An Adaptive Algorithm for Direct Frequency Estimation. H. C. So y and P. C.Ching. Tat Chee Avenue, Kowloon, Hong Kong

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1 This paper is a postprint of a paper submitted to and accepted for publication in IEE Proceedings - Radar, Sonar and Navigation and is subject to Institution of Engineering and Technology Copyright. An Adaptive Algorithm for Direct Frequency Estimation H. C. So y and P. C.Ching y Department of Computer Engineering & Information Technology, City University of Hong Kong Tat Chee Avenue, Kowloon, Hong Kong Department of Electronic Engineering, The Chinese University of Hong Kong Shatin, N.T., Hong Kong June 13, 003 Keywords: Adaptive Filter, Fast Algorithm, Frequency Estimation Abstract Based on the linear prediction property of sinusoidal signals, a new adaptive method is proposed for frequency estimation of a real tone in white noise. Using the least mean square algorithm, the estimator is computationally ecient and it provides unbiased and direct frequency measurements on a sample-by-sample basis. Convergence behavior of the estimated frequency is analyzed while its variance in Gaussian noise is derived. Computer simulations are included to corroborate the theoretical analysis and to show its comparative performance with two adaptive frequency estimators in dierent nonstationary environments. Corresponding Author H. C. So Department of Computer Engineering & Information Technology City University ofhongkong Tat Chee Avenue, Kowloon HONG KONG Tel. : (85) Fax: (85) ithcso@cityu.edu.hk 1

2 This paper is a postprint of a paper submitted I. to Introduction and accepted for publication in IEE Proceedings - Radar, Sonar and Navigation and is subject to Institution of Engineering and Technology Copyright. The copy Estimating of record the is available frequency of at aiet realdigital sinusoidlibrary. noise has applications in many areas [1]-[3] such as carrier and clock synchronization, angle of arrival estimation, demodulation of frequency-shift keying (FSK) signals, and Doppler estimation of radar and sonar wave returns. The discrete-time noisy sinusoid is usually modeled as x(n) = cos(!0n + )+q(n) = 4 s0(n)+q(n) (1) where the noise q(n) is assumed to be a white zero-mean random process while,!0 and [0 ) which represent the tone amplitude, frequency and phase of the sinusoid, respectively, are unknown. Without loss of generality, the sampling period is assigned to be one second. The task here is to nd!0 (0 ) from x(n). If the sinusoidal parameters are constant in time, classical batch techniques []-[3] include maximum-likelihood estimator [4] and eigenanalysis algorithms such as Pisarenko's harmonic retrieval method [5], which involves determining the eigenvector corresponding to the minimum eigenvalue of the covariance matrix of x(n), and MUSIC [6], canbeemployed to achieve accurate frequency estimation at the expense of a large computational cost. On the other hand, when real-time estimation is desired and/or the environment is nonstationary, such as the frequency is an abruptly changing function of time and the amplitude/phase is time-varying, fast tracking of!0 is necessary. An adaptive gradient search algorithm for unbiased sinusoidal frequency estimation in the presence of noise was rst introduced by Thompson [7]. Fundamentally, it is an on-line implementation of Pisarenko's method via a unit-norm constrained least mean square (LMS) algorithm [8]. By exploring the autocorrelation function of the received signal, Etter and Hush [9] also suggested a computationally