On Preambles With Low Out of Band Radiation for Channel Estimation

Transcription

1 On Preambles With Low Out of Band Radiation for Channel Estimation Gourab Ghatak, Maximilian Matthé, Adrish Banerjee, Senior Member, IEEE and Gerhard P. Fettweis, IEEE Fellow arxiv:68.698v cs.ni] 22 Aug 26 Abstract Existing preamble based channel estimation techniques give no consideration to the out of band (OOB) radiation of the transmit preambles which is a key aspect for novel communication schemes for future cellular systems. In this paper, preambles with low OOB radiation are designed for channel estimation. Two particular preamble design techniques are proposed and their performance is analyzed in terms of OOB radiation and estimation error. The obtained preambles are shown to have 5 to 2 db lower OOB radiation than the existing preamble based estimation techniques. As a case study, the estimated channel values are used in equalization of a MIMO GFDM system that is aimed for transmit diversity. I. INTRODUCTION Emerging applications in upcoming cellular networks will require a wireless communication scheme that can not only provide higher throughput and low latency but also have a low OOB radiation, especially for opportunistic use of vacant spectrum and fragmented spectrum allocation ] 2]. Several candidate waveforms like Universal Filtered Multi- Carrier 3] and Generalized Frequency Division Multiplexing (GFDM) 4] are being proposed as solutions to these challenges which require good channel estimation techniques for efficient performance. Applications such as aircraft to ground communication, using OFDM, also require efficient estimation of the channel. Thus there is a need for preamble designs that are efficient in channel estimation in addition to having low OOB radiation for serving multiple applications. In 5] the authors provided a survey on preamble based channel estimation schemes in Orthogonal Frequency Division Multiplexing (OFDM). In 5], the authors provided interference approximation methods (IAM-R, I, C, E-IAM-C etc.) to design preambles for channel estimation. Equi-powered and equi-spaced preambles were proved to be optimum in case of no OOB constraint by authors in 6]. However, so far, no consideration has been given to the OOB radiation of preambles designed for channel estimation. A survey of OOB reduction techniques was presented in 7] but these techniques were not used for preamble design. The main contributions of this paper are as follows: Firstly, two methods of designing preambles having low OOB power for channel estimation over an arbitrary range of frequencies are proposed. G. Ghatak is with Leti, CEA Grenoble, France (gourab.ghatak@cea.fr) A. Banerjee is with Department of EE, IIT Kanpur, India (adrish@iitk.ac.in) G. Fettweis and M. Matthé are with Vodafone Chair Mobile Communications Systems, TU Dresden, Germany. {first name.last name}@ifn.et.tu-dresden.de Secondly, reduced length (sparse) preambles are studied for practical scenarios and the effect of pinching (time domain windowing) is demonstrated. Finally, as a case study, the proposed preambles are employed in a multiple input multiple output (MIMO) system using Time Reversal-Space Time Code (TR-STC)- GFDM 8] and the results are compared with perfect channel knowledge. It is also shown that the errors of individual channels of a MIMO system are separable in terms of transmit powers from the corresponding antennas. The results are compared to preamble based estimation schemes existing in literature for OFDM/OQAM systems 5]. The rest of the paper is organized as follows: Section II defines the system model and outlines the optimization objectives. The optimization problem is converted into a convex problem. Section III contains the proposed preamble designs for channel estimation. Simulation results and a case study is presented in section IV. Finally the paper concludes in section V. II. SYSTEM MODEL AND THE OPTIMIZATION PROBLEM Consider a single input single output (SISO) channel where a known preamble p of length N is transmitted, using a cyclic prefix (CP) of length N CP, which leads to circular convolution of the channel with the preamble. The received signal ( y), transformed into frequency domain is given by: Y = W N y = H P + N where W N is a N N unitary DFT matrix, H is an N N diagonal matrix having the channel frequency response as the diagonal elements and P = W N p. N is the Fourier transform of the additive White Gaussian noise (AWGN) with zero mean and variance σ 2. Assume that channel estimation is desired over a certain range of frequencies denoted by the set K. The estimated channel values, using zero-forcing estimation, over this range of frequencies (the corresponding diagonal elements) is given by: Ĥ kk = Y kp k P k = H kk + N k k K () where (.) denotes complex conjugation. Hence, the mean squared error (MSE) for the estimation H kk is: ] χ( P K ) = E Ĥ kk H kk 2 a = E N K ] 2 P K k K σ 2 = P = σ 2 ξ (2) k K k

