An Improved Unique Word DFT- Spread OFDM Scheme for 5G Systems

Transcription

1 1 An Improved Unique Word DFT- pread OFDM cheme for 5G ystems Alphan Şahin, Rui Yang, Monisha Ghosh, Robert L Olesen {Alphanahin, RuiYang, RobertOlesen}@InterDigitalcom, monisha@uchicagoedu Abstract In this paper, we propose a new waveform based on discrete Fourier transform spread orthogonal frequency division multiplexing (DFT--OFDM), in which the tail of the DFT-- OFDM symbol contains a fixed sequence, ie, a unique word (UW) The proposed waveform, called UW DFT--OFDM, improves upon both existing zero-tail (ZT) DFT--OFDM and UW OFDM schemes by removing the impact of data symbols on the tail of the transmitted signal This is done by creating a redundant symbol vector that has 1% of the total transmitted energy approximately As a result, the proposed UW DFT--OFDM scheme keeps the advantages of UW OFDM and ZT DFT--OFDM such as the circular convolution of the channel without the use of a cyclic prefix (C), low peak-to-average power ratio (AR), and low out-ofband (OOB) emission In addition, by generating the UW sequence at the input of the DFT process, simpler receiver operation is obtained Keywords 5G, DFT--OFDM, C, unique word I INTRODUCTION Fifth generation (5G) systems will require significant improvements over the state-of-the-art OFDM based systems in order to support the increasing demands of higher traffic densities, higher spectral efficiency, very low latency, and low power Improvements of the air interface design is seen as a key method for achieving these goals To this end, we propose a novel waveform structure based on discrete Fourier transform spread orthogonal frequency division multiplexing (DFT-- OFDM) which offers a number of important advantages for a flexible and energy efficient 5G mobile broadband air interface It is well known that orthogonal frequency division multiplexing (OFDM) with a cyclic prefix (C) is an effective scheme for dealing with multipath channels that introduce intersymbol interference (II), and hence has been widely adopted in many standards such as 3G LTE and IEEE imilar schemes, such as DFT--OFDM and single carrier (C) with C have also been developed on the same principle of converting a linear convolution to a circular convolution However, in these schemes, the total transmitted energy is not fully utilized since the C, which contains information about the data and consumes extra energy when transmitted, is discarded at the receiver Various alternatives to use of a C have been studied in the literature To retain the benefits of the use of a C these alternatives also achieve a circular convolution of the channel, while reducing transmit energy One of the alternatives is zeropadding (Z) between symbols, for this approach the receiver applies an overlap-and-add operation to obtain a circular convolution [1] Other methods proposed in [2] [10] follow a different rationale for II mitigation and generate a fixed sequence, ie, a unique word (UW), at the end (or the tail) of each symbol ince the UW is predefined and deterministic, circular convolution of the channel can still be maintained The use of a UW has been thoroughly investigated for single carrier frequency domain equalization (C-FDE) systems (see eg, [4], [5], [10] and the references listed therein) and has been adopted in the IEEE 80211ad [11], where the UW is a Golay sequence that is inserted between a group of symbols in the time domain However, generating a fixed sequence in the tail of OFDM symbols is not straightforward and requires several steps [2] In the first step, the tail of each time domain OFDM symbol is forced to be a zero sequence by using a set of redundant subcarriers in the frequency domain After this zero sequence is generated at the tail, the zero tail (ZT) of the OFDM symbol is replaced by a UW At the receiver, the UW is subtracted after the equalization The main disadvantage of zeroing the tail of the OFDM symbol in this manner is that the redundant subcarriers may require significant energy One way to reduce this energy is to optimize the location of the redundant subcarriers [6] However, the optimization of the location of the redundant subcarriers depends on the input data, which is an N-hard problem and requires an exhaustive search among all of the possible solutions [2] In the literature, heuristic algorithms for optimizing the location of redundant subcarriers have been proposed, but the energy used by these redundant subcarriers may still be considerably large Another option is to spread the energy to all the subcarriers [3] However, this causes the receiver structure to be complicated Instead of generating a UW for OFDM symbols, [7-9] consider a DFT--OFDM symbol structure where the energy in the tail of the transmitted time domain symbol is db less than that of the non-tail part; this is referred to as zero tail (ZT) DFT--OFDM Note that although this is referred to as ZT, the tails of the transmitted symbols are not completely zero and may have some residual energy ZT DFT--OFDM places zeroes at the two ends of the inputs of the DFT spreading blocks However, at the output of the inverse DFT (IDFT), the tail does not contain exactly zero samples and their values change depending on the data transmitted Hence, ZT DFT--OFDM still suffers from II, especially in rich scattering environments Therefore, the scheme is inherently interference-limited due to the II in multipath channels In addition, use of ZT may not be preferable at the receiver since it may adversely affect automatic gain control Also, the peak-to-average power ratio (AR) is larger than the original DFT--OFDM structure

2 Equalization ( ) 2 Data + Estimated Data UW signal Figure 1 UW DFT--OFDM transmitter and receiver In this paper, we combine the ideas of UW OFDM and DFT- -OFDM and propose an improved UW approach based on the DFT--OFDM symbol structure The proposed method, called UW DFT--OFDM, exploits the pulse shaping structure of DFT- -OFDM to reduce the energy consumed by accommodating a UW in the tail imilar to ZT DFT--OFDM [7]-[9], the data symbols are mapped to the middle of the DFT spreading blocks In addition, instead of placing zero redundant symbols at the two ends of the DFT spreading blocks, non-zero redundant symbols which suppress the leaked energy at the tail are employed The advantages of the proposed UW DFT--OFDM scheme over UW OFDM and ZT DFT--OFDM are as follows: The proposed scheme consumes very low energy for zeroing out the tails compared to UW OFDM [2] ince the energy in the tail is suppressed, the II-limited performance of ZT DFT--OFDM [7]-[9] in multipath channels is addressed As the UW is inserted at the input of the DFT process, the receiver is able to separate the UW and data symbols without any extra operation Hence, the proposed scheme adds no additional complexity to existing receivers for DFT--OFDM Due to the C nature of DFT--OFDM and the way the redundant symbols are generated, the proposed scheme offers low AR and lower out-of-band (OOB) leakage compared to UW OFDM The rest of the paper is organized as follows In ection II, the proposed UW DFT--OFDM scheme is explained Also the transmitter and receiver structures, the method of forming the waveform, the design for the redundant symbols, and associated required conditions are described In ection III, we evaluate UW DFT--OFDM numerically and compare it with the other schemes Finally, conclusions are provided in ection IV Notations: Matrices [columns vectors] are denoted with upper [lower] case boldface letters (eg, A and [a]) The Hermitian operation and the transpose operation are denoted by ( ) H and ( ) T, respectively The Moore-enrose pseudoinverse operation and inverse operation are denoted by ( ) and ( ) 1, respectively The operation of 2 is the 2-norm of its argument The trace of a square matrix is represented by tr( ) The field of complex numbers and the field of real numbers are shown as C and R, respectively The multivariate complex Gaussian distribution is denoted by CN(μ, C), where μ is the mean vector and C is the covariance matrix I N and 0 N M are the N N identity matrix and N M zero matrix II UNIQUE WORD DFT--OFDM Consider a single user scenario consisting of a transmitter and a receiver communicating over a wireless channel as illustrated in Fig 1 Let the data symbols to be transmitted within one DFT- -OFDM symbol be the vector d C Nd 1, where N d is the number of data symbols A redundant symbol vector r C N r 1 where N r is the number of redundant symbols is first generated as explained in ection IIA below The composite vector of data and redundant symbols, [d T r T ] T is then permuted by a permutation matrix R L L before spreading via DFT spreading blocks denoted by D k C M k M k, where M k is the size of kth DFT matrix We consider K DFT spreading blocks that operate in parallel and stack them into a larger matrix C L L, where L = K k=1 M k = N d + N r As shown in Fig 1, the permutation matrix maps the data symbols in vector d to the inner inputs of each of the DFT spreading blocks, and the redundant symbols in vector r to the upper and lower ends of the inputs of each of the