Cyclostationarity-Based Detection of LTE OFDM Signals ... - IEEE Xplore

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Cyclostationarity-Based Detection of LTE OFDM Signals for Cognitive Radio Systems 1

Ala’a Al-Habashna, 1Octavia A. Dobre, 1Ramachandran Venkatesan, 2Dimitrie C. Popescu 1

Electrical and Computer Engineering, Memorial University of Newfoundland, Canada Department of Electrical and Computer Engineering, Old Dominion University, USA Email: 1{alaaa, odobre, venky}@mun.ca, [email protected]

2

Abstract—In this paper, a distinctive cyclostationarity-based feature of the Long Term Evolution (LTE) Orthogonal Frequency Division Multiplexing (OFDM) signals used in the Frequency Division Duplex (FDD) downlink transmission is proved, and further employed for their detection. This relates to the existence of the reference signals (RSs) used for channel estimation and cell search/ acquisition purposes. The analytical closed form expressions for the RS-induced cyclic autocorrelation function (CAF) and cyclic frequencies (CFs) are derived. Based on these findings, a signal detection algorithm is then developed. Simulation results show that the proposed algorithm achieves a good detection performance for low signal-to-noise ratios (SNRs), short sensing times, and under diverse channel conditions.

I. INTRODUCTION Nowadays, most of the valuable radio spectrum is allocated for licensed services, which causes a profound scarcity of this resource. Cognitive radio (CR) has emerged as a new technology tailored to address the spectrum scarcity problem. CR aims to improve the spectrum utilization through its ability to reconfigure the transmission parameters according to the surrounding electromagnetic environment. Such radios are able to access the spectrum opportunistically without interfering with the licensed/ primary users. Therefore, spectrum sensing and awareness are key aspects of the CRs. The energy detection, matched filter detection, and cyclostationarity-based feature detection represent major signal detection approaches [1]-[2]. The latter has the advantages that it is less sensitive to the noise uncertainty, relies less on the information about the signal to be detected, and can be also exploited for spectrum awareness [1]-[2]. Orthogonal frequency division multiplexing (OFDM) represents one of the main candidates for high data rate transmission for current and next generation wireless applications, being adopted by several wireless standards. Recently, OFDM signal detection has been intensively researched in the context of CR [3]-[4]. Most of the algorithms proposed for the OFDM signal detection rely on the cyclic prefix (CP)-induced cyclostationarity, viz., the existence of statistically significant peaks in the cyclic autocorrelation function (CAF) due to the CP [3]-[7]. If the parameters of the OFDM signal are unknown, this involves a search over a large delay range [4]-[5]. Although the parameters related to the CP-induced cyclostationarity are known for standard signals (note that diverse transmission modes can have different parameters) [7], cognitive users sharing the spectrum might employ the OFDM modulation with similar subcarrier frequency separation. Thus, detection of the primary user presence based on the CP-induced cyclostationarity can be unreliable. Other cyclostationarity-based methods involve the detection of cyclostationary signatures that are artificially created in the OFDM signals for detection and classification purposes [8]. The problem in

