IN this paper, we consider the estimation of the frequency

Iterative Frequency Estimation y Interpolation on Fourier Coefficients Elias Aoutanios, MIEEE, Bernard Mulgrew, MIEEE Astract The estimation of the frequency of a complex exponential is a prolem that is relevant to a large numer of fields. In this paper we propose and analyze two new frequency estimators that interpolate on the Fourier coefficients of the received signal samples. The estimators are shown to achieve identical asymptotic performances. They are asymptotically uniased and normally distriuted with a variance that is only.047 times the asymptotic Cramer-Rao ound (ACRB uniformly over the frequency estimation range. Index Terms Digital signal processing, frequency estimation, parameter estimation. I. ITRODUCTIO I this paper, we consider the estimation of the frequency of a complex exponential, s, given y s(k = Ae jπk f +θ] fs + w(k, k = 0,.. ( A is the signal amplitude, f the signal frequency and θ the initial phase. samples are used and the sampling frequency is. The noise terms, w(k, are assumed to e zero mean, complex additive white Gaussian noise with variance σ. The signal to noise ratio is given y ρ = A σ. We set, without loss of generality, A = and =. Although the noise is assumed to e white Gaussian, the derivation of the asymptotic properties of the estimators holds under weaker, more general conditions. These relaxed conditions are stated in ] for the case of real valued noise. However, their extension to the complex case is straightforward and is not explicitly carried out here. The results otained in this paper are easily extended to the more general case y replacing σ with the power spectral density of the noise at the frequency of interest. The frequency estimation prolem outlined aove is relevant to a wide range of areas such as radar, sonar and communications, and has consequently received significant attention in the literature ] and 3]. It is well known that the maximum likelihood (ML estimator of the frequency is given y the argument of the periodogram maximizer, 4]. That is, ˆf ML =arg maxy (λ} (a λ Y (λ= s(ke jπλ. ( Part of this work was carried out while E. Aoutanios was at the University of Technology, Sydney, with partial financial support from the Commonwealth of Australia through the Cooperative Research Centres Program. The authors are with School of Engineering and electronics, the University of Edinurgh, AGB Building, The King s Buildings, Mayfield Road, Edinurgh EH9-3JL, Scotland, The United Kingdom. Phone: +44 3 650 7454, fax: +44 3 650 6554, e-mail: elias@ieee.org. The Cramer-Rao ound (CRB of the frequency estimates is given y, 4], σ f = 6f s (π ρ(. (3 For, the asymptotic CRB (ACRB ecomes σ f 6f s (π ρ 3. The numerical maximization of equation ( is not a computationally simple task and may suffer from convergence and resolution prolems, ]. Therefore, it is common to estimate the frequency of a sinusoid y a two step process comprising a coarse estimator followed y a fine search, 4] 0]. The coarse estimation stage is usually implemented using the maximum in search (MBS as a coarse approximation of the periodogram maximizer, ]. This consists of calculating the -point FFT of the sampled signal and then locating the index of the in with the highest magnitude. Various fine frequency estimators have een proposed in the literature. Zakharov and Tozer, 7], present a simple algorithm that consists of an iterative inary search for the true signal frequency. However, they found it necessary to pad the data with zeros to a length of.5 in order to approach the CRB. Furthermore, the required numer of iterations depends on the resolution as well as the operating signal to noise ratio and can e quite large. Quinn, in ], 5] and 6], proposes a numer of estimators that interpolate the true signal frequency using the two discrete Fourier transform (DFT coefficients either side of the maximum in. These algorithms, however, have a frequency dependent performance that is worst for a signal frequency