DFT spreading blocks The size of the upper and lower ends of the inputs of the kth DFT spreading block are denoted by M header,k and M tail,k, respectively Thus, the number of redundant symbols is K calculated as N r = k=1 (M header,k + M tail,k ) Without loss of generality, no additional constraints on the orders of the entries of the vector r and the vector d are introduced in the permutation process In order to create the null subcarriers such as DC and the edge subcarriers at the input of the IDFT matrix F H C N N, we introduce a matrix B C N L, where N is the number of subcarriers It is worth noting that the circular pulse shaping at the transmitter [12, 13, 16] can also be achieved with the proper construction of the matrix B However, in order to obtain simpler transmitter and receiver structures, we limit the construction of matrix B such that it provides a localized mapping from the

3 3 outputs of DFT spreading matrices to the subcarriers utilized during the transmission Finally, the output of the matrix B is mapped into time domain via F H as x = F H B [ d ] (1) r A) Tail uppression and Unique Word Generation The redundant symbol vector r is generated as a function of the data vector d as follows Let r = Td s + w, (2) where s C N r 1 is the tail suppressing vector, T C N r N d is the tail suppressing precoder, and w C N r 1 is the unique word generator vector Let A be F H B Then, similar to the expressions in [3], A is partitioned into four submatrices as A = F H B = [ M 11 M 12 M 21 M 22 ], (3) where M 21 C N tail N d, M 22 C N tail N r, and N tail is the number of samples at the tail The non-tail part and the tail part of the symbol are then obtained as x = [ x non-tail x ] = [ M 11 M 12 ] [ d ], (4) tail M 21 M 22 r where x non-tail C (N N tail) 1 and x tail C N tail 1 Hence, x tail = M 21 d + M 22 s Tail suppression + M 22 w UW sequence (5) While the first part of (5) shows the tail suppression operation, the second part of (5) shows how the UW sequence is generated It is worth noting that if M 22 is not a complete matrix, ie, N tail > N r, it is not possible to generate arbitrary UW sequence via vector w However, we synthesize the UW sequence via M 22 w as this approach simplifies the receiver, which is discussed in ection II B In order to suppress the impact of data symbols on the tail, the vector M 22 s should cancel M 21 d as much as possible The optimization problem which minimizes the energy of M 21 d + M 22 s while limiting the energy of the vector M 22 s is then expressed as follows: 2 s = argmin M 21 d + M 22 s 2 st s 2 2 α (6) s where α is an energy constraint on the vector s The optimization problem in (6) is known as a least squares with a quadratic constraint (LQI) problem To solve this problem, we first examine the unconstrained least squares problem, ie, without the energy constraint If M 21 d is in the range of the columns of M 22, the problem in (6) corresponds to minimum norm problem If it is not in the range of the columns of M 22, (6) corresponds to minimum norm least square-error problem In both cases, the solution can be expressed via the pseudoinverse of M 22 as s = M 22 M 21 T d (7) If the energy constraint is introduced, (6) is equivalent to the following unconstrained problem: s = argmin M 21 d + M 22 s λ s 2, (8) s where λ is the Lagrange multiplier In this case, the solution is obtained as s = (M H 22 M 22 + λi) 1 M 22 T H M 21 d (9) A proper Lagrange multiplier can be found for the case of s 2 2 = α, by using the bi-section search algorithm [15] The tail suppression by using (7) always achieves a perfect zero tail if the rank of M 22 is greater or equal to N tail, otherwise, the tail will have non-zero values with minimum energy of M 22 s by design If M 22 is ill-conditioned (eg, the existence of guard subcarriers may cause an ill-conditioned M 22 ), the solution provided in (9) limits the energy of M 22 s and can still yield a significant tail suppression with a proper selection of λ Hence, the proposed UW DFT--OFDM scheme achieves better convolution properties than the ZT DFT--OFDM structure proposed in [7-9] which has a non-zero tail in the transmitted waveform but are not optimized to have minimal energy It is worth noting that when the zero tail is generated without DFT-spreading, the energy of the redundant symbols in generating the zero tail may turn out to be extremely high since this energy is concentrated on only a few subcarriers which have to be chosen appropriately However, in the proposed scheme, the redundant symbol energy is spread across the data vector in frequency