAX signal is provided with a probability approaching 1 with the kurtosis-based method, whereas CP-induced cyclostationarity gives around 0.95 probability when Fig. 22. Probability of correct classification of LTE signals (non-mbsfn mode with long CP) versus SNR under different channel conditions. essentially fails for [17]. However, when the observation interval is longer, e.g., 300 OFDM symbols at 0 db SNR, the classification performance of the two methods becomes comparable [18]. Figs show the simulation results for the probability of correct classification versus SNR for mobile WiMAX LTE with the non-mbsfn mode long CP (LTE1), respectively, with AWGN, ITU-R pedestrian vehicular A channels. One can notice that for all cases, the correct classification performance degrades at lower SNRs under the ITU-R vehicular channel. An increase of 2 db in SNR is needed to achieve a probability of almost 1 in such a channel. We reiterate that all previous results are obtained for a phase offset uniformly distributed in a timing offset uniformly distributed in a carrier frequency offset. Simulations were run, showed that these results are essentially identical to those achieved for the case when the signal is not affected by such impairments. Further, diverse other values were considered for the carrier frequency offset, e.g., for which the same results hold. This is in agreement with the theoretical findings, according to which the test statistics are independent of such

13 38 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 1, FEBRUARY 2012 TABLE IV NUMBER OF OPERATIONS REQUIRED TO JOINTLY DETECT AND ESTIMATE EACH CONSIDERED SIGNAL FOR 20 ms AND 30 ms OBSERVATION INTERVALS discriminate WiMAX LTE signals is different for the same observation interval, as different signal features are employed diverse steps are required (see Section IV-A). However, the computation time is short when compared to the observation time, it does not add much to the overall time needed to make a decision. signal impairments,, thus, these do not affect the classification performance. C. Computational Complexity of the Proposed Algorithm For the computational complexity of the proposed algorithms, we are interested in the computation of the test statistics. For a certain delay tested frequency, one needs the estimate of the covariance matrix the estimate of the vector to calculate the statistic. When testing multiple frequencies at a certain delay, a statistic is calculated for each frequency their sum is used as a test statistic. We first obtain the number of complex multiplications additions needed to calculate. Then, based on the number of delays tested frequencies at each delay which are used for considered signals, one can obtain the number of complex operations required to obtain the performance presented in Section V-B for a certain observation interval (number of samples). When using the time domain decimation FFT, based on (14), (17), (18), the expression of in Section IV-B, one can find that the number of complex multiplications additions needed to calculate is, respectively. Given that is usually a fraction of, e.g., the order of complexity of the algorithm can be given as where represents the big notation. With the parameters considered for the simulation setup an observation interval of ms, respectively, the numbers of complex operations required with the proposed algorithm are listed in Table IV. Note that with a microprocessor that can execute up to billion floating point operations (flops) per second, 7 the highest number of complex computations required to make a decision (as considered in Table IV) can be performed in approximately 2.3 ms (this is obtained assuming that 1 flop involves either a real multiplication or a real addition [30]). Apparently, there is a tradeoff between complexity performance, according to results in Sections V-B C. Also, the number of operations needed to 7 [Online]. Available: VI. CONCLUSION AND FINAL REMARKS In this paper, the mathematical models second-order cyclostationarity of OFDM-based signals employed in two popular wireless technologies, namely, the mobile WiMAX third Generation Partnership Project Long Term Evolution (3GPP LTE), are derived. Furthermore, the second-order cyclostationarity of these signals is exploited to develop an algorithm for their classification. The proposed algorithm provides a good performance at low SNRs, with short observation intervals, under diverse channel conditions. In addition, it does not require carrier, waveform, symbol timing recovery, is immune to