Secondary Transceiver Design in the Presence of Frequency Offset between OFDM-based Primary and Secondary Systems

Transcription

1 Secondary Transceiver Design in the Presence of Frequency Offset between OFDM-based Primary and Secondary Systems Zhikun Xu and Chenyang Yang School of Electronics and Information Engineering, Beihang University 37 Xueyuan Road, Haidian District, Beijing, P. R. China Abstract In a cognitive network where both rimary and secondary systems are Orthogonal Frequency Division Multilexing modulated, carrier frequency offset between the two systems is inevitable and may cause harmful interference. In this aer, we jointly otimize the transceiver for the secondary system considering the frequency offset between secondary transmitter (ST) and rimary receiver (PR). We first derive unified interference constraints with different information of inference channels known at ST, and formulate the transceiver design roblem based on imum mean square criterion as a convex otimization roblem. To reveal the structure of secondary transceiver, we derive closed-form re-rocessors in two secial cases. It is shown from the simulations that with the increase of frequency offset, the erformance of rimary system degrades evidently when its bandwidth is smaller than that of secondary system, whereas the erformance of secondary system using the roosed transceiver imroves due to frequency diversity. I. INTRODUCTION Cognitive radio (CR) technology has been identified as an efficient means to reuse the scarce sectrum resources recently [1]. Under the co-existence constraints in the licensed band, underlay sectrum sharing strategy can rovide large network caacity for secondary users when their bandwidths exceed those of rimary systems [2]. Orthogonal Frequency Division Multilexing (OFDM) is a cometitive candidate for CR systems due to its high sectrum flexibility [3, 4]. Assug that the central frequency and bandwidth of a rimary system can be estimated by sectrum sensing techniques, various issues of designing overlay CR OFDM transceivers have been studied [3 5]. On the other hand, OFDM techniques have been alied in many wideband systems [6]. If we know the rimary system is OFDM modulated and we know its system arameters, it is ossible to design an OFDM-based CR system in the underlay mode. The caacities of CR systems over flat fading channels with various interference constraints have been analyzed in [7, 8]. Their results imly that OFDM secondary systems can oerate in the same sectrum band with OFDM rimary systems over frequency selective channels, when the rimary and secondary systems have the same subcarrier sacing and are synchronous. Using these assumtions imlicitly, [9] otimizes the ower and subcarrier allocation, and [10] designs the transmit covariance matrix for OFDM secondary systems. When rimary and secondary systems are non-cooerative, carrier frequency synchronization between the two systems is hard to achieve while symbol tig synchronization is easier to imlement. It is well known that the OFDM system is very sensitive to the frequency offset between the oscillators of its transmitter and receiver [6]. The frequency offset between secondary transmitter (ST) and rimary receiver (PR) may degrade the erformance of rimary system significantly. Therefore, it is critical to design an OFDM-based secondary system considering the frequency offset between rimary and secondary transceivers. As far as the authors know, this has not been addressed in the literature. In this aer, both rimary and secondary systems are suosed to be OFDM-based time division dulexing (TDD) systems. It is assumed that the rimary system s arameters such as training sequence, central frequency and subcarrier sacing are known at the ST [4]. Two tyes of channel state information (CSI) of the interference channel from ST to PR are considered, including the instantaneous CSI and the statistical CSI. Linear transceiver for the secondary system are designed in the resence of frequency offset between ST and PR using imum mean square error (MMSE) criterion under the transmission ower constraint and interference constraints. II. SYSTEM MODEL Consider an OFDM-based secondary system coexists with an OFDM-based rimary system in the same sectrum band, as shown in Fig. 1. The subcarrier sacings of the two systems are assumed to be the same. The number of available subcarriers is N s. The number of subcarriers used by the rimary system is and that used by the secondary system is N s, where N s. The frequency offset between ST and PR, δ f, is assumed to be erfectly known at ST which can be estimated by using the received training sequences from PR. We also assume that the rimary and secondary systems are erfectly synchronized between their own transceivers in both symbol tig and /10/$ IEEE