via multiple DFT-spread blocks In addition, as we will discuss in ection IIC, specific inputs of DFT-spread blocks, which affects the tail significantly, are used for the redundant symbols Therefore, the proposed scheme tends to consume very low energy for the redundant symbol vector Note that the consumed energy for the redundant symbols depends on the parameters that affect the matrix T, such as N tail, N r, and λ B) Receiver tructure Let the channel impulse response (CIR) between the transmitter and the receiver be the vector h = [h 0 h 1 h L ] where L + 1 is the number of taps Then, the received signal vector y can be expressed as y = Hx + n, (10) where H C N N is the circular convolution matrix that models the interaction between the transmitted signal x and the channel h and n C N N CN(0 N 1, σ 2 I N ) is the additive white Gaussian noise (AWGN), with variance σ 2 Note that when the transmitted signal passes through a multipath channel, the previous symbol leaks into the current symbol ince the inserted UW is the same for each symbol, this can be expressed as a circular convolution as in (10) At the receiver, the operations applied at the transmitter are reversed by considering the impact of the communication channel The receiver operation can be expressed as [d T r T] T = T H QB H Fy, (11)

4 Amplitude 4 5 () Dirichlet sinc function Instaneous ower [db] ZT DFT--OFDM [7-9] -30 roposed UW DFT--OFDM -35 Figure 2: Illustration of Dirichlet sinc functions of DFT--OFDM (DFT size and IDFT size are 8 and 64, respectively) where d C N d 1 is the estimated data symbol vector, r C N r 1 is the estimated redundant vector and Q C L L is the equalizer, which equalizes the convolution of physical channel and any filtering that may be used via B Equalizer Q can be derived using the minimum mean square error (MME) or zero forcing (ZF) criteria Note that a further refinement on receiver can be obtained by exploiting the fact that s = Td [2, 3] However, such an approach yields only very slightly improved performance compared to the receiver in (11), with added complexity, and hence will not be considered in this paper It is important to emphasize that the UW sequence in (5) is generated before the DFT spread blocks via (2) Therefore, the receiver does not need to explicitly remove the UW as is done in [2] Note that the vector r is discarded for data demodulation at the output of permutation matrix T as illustrated in Fig 1 C) arameter Constraints Consider a single DFT-spread block where its output is mapped to the matrix F H in a localized manner as illustrated in Fig 2 Then, the resulting signal at the output of F H corresponds to a modulated signal which is generated based on the circular convolution of the interpolated modulation symbol sequence with a Dirichlet sinc function The upsampling ratio for kth DFT spreading block is expressed as NM k In the proposed scheme, we exploit this structure of the DFT--OFDM symbols and do not allow the main lobe of the Dirichlet sinc function to appear in the tail in order to reduce the energy in the tail part of DFT-- OFDM symbol Thus, the lower-end of the DFT spreading blocks should not be used for the data symbols and the conditions given by and M tail,k M k N tail N Time (12) M header,k 1, (13) must hold true, which is also considered in [7-9] For example, if there are 512 subcarriers and the last 64 samples are for the tail, at least M tail,k = 7 symbols and M header,k = 1 symbols -40 Figure 3: Time samples ample should not be used for the modulation symbols when M k = 56 In addition, as previously stated, the rank of M 22 should be greater or equal to N tail in order to achieve a perfect zero sequence at the tail as (7) yields at least one exact solution Otherwise, the tail is not canceled exactly, but it is still suppressed by (9) As illustrated in Fig 2, UW DFT--OFDM can be considered as the superposition of the data signal and the UW signal ince the data signal starts from the zero sample and ends with a zero sample due to the suppressed tail in the proposed scheme, the proposed scheme allows zeroth-order continuity between UW DFT--OFDM symbols In addition, the existence of the UW does not hurt the continuity of the waveform since the Dirichlet sinc functions that construct the UW repeat in time due to the back-to-back DFT--OFDM symbols and provide continuity Thus, the conditions in (12) and (13) also mitigate OOB leakage due to the smoother transitions between symbols III NUMERICAL REULT In this section, we compare the proposed UW DFT--OFDM scheme with and ZT DFT--OFDM [7-9] through