phase, frequency, timing offsets. The algorithm can be implemented in real time, with a tradeoff between complexity performance, i.e., a longer observation interval leads to an improved performance but also to an increase in the number of operations. However, the processing time is short when compared with the observation time, it does not add much to the overall decision time. In further work, we will investigate other efficient methods for the implementation of the proposed algorithm, e.g., the Goertzel method, as well as classification of single carrier signal employed by the WiMAX LTE stards. APPENDIX A SECOND-ORDER SIGNAL CYCLOSTATIONARITY: DEFINITIONS A rom process is said to be second-order cyclostationary if its mean autocorrelation are almost periodic functions of time [31]. The latter is expressed as a Fourier series as [31] (19) where is the CAF at CF delay, represents the set of CFs. The CAF can be estimated based on samples, as [31] (20) APPENDIX B DERIVATION OF THE ANALYTICAL EXPRESSIONS FOR THE CAF AND CFS CORRESPONDING TO THE MOBILE WiMAX OFDM SIGNALS Using the signal model in (1.a) (1.c), one can express the autocorrelation function of as the sum of autocorrelation functions corresponding to the signal components,

14 AL-HABASHNA et al.: SECOND-ORDER CYCLOSTATIONARITY OF MOBILE WIMAX AND LTE OFDM SIGNALS 39 signal noise, only noise. We expect that non-zero significant values of are attained at certain delays, for which we subsequently study its representation as a Fourier series, further determine the expressions for the CAF at CF these delays, set of CFs,. Case (1): of mobile WiMAX signals is non-zero at delays equal to zero (due to the correlation of the signal with itself) (due to the CP). Assuming that the symbols on each subcarrier are i.i.d. mutually independent for different subcarriers, one can easily see that only are non-zero at such delays, this occurs when. Based on the above considering the data pilot symbols separately in each OFDM symbol in the DL UL subframes, one can show that, respectively, at which we expect have non-zero values due to the structure of the preamble its repetition every frame. Without considering its particular position in the frame, a preamble symbol can be expressed in time domain as (23) Without loss of generality, we consider a rectangular window. According to [23], such a preamble consists of three consecutive sequences that are highly correlated waveforms, can be written as (24) where with. As such, one can see that there is a non-zero autocorrelation of at delays equal to given by (21) where denotes convolution. By expressing as using that in (21), applying the Fourier transform, employing the convolution theorem the identity one can further obtain (25) Similarly, one can show that there is a non-zero autocorrelation of at delays equal to as given in (25). Furthermore, by considering the other components of the mobile WiMAX signal within a single frame, one can show that the only non-zero component of the autocorrelation function at delays equal to, respectively, is due to the preamble. With multiple frames based on the time-domain repetition property of the preamble, one can further show that the values of for delays equal to integer are equal to those at respectively. With integer, can be shown to be (22) As one can easily see from (22), is non-zero only for with as an integer. By using the inverse Fourier transform of (4), one can further observe that has a Fourier series representation. This means that the CF domain is discrete, the CAF at CF delay, the set of CFs are given as in (2) (3), respectively. Case (2): We investigate for delays equal to integer multiples of plus (26)

15 40 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 1, FEBRUARY 2012 By following the same steps as in Case (1), one can further show that the CAF at the above delays the set of CFs are given as in (4) (3), respectively. Case (3): We investigate for delays equal to at which we expect have non-zero values due to the transmission of boosted pilot symbols every th th subcarriers in the DL UL OFDM symbols, respectively. As in Case (2), we consider a single DL OFDM symbol, which includes both data pilot subcarriers; without loss of generality, this is a DL symbol, a rectangular window is assumed. The pilot symbols are transmitted every th subcarrier, are boosted over the data symbols. The time domain representation of such a DL OFDM symbol is (27) where is the (data/pilot) symbol transmitted on the th subcarrier. The DL OFDM symbol can be further expressed as the sum of two