2 Fig. 1. PT ST h ss h h s PT: rimary transmitter PR: rimary receiver ST: secondary transmitter SR: secondary receiver Network structure consisting of a rimary link and a secondary link carrier frequency, which can be easily achieved through traditional methods. We also assume that the ST and PR are synchronized in symbol tig, which can be accomlished with the following method. The ST first obtains the ositions of rimary users through GPS or other ositioning methods, and then estimates transmission time from PT to PR and from ST to PR. Based on above time information, ST can send the signal at roer time to synchronize with PR. Let h ss, h s and h denote the channels from ST to SR, from ST to PR and from PT to PR, with the numbers of resolvable aths being L ss, L s and L, resectively. We assume that h ss is erfectly known by the secondary transceiver, and different tyes of channel information from ST to PR are known by ST, i.e., the instantaneous CSI, h s, and the statistical CSI, which is the covariance matrix, R s. The CSI can also be obtained by making use of the training sequences from PR. A. Signal Model of Secondary Transceiver At the ST, the data symbols d 0,d 1,,d Ns 1 are first serial-arallel converted then re-rocessed by a matrix B. After its outut signal x f s assing an inverse discrete Fourier transform (IDFT) and inserting a cyclic refix (CP), an OFDM symbol is generated. An OFDM symbol without CP can be exressed as x s = F H x f s = F H Bd, (1) where d =[d 0,d 1,,d Ns 1] T, and F is the DFT matrix. Assug that E d [dd H ]=I Ns, where E x [ ] denotes exectation over random vector x and I Ns denotes N s N s identity matrix, the transmission ower constraint can be exressed as PR E d [Tr(x s x H s )] = Tr(BB H ) P t. (2) When the secondary transceiver are synchronous in both symbol tig and carrier frequency, the discrete received OFDM signal after removing CP is SR y s = H ss x s + u + n = H ss F H Bd + ñ, (3) where u denotes the interference signal from PT to SR, n CN(0,σn), 2 i.e., n is the additive white Gaussian noise (AWGN) with zero mean and variance σn, 2 and ñ = u + n reresents the total interference at SR. H ss is a N s N s circulant matrix whose first column is [h ss 0,h ss 1,,h ss L, 0,, ss 1 0]T. {h ss i }Lss 1 i=0 are the coefficients of h ss. H ss can be decomosed as H ss = F H Λ ss F where the diagonal entries of diagonal matrix Λ ss are the frequency resonses of h ss [11, Cha. 3]. The SR uses a ost-rocessor in frequency domain to estimate the transmitted data, which is where G is a N s N s matrix. d = GFy s, (4) B. Interference Constraints at Primary Receiver To rotect the rimary system, the interference at PR should be blow a certain threshold. A reasonable constraint for an OFDM-based rimary system is to restrict the interference ower on each subcarrier that the system uses. When the frequency offset exists between ST and PR, the discrete interference signal in time domain received by the PR can be exressed as [11, Cha. 4] u s n = 1 N e j2π nδ f s 1 Nsfs λ s nk ej2π Ns,n=0,,N s 1, Ns k=0 k xs,f k where x s,f k is the kth element of x f s, λ s k denotes the frequency resonse value of h s on the kth subcarrier, and f s reresents the subcarrier sacing. Considering that x f s = Bd and Λ s = FH s F H, we can rewrite the interference signal in matrix form as u s = Δ f F H Λ s x f s = Δ f H s F H Bd, where Δ f = diag{1,e j2π δ f Nsfs,,e j2π (N s 1)δ f Nsfs }, Λ s = diag{λ s 0,λs 1,,λs N s 1 }, and diag{x 1,...,x N } denotes a N N diagonal matrix. Then, the interference imosed on the ith subcarrier of the rimary system is u s,f i = e H i Fu s = e H i FΔ f H s F H Bd i Γ, (5) where e i denotes a column vector with 1 in the ith osition and 0 in other ositions, and Γ denotes the set of subcarrier ositions used by the rimary system. According to the different interference channel information that ST can obtain, we consider the following two kinds of interference constraints at PR: 1. When the instantaneous CSI, h s, is available, the interference must satisfy E d [ u s,f i 2 ] Pi ICSI i Γ. Uon substituting (5) and after some maniulations, the interference constraints can be written as Tr(a i a H i BB H ) P ICSI i i Γ, (6) is the interfer- where a i = FH H sδ H f FH e i, and Pi ICSI ence threshold.