simple algorithm for nonstationary frequency estimation. The idea is to maximize the mean square dierence between x(n) and its delayed version using an adaptive time delay estimator (ATDE) [10] and the frequency estimate is given by over the estimated delay. Since the delay of the ATDE is restricted to be an integral multiple of the sampling interval, the algorithm cannot give accurate frequency estimate particularly for large!0. An improvement to[9]was made by providing fractional sample delays in the ATDE with the use of Lagrange interpolation [11]. However, the frequency estimate of the modied method is still biased because the Lagrange interpolator cannot perfectly model subsample delays for sinusoidal signals. Basically, nite length fractional delay lters are never ideal for noninteger delays [1]-[13]. Other recent adaptive frequency estimators include constrained pole-zero notch ltering [14], Pisarenko's method combined with Kamen's pole factorization [15] and adaptive IIR-BPF [16], which is an LMS-style linear prediction algorithm with an IIR bandpass lter for noise reduction. In this paper, a new unbiased frequency estimation approach based on linear prediction of sinusoidal signals is proposed. Starting from the property that a pure sinusoid is predictable from its past two sampled values, a cost function whose minimum exactly corresponds to the sinusoidal frequency in

3 This paper white is noise a postprint is developed of a paper in Section submitted II. The LMS to and algorithm accepted is then for publication applied to minimize IEE Proceedings the cost function - Radar, Sonar and and Navigation the frequency and estimate is subject is updated to Institution explicitly of on Engineering a sample-by-sample and Technology basis. Performance Copyright. measures of the estimator, viz. convergence behavior and variance of the estimated frequency, are also analyzed. Simulation results are presented in Section III to corroborate the theoretical analyses and to evaluate the frequency estimation performance of the algorithm by comparing with the adaptive Pisarenko's algorithm and adaptive IIR-BPF. Finally, conclusions are drawn in Section IV. II. Direct Frequency Estimator (DFE) It is easy to verify that s0(n) obeys the following simple recurrence [17]: With the measurement x(n), we can predict s0(n) using s0(n) =cos(!0)s0(n ; 1) ; s0(n ; ) () ^s0(n) =cos(^!0)x(n ; 1) ; x(n ; ) (3) where ^!0 represents an estimate of!0. Dening the error function as e(n) 4 = x(n) ; ^s0(n) (4) It can be shown that the mean square error function Efe (n)g can be calculated as Efe (n)g = 4(cos(^!0) ; cos(!0)) s +(+cos(^!0)) q (5) where s = = denotes the tone power while q is the noise variance. Apparently, minimizing Efe (n)g with respect to ^!0 will not give the desired solution because of the noise component. Notice that if an estimate of q is available [18], then unbiased frequency estimation can still be attained with the use of Efe (n)g. To remove the eect of noise without knowing the noise power, we employ a cost function Ef (n)g which is expressed as Ef (n)g = Efe (n)g ( + cos(^!0)) = (cos(^! 0) ; cos(!0)) s + cos(^!0) + q (6) It is worthy to note that (6) can be considered as an alternate form of the modied mean square error suggested in [19] but there was no theoretical analysis of their frequency estimator. The advantages of using (6) are that we can obtain direct frequency measurements and derive the estimator performance in a simpler way. Investigating the rst and second derivatives of (6) shows that for!0 (0 ), the performance surface Ef (n)g has a unique minimum at ^!0 =!0 with the value of q, but it also has a maximum when ^!0 <=3or^!0 > =3. This suggests minimization of Ef (n)g can be achieved via gradient search methods if the initial value of ^!0 is chosen between =3 and =3. In our study, 3