parts: a first part which corresponds to symbols over all subcarriers, whose variance is a second part which corresponds to symbols only over the pilot subcarriers, whose variance is i.e., with (28) (29) where are the symbols with variance, respectively, is the number of subcarriers used for pilot transmission in the considered DL OFDM symbol, is the position of the first subcarrier where a pilot symbol is transmitted. Apparently, Case (3) can be treated similarly to Case (2), where is replaced by,, with, respectively. By following the same procedure as in Case (2), one can obtain the expressions for the CAF set of CFs in (5) (3), respectively. Case (4): We further investigate for delays equal to integer multiples of. At these delays, one can easily show that the only non-zero term corresponds to when 8 Note that, for simplicity, we assume that one set of pilot symbols are transmitted in an OFDM symbol. In case that another set is transmitted with different distribution, simply, another term e.g., s (n); can be added to represent any additional set. The analysis for s (n) also applies for s (n); with CAF being the summation of results for all terms. which is due to the time repetition of the preamble every frame. Thus, for such delays becomes (30) where. Note that at these delays unless (otherwise the pulses do not overlap, yielding zero product). Based on the above observation, by using that after some mathematical manipulations, (30) can be re-expressed as. Furthermore, by following the same steps as in Case (1), one obtains the CAF set of CFs as in (6) (3), respectively. APPENDIX C DERIVATION OF THE ANALYTICAL EXPRESSIONS FOR THE CAF AND CFS CORRESPONDING TO THE LTE OFDM SIGNALS By using the signal model in (7), the autocorrelation function of can be expressed as the sum of the autocorrelation functions corresponding to any two signal components, signal noise components, noise component. We expect that non-zero significant values of are attained at certain delays, for which we subsequently study its representation as a Fourier series, determine the expressions for the CAF at CF these delays, set of CFs,. Case (1): We expect that non-zero significant values of are attained for delays equal to zero (due to the correlation of the signal with itself) (due to the existence of the CP). Assuming that the symbols on each subcarrier are i.i.d. mutually independent for different subcarriers, considering that the number of RS subcarriers is equal for the symbol variances satisfy by following the same procedure as in Case (1) for WiMAX signals, one can show that (31)

16 AL-HABASHNA et al.: SECOND-ORDER CYCLOSTATIONARITY OF MOBILE WIMAX AND LTE OFDM SIGNALS 41 the CAF at CF delay the set of CFs are given, respectively, in (8) (9). Case (2): We expect that non-zero significant values of are also attained for delays equal to integer multiples of (due to the repetition of the RS every frame). At these delays, one can show that the only non-zero terms in are due to the repetition of the RS, correspond to when with as an integer as the number of OFDM symbols in the transmission frame. By taking this into account following the same procedure as for Case (1) of WiMAX signals, after several mathematical manipulations, one can show that (32) where represents the number of RS subcarriers, which is equal in. Hence, the CAF at CF delay the set of CFs are given, respectively, as in (8) (9). Similarly, one can show that for the LTE signals with long CP, non-zero is attained for delays equal to zero [Case (1)] integer multiples of [Case (2)]. The CAF at CF delay the set of CFs can be similarly shown to be given as per Section III-B. APPENDIX D EFFECT OF PHASE, FREQUENCY, AND TIMING OFFSET ON TEST STATISTICS According to results provided in Sections II.C III-B, the phase offset does not affect the CAF CFs of the mobile WiMAX LTE OFDM signals. Hence, the CAF-based test statistics used for classification, which are given in Section IV-B, are independent of the phase offset. On the other h, the frequency timing offsets,, affect the phase of CAF at CF delay by, respectively; the CAF magnitude CFs are not affected. To emphasize the effect of on the CAF phase, we express CAF as, where represents the CAF of the signal unaffected by these signal impairments,. As such, when the signal is affected by becomes, by using the expression of (17) (18), one can see that the effect on is multiplication by while there is no effect on. 