3 and 2. When the statistical CSI, namely, the covariance matrix of h s, R s, is known, the interference constraints are E d,hs [ u s,f i 2 ] Pi SCSI i Γ. After some maniulations (see [12] for details), the interference constraints can be derived as Tr(A i R s A H i BB H ) P SCSI i i Γ, (7) where the nth column of N s L s matrix A i is FΠ n 1,H Δ H f FH e i, Π =[e 1, e 2,, e Ns 1, e 0 ], and Pi SCSI is the interference threshold. Define { ai a H i, case 1 Ψ i A i R s A H i, case 2 P th i { P ICSI i, case 1 Pi SCSI, case 2, then (6) and (7) can be exressed in a unified form Tr(Ψ i BB H ) P th i i Γ. (8) III. PROBLEM FORMULATION In this section, we jointly design the re-rocessing matrix B at ST and the ost-rocessing matrix G at SR. The estimation error of data symbol is e = d d =(GFH ss F H B I Ns )d + GFñ. Then the mean square error (MSE) is Tr(E[ee H ]) = Tr(G(H ss F H BB H FH H ss + Rñ)G H GH ss F H B (GH ss F H B) H + I Ns ), (9) where Rñ = E[ññ H ] is the covariance matrix of the total interference at SR. Considering the constraints (2) and (8), the roblem to jointly design B and G based on MMSE criterion can be formulated as B,G Tr(E[eeH ]) Tr(BB H ) P t Tr(Ψ i BB H ) P th i i Γ. By imizing the objective function with resect to G when B is given, we can easily obtain G = B H FH H ss(h ss F H BB H FH H ss + Rñ) 1 F H. (10) After substituting (10) into (9) and defining a correlation matrix U = BB H, the otimization roblem becomes U Tr(Rñ(H ss F H UFH H ss + Rñ) 1 ) Tr(U) P t Tr(Ψ i U) Pi th i Γ U 0. (11) By using the Schur s comlement method introduced in [13], the nonconvex roblem shown in (11) can be transformed into the following semidefinite rogramg roblem W,U Tr(R ñw) Tr(U) P t Tr(Ψ i U) Pi th i Γ [ W INs I Ns H ss F H UFH H ss + Rñ ] 0 U 0. (12) We can obtain the otimal U by using the interior oint method [14]. Let the eigenvalue decomosition of U be QΛ Q H, then we can obtain the otimal re-rocessing matrix B = QΛ 1/2, where Q is a recoding matrix and Λ is a ower allocation matrix. The corresonding otimal ostrocessor matrix G can then be comuted by substituting B into (10). IV. THE IMPACT OF THE FREQUENCY OFFSET ON SECONDARY TRANSCEIVER In this section, we will find the closed-form solutions of the roblem (12) in two secial cases to gain more insight on the transceiver structure when the frequency offset between ST and PR exists. A. No Frequency Offset Exists between ST and PR When there is no frequency offset between ST and PR, the otimal structure of secondary system is given by the following theorem. Theorem 1: When δ f =0and the total interference at SR is white, the otimal U is a diagonal matrix and the otimal recoder Q = I Ns. Proof: The roof is shown in [12]. Remark 1: Since the otimal recoder Q = I Ns, we can obtain the otimal re-rocessor B = Λ 1/2. It is not hard to derive the ost-rocessor G = Λ 1/2 Λ H ss(λ ss Λ Λ H ss + σñ 2I N s ) 1 from (10), where σñ 2 is the variance of the total interference at SR. This indicates that when there is no frequency offset between ST and PR, the otimal rocessing of secondary system is to allocate ower on each subcarrier at the transmitter and to use one-ta MMSE equalization at the receiver, which is the same as the rocessing in traditional OFDM systems without interference constraints [13]. B. Frequency Offset Exists between ST and PR When there exists frequency offset between ST and PR, the closed-form solutions of the roblem shown in (12) can not be obtained in general cases. To gain some insight of the imact of δ f on the structure and erformance of secondary system, we consider a secial case as follows. Theorem 2: Assume that = N s 1 and the instantaneous CSI, h s, is known by the ST. When the interference thresholds Pi th =0, i Γ, the otimal recoder Q = c Λ 1 s FΔ H f FH e i0 Λ 1 s FΔ H f FH e i0, (13)

4 where i 0 is the index of the subcarrier not being used by the rimary system and c is an arbitrary comlex number with unit amlitude. If the total interference at SR is white, the MSE of the secondary system is Tr(E[ee H 1 ]) = N s 1+. (14) 1+ Pt ΛssΛ 1 s FΔ H f FH e i0 2 σñ 2 Λ 1 s FΔ H f FH e i0 2 Proof: The roof is shown in [12]. Remark 2: It is shown from Theorem 2 that the otimal Q is no longer an identity matrix, and the MSE of the secondary system deends on δ f, which is different from the imact of frequency offset between traditional OFDM transceiver. BER N =8 NIT= 25dB N =8 NIT= 15dB =8 NIT= 5dB N NIT= 25dB N NIT= 15dB NIT= 5dB No Inf V. SIMULATION RESULTS In this section, we first show the imact of the frequency offset on the rimary system by simulating the erformance of rimary system when ST does not know the frequency offset. Then we evaluate the erformance of secondary system when the roosed transceiver are used. In the simulation, we assume that PT is far away from SR and does not induce interference to SR for simlicity. 