2 2 where ξ is the noise enhancement factor (NEF) and P K denotes the part of the preamble in the allocated frequency range and N K is the part of noise in K. The division in (a) is done element wise. Proposition : The MSE given by the above equation is the minimum possible variance in the estimation error. Proof: See Appendix A. Without any OOB constraint, authors in 6] showed that an optimum preamble which s the NEF under total power constraints, uses pilot tones that are equally spaced and equipowered. With the OOB constraint, the present optimization problem can be stated as: k K P k NEF (ξ) P H P TP Preamble Power (S Z) H (S Z) ɛ OOB Power where Z = W U UCW H N P is the over-sampled transmit signal in the frequency domain. W H N transform the preamble in time domain. The matrix C performs the CP insertion in the time domain signal W H NP. The matrix U performs zero padding by a factor of L which corresponds to interpolation in the frequency domain. The zero-padded signal with CP is then transformed into frequency domain by W H U. The matrix S selects the OOB region samples from Z. T P and ɛ are respectively the constraints on total power and the OOB power of the preamble. Z is used to approximate the continuous time spectrum of the preamble. The fractional OOB radiation is defined as the ratio between the amount of energy in the frequency range outside the allocated bandwidth and the amount of energy in the total bandwidth given by: O = f OOB P (f)df f P (f)df (SZ)H (SZ) P H P Eq. (3) cannot be solved by standard solver software due to quadratic vector variables in the denominator of the objective function. Lemma : The problem of (3) can be converted into a semi definite program (SDP) given by: t ] dia( PK ) I I dia( t) P H P TP, (SZ) H (SZ) ɛ Where stands for positive definiteness. dia( a) denotes a diagonal matrix with diagonal entries as the elements of a vector a. Proof: Define a vector variable t of length equal to L K (number of components in P K ). Now instead of minimizing the objective function i.e. k K P is k made to be less than each element of t i.e. t k. The problem each element (3) (4) (5) can be restated as: t H t t k, k K P 2 T P, (SZ) H (SZ) ɛ This is an epigraph form of the problem 9]. In order to justify this formulation, a relaxation is made in terms of the allowable values of the preambles: The preambles are assumed to be real and the preambles within the range of estimation are positive. The problem can be further modified by arranging the components of t and into diagonal matrices as: norm( t) dia( t) dia( P K ) P 2 T P, (SZ) H (SZ) ɛ To convert the problem into a convex optimization problem we take help of a property of Schur s complement which states that for any symmetric matrix: ] A B X = B T C (6) (7) X A and C B T A B (8) Comparing with parameters of Schur s complement we have: A = dia( P K ), B = I and C = dia( t). Using (8) for these values completes the proof. The problem can also be formulated as follows, where the optimization aims to the OOB radiation, constraining the overall MSE: (SZ) H (SZ) ] dia( PK ) I I dia( t) P H P TP, t ξ Where ξ is the constraint on the MSE. This formulation of the problem can be applied in scenarios where the channel estimation accuracy is needed to be over a specified threshold. III. PREAMBLE DESIGNS Two preamble design techniques are proposed to obtain the preambles for estimation of the channel over a small range of frequencies: ) All Frequency Components as Variable (AFV): where all the frequency components of the preamble (of total length equal to the entire bandwidth) are specified as variables, and 2) Estimation Frequency Components as Variable (EFV): where all the preamble values outside K are specified to be zero (note that the total length is still equal to the entire bandwidth). In order to estimate the channel for a wider range of frequencies, the preamble obtained by either of the two methods is juxtaposed to positions where the channel estimation is desired. For example, after obtaining a preamble P for estimation (9)

3 3 of a block of M frequency samples using either of the two methods, in order to have a preamble for estimation for the entire bandwidth of length N, the overall preamble is designed as: P O n] = N/M β= P n βm] M () where.] M denotes modulo M operation. As the problem in (5) is a convex SDP, optimal solutions can be obtained using standard solver software. cvx solution 9] for AFV provides preambles with the minimum OOB radiation for a given MSE constraint. However, as the preamble extends outside K, the juxtaposition, using (), results in some cancellation of the preamble in the overlapping parts which increases the MSE. On the other hand, EFV performs better in terms of MSE for large scale juxtaposition as there are no overlapping parts. However, as only fewer variables are available, there are lesser degrees of freedom for the optimization problem. This leads to sub-optimal fractional OOB performance. The method of juxtaposition starting with smaller preambles provides more flexibility in the sense that preambles for estimating any range of frequencies can be obtained without having to solve a new optimization