A Least-Squares Based Data-Aided Algorithm for Carrier Frequency Estimation

Transcription

1 A Least-Squares Based Data-Aided Algorithm for Carrier Frequency Estimation Wolfgang Steinert, Lothar Friederichs, Susanne Godtmann, André Pollo, iels Hadaschi, Gerd Ascheid and Heinrich Meyr AUDES Telecommunications Consulting GmbH, P.O. Box 14, 7150 Bacnang, Germany Institute for Integrated Signal Processing Systems, RWTH Aachen University, Templergraben 55, 5056 Aachen, Germany Abstract A new low-complexity algorithm for data-aided carrier frequency estimation in digital transmission systems is presented. The algorithm is derived from the least-squares LS criterion and achieves an estimation accuracy very close to the theoretical limit, even at signal-to-noise ratios SR as low as 0 db. In conjunction an FFT, it provides an estimation range of nearly ±50% of the symbol rate. I. ITRODUCTIO Modern digital communication systems operating in burst transmission mode rely on sufficiently accurate recovery of carrier frequency and phase. Fast carrier frequency acquisition is particularly important for satellite TDMA systems, interactive multimedia systems lie DVB-RCS, as well as for terrestrial or satellite-based mobile communication systems operating in burst mode. Synchronization algorithms operate in the data-aided DA or in the non-data-aided DA mode. When using DA algorithms, the estimation process is solely based on nown pilot symbols. In contrast, DA estimators exclusively exploit information that unnown data symbols convey about the channel. For a given number of channel observations and low SR, DA estimation is much more accurate than DA estimation [1]. Several synchronization algorithms have been proposed for DA carrier frequency recovery e.g. [1] [4]. References [] and [5] provide a detailed performance comparison. As yet, the most promising approaches rely on calculating or approximating the autocorrelation of the received unmodulated symbols. This concept was first introduced by Fitz in 1991 [] and later improved by Luise and Reggiannini L&R algorithm [4]. ote that the cited publications are not recent. evertheless, they still seem to represent the state-of-the-art in DA carrier frequency estimation. The main criteria to assess the performance of synchronization algorithms are the estimation accuracy, the estimation range and the SR threshold. Even at low SR, the L&R algorithm [4] and the approach proposed by Mengali [5] This wor has been financed by BMBF, the German Federal Ministry of Education and Research and supported by DLR, the German Aerospace Center. Special thans go to Dr. M. Keller. S. Godtmann thans the Deutsche Teleom Stiftung for its financial support. provide an estimation accuracy close to the theoretical limit, the so-called Cramér-Rao bound CRB. The estimation range of the L&R algorithm is rather small but may be increased at the expense of estimation accuracy, whereas the algorithm by Mengali offers an estimation range of about ±0% of the symbol rate. As far as the SR threshold is concerned, the Mengali algorithm is outperformed by the L&R algorithm. evertheless, the threshold of both algorithms is rather low. However, a main drawbac of all these approaches is their vast computational complexity originating from the calculation of the autocorrelation of the received unmodulated symbols. The algorithm proposed in this paper refrains from calculating the autocorrelation and, thus, benefits from significantly reduced computational complexity as compared to earlier algorithms of similar performance. Although the estimation range is slightly reduced, the SR threshold is improved and the estimation accuracy is close to the Cramér-Rao bound. Furthermore, it is shown that the minor drawbac of the proposed algorithm, its reduced estimation range, can be easily overcome by performing a coarse estimation based on a Discrete-Fourier-Transform DFT prior to the considered algorithm. Doing so extends the estimation range to almost ±50% of the symbol rate. It is important to mention that the overall computational complexity is still lower than that of the L&R algorithm and the approach introduced by Mengali [5]. After introducing the transmission model in Section II, we present the derivation of the proposed algorithm based on the least-squares LS criterion. In order to illustrate the performance of the proposed algorithm, simulation results are presented in Section IV. In Section V, the computational complexity of the proposed algorithm is analyzed and compared to other approaches. II. TRASMISSIO MODEL We consider the transmission of consecutive pilot symbols taen from an M-ary PSK modulation alphabet. As purely DA estimation is considered in this paper, the signal model does not need to tae data symbols into account. Under the assumption of perfect symbol timing, the received unmodulated baseband signal after matched filtering and sam-