4 This paper the is computationally a postprint of attractivea paper submitted LMS algorithm to and isaccepted utilized tofor estimate publication!0 iteratively. in IEE Proceedings From (6), the - Radar, Sonar and instantaneous Navigation value and of is subject Ef (n)g, to Institution (n), is of Engineering and Technology Copyright. e (n) (n) = (7) ( + cos(^!0(n))) where ^!0(n) denotes the estimate of!0 at time n. Note that (n) is in fact an estimate of q ^!0(n)!!0. The stochastic gradient estimate is computed by dierentiating (n) with respect to ^!0(n) and is given ^!0(n) = sin(^!0(n)) ( + cos(^!0(n))) e(n)((x(n)+x(n ; )) cos(^! 0(n)) + x(n ; 1)) (8) Since the term sin(^!0(n))=( + cos(^!0(n))) is positive for ^!0(n) (0 ), it does not aect the sign of the gradient estimate. As a result, the LMS updating equation for the direct frequency estimator (DFE) can be simplied as ^!0(n +1)= ^!0(n) ; e(n)((x(n)+x(n ; )) cos(^!0(n)) + x(n ; 1)) (9) where is a positive scalar that controls convergence rate and ensures system stability of the adaptive algorithm. To reduce computation, value of the cosine function is retrieved from a pre-stored cosine vector of the form [1 cos(=l) cos((l ; 1)=L)] where L is the vector length. Notice that when L increases, the frequency resolution increases but a larger memory will be needed. as The method is computationally ecient because only 5 multiplications, 5 additions and 1 look-up operation are required for each sampling interval. Taking the expected value of (9), the learning behavior of the frequency estimate is evaluated as Ef^!0(n +1)g;Ef^!0(n)g = Ef( cos(!0n + ) ; cos(!0(n ; 1) + )cos(^!0(n)) + cos(!0(n ; ) + )+q(n) ; cos(^!0(n))q(n ; 1) + q(n ; ))(( cos(!0n + )+ cos(!0(n ; ) + )+q(n) +q(n ; )) cos(^!0(n)) + cos(!0(n ; 1) + )+q(n ; 1))g = ; s Ef(1 + cos(! 0)) cos(^!0(n)) + cos(!0) ; ( cos(!0)+cos(!0)) cos (^!0(n)) ; cos(^!0(n)) + (cos(!0)+1)cos(^!0(n)) + cos(!0)g; q Efcos(^! 0(n)) ; cos(^!0(n)) + cos( ^!0(n))g = ; Ef(cos(^! s 0(n)) cos(!0) ; cos(!0)cos(^!0(n)))g = ; ^!0 (n) ;!0 3(^!0 (n)+!0) ^!0 (n)+!0 3(^!0 (n) ;!0) sin E sin + sin sin s ; 3(^!0 (n)+!0) ^!0 (n)+!0 (^!0(n) E ;!0) sin + 3 sin s ; 3(Ef^!0 (n)g +!0) Ef^!0 (n)g +!0 (Ef^!0(n)g;!0) sin +3sin s ) Ef^!0(n +1)gEf^!0(n)g 1 ; s g(ef^! 0(n)g) + s! 0g(Ef^!0(n)g) (10) 4

5 This paper where is a postprint of a paper submitted to and accepted for publication in IEE Proceedings - Radar, Sonar and Navigation and is subject to Institution of Engineering g(ef^!0(n)g) =sin 4 3(Ef^!0 and Technology (n)g +!0) Ef^!0 Copyright. (n)g +!0 + 3 sin A closed form expression for Ef^!0(n)g is not available because the geometric ratio ; 1 ; s g(ef^! 0(n)g) is changing at each iteration, but the convergence trajectory can be easily acquired using (10) by brute force. Nevertheless, some observations can be made from (10). First, the mean convergence rate of ^!0(n) is independent of the noise level. To ensure convergence and stability, should be chosen so that j1 ; s g(ef^! 0(n)g)j < 1 is satised. Since 0 <g(ef^!0(n)g) < 4, the bound for can thus be computed from j1 ; 4 sj < 1, which gives 0 <<1=( s). In addition, the algorithm has a time-varying time constant of1=( s g(ef^! 