problem each time. A. Complexity Analysis For an SDP, the infeasible path following algorithm of cvx has O(n ln ɛ ) complexity for an ɛ-optimal problem ]. The AFV method uses all the frequency components as variables and hence the computational complexity increases with increase in the number of subcarriers while keeping the number of subsymbols constant and carrying out the initial estimation over one subsymbol before juxtaposition. However in EFV, estimating over one subcarrier keeping the number of subsymbols constant results in constant computational complexity with respect to increasing the number of subcarriers. Thus to estimate the channel for K frequency components out of a total bandwidth of N frequency components, the complexity of the AFV method is O(N ln ɛ ) whereas the complexity of the EFV method is O( K ln ɛ ) where K denotes the number of components in K. B. Pinching To further reduce the fractional OOB radiation, a particular pinching technique as employed in 4] is used. A pinching prefix and a suffix 4] is introduced and the resultant preamble with the overhead (CP + pinching) in the time domain is multiplied by a raised-cosine window function vector ( w) as in 4] to provide a smooth fade-in and fade-out. L W is the length of the pinching window. In the formulation, the pinching is integrated into the optimization problem Eq. (5) by using a pinching matrix T = dia( w). The modified problem has Z = W U UTCW H NP. C. Optimal Reduced Length (Sparse) Preambles Coherence bandwidth refers to the range of frequencies over which the channel can be considered constant. Thus, one estimated sample of the channel per coherence bandwidth can suffice the estimation of the full resolution channel. However, for equalization, estimated channel values for the entire bandwidth is needed. Once the sparse estimation is obtained, a DFT-based interpolation technique calculates the remaining channel coefficients. Let N g denote the length of the full resolution bandwidth. The channel in the time domain has length L C, a-priori knowledge of which is assumed (given by N CP ). Let that the estimation is done over a set of frequencies K. A sub-matrix of W N is defined as: F T = W N (K ; : L C ). That is, the matrix F T consists of the those rows of W N that correspond to the index of the sparse channel and the number of columns is kept same as the channel length. The leastsquared (LS) estimation of the channel in time domain is given by: ĥls = F + T Ĥ where F + T denotes the pseudo-inverse of F T and Ĥ is a vector containing the estimated values in frequency domain by Eq. (). Lemma 2: The MMSE estimator is given by: ( )] σ ĥ MMSE = F H T F T F H 2 T + dia HL () P K where dia(σ 2 / P K ) denotes a diagonal matrix with the diagonal entries as σ 2 / P i where i K i.e. the division is done element wise. HL is the Fourier transform of ĥls: Proof: Let H L,k = (F T h)k + N k / P k where k K be the elements of H L. K is the number of components of K. The MMSE estimate of the channel is given by: C HL HC H HL L where C HL H is the cross co-variance matrix of the LS estimate and the channel. C HL is the auto co-variance matrix of the LS N estimate. Let r be a vector with elements: r i = i, i K P, i then, C HL = E (F T h + r)(ft h + r) H] = F T F H T + dia(σ 2 / P ] K ) The last step comes from assuming E( h h H ) = I. This is assumed since there is no other a-priori information about the power delay profile. C HL H = E h(ft h + r) H ] = F H T (2) Using (2) and the value of C HL completes the proof. Ĥ F ULL = W Ng ĥ then gives the full resolution estimate of the channel in frequency domain. IV. RESULTS AND DISCUSSION A. Case Study: MIMO TR-STC GFDM A TR-STC-GFDM system was introduced in 8] which exploits transmit diversity by using two transmit antennas. Consequently, two unknown channels per receive antenna need to be estimated. In order to simultaneously estimate both channels, two length-k preambles are designed that contain a comb-type frequency allocation given by: P = P ] P ]... P K/2 ] ] (3) P 2 = P 2] P 2]... P 2K/2 ]] (4) i.e. Pi n] n K i where i =, 2. The convex optimization is carried out with these allocations and each

4 4 Spectrum (db) Without Pinching With Pinching OOB Selection Normalized Frequency Fig. : Oversampled Spectrum of AFV Fractional OOB (db) MSE (db) SNR = 3 db SNR = 25 db SNR = 2 db SNR = 5 db Fig. 3: Dependence of Fractional OOB Power on MSE Spectrum (db) Without Pinching OOB Selection With Pinching Normalized Frequency Fig. 2: Oversampled Spectrum of EFV preamble is separately optimized. The received symbol at the r th antenna is given by: Y r = H r P + H 2r P2 + N r (5) Where H r and H 2r are the channel matrices for the channels from each transmit antenna to r th receive antenna respectively. For simplifying the notations, we drop the subscript r. Proposition 2: The error variance in the estimation of the channel i {, 2} depends only on the power in the preamble transmitted by antenna i given by: var(dia(h i )) σ 2 trace(e P i ) E P i = M P i M H P i is the power matrix where M P i = dia( P i ) is a