this case is that the embedding of these signatures cause additional overhead and data rate reduction. Other methods rely on the existence of pilot symbols for channel estimation or synchronization [9], assuming that these symbols are replicated according to a predefined time/frequency distribution. However, in many cases of practical interest, the pilot symbols are randomly generated, which yields zero correlation. The drawbacks of the previous methods make it necessary to further study the cyclostationarity-based features of existing OFDM standard signals. The third Generation Partnership Project Long Term Evolution (3GPP LTE) is one of the standards that employ the OFDM as the physical layer for its downlink traffic [10]-[11]. LTE is the next step forward in 3GPP cellular technology with increased user data rates and network capacity. In this paper, we study the second-order cyclostationarity of LTE OFDM signals (associated with the downlink transmission) induced by the reference signals (RSs) that repeat every frame. We model the signals and derive the analytical closed-form expressions for their RS-induced CAF and cyclic frequencies (CFs). Based on these results, we propose an algorithm for the detection of the LTE OFDM signals in the operating bands where the frequency division duplexing (FDD) downlink transmission is employed. The proposed algorithm provides a good detection performance at low signal-tonoise ratios (SNRs), short sensing times, and under diverse channel conditions. To the best of our knowledge, no previous work has been done on the detection of the primary users which employ LTE OFDM signals. The rest of the paper is organized as follows. Section II presents the model of the LTE OFDM signals used in the FDD downlink transmission. The second-order signal cyclostationarity is discussed in Section III, while the proposed signal detection algorithm is introduced in Section IV. Simulation results are shown in Section V, followed by conclusions and final remarks in Section VI. Derivation of the CAF and CFs of the LTE OFDM signals is provided in the Appendix. II. LTE OFDM SIGNAL MODEL A. Frame structure Fig. 1 shows the FDD downlink frame structure used in the LTE systems [10]. The frame time duration is 10 ms, and each frame is divided into 20 slots, with the slot duration equal to 0.5 ms. DL DL OFDM symbols, where N symb depends on Each slot contains N symb the CP length and useful symbol duration (equal to the inverse of the subcarrier frequency spacing) parameters of the OFDM signal. The LTE standard 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

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form a Multicast Broadcast Single Frequency Network (MBSFN) [10]-[11]. Here we consider the case where a single operational mode is employed in a cell, i.e., either MBSFN or non-MBSFN [10]-[11]. The MBSFN mode uses either Δf = 7.5 kHz or 15 kHz DL DL = 3 and N symb = 6, subcarrier spacing and long CP, with N symb respectively. On the other hand, the non-MBSFN mode employs Δf = 15 kHz subcarrier spacing and either short or long CP. With DL DL = 6, while with the short CP N symb = 7. The ratio the long CP N symb between the CP length and the useful OFDM symbol duration (Tcp / Tu ) equals 1/4 for long CP, while for short CP this is 10/128 for the first OFDM symbol in the slot and 9/128 for the remaining OFDM symbols in the slot.

Fig. 1. The FDD downlink frame structure in the LTE systems.

C. Reference signals RSs are embedded in the resource blocks of the transmission frame for channel estimation and cell search/ acquisition purposes [10]. Overall, there are 512 possible RSs. An RS is assigned to each cell of the network and acts as a cell identifier. Therefore, the RS repeats each downlink frame. Here, we study two types of RSs: the cell-specific RS associated with the non-MBSFN mode and the MBSFN RS associated with the MBSFN mode. Note that the terminology used here is according to [10]. The RSs are interspersed over the resource elements, being usually transmitted on some of the subcarriers of one or two non-consecutive symbols in each slot. An example is provided in Fig. 3, which shows the distribution of the cell-specific RS for long CP over one resource block and DL =6 OFDM symbols per slot and two consecutive slots ( N symb RB Nsc =12 subcarriers per resource block). As one can notice from this figure, the cell-specific RS is transmitted on the first and seventh subcarriers of the first OFDM symbol and on the fourth and tenth subcarriers of the fourth OFDM symbol in each slot. For the distribution of other RSs, the reader is referred to [10]. The RSs are usually binary pseudo-random sequences which are binary phase shift keying (BPSK) modulated.

B. Slot structure and resource grid The slot structure and associated resource grid used in the FDD downlink frame are illustrated in Fig. 2.

Fig. 3. Resource element mapping of cell-specific RS [10].

Fig. 2. The slot structure and resource grid in the FDD downlink frame [10].

