The topic of low computational complexity frequency estimation of a single complex sinusoid corrupted by additive white Gaussian noise has received significant attention over the last decades due to the wide applicability of such estimators in a variety of fields. In this letter, we propose a computationally fast and statistically improved hybrid phase-based estimator that outperforms other recently proposed approaches, lowering the signal-to-noise ratio at which the Cramér–Rao lower bound is closely followed

Andreas Jakobsson (1624294)

Jonathon Chambers (1251609)

M.D. Macleod (7210166)

Zhang Zhuo (7215596)

English

Loughborough University Institutional Repository

   This item was submitted to Loughborough’s Institutional Repository (https://dspace.lboro.ac.uk/) by the author and is made available under the following Creative Commons Licence conditions.      For the full text of this licence, please go to: http://creativecommons.org/licenses/by-nc-nd/2.5/  IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 9, SEPTEMBER 2005 657A Hybrid Phase-Based Single Frequency EstimatorZhuo Zhang, Student Member, IEEE, Andreas Jakobsson, Member, IEEE, Malcolm D. Macleod, Member, IEEE, andJonathon A. Chambers, Senior Member, IEEEAbstract—The topic of low computational complexity frequencyestimation of a single complex sinusoid corrupted by additivewhite Gaussian noise has received significant attention over thelast decades due to the wide applicability of such estimators ina variety of fields. In this letter, we propose a computationallyfast and statistically improved hybrid phase-based estimator thatoutperforms other recently proposed approaches, lowering thesignal-to-noise ratio at which the Cramér–Rao lower bound isclosely followed.Index Terms—Classification theory and methods, computation,detection, estimation, fast algorithms, transforms.I. INTRODUCTIONI N a wide variety of application areas, such as biomedicine,communications, and radar, one often encounters a need tofind a low computational complexity estimate of the dominantfrequency component of data that are assumed to consist of asingle complex sinusoid corrupted by additive white Gaussiannoise, and the topic has, as a result, attracted significant in-terest over the last decades (see, e.g., [1]–[14] and the referencestherein). The problem can be briefly stated as follows; considerthe data sequence(1)where , , and denote the deterministic butunknown amplitude, frequency, and initial phase, respectively,of a complex sinusoid. Further, is a circular zero-meancomplex Gaussian white noise with variance . Then, giventhe sequence , for , the problem is simplyto estimate with a low computational complexity and sta-tistically efficient estimator. In [2], Rife and Boorstyn derivedthe maximum likelihood (ML) estimator of and proposeda statistically efficient approximative ML approach involvingboth a combined coarse and fine search using the fast Fouriertransform (FFT) algorithm. However, zeropadding is often re-quired to obtain sufficient resolution, requiringoperations, where is the size of the desired frequency grid,with typically . Further, an iterative linear predic-tion (ILP) approach requiring operations wassuggested in [15], showing similar performance to that of theManuscript received February 8, 2005; revised April 14, 2005. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Dr. Israel Cohen.Z. Zhang and J. A. Chambers are with the Centre of Digital Signal Pro-cessing, Cardiff School of Engineering, Cardiff University, Cardiff CF24 0YF,U.K. (e-mail: zhuo.zhang@ieee.org; ChambersJ@cardiff.ac.u).A. Jakobsson is with the Department of Electrical Engineering, Karlstad Uni-versity, SE-651 88 Karlstad, Sweden (e-mail: andreas.jakobsson@ieee.org).M. D. Macleod is with QinetiQ Ltd, St. Malvern WR14 3PL, U.K. (e-mail:m.macleod@signal.qinetiq.com).Digital Object Identifier 10.1109/LSP.2005.853043ML. A variety of phase-based methods requiring onlyoperations have been developed. In [3], for example, Tretterproposed a phase-based approach simplifying the problem to alinear regression on the phase. The method is based on a phaseunwrapping algorithm, requiring a very high signal-to-noiseratio (SNR) (SNR ), here defined as SNR , towork well. Later, Kay proposed a modified version of Tretter’salgorithm avoiding the use of the phase unwrapping algorithm[4]. The method, here termed Kay’s weighted phase average(KWPA) estimator, can be shown to attain the Cramér–Raolower bound (CRLB) for sufficiently high SNR, but the methodis in general biased, and the SNR for which the CRLB isachieved depends on the underlying frequency [6], [8], [12]. Asa result, the focus of recent contributions has mainly been aimedto reduce the SNR threshold [10], the frequency dependency ofthe threshold [7], or both [11], [14]. In this letter, we proposea hybrid method combining the ideas in [10], [11], and [14]to show the performance close to that of ML or ILP but onlyrequiring operations. The hybrid estimate is based onan initial coarse estimate of the unknown frequency using theuniformly weighted linear predictor (UWLP) method [1], [4];this estimate is used to remove the frequency dependence of theSNR threshold. This SNR threshold is then further reduced viaa combination of using an averaging filter, as suggested in [10],and an outlier removal scheme, as proposed in [14]. Finally, arefined frequency estimate is formed along the lines proposedin [10] and [11].The remainder of this letter is organized as follows: In thefollowing section, the proposed hybrid method is introduced, in-cluding a discussion of the different involved algorithms. Then,Section III contains numerical simulations illustrating the per-formance of the hybrid estimator. Finally, Section IV containsour conclusions.II. HYBRID PHASE-BASED FREQUENCY ESTIMATORAs suggested in [3], the data model in (1) can be written as(2)where is a complex white sequence.Let and denote the real and the imaginary parts of, respectively. Then, for high SNR(3)allowing the approximation(4)where . Most of the recent phase-basedapproaches exploit this approximation, allowing the phase to1070-9908/$20.00 © 2005 IEEEAuthorized licensed use limited to: LOUGHBOROUGH UNIVERSITY. Downloaded on January 8, 2010 at 07:27 from IEEE Xplore.  Restrictions apply. 658 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 9, SEPTEMBER 2005be approximately estimated from the difference of the adjacentphase values, i.e.,(5)where denotes the complex conjugate, suggesting theUWLP frequency estimator [1](6)The UWLP estimator is unbiased but statistically inefficientwith variance [4], [8]SNR (7)As suggested in [7] and [14], we will here use the UWLP esti-mate to form a downshifted signal to remove the frequencydependence of the SNR threshold, i.e.,(8)To further reduce the SNR threshold, Kim et al. proposed av-eraging the downshifted signal using a simple -tap averagingfilter, forming [10](9)Such an averaging can be shown to lower the SNR threshold upto dB but can only be applied to the downshiftedsignal as it restricts the frequency range to . Sim-ilar to (5), the adjacent phase difference of (9) can be formed as(10)where is given by (23) for a general . It is worth notingthat the noise process will now be colored due to the av-erage filtering [13].As shown in [5], the SNR threshold behavior of the phase-based frequency estimators is affected by cumulative phaseerrors resulting from the effect of the additive noise. This effectcan be countered for by introducing an outlier detection scheme.Recently, an effective scheme was proposed in [14], whereoutliers are detected if , , and, with(11)where and .Thus, the outliers can be removed as follows:if outlier detectedotherwise(12)for . Here, is a user parameter selected tominimize the mean-squared error (MSE). We note that for large, some outliers are not detected, increasing the MSE. Similarly,for small , some events that are not outliers are wrongly de-tected as outliers, causing errors and increasing the MSE. There-fore, there is an optimum value of that can be determined em-pirically. We remark that, according to our experience, the per-formance of the resulting estimator is only slightly affected byvarying [16]. After the SNR threshold reduction using (10)and (12), further improvement can be achieved by taking intoaccount the coloration of the noise term in (10). This