0(n)g)). Considering when ^!0(n)!!0, it is found that g(ef^!0(n)g) will approach zeroat!0 =0or!0 =, which implies that the steady state learning rate of ^!0(n) isfairlyslow if the frequency is close to one of these extreme values. Assuming that q(n) is of Gaussian distribution and using (9) again, the steady state frequency variance of the DFE algorithm, denoted by var(^!0), is derived as (See Appendix I) where SNR = s = q q var(^!0) 4 = lim n!1 Ef(^! 0(n) ;!0) g q SNR sin(!0) cos(4!0 ) +cos(!0) +1 (11) is the signal-to-noise ratio. It can be seen that var(^!0) is proportional to and and inversely proportional to SNR. Investigating the term (cos(4!0)=( + cos(!0)) + 1)= sin(!0) reveals that the frequency variance approaches its minimum value of 0:3 q =SNR if! 0! 0:8 or!0! 0:7 while it has a large value when!0 is close to 0 or. At!0 = 0 and!0 =, var(^!0)!1, which means the unbiasedness of the DFE excludes these two frequencies. Moreover, the variance of ^!0(n) has zero value in the absence of noise. From (10) and (11), the choice of should be a tradeo between a fast convergence rate and a small variance, as in the standard LMS algorithm [8]. We also note that the estimation performance of the DFE is relatively poor when!0 iscloseto0or because both the convergence time and variance are large. On the other hand, the steady state variance of the noise power estimate using (7) is given by var( ) 4 = lim n!1 Ef( (n) ; q) g 4 q (1) It is interesting to note that var( ) does not depend on and!0. III. Simulation Results Computer simulations had been conducted to evaluate the sinusoidal frequency estimation performance of the DFE in the presence of white Gaussian noise for dierent nonstationary conditions. Comparisons with two LMS-style frequency estimators which are claimed to provide unbiased estimation, namely, the adaptive Pisarenko's algorithm [7] and adaptive IIR-BPF [16] were also made. For each iteration, 5

6 This paper [7] requires is a postprint 10 multiplications, of a paper submitted 4 divisions, to 5and additions, accepted 1 look-up for publication and 1 square-root in IEE Proceedings operation while - Radar, Sonar and [16] Navigation needs 6 multiplications, and is subject 6 additions to Institution and 1 of look-up Engineering operation. and The Technology signal power Copyright. was unity and =0:1 which corresponded to a SNR of 10dB. The length of the cosine vector L was chosen to be q 1000 and this provided a frequency resolution of =1000 rad/s. The initial frequency estimates of all three methods were set to be 0:5 rad/s and the 3-dB bandwidth coecient in [16] was =0:5. In the rst example, and were unknown constants while!0 was a piecewise constant function. The tone amplitude was equal to p and the phase parameter [0 ) was a uniform random variable for each trial. The actual frequency had a value of 0:95 rad/s during the rst 4000 iterations and then changed instantaneously to 0:55 rad/s and to 0:3 rad/s, at the 4000th and the 8000th iteration, respectively. The step size parameters of the DFE and adaptive Pisarenko's algorithm were chosen to be 0.00 while that of the adaptive IIR-BPF was Figure 1 shows the trajectories for the frequency estimates of the three algorithms in tracking this time-varying frequency. These results provided were averages of 00 independent runs. It can be seen that ^!0(n) converged to the desired values at approximately the 000th, the 5300th and the 9000th iteration. The convergence time for!0 = 0:95 rad/s almost doubled that of!0 = 0:3 rad/s because upon convergence, the term sin(3(ef^!0(n)g +!0)=) + 3 sin((ef^!0(n)g +!0)=) approached 0.16 and 3.4 in the former and the latter case, respectively. In addition, we observe that (10) had predicted the learning behavior of the frequency estimate accurately. On the other hand, the adaptive Pisarenko's algorithm also estimated the step-changing frequency accurately but with dierent convergence behaviors, while the adaptive IIR-BPF was incapable of tracking the true frequency after the 4000th iteration. A comprehensive testwas then performed for a wide range of!0 [0:05 0:95], and the steady state mean square frequency errors (MSFEs) of the three algorithms were measured and plotted in Figure. In order to provide a fair comparison, was xed to be 0.00 while we adjusted the step sizes of [7] and [16] such that their convergence times were approximately identical for each tested frequency. It is seen that the measured MSFEs of the DFE agreed with their theoretical values particularly when!0 was close to 0:5 rad/s. Furthermore, the value of var(^!0) was bounded by : 10 ;5 rad /s for!0 [0: 0:8] rad/s. Interestingly, the frequency dependence of the MSFEs was similar to that of the adaptive Pisarenko's method but the DFE had smaller variances for all cases. Although [16] gave the best performance for!0 < 0: and!0 > 0:8, ithadmuch larger MSFEs for other frequencies, particularly when!0 was close to 0:5 rad/s, and it failed to work at this frequency. Figure 3 shows the estimated noise power using (7) and its variance for dierent frequencies. Along the frequency axis, the estimated noise power and variance uctuated around their nominal values, with minimum and maximum values of 9:90 10 ; and 1:01 10 ;1, and 1:88 10 ; and :08 10 ;1, respectively. This implies that for all frequencies, (7) estimated q accurately while the mean square errors of the noise power estimates agreed with (1), and as expected, their frequency dependence was negligible. Figure 4 demonstrates the carrier frequency estimation performance for a noisy binary phase-shift 6

7 This paper keying is a (BPSK) postprint signal of a where paper its submitted amplitudeto asand well accepted as phase for werepublication nonstationary. in IEE The Proceedings baud rate and - Radar, Sonar and the Navigation carrier frequency and is of subject the BPSK to Institution signal was of selected Engineering as and 0:05 rad/s Technology and 0:5 rad/s, Copyright. respectively, and thus there were 40 samples for each symbol. In this example, =0:0 was used to achieve fast convergence at the expense of a larger variance and the results was based on 100 independent runs. We can see that the DFE algorithm converged at approximately the 100th iteration and an accurate estimate of the carrier frequency was obtained. IV. Conclusions A computationally attractive algorithm, called the DFE, has been proposed for tracking the frequency of a real sinusoid embedded in white noise. Using an LMS-style method, the frequency estimate is adjusted directly on a sample-by-sample basis. Learning behavior and mean square error of the estimated frequency in Gaussian noise are derived and veried by computer simulations. It is shown that the DFE gives unbiased frequency estimates in dierent nonstationary conditions and has high frequency estimation accuracy when the frequency is neither close to 0 nor. In addition, the DFE outperforms two existing LMS-style frequency estimators in terms of estimation accuracy, computational complexity and/or tracking capability. Appendix I The steady state mean square error of ^!0(n) is derived as follows. Subtracting!0 from both sides of (9), squaring both sides, taking expectation and then considering n!1yields lim n!1 Ef(^! 