2 pling can be modelled as z r a e jπ ft+ϑ + n, T being the symbol duration. The transmitted and the received symbol at sampling instant are denoted as a and r, respectively. Unit energy symbols are assumed, i.e. E s E a 1. Moreover, n are the samples of complex-valued AWG independent real and imaginary parts, each having zero-mean and variance 0 /E s, where 0 denotes the one-sided power spectral density of the noise process. In addition to the noise component, a phase offset ϑ and a frequency offset f are introduced by the physical channel and the oscillators. Phase and frequency offset are assumed to be constant for the duration of one burst. III. DERIVATIO OF THE ALGORITHM The objective is to find an estimate of the unnown normalized frequency offset ft. First, we consider the noise free case, i.e. z e jπ ft+ϑ. Let ẑ be defined as ẑ e jπ ˆfT+ˆϑ. According to the least-squares criterion, the positions of ẑ are chosen such that the sum of all distances z ẑ is minimized. The sum can be expressed as S z ẑ z ẑ z ẑ z z ẑ + z ẑ + ẑ cosπ f ˆfT + ϑ ˆϑ. 1 In order to minimize 1, the partial derivatives of S are taen respect to ˆϑ and ˆf and subsequently set to zero. After some algebra, we obtain S Im e ˆϑ jεϑ e jπε ft 0 ˆf S Im e jεϑ e jπε ft 0, ε f f ˆf and ε ϑ ϑ ˆϑ. The sums in and can be expressed as magnitude and phase as follows: e jπε ft A e jα 4 e jπε ft B e jβ, 5 A,B,α,β R. Applying the rule n x x1 xn 1 x to 4 yields A sinπε ft sinπε f T, α + 1 πε f T. 6 Since the magnitude A is positive by definition, the condition ε f T <1/ has to be met. Under the assumption that the estimation error ε f is small, the following approximation holds: e jπε ft 1 + j πε f T. 7 According to the rules for power sums, B and β can then be approximated as B + 1, β + 1 πε f T. 8 Equating and inserting 4 and 5 yields: A Im e jα+ε ϑ B Im e jβ+ε ϑ. 9 In order to eliminate ε ϑ and therefore the dependency on ϑ and in order to ensure that the expressions in 9 is equal zero, we subtract their arguments: β α 1 Solving 10 for ε f T yields ε f T πε f T. 10 β α. 11 π 1 Due to 4 and 5, α and β can be directly obtained from the channel observations z. Thus, by applying 11 it is possible to obtain an initial estimate ˆε 0 f T for ε ft ˆε 0 f T π 1 arg z e jπ ˆf 0 T z e jπ ˆf 0 T. 1 It should be stressed that 1 is an estimate of the estimation error incurred by choosing ˆf 0. The first estimate for the frequency offset ˆf 1 can then be calculated as ˆf 1 ˆε 0 f + ˆf 0. Without loss of generality, we set ˆf 0 to zero. Due to the approximation in 7, the characteristic of the algorithm is non-linear. Therefore, the estimate ˆε 0 f is strongly biased for large ε f. In order to circumvent this problem, the algorithm is applied iteratively and the observations z are precorrected before each iteration. The n-th frequency estimate can then be obtained as n 1 ˆf n ˆf 0 + i0 ˆε i f

3 ˆε ft Unbiased Estimation ε f T Figure 1. ˆε n f T π 1 arg where z n Characteristic of the proposed estimator. z n is defined as z n z e jπ ˆf n T. z n In order to illustrate the non-linear characteristic of the algorithm, ˆε f versus ε f is shown for different numbers of pilot symbols in Fig. 1. As the maximum normalized frequency estimation error ε f T is restricted to values smaller than 1/ see Eq. 6, the estimation range of the algorithm is limited to ft <1/. Since the characteristic in Fig. 1 is strictly monotonically increasing, the algorithm converges for the noise free case when applied iteratively. The smaller the initial frequency offset f and the longer the pilot symbol sequence, the fewer iterations are required. For a typical scenario, simulations show that three iterations are sufficient. As the basic concept of our approach is to adjust ˆf such that the estimates ẑ best fit the observations z, the algorithm will be denoted as Best Angular Fit BAF estimator in the sequel. Even though the BAF algorithm has an iterative nature, it becomes immediately apparent that its complexity is almost negligible in comparison to the calculation of the autocorrelation required by e.g. [], [4] and [5]. More detailed results are provided in Section V. It is shown in the appendix that the algorithm approaches the Cramér-Rao bound for high SR and small ε f T. IV. SIMULATIO RESULTS For the analysis of the estimation performance, we resort to the mean square estimation error. In the case of unbiased estimation, the of ˆf is lower bounded by the Cramér-Rao bound CRB, i.e.., As we only consider the case that the entire transmitted sequence is nown at receiver side DA estimation, according to [6], the CRB is given as: π T E s In order to better assess the performance of the proposed approach, it should be compared to the other discussed algorithms, i.e. [], [4] and [5]. As reference [4] is an improvement of the algorithm proposed by Fitz [], we do not further consider []. The L&R algorithm [4] is given as M 1 ˆfT πm + 1 arg R, R 1 1 z m z m, 1 M. 14 m It is stated in [4] that the optimum choice for M is M /. The Mengali algorithm also relies on the calculation of 14 but includes an additional smoothing function. Please refer to [5] for more detailed information. Fig. depicts the versus the SR for a fixed normalized frequency offset of ft It can be seen that the SR threshold of the proposed BAF approach is slightly lower than that of the widely used L&R algorithm. Moreover, both algorithms perform close to the CRB, even for low SR and small numbers of pilot symbols. The algorithm proposed by Mengali has a rather high SR threshold. However, it can be lowered by providing a longer sequence of pilot symbols. When the is plotted versus the initial frequency offset f T, we can see that the estimation range of the L&R algorithm is approximately twice as wide as the estimation range of the proposed approach. Exemplarily, results BAF Iterations L&R Mengali E s / 0 Figure. vs. SR, 16, ft 7 10.