simulations We set N = 512 subcarriers, L = 448 (ie, 32 guard subcarriers at the edges of the band), and a C length of 64 samples The length of UW is fixed as N tail = 64 samples We consider K = 8 identical DFT spreading blocks where M k = 56 for k = 1,2,,8 For each DFT-spread block, we set M tail,k = 7 and M header,k = 1, which yields N r = K (M tail,k + M header,k ) = 64 redundant symbols The vector s is calculated based on (9) and the Lagrange multiplier λ is set to 001 The UW generator vector w is generated based on a Golay sequence, ie, Ga64, defined in [11] and π2 binary phase shift keying (BK) modulation is employed for the sequence The multipath channel is modeled as an exponential Rayleigh fading channel with L = N tail + 1 independent taps and power delay profile (D) where the unnormalized power of the lth tap is expressed as exp( τl), where τ corresponds to the decay rate Note that τ = 0 yields a uniform D

5 ower spectral density [dbmhz] Figure 4: ower spectrum A) Time amples In Fig 3, the energy distribution in time is provided for C OFDM, ZT DFT--OFDM, and the proposed UW DFT-- OFDM scheme In order to compare the tails of schemes, the UW signal is not included As shown in Fig 3, utilizes samples in time, while the ZT DFT--OFDM and the proposed UW DFT--OFDM scheme both utilize 512 samples UW DFT--OFDM achieves suppressed samples at the tails of DFT--OFDM symbols compared to the ZT DFT-- OFDM ince the power of the tail part is 15 db lower than that of the non-tail part for ZT DFT--OFDM and changes depending on the data, II can be a limiting factor for this scheme in a rich scattering environment In contrast, the proposed scheme obtains a tail which is db lower than the non-tail part and improves the tail characteristics of the scheme B) OOB Leakage In Fig 4, the power spectrum for, ZT DFT-- OFDM, and the proposed scheme are provided The Welch s averaged periodogram method is utilized to estimate the power spectrum ince the upper-end of the DFT-spreading blocks are set to zero for the ZT DFT--OFDM and small values generated based on data for the proposed scheme, the symbol transitions are smoothed for both schemes Therefore, the ZT DFT-- OFDM and the proposed scheme both yield better OOB leakage performance compared to the The proposed scheme exhibits slightly better OOB leakage performance compared to the ZT DFT--OFDM since the proposed scheme achieves a better continuity due to the suppressed tail It is also notable that the impact of the UW appears as ripples in the main lobe as shown in Fig 4 since the UW is a non-zero constant sequence C) AR erformance Ripples due to UW ZT DFT--OFDM [7-9] roposed UW DFT--OFDM ubcarrier Index In Fig 5, we evaluate the AR performance using the complementary cumulative distribution function (CCDF) The dynamic range of the ZT DFT--OFDM signal is slightly higher than since it does not introduce a non-zero UW at the tail On the other hand, existence of a non-zero UW decreases the dynamic range of the signal and the proposed scheme r(ar > AR 0 ) ZT DFT--OFDM 10-3 roposed UW DFT--OFDM AR (db) 0 Figure 5: AR performance exhibits a better AR performance as compared to ZT DFT-- OFDM due to its single carrier nature D) Overhead Analysis The schemes offer similar resource usage efficiencies The C OFDM scheme has an efficiency of (512 64)( ) = 777% The proposed scheme and the ZT DFT--OFDM scheme utilize 64 redundant symbols and 64 guard subcarriers and provide ( )512 = 75% efficiency E) BER erformance In order to achieve a fair comparison of the BER performance of the different schemes, identical symbol energy is considered in the simulations, ie, the energy per symbol is normalized by considering the energy consumed by C for the scheme, and the redundant subcarriers for the proposed scheme For all the schemes, we consider an MME-FDE which allows the receiver to the exploit the path diversity for C systems [14] No channel coding is considered in the simulations In Fig 6, the BER performance is evaluated in AWGN channel Although the UW scheme allows the receiver to exploit all the symbol energy, some portion of the symbol energy is lost for the scheme as the C is discarded Therefore, while ZT DFT--OFDM attains the Gaussian BER bound, a shift is observed for the proposed scheme Yet, they both effectively utilize the symbol energy As shown in Fig 6, the energy consumed for the tail suppression causes only 005 db shift from the Gaussian bound (ie, approximately 1% extra energy for the redundant symbols) This value can also be analytically verified by calculating tr(tt H )N d 100 In Fig 7, the BER performance is investigated by considering the impact of the multipath channel When τ = 03, the OFDM and the proposed schemes yield a better BER performance and the ZT DFT--OFDM scheme saturates at a high NR This may be understood by noting that the ZT DFT--OFDM scheme does not exactly maintain the circular convolution of the channel and II dominates the noise The saturation becomes more apparent when the frequency selectivity is increased by setting τ = 0, as shown in Fig 7(b) While the proposed scheme enjoys the path diversity via MME-FDE and yields better BER performance