can beachieved using the suggested four-channel filter bank (FCFB)method1 in [11] and [13], whereby the frequency correction termcan be found as(13)where(14)with . Combined with the coarseestimate, the hybrid frequency estimate is found as(15)In summary, the proposed hybrid estimator is found by ini-tially forming the downshifted signal in (8) using the UWLPestimate in (6). Then, the phase difference of the filtered signalis formed using (10), followed by the outlier removal scheme in(12). Finally, the refined frequency estimate is formed as (15),using (13). It is worth stressing that the hybrid method differsfrom previously suggested approaches in that it combines all theabove steps; the FCFB method does not include the outlier re-moval scheme in (12). Similarly, the method proposed in [14],hereafter termed the outlier removal estimator (ORE), does notinclude the filtering in (10).III. SIMULATIONSIn this section, we will briefly examine the performance of theproposed estimator. Initially, we consider data samplescontaining a single complex sinusoid with frequencyand examine the estimated MSE as a function of the SNR. Fig. 1illustrates the MSE for the proposed hybrid estimator, usinga tap averaging filter and outlier detection threshold, as compared to the UWLP approach [1], the FCFBapproach following [11], the ORE approach [14], the ILP ap-proach using three iterations [15], and the corresponding CRLBas given in [2]. As is clear from the figure, the hybrid estimatorshows similar but slightly better performance than the ILP es-timator; however, the latter requires operations,whereas the former only requires operations. We note thatthe hybrid estimator is statistically improved, closely followingthe CRLB at a lower SNR threshold than the other examinedmethods. Further, the hybrid method uniformly yields alower MSE than the other methods. It is worth noting that the1It is worth noting that the FCFB applies a different set of weights than thoseused in the KWPA approach.Authorized licensed use limited to: LOUGHBOROUGH UNIVERSITY. Downloaded on January 8, 2010 at 07:27 from IEEE Xplore.  Restrictions apply. ZHANG et al.: HYBRID PHASE-BASED SINGLE FREQUENCY ESTIMATOR 659Fig. 1. MSE of the examined estimators as a function of the SNR.Fig. 2. MSE of the examined estimators as a function of the underlyingfrequency, for SNR = 6 dB.hybrid estimator will suffer some performance degradation dueto the introduced averaging in (9), as pointed out in [10]. Thisexplains why the MSE of the proposed hybrid estimator cannotexactly reach the CRLB, as shown in Fig. 1 and in other figures.As is well known, the performance of single frequency estima-tors is often affected by the underlying frequency. Figs. 2 and3 illustrate how the MSE varies as a function of the frequencyof the sinusoid for SNR dB and SNR dB, respec-tively. As seen in the figures, the hybrid estimator is uniformlyachieving a lower MSE than the other approaches and is essen-tially independent of the true frequency. Further, it is clear thatthe FCFB approach is significantly affected by the frequency,whereas the ORE approach is showing a similar robustness asthe hybrid approach, although with a somewhat worse perfor-mance. Here, all the simulation results have been obtained usingMonte Carlo simulations.Fig. 3. MSE of the examined estimators as a function of the underlyingfrequency, for SNR = 4 dB.IV. CONCLUSIONIn this letter, we have proposed a low computationalcomplexity hybrid phase-based single frequency estimatorcombining previously proposed SNR threshold reductionapproaches with a recent outlier removal scheme. Numericalsimulations indicate that the proposed estimator achieves alower MSE than other available techniques, closely followingthe CRLB at a lower SNR threshold. Furthermore, in contrastto many other techniques, the hybrid estimator’s performanceis found to be essentially independent of the true frequency.APPENDIXIn this Appendix, we derive an expression for the noiseprocess , given in (10), for a general . Let .Then, using (2), can be expressed as(16)for . Introduce(17)(18)ThenThus, the argument of can be expressed asAuthorized licensed use limited to: LOUGHBOROUGH UNIVERSITY. Downloaded on January 8, 2010 at 07:27 from IEEE Xplore.  