coinciding with a in center. This results in a degradation in performance when they are implemented iteratively, 0], chapter 5. In this paper we present two new frequency estimators that elong to a family of interpolators amenale to iterative implementation, 0], chapter 6. The first algorithm, denoted Alg, employs an error functional independently suggested in 0], pp. 9, and ]. It uses two complex DFT coefficients calculated midway etween the standard DFT coefficients. The second algorithm, Alg, was suggested in 0], pp. 37, and works on the magnitudes of the DFT coefficients. We analyze the new algorithms and show that they have an asymptotic variance that is only.047 times the ACRB. The theoretical results are then verified y simulation. The paper is organized as follows: In section II we present the details of the frequency estimators including the motivation ehind them. In section III we proceed to analyze

TABLE I ITERATIVE FREQUECY ESTIMATIO BY ITERPOLATIO O FOURIER COEFFICIETS ALGORITHM Let S = F F T (s and Y (n = S(n, n = 0... Find ˆm = arg maxy (n} n Set ˆ0 = 0 Loop: Finally for each i from to Q do X p = s(ke jπk ˆm+ˆ i +p, p = ±0.5 ˆ i = ˆ i + h(ˆ i h(ˆ i = } Re X0.5 + X 0.5, for Alg X 0.5 X 0.5 or h(ˆ i = X 0.5 X 0.5, for Alg X 0.5 + X 0.5 ˆf = ˆm + ˆ Q L the algorithms and derive their asymptotic performances. The convergence properties are also discussed. Section IV shows the simulation results while section V gives the concluding remarks. II. THE ITERATIVE FREQUECY ESTIMATOR The algorithms are summarized in tale I. The coarse search returns the index, ˆm, of the in with the largest magnitude. Two DFT coefficients at the in edges are then calculated and used to interpolate the true frequency. The motivation ehind each algorithm is easily seen y examining the noiseless case. Assuming that ˆm is the index of the true maximum, i.e. ˆm = m, the frequency of the signal can e written as f = ˆm + (4 is a residual in the interval 0.5, 0.5]. The suscript indicates the dependence of the various parameters on. In the rest of the paper, unless the dependence on needs to e emphasized, we drop the suscript for the sake oimplicity of the notation. The goal of the estimator, then, is to otain an estimate of, say ˆ. Consider the DFT coefficients, X p = ˆm+p jπk s(ke, p = ±0.5. (5 Sustituting the expression of the sinusoidal signal into (5 and carrying out the necessary manipulations we otain X p = e jθ + ejπ p ejπ + W p, (6 the terms W p are the Fourier coefficients of the noise. ow for ( p, equation (6 ecomes X p = p + W p (7 with given y jθ + ejπ = e. jπ At this point we ignore the noise terms and proceed to examine the interpolation function of the proposed algorithms. Denote the ratio in the expression of h( in Alg y β. Sustituting the expressions for X p into β and simplifying yields, β = 0.5 + 0.5 =. +0.5 +0.5 Hence, ˆ = β/ can e used as an estimator for the residual frequency. However, as we will see in section III-A, it is necessary to take the real value of β in order to otain a real valued estimate of. In a similar way, the motivation ehind Alg is estalished as follows; the magnitude of X p is X p = p. Since 0.5, the error mapping for Alg ecomes 0.5 0.5 + 0.5+ 0.5+ =. Again, we find that ˆ = h( can e used as an estimator for. ote that the ias resulting from the approximation used in going from (6 to (7 is of order. In the following section, we examine the noise performance of the estimators. We show that they are asymptotically uniased and normally distriuted. A. Asymptotic Performance III. THEORETICAL AALYSIS The motivation ehind each estimator was estalished in the previous section y examining the noiseless case. We will, now, include the noise terms and derive the asymptotic properties of the estimates. We adopt an analysis strategy similar to that used in ] and show that oth algorithms are asymptotically uniased and normally distriuted. In the case that the noise is assumed Gaussian, the Fourier coefficients of the noise terms are independent zero mean Gaussian with variance σ. However, it was shown in 3] and 4] that, given the relaxed assumptions mentioned in the introduction, the noise Fourier coefficients converge in distriution and W (λ lim sup sup, almost surely. λ ln ( Thus, the terms W p are O ln( (for the order notation refer to 5], pp. 4-48. ow, we have that as, ˆm m almost surely (a.s., ]. In fact, we can show that for < 0.5, P ˆm = m } as. If = 0.5, P ˆm = m} = P ˆm = m + } = 0.5 almost surely and either in is an