The slot can be represented as a two dimensional resource grid DL OFDM symbols in time domain and consisting of N symb DL RB DL K = N RB N sc subcarriers in frequency domain, with N RB as the RB number of resource blocks and Nsc as the number of subcarriers in a resource block. Note that K represents the total number of subcarriers in an OFDM symbol. A resource block is defined as DL N symb consecutive OFDM symbols in time domain and NscRB consecutive subcarriers in frequency domain. NscRB equals 12 and 24 for the LTE signals with Δf = 15 kHz and 7.5 kHz subcarrier DL spacing, respectively. N RB then depends on the signal bandwidth; for possible values of this parameter the reader is referred to [11]. The smallest entity of the resource grid is called resource element; a DL resource block consists of N symb × N scRB resource elements.

D. Signal model According to the above description, we provide a general model for the LTE OFDM signal with any RS distribution. The continuous-time signal affected by additive Gaussian noise can be expressed as, ∞

∑

r (t ) = a[

K /2

∑

l =−∞ k =− K / 2, k ≠ 0 l mod N z ∈Z1 k ∉ A1 ∞

K /2

∑

+

∑

l =−∞ k =− K /2, k ≠ 0 l mod N z ∈Z1 k∈ A1 ∞

∑

+ a[

ck ,l g (t − lT )e j 2π k Δf (t −lT )]

K /2

∑

l =−∞ k =− K /2, k ≠ 0 l mod N z ∈Z 2 k∉ A2

+

∞

∑

K /2

∑

l =−∞ k =− K /2, k ≠ 0 l mod N z ∈Z 2 k ∈ A2

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bk ,l g (t − lT )e j 2π k Δf (t −lT )

bk ,l g (t − lT )e j 2π k Δf (t −lT )

ck ,l g (t − lT )e j 2π k Δf (t −lT )]


K /2

∑

l =−∞ k =− K /2, k ≠ 0 l mod N z ∉Z1 , Z 2

bk ,l g (t − lT )e j 2π k Δf (t −lT ) + w(t ), (1)

where a is the amplitude factor equal to 1/ K , N z is the repetition period for the RS distribution (in number of OFDM DL DL symbols), which equals either N symb or 2 N symb , Z1 and Z 2 are the sets of the OFDM symbols in which the RS is transmitted, A1 and A2 are the sets of subcarriers on which the RS is transmitted in the OFDM symbols belonging to Z1 and Z 2 , respectively, T is the OFDM symbol period equal to the useful symbol duration plus the CP length, g (t ) is the impulse response of the transmit and the receive filters in cascade, bk ,l and ck ,l are the data and RS symbols transmitted on the kth subcarrier and within the lth OFDM symbol, respectively, and w(t ) is the additive zero-mean Gaussian noise. Note that the data symbols bk ,l are taken either from a quadrature amplitude modulation (QAM) or PSK signal constellation, while the RS symbols ck ,l are taken from the BPSK signal constellation. Both data and RS symbols are zero-mean independent and identically distributed (i.i.d.) random variables. To ease the understanding of the expressions for the signal model in (1), here we provide a brief explanation. In all the terms on the right hand-side of (1), the inner summation is over the number of subcarriers, while the outer summation is over the OFDM symbol index. The first four terms model the OFDM symbols which include RS, while the fifth term expresses the remaining OFDM symbols. In the OFDM symbols with RS, there can be only two different assignments of the RS symbols on subcarriers. As such, the second and fourth terms in (1) represent the subcarriers where the RS is induced for the two possible assignments, whereas the first and third terms represent the data subcarriers within the same OFDM symbols. A similar model can be found for the LTE signal with short CP, by considering the longer duration of the first OFDM symbol in the slot; this is not presented here due to the lack of space. The discrete-time signal, r (n) = r (t ) |t = nf −1 , is obtained at the s receive-side by sampling r (t ) at rate f s .