diagonal matrix with diagonal entries as the preamble values. Proof: The proof follows from CRLB for two antennas, similar to Appendix A. B. Estimation Error and OOB Performance The oversampled spectrum of the preamble including the cyclic prefix with and without pinching is shown in Fig. and Fig. 2. The preamble for allocated frequencies (not shown here) contains non-zero elements outside the frequency region of the allocated frequencies and are no longer equipowered. The problem in (9) is solved with ξ as given in Table I at SNR of 24 db. From Fig. 3 it can be seen that for each signalto-noise ratio (SNR), the fractional OOB initially decreases with increasing MSE. This is due to the fact that as the MSE increases, the preambles have greater range of values they can take and that results in smaller preamble values to make the fractional OOB lesser. Increasing the MSE over a certain threshold makes the optimization fractional OOBconstrained rendering it independent of MSE. The fractional Parameter Value Parameter Value K 32 M 5 L C N = N g 6 N CP 2 L W 6 T P ɛ ξ. K 76,..., 8 K K K 2 K 9,..., 23 K 2,..., 24 L 8 Channel taps e.5t, t =,... Modulation QPSK TABLE I: Simulation Parameters OOB radiation for AFV and EFV is given in Table II. From Fig. 4 it can be seen that the for the estimation in the entire bandwidth, EFV gives a 3 db SNR gain over AFV. So there is a trade-off between the MSE and the fractional OOB power in the two methods. C. Effect of Pinching In case of AFV, Fig. shows that pinching increases the fractional OOB radiation. This is because in case of unconstrained design like AFV, the optimization problem in itself gives the optimal preambles. The pinching introduces additional design constraints that lead to higher fractional OOB values. However, in case of pinching in EFV, Fig. 2 shows that the pinching effectively reduces the fractional OOB radiation. This is due to the inherent nature of the pinching scheme i.e. pinching in a constrained design like EFV improves the performance. D. Comparison with Other Preamble Design Techniques In 5], it was shown that the interference approximation method (IAM-C) provides preambles similar to the optimum preambles without any OOB constraint, i.e. equi-powered and equi-spaced as also found in 6]. From 5] for the SISO MSE in db SNR in db Fig. 4: MSE vs SNR for Estimation in Entire Bandwidth AFV EFV

5 5 AFV FV Without Pinching db db With Pinching db -4.2 db TABLE II: Fractional OOB of Two Methods SER PCK LS MMSE SNR in db Fig. 5: SER vs SNR channel, apart from CP-OFDM, all methods (IAM-R, I, C, E-IAM-C etc.) reach an error floor around SNR of 2 db. This is due to an approximation of that the channel frequency response is almost constant over a time-frequency neighborhood which is not true specially at high SNR. The performance of the CP-OFDM technique is comparable to the proposed schemes in this paper but it suffers from a very large OOB radiation itself. Comparing our work, with the OOB reduction literature survey presented in 7], we observe that the different blocks of 7] i.e. data domain cancellation symbols, time domain windowing etc. of the unified framework for OOB reduction is simultaneously performed by the optimization problem proposed for preamble design in this paper. In Fig. 5 a comparison of the symbol error rate (SER) performance of the estimators in a 2 2 MIMO GFDM show that the MMSE estimator performs better than the LS estimator. For reference, the BER curve with perfect channel knowledge (PCK) has also been shown in the figure which has a 3dB gain as compared to the MMSE estimator. V. CONCLUSIONS From the studies carried out in this paper, it can be established that the fractional OOB radiation constraint effectively changes the structure of the optimum preambles. The obtained preambles are not only non-equipowerd but also the non zero values extend into regions outside the frequency range of estimation. In this paper we have designed preambles that have 45 db lesser fractional OOB radiation compared to the studies carried out without considering the OOB radiation constraint. Furthermore, the proposed preambles perform better than the IAM schemes for OFDM/OQAM. The obtained optimum preambles are used to estimate two channels simultaneously by considering a 2 2 MIMO system which can be extended to any number of transmit antennas. The estimated channel values are used for equalization in TR-STC GFDM. The minimum mean squared error (MMSE) estimator performs better than the least squares (LS) estimator. Both the LS and MMSE estimator requires a-priori knowledge of the channel length which in turn should be at most equal to the length of the CP. APPENDIX A. Cramer Rao Lower Bound for SISO The expression for the received signal is reformulated as: Y = MP VH + N, where M P = dia( P ) is a diagonal matrix with diagonal entries as the preamble values. And VH = dia(h) is a vector containing the diagonal values of H. The likelihood function then is: Λ = exp (πσn 2 )N σ Y M 2 P VH ] H Y ] M P VH ]. 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