4 for 16 are shown in Fig.. The graph is truncated at ft 0.1. It should be stressed that the estimation range of the Mengali algorithm is significantly wider about ±0%. In order to overcome the limited estimation range of the proposed approach, it is possible to pre-correct the received sequence the imprecise estimate emerging from a Discrete- Fourier-Transform DFT [7]. The DFT can be realized as a length- zero-padded Fast-Fourier-Transform FFT. It should be stressed that the of an estimate produced by the FFT does not even come close to the CRB. However, provided that the pilot sequence is not too short, it is sufficiently precise to guarantee a residual frequency offset in the estimation range ft <1/ of the proposed algorithm. Thus, the estimation range can be widened to ft < 0.5. Fig. 4 depicts the results for an initial frequency offset ft 0.4. In order to guarantee comparability, the received sequence is, of course, also pre-corrected prior to applying the L&R algorithm and the Mengali algorithm. It can be seen that the proposed approach and the L&R algorithm perform close to the CRB, even for low SR, e.g. E s / 0 0 db. It should be stressed that the observable threshold value of approximately E s / 0 0 db for the L&R algorithm and the BAF algorithm results from the FFT and does not originate from the algorithms. V. COMPLEXITY OF ALGORITHM It should be stressed that the complexity of the proposed approach is significantly lower than that of most existing algorithms, as e.g. [], [4] and [5]. The number of complex multiplication required for the algorithms [], [4] and [5] is of the order O. In contrast, the proposed algorithm only has an I-fold complexity of O, I being the number of iterations. As three iterations are usually sufficient, I is much smaller than so that the complexity is approximately of order O BAF Iterations L&R Mengali ft Figure. vs. f, 16, E s/ 0 5 db FFT + BAF Iterations FFT + L&R FFT + Mengali E s / 0 Figure 4. vs. SR,, ft 0.4. As already indicated in the previous section, pre-applying a length- FFT widens the estimation range of the proposed approach to its maximum ft <0.5. Even together the complexity O log of this FFT, the proposed algorithm is still less computationally complex than [], [4] and [5]. VI. COCLUSIO In this paper, we present a low-complexity algorithm for data-aided carrier frequency estimation. The algorithm has been derived on the basis of the least-squares criterion. It has been shown analytically and demonstrated by simulations that the approach performs close to the theoretical limit. In combination an FFT, the maximum estimation range of ft <0.5 is achieved. The complexity of the algorithm is very small compared to other approaches nown from the literature that exhibit similar performance. APPEDIX We compute the error variance of the estimator ˆε f T ˆβ π 1 ˆα ˆα arg e jπε f T + n, ˆβ arg e jπε ft + n. For the sae of simple notation, we write 15 ˆα args α + α 16 ˆβ args β + β, 17

5 the following definitions: S α e jπε ft, α n S β e jπε ft, β n. As the the estimate ˆε f T is biased for large ε f T, it is assumed that ε f T 1. Due to this assumption and according to the equations 6 and 8, S α and S β can then be approximated as: S α e jα + 1 S β e jβ. ormalization of 16 and 17 yields ˆα arg e jα + α arg e jα + S α n, 18 ˆβ arg e jβ + β arg e jβ + n. 19 S β + 1 For large E s / 0 or large, we can approximate 18 and 19 as: ˆα α + Im n, ˆβ β + Im n. + 1 Inserting the latter expressions into 15 then yields ˆε f T π 1 β α+ Im n + 1 Im n. 0 The error variance of the unbiased estimate ˆε f T in 0 can finally be calculated as E ˆε ft 9 π E s / E s / 0 π 1 + 1, which coincides the CRB given by 1. REFERECES [1] H. Meyr, M. Moeneclaey, S. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing, 1st ed. ew Yor, Y: John Wiley & Sons, [] U. Mengali, A.. D Andrea, Synchronization Techniques for Digital Receivers. ew Yor and London: Plenum Press, [] M. P. Fitz, Planar Filtered Techniques for Burst Mode Carrier Synchronization, in Proceedings of IEEE Globecom, Phoenix, Arizona, USA, Dec [4] M. Luise, R. Reggiannini, Carrier Frequency Recovery in All-Digital Modems for Burst-Mode Transmissions, IEEE Transactions on Communications, vol. 4, no. //4, pp , Feb.-Apr [5] U. Mengali, M. Morelli, Data-Aided Frequency Estimation for Burst Digital Transmission, IEEE Transactions on Communications, vol. 45, no. 1, pp. 5, Jan [6] J.A. Gansman, J.V. Krogmeier, M.P. Fitz, Single Frequency Estimation on-uniform Sampling, in Proceedings of 0th Asilomar Conference on Signals, Systems and Computers, vol. 1, Pacific Grove, California, USA, ov. 1996, pp [7] L. C. Palmer, Coarse Frequency Estimation Using the Discrete Fourier Transform, IEEE Transactions on Information Theory, vol. 0, no. 1, pp , Jan