Furthermore, for the discrete-time signal, r (n), the CAF at CF α and delay τ is given by (under the assumption of no aliasing) [12],

Rrα (τ ) = R rα (τ ) , where α = α f s

s∈Z1

× ∑ g ( n) e

s∈Z 2

(5)

,

n

κ = {α : α = ν Dz−1 , ν integer},

(6)

Dz = Tz f s is the number of samples over Tz , with Tz as the time period corresponding to N z OFDM symbols, D = Tf s is the number of samples over T , σ c2 is the variance of the RS symbols, and K r is the number of reference signal subcarriers. Based on (5) and (6), the CAF magnitude is presented in Figures 4 to 7 for both MBSFN and non-MBSFN modes. 0.06

0.05

0.04

0.03

0.02

0.01

-1

-0.5

0

0.5

1

α

1.5 -3

x 10

Fig. 4. The CAF magnitude versus CF at delay equal to DF for the LTE OFDM signals with non-MBSFN mode (cell-specific RS) and long CP. 0.05 0.045 0.04

∑ Rrα (τ)e j 2πα t ,

(2)

α ∈κ

I /2

∫

Rr (t , t + τ )e − j 2πα t dt ,

and κ = {α : R rα (τ ) ≠ 0} represents the set of CFs.

α

|R (DF)|

0.03 0.025 0.02 0.015

0.005

(3)

−I / 2

0.035

0.01

where R rα (τ ) is the CAF at CF α and delay τ , defined as I →∞

− j 2πα n

∑ e − j 2πα sD]

where α belongs to the set

0 -1.5

A random process, r (t ), is said to be second-order cyclostationary if its mean and autocorrelation are almost periodic functions of time [12]. The latter is expressed as a Fourier series as [12]

R rα (τ ) = lim I −1

and τ = τ f s .

2

A. Definitions

R r (t , t + τ ) =

(4)

Rrα (τ ) = Dz−1a 2σ c2 K r [ ∑ e − j 2πα sD +

III. SECOND-ORDER CYCLOSTATIONARITY OF THE LTE OFDM SIGNALS In this section we first provide a brief overview of the secondorder signal cyclostationarity, followed by the results we derived for the LTE OFDM signal used in the FDD downlink transmission (see Appendix for derivations).

−1

B. The second-order cyclostationarity of the signals of interest According to the derivations in the Appendix, the discretetime LTE OFDM signal, r (n), exhibits RS-induced secondorder cyclostationarity. The CAF at CF α and delays equal to integer multiples of the frame duration, DF (in number of samples) is given by

α

∞

∑

|R (DF)|

+a

0 -1.5

-1

-0.5

0

α

0.5

1

1.5 -3

x 10

Fig. 5. The CAF magnitude versus CF at delay equal to DF for the LTE OFDM signals with non-MBSFN mode (cell-specific RS) and short CP.

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0.14

0.12

0.08

α

|R (DF)|

0.1

0.06

0.04

0.02

0 -1.5

-1

-0.5

0

0.5

1

α

1.5 -3

x 10

Fig. 6. The CAF magnitude versus CF at delay equal to DF for the LTE OFDM signals with MBSFN mode (MBSFN RS) and Δf = 15 kHz.

- The CAF of the received signal, r (n) , is estimated (from N s samples) at tested frequency α and delay τ , and a vector ˆ α is formed as R r

0.14

0.12

0.1

ˆ α = [Re{Rˆ α ( τ )} Im{Rˆ α ( τ )}], R r r r

0.08

(7)

where Re{} ⋅ and Im{} ⋅ are the real and imaginary parts, respectively. - A statistic Ψ αr is computed for the tested frequency α and delay τ as

α

|R (DF)|

regardless of the mode, it can be used to detect the presence of any of the LTE OFDM signals employed in the FDD downlink transmission. As such, if we detect that α = 0 is a second-order CF for the delay equal to DF , we decide that an LTE OFDM signal is present in the corresponding frequency band. Note that the number of samples over a frame, DF , is known at the receive-side, as being equal to TF f s , with TF as the frame time duration ( TF = 10 ms and f s is selected based on an estimate of the signal bandwidth). For the second-order CF detection we use the statistical test developed in [13], which is briefly presented as follows. With this test, the presence of a CF is formulated as a binary hypothesis-testing problem, i.e., under hypothesis H 0 the tested frequency α is not a CF at delay τ , and under hypothesis H1 the tested frequency α is a CF at delay τ . The test consists of the following three steps:

0.06

0.04

0.02

0 -1.5

ˆ −1 R ˆα∑ ˆ α †, Ψ αr = N s R r r -1

-0.5

0

0.5

1

α

1.5 -3

x 10

Fig. 7. The CAF magnitude versus CF at delay equal to DF for the LTE OFDM signals with MBSFN mode (MBSFN RS) and Δf = 7.5 kHz.

The OFDM parameters are set as per Section V. As one can see from these figures, the CAF magnitude is higher for the case of MBSFN RS. This can be easily explained, as the MBSFN RS is induced on more subcarriers when compared with the cellspecific RS (see [10] for the RS distributions). Also, due to diverse RS distributions for different transmission modes, the non-zero CF values depend on the mode. We should also mention that simulation results for the CAF estimates agree with the theoretical findings presented in Figures 4 to 7; these are not shown due to the lack of space. Similar to the generic OFDM signals, the LTE OFDM signals exhibit a CP induced cyclostationarity (for this type of cyclostationarity exhibited by generic OFDM signals, see, e.g., [3]). However, this information can be unreliable for detection, as the cognitive users sharing the spectrum may employ the OFDM modulation with the same subcarrier frequency separation. Therefore, in this work, we exploit the RS-induced cyclostationarity for the detection of such signals. IV. PROPOSED ALGORITHM FOR THE LTE OFDM SIGNAL DETECTION According to the results presented in Section OFDM signals exhibit RS-induced cyclostationarity, with the CAF Rrα (τ ) ≠ 0 at CF α τ = DF for all transmission modes. As this

III, the LTE second-order = 0 and delay result holds

(8)

where the superscripts −1 and † denote the matrix inverse and transpose, respectively, and ∑ˆ is an estimate of the covariance matrix of Rˆ αr . For the covariance estimators one can see, e.g., [13]. - The test statistic Ψ αr is compared against a threshold, Γ , for decision making. If Ψ αr ≥ Γ , we decide that the tested frequency α is a CF at delay τ ; otherwise it is not. The threshold Γ is set for a given asymptotic probability of false alarm, which is defined as the asymptotic probability to decide that the tested frequency α is a CF at delay τ, when this is actually not. This can be expressed as Pr{Ψ αr ≥ Γ | H 0 }. By using that the statistics Ψ αr has an asymptotic chi-square distribution with two degrees of freedom under the hypothesis H 0 [13], the threshold Γ is obtained from the tables of the chi-squared distribution for a given value of this probability. V. SIMULATION RESULTS The performance of the algorithm proposed for the detection of the LTE OFDM signals used in the FDD downlink transmission is investigated here. The signal is simulated with 5 MHz double-sided bandwidth. The data subcarriers are modulated using 16-QAM, while the RS subcarriers are modulated using BPSK. Unit variance constellations are considered. A raised root cosine pulse shape window with 0.025 roll-off factor is employed at the transmit-side. We consider the additive white Gaussian noise (AWGN), and ITU-R pedestrian and vehicular A channels. For the delay profile of the fading channels the reader is referred to [14]. The maximum Doppler frequencies equal 7.28 Hz and 145.69 Hz for the pedestrian and vehicular fading channels, respectively. At the receive-side we use a filter to remove the out-of-band noise, and set

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1

the SNR at the output of this filter. The sampling rate is 7.68 MHz. The probability of detection Pd , is used as a performance measure; this is estimated based on 1000 Monte Carlo trials. Unless otherwise mentioned, the sensing time equals 2TF , the channel is AWGN, and Pfa = 0.05.