1 In both systems, the received SNRs are set to be 20 db, the variances of noises are assumed to be identical, and BPSK modulation is emloyed. Because the comlexity of solving roblem (12) increases raidly with the subcarrier number, we only simulate small number of subcarriers, but this will not change the conclusion. To understand the imact of the bandwidth, we consider two cases for the rimary system, and =8, while the number of subcarriers used by the secondary system N s. We consider frequency selective channels with five resolvable aths, i.e., L ss = L s = L = 5. Each element of h ss, h s and h is indeendent and identically distributed subjecting to CN(0,ρ ss /L ss ), CN(0,ρ s /L s ) and CN(0,ρ /L ), resectively. ρ ss, ρ s and ρ are the large-scale fading gains of the corresonding channels. Without loss of generality, we assume ρ ss =1and ρ =1. In the legend, NMSE means the normalized MSE, ICSI and SCSI resectively stand for the instantaneous CSI and statistical CSI. The simulation results are averaged over 1000 Monte Carlo tests. Once the SNR is given, the erformance of both systems deends on the normalized interference threshold, which is NsP th i defined as NIT i = ρ sp t, i Γ. In the simulations, the interference threshold on each subcarrier Pi th is assumed to have the same value P th and is equal to the variance of noise at PR. The corresonding NIT i is denoted as NIT. We use NIT to reflect the imact of the large-scale fading gain ρ s or the distance between ST and PR on their erformance. We first analyze the erformance of rimary system when δ f is not re-comensated by ST and the instantaneous CSI is known at ST. The case when the statistical CSI is known is almost the same and thus is omitted here. Figure 2 illustrates 1 If the interference exists and is not white, a whitening filter can be alied at SR first. This filter can then be incororated into the channel matrix H ss as did in [10] Fig. 2. BER of rimary system vs. δ f /f s when δ f is not re-comensated by ST and the instantaneous CSI is known at ST the BER of rimary system versus δ f under different interference constraints. We also rovide the result when there is no interference at PR for reference, which is shown as No Inf in the legend. All results when overla. From the figure, we can see that the erformance of rimary system degrades with the increase of δ f when =8, while the erformance remains invariant when. This is because when the bandwidths of the two systems are identical, the PR sees the interference from ST with a flat sectrum whose energy does not deend on the frequency offset. When the bandwidth of secondary system exceeds that of rimary system, more interference is introduced to the sideband of the rimary system and thus the interference to rimary system increases as δ f rises. We can also observe that the BER increases with the decrease of NIT (i.e., the increase of ρ s ) when =8. This indicates that the rimary system is more sensitive to the frequency offset when the interference constraint is tight. Consequently, we can not ignore the frequency offset between ST and PR when the secondary system has larger bandwidth than that of the rimary system. We next analyze the imact of δ f on the erformance of secondary system. The NIT is set to be 5dB. Figure 3 shows the NMSE and BER of secondary system when the instantaneous CSI is known. The henomenon is similar when the statistical CSI is known, and is not shown here. We can see from the figure that when δ f =0, the NMSE/BER erformance of the otimal scheme is the same as that of ower-allocation-only scheme, which validates Theorem 1. When δ f 0, the otimal scheme outerforms the ower-allocation-only scheme. The erformance of the otimal scheme even imroves with large δ f, which is quite different from the imact of the frequency offset between traditional OFDM transceiver on its erformance 2.As increases, the erformance of secondary system degrades since more interference constraints are intro- 2 In a traditional OFDM system, after the frequency offset is erfectly estimated, its imact can be eliated comletely by frequency correction at the receiver, thus the erformance is indeendent of frequency offset.