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F. van Loan, Matrix Computations. Baltimore, MD: Johns Hopkins Univ. Press, [31] C. M. Spooner W. A. Gardner, The cumulant theory of cyclostationary time-series, Part I: Foundation Part II: Development applications, IEEE Trans. Signal Process., vol. 42, no. 12, pp , Dec Ala a Al-Habashna received the B.S. degree (with excellent assessment) in computer engineering from Mu tah University, Karak Governorate, Jordan, in 2006, the M.S. degree in electrical computer engineering from Memorial University of Newfoundl, St. John s, NL, Canada, in He received the fellowship of school of graduate studies at Memorial University of Newfoundl. He is currently with Stratos Global, St. John s, as a Technical Communications Specialist. His research interests include signal detection classification, cognitive radio systems, communication networks architecture protocols, communication networks security. Octavia A. Dobre (M 04 SM 07) received the Dipl.Ing. Ph.D. degrees in electrical engineering from Politehnica University of Bucharest (formerly the Polytechnic Institute of Bucharest), Bucharest, Romania, in , respectively. In 2000 she was the recipient of a fellowship at Westminster University, U.K., in 2001 she held a Fulbright fellowship at Stevens Institute of Technology, Hoboken, NJ. Between , she was a Research Associate with the Department of Electrical Computer Engineering, New Jersey Institute of Technology, Newark. In 2005, she joined the Faculty of Engineering Applied Science at Memorial University of Newfoundl, St. John s, NL, Canada, where she is currently an Associate Professor. Her research interests include blind signal recognition, cognitive radio systems, cooperative communications, network coding, resource allocation in emerging wireless networks. Dr. Dobre is an Associate Editor for the IEEE COMMUNICATIONS LETTERS has served as the technical program chair for the signal processing multimedia symposium of the IEEE Canadian Conference on Electrical Computer Engineering (CCECE) 2009 the signal processing for communications symposium of the International Conference on Computing, Networking, Communications (ICNC) Ramachran Venkatesan (M 78 SM 92) received the M.S. doctoral degrees from the University of New Brunswick, Saint John, NB, Canada. He is the Dean Pro Tempore a Professor of Computer Engineering in the Faculty of Engineering Applied Science of Memorial University of Newfoundl, St. John s, NL, Canada. His research interests include architecture application of parallel processing structures, wireless sensor networks, underwater acoustic communications, error control codes, digital design. He is member of the Professional Engineers Geoscientists of Newfoundl Labrador. For over ten years he has held several academic administrative positions including the Chair of Electrical Computer Engineering, Associate Dean of Graduate Studies Research, Associate Dean of Undergraduate Studies, Acting Dean of Engineering. Dimitrie C. Popescu (S 98 M 02 SM 05) received the Engineering Diploma M.S. degrees in electrical engineering from the Polytechnic Institute of Bucharest, Bucharest, Romania, in 1991, the Ph.D. degree in electrical computer engineering from Rutgers University, New Brunswick, NJ, in From 2002 to 2006, he was with the Department of Electrical Computer Engineering, The University of Texas at San Antonio. He has also worked for AT&T Labs, Florham Park, NJ, on signal processing algorithms for speech enhancement, for Telcordia Technologie, Red Bank, NJ, on wideb CDMA systems. He is currently with the Department of Electrical Computer Engineering, Old Dominion University, Norfolk, VA. He is the coauthor of a monograph book on Interference Avoidance for Wireless Systems (Kluwer, 2004). His current research interests are in the areas of wireless communications cognitive radio systems include spectrum management dynamic spectrum access, transmitter/receiver optimization to support quality of service, spectrum sensing modulation classification. Dr. Popescu has served as the Technical Program Chair for the vehicular communications track of the IEEE Vehicular Technology Conference (VTC) 2009 Fall, the Finance Chair for the IEEE Multiconference on Systems Control (MSC) 2008, a Technical Program Committee member for the IEEE Global Telecommunications Conference (GLOBECOM), the IEEE International Conference on Communications (ICC), the IEEE Wireless Communications Networking Conference (WCNC), VTC conferences. He was the recipient of the Second Prize Award at the AT&T Student Research Symposium (an ACM regional competition) in 1999 for his work on interference avoidance dispersive channels.