1

MBSFN

non-MBSFN

Pd

0.6 0.4

MBSFN, Δf= 7.5 kHz MBSFN, Δf= 15 kHz non-MBSFN, long CP non-MBSFN, short CP

0.2 -8

0.4 AWGN Channel Pedestrian A Channel Vehicular A Channel

0.2 0 -10

-4 -2 0 2 SNR (dB) Fig. 11. Probability of detection vs. SNR, for the LTE OFDM signals with non-MBSFN mode and short CP under different channel conditions.

0.8

0 -10

0.6 Pd

Fig. 8 shows the probability of detection versus SNR for the investigated LTE OFDM signals. As one can notice, the best performance is obtained with the MBSFN RS (for both 15 kHz and 7.5 kHz subcarrier frequency spacing), followed by the cellspecific RS with long CP, and the cell-specific RS with short CP.

0.8

-6

-4 -2 0 SNR (dB) Fig. 8. Probability of detection vs. SNR for the LTE OFDM signals.

2

1 0.8

Pd

0.6

-8

-6

Note that this is in agreement with the CAF magnitudes in Figures 4 to 7. Further results are presented for the cell-specific RS with short CP, as the worst performance is reached in this case. Fig. 9 depicts the probability of detection versus SNR for different sensing times. As expected, the detection performance improves with an increase in the sensing time. The receiver operating characteristic is plotted in Fig. 10 at different sensing times and SNRs. With 3TF sensing time, the detector has a very good performance at -6 dB SNR, while the performance degrades with 2TF sensing time at the same SNR. A similar performance is attained with 2TF sensing time at higher SNR (-4 dB). The probability of detection versus SNR is shown in Fig. 11 for the AWGN, and ITU-R pedestrian and vehicular A fading channels. It is noteworthy that an increase in the performance degradation can be noticed in the vehicular A channel at lower SNRs, when compared with the AWGN and pedestrian A channels.

0.4

VI. CONCLUSION 3 TF (30 ms)

0.2

2 TF (20 ms) 0 -10

-8

-6

-4 -2 0 2 SNR (dB) Fig. 9. Probability of detection vs. SNR for the LTE OFDM signals with non-MBSFN mode and short CP, for different sensing times. 1 0.8

The RS-induced second-order cyclostationarity of the LTE OFDM signals used in the FDD downlink transmission is studied in this paper, and closed from expressions for the CAF and CFs are derived. This distinctive characteristic is further exploited for signal detection. An RS-induced cyclostationarity-based algorithm is proposed for the detection of the previously mentioned LTE signals, which provides a good performance with reasonable short sensing time, at low SNR, and under diverse channel conditions. In future work we plan to extend our algorithm to handle the detection of the LTE FDD downlink signals when both MBSFN and non-MBSFN modes can be used in a cell, LTE FDD uplink signals, as well as LTE time division duplex signals.

0.6 Pd

REFERENCES D. Cabric and R. W. Brodersen, “Physical layer design issues unique to cognitive radio systems,” in Proc. IEEE PIMRC, 2005, pp. 759 – 763. [2] T. Yucek and H. Arslan, “A survey of spectrum sensing algorithms for cognitive radio applications,” IEEE Communication Surveys and Tutorials, vol. 11, pp. 116-130, 2009. [3] M. Oner and F. Jondral, “On the extraction of the channel allocation information in spectrum pooling system," IEEE Journal on Selected Areas in Communications, vol. 25, pp. 558-565, 2007. [4] A. Punchihewa, Q. Zhang, O. A. Dobre, C. Spooner, S. Rajan, and R. Inkol, "On the cyclcostationarity of OFDM and single carrier linearly digitally modulated signals in time dispersive channels: [1]

0.4 2 TF , -4 dB SNR 3 TF , -6 dB SNR

0.2

2 TF , -6 dB SNR 0 0

0.2

0.4

Pfa

0.6

0.8

1

Fig. 10. The receiver operating characteristic for LTE OFDM signals with nonMBSFN mode and short CP, for diverse sensing times and SNRs.