3 acceptale choice. The same argument applies for = 0.5. Thus, as, = ˆm f 0.5, 0.5] a.s. Turning our attention to Alg and sustituting the expression for X p, shown in equation (7, into β yields, after some simplifications, β = + 0.5 (W 0.5 + W 0.5 + 0.5 (W 0.5 W 0.5. (8 ( Since W p are O ln( as is O(, the term ( involving in the denominator of (8 is of order O ln(. Hence, for large β= + 0.5 ] (W 0.5 + W 0.5 0.5 (W 0.5 W 0.5 + O ( ln ]. (9 Expanding and simplifying, yields β = } + 0.5 ( W0.5 + ( + W 0.5 Re } +j 0.5 ( W0.5 + ( + W 0.5 Im +O ( ln (0 Re } and Im } are respectively the real and imaginary parts of. We clearly see that the real part of β is a noisy estimate of. This clarifies the use of the real part of β as an estimator for. Thus we set ˆ = Reβ}. In fact, taking the real part asymptotically improves the estimation variance y 3dB. Equation (0 implies that the distriution of ˆ asymptotically follows that of the noise coefficients W p. Hence, ˆ is asymptotically uniased and normally distriuted. The asymptotic variance of the estimator is given y varˆ] = ( 0.5 ( 4 varre W 0.5 }] +( + varre W 0.5 }] } = σ 4 π ( 0.5 ( 4 + cos (π, ( the second equality follows from the fact that, under the Gaussianity assumption, varre W 0.5 }] = varre W 0.5 }] = σ /, and = cos (π (π ( The performance of Alg can e otained in a similar fashion. Let Y p = X p. Thus, Y p = p + p W p. (3 The second factor in the aove expression can e expanded as follows + p W p = ( p + W p p } Wp Re (4 Upon examination of the two terms under the square root, we find that their relative orders change as 0.5. This leads us to divide the interval 0.5, 0.5] into two regions, and, defined for some a > 0 and ν > 0, as shown and = ; 0.5 a ν} (5 = ; 0.5 a ν 0.5 }. (6 For, Re W p /} is of order O( ln, as W p / is O( ln. Therefore, ignoring the lower order term involving W p and using the fact that for x, + x = + x/ + O(x, we otain Y p = p p Re Wp }] + o(. (7 Sustituting Y p into the error mapping for Alg and carrying out the analysis in a similar way as was done for Alg, we find that ˆ= + 0.5 ( Re +( + Re W0.5 } W 0.5 }]. (8 This result is similar to the estimator expression of Alg otained y taking half the real part of (0. In fact for the performances of the two algorithms are statistically equivalent since is a complex constant and does not affect the statistics of the noise coefficients W p. ow, turning our attention to region, we find that the estimator is iased. We consider here the case 0.5, the other case is similar. As 0.5, the orders of the terms in the expression of Y 0.5 are preserved. However, looking at Y 0.5 we find that there is a value of close to 0.5 after which the term in W 0.5 starts to dominate that in ReW 0.5 }. The estimator then ecomes iased since E W 0.5 ] 0. We take this value of to e the oundary etween regions and. Let ζ = 0.5. ow the term involving W 0.5 is of order O(ζ ln, as that in ReW 0.5 } is O(ζ ln. As a definition, we take a quantity Q to dominate another quantity Q if Q /Q = o(. A function that satisfies this requirement is φ = / ln. ote that this choice of φ is aritrary and any other function that is o( could have een used. Using this definition, we find that the term in W 0.5 dominates that in ReW 0.5 } when ζ =. Thus, the oundary etween and is given y 0.5. The resulting ias of the estimator for is O(ln. On the other hand, the width of region is o(. As vanishes faster than the growth rate of the ias, the asymptotic result of region holds and the algorithm is asymptotically uniased with a performance that is identical to that of Alg.