0(n) ;!0)e(n)((x(n)+x(n ; )) cos(^!0(n)) + x(n ; 1))g = lim n!1 Efe (n)((x(n)+x(n ; )) cos(^!0(n)) + x(n ; 1)) g (A:1) Suppose is chosen suciently small such that ^!0(n)!!0 upon convergence. The component which involves both signal and noise in the RHS of (A.1) is approximated as Ef(q(n) ; cos(!0)q(n ; 1) + q(n ; )) ( cos(!0n + )) cos(!0)+ cos(!0(n ; ) + )) cos(!0) + cos(!0(n ; 1) + )) g = (q (n)+4cos (!0)q (n ; 1) + q (n ; )) (cos (!0n + )cos (!0) + cos (!0(n ; ) + )cos (!0)+cos (!0(n ; 1) + ) + cos(!0n + )cos(!0(n ; ) + )cos (!0)+cos(!0n + )cos(!0(n ; 1) + ) cos(!0) + cos(!0(n ; 1) + ) cos(!0(n ; ) + )cos(!0)) = s q (cos(!0)+) 3 (A:) 7

8 This paper Furthermore, is a postprint the component of a paper due submitted to noise to only and in accepted the RHS of for (A.1) publication can be estimated in IEE Proceedings as - Radar, Sonar and Navigation and is subject to Institution Ef(q(n) ; cos(!0)q(n ; 1) + q(n ; )) of Engineering and Technology Copyright. (cos(!0)q(n)+cos(!0)q(n ; ) + q(n ; 1)) g = Ef(q (n)+4cos (!0)q (n ; 1) + q (n ; ) ; 4cos(!0)q(n)q(n ; 1) + q(n)q(n ; ) ;4cos(!0)q(n ; 1)q(n ; ))(cos (!0)q (n) + cos (!0)q (n ; ) + q (n ; 1) + cos (!0)q(n)q(n ; ) + cos(!0)q(n)q(n ; 1) + cos(!0)q(n ; 1)q(n ; ))g = 4 q (cos(! 0)+) (A:3) On the other hand, lim n!1 Ef(^! 0(n) ;!0)e(n)((x(n)+x(n ; )) cos(^!0(n)) + x(n ; 1))g = lim n!1 Ef(^! 0(n ; ) ;!0)e(n)((x(n)+x(n ; )) cos(^!0(n)) + x(n ; 1))g X ; lim n!1 Efe(n ; i)((x(n ; i)+x(n ; i ; )) cos(^! 0(n ; i)) + x(n ; i ; 1)) i=1 e(n)((x(n) +x(n ; )) cos(^!0(n)) + x(n ; 1))g lim n!1 Ef(^! 0(n) ;!0)e(n)((x(n)+x(n ; )) cos(^!0(n)) + x(n ; 1))g X ; lim n!1 Efe(n ; i)((x(n ; i)+x(n ; i ; )) cos(^! 0(n)) + x(n ; i ; 1)) i=1 e(n)((x(n) +x(n ; )) cos(^!0(n)) + x(n ; 1))g (A:4) With the use of (10), the rst term of (A.4) can be evaluated as lim n!1 Ef(^! 0(n) ;!0)e(n)((x(n)+x(n ; )) cos(^!0(n)) + x(n ; 1))g = lim s n!1 Ef((^! 0(n) ;!0))(sin( ^! 0(n) ;!0 ) sin( 3(^! 0(n)+!0) ) + sin( ^! 0(n)+!0 )sin( 3(^! 0(n) ;!0) ))g lim s n!1 Ef(^! 0(n) ;!0) (sin( 3(^! 0(n)+!0) 0(n)+!0 ))g var(^! s 0)(sin(3!0)+3sin(!0)) (A:5) It can also be shown that lim n!1 Efe(n ; 1)((x(n ; 1) + x(n ; 3)) cos(^! 0(n)) + x(n ; )) and e(n)((x(n)+x(n ; )) cos(^!0(n)) + x(n ; 1))g Ef(q(n ; 1) ; cos(!0)q(n ; ) + q(n ; 3))((x(n ; 1) + x(n ; 3)) cos(!0)+x(n ; )) (q(n) ; cos(!0)q(n ; 1) + q(n ; ))((x(n)+x(n ; )) cos(!0)+x(n ; 1))g = ;4 s q cos (!0)(cos(!0)+) + 4 q(4 cos 4 (!0) ; 1 cos (!0)+1) (A:6) lim n!1 Efe(n ; )((x(n ; ) + x(n ; 4)) cos(^! 0(n)) + x(n ; 3)) 8

9 This paper is e(n)((x(n)+x(n a postprint of a paper ; )) cos(^!0(n)) submitted + to x(n and ; 1))g accepted for publication in IEE Proceedings - Radar, Sonar and Navigation and is subject to Institution of Engineering and Technology Copyright. Ef(q(n ; ) ; cos(!0)q(n ; 3) + q(n ; 4))((x(n ; ) + x(n ; 4)) cos(!0)+x(n ; 3)) (q(n) ; cos(!0)q(n ; 1) + q(n ; ))((x(n)+x(n ; )) cos(!0)+x(n ; 1))g = ( cos (!0) ; 1)(cos(!0)+) + 4 cos (!0) (A:7) s q q Substituting (A.)-(A.7) into (A.1) and after simplication, we get (11). References [1] S.M.Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Englewood Clis, NJ: Prentice-Hall, 1993 [] P.Stoica and R.Moses, Introduction to Spectral Analysis, Upper Saddle River, NJ: Prentice-Hall, 1997 [3] B.G.Quinn and E.J.Hannan, The estimation and tracking of frequency, Cambridge, New York: Cambridge University Press, 001 [4] R.J.Kenec and A.H.Nuttall, "Maximum likelihood estimation of the parameters of tone using real discrete data," IEEE Journal Oceanic Engineering, vol.1, no.1, pp.79-80, 1987 [5] V.F.Pisarenko, "The retrieval of harmonics by linear prediction," Geophys. J. Roy. Astron. Soc., pp , 1973 [6] P.Stoica and A.Eriksson, "MUSIC estimation of real-valued sine-wave frequencies," Signal Processing, vol.4, pp , 1995 [7] P.A.Thompson, "An adaptive spectral analysis technique for unbiased frequency estimation in the presence of white noise," Proc. 13th Asilomar Conf. Circuits, Syst., Comput., Pacic Grove, CA, pp , Nov [8] B.Widrow et al, "Stationary and nonstationary learning characteristics of the LMS adaptive lter," Proc. IEEE, vol.64, no.8, pp , 1976 [9] D.M.Etter and D.R.Hush, "A new technique for adaptive frequency estimation and tracking," IEEE Trans. Acoust. Speech, Signal Processing, vol.35, no.4, pp , April 1987 [10] D.M.Etter and S.D.Stearns, "Adaptive estimation of time delays in sampled data systems," IEEE Trans. Acoust. Speech, Signal Processing, vol.9, no.3, pp , June 1981 [11] S.R.Dooley and A.K.Nandi, "Fast frequency estimation and tracking using Lagrange interpolation," Electronics Letters, vol.34, no.0, pp , Oct

10 This paper [1] is G.D.Cain, a postprint N.P.Murphy of a paper and submitted A.Tarczynski, to and "Evaluation accepted for of publication several FIR in fractional-sample IEE Proceedings delay - Radar, Sonar and Navigation lters," and is subject Proceedings of ICASSP, to Institution pp.61-64, of Engineering 1994 and Technology Copyright. [13] T.I.Laakso, V.Valimaki, M.Karjalainen and U.K.Laine, "Splitting the unit delay," Signal Processing Magazine, vol.13, no.1, pp.30-60, 1996 [14] G.Li, "A stable and ecient adaptive notch lter for direct frequency estimation," IEEE Trans.Signal Processing, vol.45, no.8, pp , August 1997 [15] A.Bencheqroune, M.Benseddik and A.Hajjari, "Tracking of time-varying frequency of sinusoidal signals," Signal Processing, vol.78, pp , 1999 [16] M.Sheu, H.Liao, S.Kan and M.Shieh, "A novel adaptive algorithm and VLSI design for frequency detection in noisy environment based on adaptive IIR lter," Proc. of IEEE International Symposium on Circuits and Systems, Sydney, Australia, vol.4, pp , May 001 [17] R. Prony, "Essa: Experimentale et analytique," J.Ecole Polytechnique, Paris, pp.4-76, 1795 [18] J.R.Treichler, "-LMS and its use in a noise-compensating adaptive spectral analysis technique," Proc. Int. Conf. Acoust. Speech, Signal Processing, pp , April 1979 [19] S.Jaggi and A.B.Martinez, "A modied autoregressive spectral estimator for a real sinusoid in white noise," Proceedings of Southeastcon, pp ,

11 This paper is a postprint of a paper submitted to and accepted for publication in IEE Proceedings - Radar, Sonar and Navigation 1 and is subject to Institution of Engineering and Technology Copyright. frequency estimate ( π rad/s) theoretical calculation from (9) proposed adaptive Pisarenko adaptive IIR-BPF true frequency iteration number Figure 1: Frequency estimates for step changes in frequency at SNR = 10dB var( ) rad /s ω 0 1E-03 3E-04 1E-04 theoretical calculation from (10) proposed adaptive Pisarenko adaptive IIR-BPF 3E-05 1E frequency ( π rad/s) Figure : Frequency variances at SNR = 10dB 11

12 This paper is a postprint of a paper submitted to and accepted for publication in IEE Proceedings - Radar, Sonar and Navigation and is subject to Institution of Engineering and Technology Copyright var( ζ ) true noise power estimated noise power using (6) theoretical variance of measured variance of ζ (n) ζ (n) estimate of noise power frequency ( π rad/s) Figure 3: Estimate of noise power and var( )atsnr=10db 0.5 frequency estimate ( π rad/s) estimated frequency true frequency iteration number Figure 4: Frequency estimate of DFE for an BPSK signal at SNR = 10dB 1