5 NMSE =8 (a) NMSE Otimal Scheme PA Only Scheme (b) BER 10 2 NMSE SCSI ICSI N SCSI N =8 ICSI N =8 BER Otimal Scheme 10 3 N =8 PA Only Scheme NIT(dB) Fig. 3. Performance of secondary system with different schemes when the instantaneous CSI is known, (a)nmse vs. δ f /f s,(b)ber vs. δ f /f s Fig. 4. NMSE of secondary system vs. NIT when δ f =0.5f s duced, whereas the BER degrades slower than the NMSE. An intuitive exlanation is that when δ f 0, the otimal recoder Q is no longer an identity matrix, which indicates that each data symbol is transmitted on several subcarriers rather than one subcarrier. As a result, the frequency diversity gain is obtained and the BER reduces. Finally, we analyze the imact of interference constraints on the erformance of secondary system. δ f is set to be 0.5f s. The NMSE of secondary system versus NIT is shown in Fig. 4. The BER erformance is similar and is omitted here due to the lack of sace. It is shown that when NIT 0, i.e., ρ s, the NMSE of the secondary system aroaches to 1 when, since the ower on all of its subcarriers is forced to zero. The erformance with the instantaneous CSI known is better than that with the statistical CSI known when =8. As NIT increases, the interference constraints become looser and looser and thus the erformance of secondary system imroves. Meanwhile, the ga between the two CSI conditions reduces when =8. When NIT, i.e., ρ s 0, the interference constraints become inactive and the erformance with the instantaneous CSI known is identical to that with the statistical CSI known. VI. CONCLUSION In this aer, we have designed the linear MMSE transceiver under the transmission ower constraint and interference constraints for an OFDM secondary system co-existing with an OFDM rimary system, when frequency offset between ST and PR exists. The otimal solution can be obtained by using convex otimization techniques. When there is no frequency offset between ST and PR, the otimal rocesser for secondary system is ower allocation at ST and one-ta MMSE equalization at SR. When frequency offset between ST and PR exists, the ST needs to use both ower allocation and re-coding. It is shown from the simulations that when the two systems have identical bandwidth, the rimary system is not affected by the frequency offset. When the bandwidth of secondary system exceeds that of rimary system, the erformance of rimary system degrades and the rimary system is more sensitive to the frequency offset when the interference constraint is tight. By using the roosed transceiver, the erformance of rimary system is not affected by the frequency offset since the interference constraints are met, whereas the erformance of secondary system even imroves due to the frequency diversity introduced by the recoding. REFERENCES [1] I. F. Akyildiz, W.-Y. Lee, M. C. Vuran, and S. Mohanty, Next generation dynamic sectrum access/cognitive radio wireless networks: A survey, Comuter Networks, vol. 50, no. 13, , Set [2] D. Zhang, Z. Tian, and G. Wei, Satial caacity of narrowband vs. ultrawideband cognitive radio systems, IEEE Trans. Wireless Commun., vol. 7, no. 11, , Nov [3] T. A. Weiss and F. K. Jondral, Sectrum ooling: An innovative strategy for the enhancement of sectrum efficiency, IEEE Commun. Mag., vol. 42, no. 3,. 8 14, Mar [4] H. A. Mahmoub, T. Yucek, and H. Arslan, OFDM for cognitive radio: Merits and challenges, IEEE Wireless Commun., vol. 16, no. 2,. 6 15, Ar [5] J. Ma, G. Y. Li, and B. Juang, Signal rocessing in cognitive radio, in Proc. IEEE, vol. 97, no. 5, May 2009, [6] T. Hwang, C. Yang, G. Wu, S. Li, and G. Y. Li, OFDM and its wireless alications: A survey, IEEE Trans. Veh. Technol., vol. 58, no. 04, , May [7] A. Ghasemi and E. S. Sousa, Fundamental limits of sectrum-sharing in fading environments, IEEE Trans. Wireless Commun., vol. 6, no. 2, , Feb [8] L. Musavian and S. Aissa, Caacity and ower allocation for sectrumsharing communications in fading channels, IEEE Trans. Wireless Commun., vol. 8, no. 1, , Jan [9] A. Attar, O. Holland, M. Nakhai, and A. Aghvami, Interference-limited resource allocation for cognitive radio in orthogonal frequency division multilexing networks, IET Commun., vol. 6, no. 6,. 6 15, Ar [10] R. Zhang and Y. C. Liang, Exloiting multi-antennas for oortunistic sectrum sharing in cognitive radio networks, IEEE J. Select. Areas Commun., vol. 2, no. 1, , Feb [11] Y. G. Li and G. L. Stuber, Orthogonal Frequency Division Multilexing for Wireless Communications. Sringer, [12] Z. Xu and C. Yang, Secondary transceiver design in the resence of frequency offset between OFDM-based rimary and secondary systems, Journal aer in rearation. [13] Z. Q. Luo, T. Davidson, G. Giannakis, and K. Wong, Transceiver otimization for block-based multile-access through ISI channels, IEEE Trans. Signal Process., vol. 52, no. 4, , Ar [14] L. Vandanberghe and S. Boyd, Semidefinite rogramg, SIAM Review, vol. 31, no. 1, , Mar