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[5]

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

Theoretical developments and application," IEEE Transactions on Wireless Communications, accepted to publication March 2010. A. Dobre et al., “On the cyclostationarity of OFDM and single carrier linearly digitally modulated signals in time dispersive channels with applications to modulation recognition,” in Proc. IEEE WCNC, 2008, pp. 1284-1289. A. Bouzegzi, P. Jallon, and P. Ciblat, “A second order statistics based algorithm for blind recognition of OFDM based systems,” in Proc. IEEE GLOBECOM, 2008, pp. 1-5. A. Al-Habashna, O. A. Dobre, R. Venkatesan, and D. C. Popescu, “Joint signal detection and classification of mobile WiMAX and LTE OFDM signals for cognitive radio,” submitted to IEEE Asilomar 2010. P. D. Sutton, K. E. Nolan, and L. E. Doyle, “Cyclostationary signatures in practical cognitive radio applications,” IEEE Journal on Selected Areas in Communications, vol. 26, pp. 13-24, 2008. F.-X. Socheleau, P. Ciblat, S. Houcke, “OFDM system identification for cognitive radio based on pilot-induced cyclostationarity,” in Proc. IEEE WCNC, 2009, pp. 1-6. 3GPP TS 36.211: Evolved Universal Terrestrial Radio Access (E-UTRA); Physical channels and modulation. 3GPP TS 36.101: Evolved Universal Terrestrial Radio Access (E-UTRA); User Equipment (UE) radio transmission and reception. C. M. Spooner and W. A. Gardner, “The cumulant theory of cyclostationary time-series, part I: foundation and part II: development and applications," IEEE Trans. Sig. Proc., vol. 42, pp. 3387-3429, 1994. A. V. Dandawate and G. B. Giannakis, “Statistical tests for presence of cyclostationarity,” IEEE Trans. Signal Processing, vol. 42 , pp. 2355–2369, 1994. A. F. Molisch, Wireless Communications. Wiley, 2005. H. L. van Trees, Detection, Estimation, and Modulation Theory. Wiley, 2001.

APPENDIX: DERIVATION OF THE RS-INDUCED CAF AND CFS FOR THE LTE OFDM SIGNALS By using the signal model in (1), the autocorrelation function of r (t ) can be expressed as the sum of the autocorrelation functions corresponding to any two signal components, signal and noise components, and noise component, ∞

∞

∑

R r (t ,τ ) = a 2

K /2

∑

K /2

∑

∑

l1 =−∞ l2 =−∞ k1 =− K / 2, k1 ≠ 0 k2 =− K / 2, k2 ≠ 0 l1 mod N z ∈Z1 l2 mod N z ∈Z1 k1∉ A1 k2 ∉A1

E (bl1 , k1 bl∗2 , k2 )

× g (t − l1T )e j 2π k1Δf ( t −l1T )g * (t − l2T + τ )e − j 2π k2 Δf ( t −l2T +τ ) +a 2

∞

∞

∑

K /2

∑

∑

K /2

∑

l1 =−∞ l2 =−∞ k1 =− K / 2, k1 ≠ 0 k2 =− K / 2, k 2 ≠ 0 l1 mod N z ∈Z1 l2 mod N z ∈Z1 k1∈A1 k2 ∈ A1

× g (t − l1T )e

j 2π k1Δf ( t − l1T ) *

g (t − l2T + τ )e

R r (t ,τ ) = a 2σ c2

×

∑

∑

l1 =−∞ l2 =−∞ l1 mod N z ∈Z1 l2 mod N z ∈Z1 l2 = l1 + μ N F

K /2

∑

k =− K / 2, k ≠ 0 k ∈A1

e

g (t − l1T ) g * (t − l2T + τ )

e − j 2π k Δf (τ +l1T −l2T ),

(10)

where σ c2 is the variance of the RS symbols. Note that in this case, g (t − l1T ) g ∗ (t − l2T + τ ) = 0 unless l2 − l1 = N FτTF−1. Based on this and after some mathematical manipulations, (10) can be rewritten as ∞