6 6 Gausian Bound ZT DFT--OFDM roposed UW DFT--OFDM BER 6 BER E N (db) b 0 Figure 6: BER performance in AWGN channel, 4QAM for high NR regime, the performance for ZT DFT--OFDM degrades drastically as shown in Fig 7(b) IV CONCLUDING REMARK In this study, we proposed a UW-based DFT--OFDM scheme The proposed scheme has two major advantages over UW OFDM and ZT DFT--OFDM: (i) it consumes very low energy for zeroing out the symbol tails It is demonstrated that the tail of DFT--OFDM can be suppressed by consuming 1% extra total energy approximately, and (ii) the UW is generated at the input of the DFT process which allows a simpler receiver structure than the one for UW OFDM Compared to ZT DFT-- OFDM, the proposed scheme exhibits better tail characteristics, lower BER in rich scattering environment, reduced AR and OOB leakage Multiple users can be accommodated by the use of multiple DFT-spread blocks, which also facilitates frequency selective link adaptation Future work will consider multi-user scenarios, UW design, and coded systems REFERENCE [1] B Muquet, Z Wang, G Giannakis, M de Courville, and Duhamel, Cyclic prefixing or zero padding for wireless multicarrier transmissions? IEEE Trans Commun, vol 50, no 12, pp , Dec 2002 [2] M Huemer, C Hofbauer, A Onic, and J B Huber, Design and analysis of UW OFDM signals, {AEU} - International Journal of Electronics and Communications, vol 68, no 10, pp , 2014 [3] M Huemer, C Hofbauer, and J Huber, Non-systematic complex number R coded OFDM by unique word prefix, IEEE Trans ig roc, vol 60, no 1, pp , Jan 2012 [4] M Huemer, H Witschnig, and J Hausner, Unique word based phase tracking algorithms for CFDE-systems, in roc IEEE Global Telecommunications Conference, vol 1, Dec 2003, pp [5] J Coon, M andell, M Beach, and J McGeehan, Channel and noise variance estimation and tracking algorithms for unique-word based singlecarrier systems, IEEE Trans Wireless Commun, vol 5, no 6, Jun 2006 [6] H teendam, On the selection of the redundant carrier positions in UW OFDM, IEEE Trans ig roc, vol 61, no 5, pp , Mar 2013 [7] G Berardinelli et al, On the potential of OFDM enhancements as 5G waveforms, in roc IEEE Veh Tech Conf, May 2014, pp 1 5 [8] G Berardinelli, F Tavares, T orensen, Mogensen, and K ajukoski, On the potential of zero-tail DFT-pread-OFDM in 5G networks, in roc IEEE Veh Tech Conf, ep 2014, pp 1 6 BER Rayleigh Bound ZT DFT--OFDM 10-3 roposed UW DFT--OFDM E N (db) b 0 a) 64QAM, τ = 03 Rayleigh Bound ZT DFT--OFDM roposed UW DFT--OFDM E N (db) b 0 b) 64QAM, τ = 0 Figure 7: BER performance in Rayleigh channel [9] G Berardinelli, F Tavares, T orensen, Mogensen, and K ajukoski, Zero-tail DFT-spread-OFDM signals, in roc IEEE Global Telecommunications Conference Workshops, Dec 2013, pp [10] D Falconer, Ariyavisitakul, A Benyamin-eeyar, and B Eidson, Frequency domain equalization for single-carrier broadband wireless systems, IEEE Commun Mag, vol 40, no 4, pp 58 66, Apr 2002 [11] Wireless LAN Medium Access Control (MAC) and hysical Layer (HY) pecifications Amendment 3: Enhancements for Very High Throughput in the 60 GHz Band, IEEE td 80211ad-2012, pp 1 628, Dec 2012 [12] N Michailow and G Fettweis, Low peak-to-average power ratio for next generation cellular systems with generalized frequency division multiplexing, in roc IEEE Intel ign roc and Comm yst, Nov 2013, pp [13] N Michailow, I Gaspar, Krone, M Lentmaier, and G Fettweis, Generalized frequency division multiplexing: Analysis of an alternative multi-carrier technique for next generation cellular systems, in roc International ymp on Wireless Commun ys, Aug 2012, pp [14] F ancaldi et al, ingle-carrier frequency domain equalization, IEEE ig roc Mag, vol 25, no 5, pp 37 56, ep 2008 [15] Boyd and L Vandenberghe, Convex Optimization Cambridge University ress, 2004 [16] I Gaspar, A Festag, and G Fettweis, ynchronization using a seudo- Circular reamble for Generalized Frequency Division Multiplexing in Vehicular Communication, in roc IEEE Veh Tech Conf, ep 2015, pp 1 6