Restrictions apply. 660 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 9, SEPTEMBER 2005implying that the phase difference from adjacent samplescan be expressed as(19)where for(20)Then, using the approximation in (3), (20) can, for high SNR,be approximated asIm (21)where Im denotes the imaginary part of . We note thatusing a first-order Taylor expansion(22)as is small due to the downshifting, implying thatIm (23)We note that for , (23) yields the expression given in [10],i.e.,(24)REFERENCES[1] G. W. Lank, I. S. Reed, and G. E. Pollon, “A semicoherent detection andDoppler estimation statistic,” IEEE Trans. Aerosp. Electron. Syst., vol.AES-9, no. 2, pp. 151–165, Mar. 1973.[2] D. C. Rife and R. R. Boorstyn, “Single-tone parameter estimation fromdiscrete-time observations,” IEEE Trans. Inf. Theory, vol. IT-20, no. 5,pp. 591–598, Sep. 1974.[3] S. A. Tretter, “Estimating the frequency of a noisy sinusoid by linearregression,” IEEE Trans. Inf. Theory, vol. IT-31, no. 6, pp. 832–835,Nov. 1985.[4] S. Kay, “A fast and accurate single frequency estimator,” IEEE Trans.Acoust., Speech, Signal Process., vol. 37, no. 12, pp. 1987–1990, Dec.1989.[5] S. W. Lang and B. R. Musicus, “Frequency estimation from phase dif-ferences,” in Proc. ICASSP, Glasgow, U.K., 1989, pp. 2140–2143.[6] B. C. Lovell and R. C. Williamson, “The statistical performance of someinstantaneous frequency estimators,” IEEE Trans. Signal Process., vol.40, no. 7, pp. 1708–1723, Jul. 1992.[7] V. Clarkson, “Efficient single frequency estimators,” in Int. Symp. SignalProcess. Appl., Gold Coast, Australia, 1992, pp. 327–330.[8] V. Clarkson, P. J. Kootsookos, and B. G. Quinn, “Analysis of the variancethreshold of Kay’s weighted linear predictor frequency estimator,” IEEETrans. Signal Process., vol. 42, no. 9, pp. 2370–2379, Sep. 1994.[9] P. Händel, “On the performance of the weighted linear predictor fre-quency estimator,” IEEE Trans. Signal Process., vol. 43, no. 12, pp.3070–3071, Dec. 1995.[10] D. Kim, M. J. Narasimha, and D. C. Cox, “An improved single frequencyestimator,” IEEE Signal Process. Lett., vol. 3, no. 7, pp. 212–214, Jul.1996.[11] M. L. Fowler and J. A. Johnson, “Extending the threshold and fre-quency range for phase-based frequency estimation,” IEEE Trans.Signal Process., vol. 47, no. 10, pp. 2857–2863, Oct. 1999.[12] B. G. Quinn, “On Kay’s frequency estimator,” J. Time Series Anal., vol.21, no. 6, pp. 707–712, Nov. 2000.[13] M. L. Fowler, “Phase-based frequency estimation: A review,” Digit.Signal Process., vol. 12, no. 2, pp. 590–615, Oct. 2002.[14] M. D. Macleod, “An improved fast estimator of the frequency of a singlecomplex tone,” in Proc. Sixth IMA Int. Conf. Math. Signal Process.,Chicester, U.K., 2004, pp. 155–158.[15] T. Brown and M. M. Wang, “An iterative algorithm for single-fre-quency estimation,” IEEE Trans. Signal Process., vol. 50, no. 11, pp.2671–2682, Nov. 2002.[16] Z. Zhang, A. Jakobsson, M. D. Macleod, and J. A. Chambers, “Statis-tically and Computationally Efficient Frequency Estimation of a SingleTone,” Dept. Elect. Eng., Karlstad Univ., Karlstad, Sweden, Tech. Rep.EE-2005-02, 2005.Authorized licensed use limited to: LOUGHBOROUGH UNIVERSITY. Downloaded on January 8, 2010 at 07:27 from IEEE Xplore.  Restrictions apply. 

A hybrid phase-based single frequency estimator

The topic of low computational complexity frequency estimation of a single complex sinusoid corrupted by additive white Gaussian noise has received significant attention over the last decades due to the wide applicability of such estimators in a variety of fields. In this letter, we propose a computationally fast and statistically improved hybrid phase-based estimator that outperforms other recently proposed approaches, lowering the signal-to-noise ratio at which the Cramer-Rao lower bound is closely followed. © 2005 IEEE

Zhang, Z

Jakobsson, A

Macleod, MD

Chambers, JA

Surrey Research Insight

https://repository.lboro.ac.uk/articles/A_hybrid_phase-based_single_frequency_estimator/9576485/files/17210630.pdf

A hybrid phase-based single frequency estimator

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