R r (t ,τ ) = a 2σ c2[ K1

∑

∞

2

g (t − lT ) +K 2

l =−∞ l mod N z ∈Z1

∑

2

g (t − lT ) ], (11)

l =−∞ l mod N z ∈Z 2

where K1 and K 2 are the number of subcarriers in A1 and A2 respectively. Given that K1 = K 2 = K r , (11) becomes 2 Rr (t ,τ ) = a 2σ c2 K r g (t ) ⊗ [

∞

∑

l =−∞ l mod N z ∈Z1

∑

2

s∈Z1

∞

∑ ∑

s∈Z 2

∑

δ (t − lT )]

l =−∞ l mod N z ∈Z 2

∞

= a 2σ c2 K r g (t ) ⊗ [ ∑ +

∞

δ (t − lT ) +

δ (t − lT − sT )

l =−∞ l mod N z = 0

δ (t − lT − sT )]

l =−∞ l mod N z = 0

= a 2σ c2 K r g (t ) ⊗ [ ∑ 2

∞

∑ δ (t −ν Tz − sT )

s∈Z1 ν =−∞

∞

+∑

(12)

∑ δ (t −ν Tz − sT )].

s∈Z 2 ν =−∞

By taking the Fourier transform of (12), and using the convolution theorem and the identity ℑ{∑ν δ (t − ν T )} = T −1 ∑ν δ (α − ν T −1 ) ,

s∈Z1

∞

∫

2

g (t ) e

− j 2πα t

dt

∞

∑ e − j 2πα sT ]

s∈Z 2

∑ δ (α −ν

ν =−∞

−∞

(9)

− j 2π k Δf (τ + l1T − l2T )

∑

k =− K / 2, k ≠ 0 k ∈A2

×

We further investigate (9) for delays equal to integer multiples of the frame time duration. At these delays, one can show that the only non-zero terms are due to the repetition of the RS, and correspond to E (cl1 , k1 cl∗2 , k2 ) ≠ 0 when k1 = k2 (= k ) and l2 − l1 = μ N F , with μ as an integer and N F as the number of the OFDM symbols in the transmission frame. Thus, (9) becomes ∞

g (t − l1T ) g * (t − l2T + τ )

ℑ{R r (t ,τ )} = Tz−1a 2σ c2 K r [ ∑ e − j 2πα sT +

where E (.) is the expectation operator.

∞

K /2

×

− j 2π k2 Δf ( t − l2T +τ )

+.... + E ( w(t ) w (t + τ )),

∞

∑

l1 =−∞ l2 =−∞ l1 mod N z ∈Z 2 l2 mod N z ∈Z 2 l2 =l1 + μ N F

E (cl1 , k1 cl∗2 , k2 )

*

∞

∑

+a 2σ c2

(13)

Tz−1 ).

As one can notice from (13), ℑ{R r (t ,τ )} is non-zero only at α = ν Tz−1 , with ν as an integer. By using the inverse Fourier transform of (13), one can easily see that the autocorrelation function has a Fourier series representation. Furthermore, it is straightforward that the CAF at CF α and delay τ , and the set of CFs, κ , are given respectively as R r α (τ ) = Tz−1a 2σ c2 K r [ ∑ e − j 2πα sT + s∈Z1

×

∞

∫

2

g (t ) e

∑ e − j 2πα sT ]

s∈Z 2

(14)

− j 2πα t

dt ,

−∞

and κ = {α : α = ν Tz−1 , ν integer}.

(15)

The results in (5) and (6) can be easily derived using (4), (14), and (15). 978-1-4244-5638-3/10/$26.00 ©2010 IEEE