4 Finally, we have the following theorem: Theorem : Let ˆ e given y the error functionals of Alg or Alg (with for Alg and let ˆf e defined as ˆf = ˆm + ˆ ( then σ ˆf f is asymptotically standard normal with σ given y σ = π ( 0.5 ( 4 + 4 3 ρ cos. (π A useful indicator of the algorithm performance is the ratio of its asymptotic variance to the ACRB. This is, ( R = π4 0.5 ( 4 + 6 cos. (9 (π The error functionals are then seen to have identical performances. The ratio of the variance of the estimates, for SRs aove the threshold, is dependent on, ut independent of the SR. Furthermore, it has a minimum of.047 for = 0. B. Iterative Implementation In the previous section, we showed that the performance of the interpolation functions of oth estimators depend on the true signal frequency. The iterative procedure of tale I reduces this frequency dependence and improves the performance of the algorithm. The estimate of the residual otained at each iteration is removed from the signal and the estimator reapplied to the compensated data. In this section we show that the estimators are well ehaved and the procedure converges in two iterations. This allows for a computationally efficient algorithm with a performance that is only marginally aove the CRB. We will only consider the iterative estimator constructed using Alg. A similar argument can e constructed for Alg, 0], pp. 94-99. Let the true value of the residual e denoted y 0. ow h( can e written as (0 h( = sin ( π sin ( π ( ] + O ln. (0 Expanding h( into a Taylor series aout 0 gives ] h( = ( 0 h ( 0 + O( ln h π ( 0 = sin ( π ( = + O ( ] + O ln ( ln. ( The estimation function ψ( = + h( ecomes ψ( = 0 + ( 0 O( ln. (3 ow for any, 0.5, 0.5], we have ψ( ψ( = O( ln α with α <. Thus, the iterative procedure constitutes a contractive mapping on -0.5,0.5]. Also, it has a unique fixed point at 0. That is, ψ( 0 = 0. Consequently, the fixed point theorem, 6], pp. 33, ensures that the algorithm of tale I converges to the fixed point. The residual input to the algorithm at the i th iteration is 0 ˆ i. Let the variance expression of the estimator, shown in equation (, e denoted y g(. The variance of the estimate, ˆ i, at the i th iteration is given y g( 0 ˆ i. Thus, the limiting variance of the estimator is varˆ ] = lim i g( 0 ˆ i =g(0, (4 the last result follows from the fact that lim i ˆi = 0 and g( is continuous on 0.5, 0.5]. ow we turn our attention to the stopping criteria. The CRB for, which is O(, forms a lower ound on the estimation variance and no further gain is achievale once the residual frequency is of order lower than it. Therefore, it is reasonale to stop the estimator once the residual is o(. Let this iteration numer e Q. Starting with an initial estimate, ˆ 0 = 0, and using equation (3, the estimate after the first iteration, ˆ is given y, ] ˆ = 0 + O( ln and the residual is ˆ 0 = O( ln. This is still of order higher than the CRB. Looking at the estimate after the second iteration, we have ˆ = 0 + O( ln ] (5 and the residual is ˆ 0 = O( ln, which is now o(. Thus, only two iterations are needed for the residual to ecome of lower order than the CRB. We say that the algorithm has converged after two iterations. These results are summarized y the following theorem. Theorem : The iterative procedure defined using Alg or Alg, as shown in tale I, converges with the following properties The fixed point of convergence is the true offset, 0 The procedure takes iterations for the residual error to ecome o( and The limiting ratio of the variance of the estimator to the asymptotic CRB is π 4 /96 or.047 uniformly over the interval -0.5,0.5]. At this stage, we note that, as mentioned in the introduction, the results derived in this paper are valid under the more general and relaxed noise assumptions stated y Quinn in ]. IV. SIMULATIO RESULTS The algorithms presented aove were implemented and simulated. The numer oamples used in the simulation was = 04. Fig. displays the theoretical and simulation results on the ratio of the variance of the estimates to the ACRB versus for one and two iterations. The signal to noise ratio for this simulation was set to 0dB. We see that for

5 Ratio of Estimator Variance to Asymptotic CRB 7 6 5 4 3 Theoretical Curve Alg: Simulation Curve, Q= Alg: Simulatiojn Curve, Q= Alg: Simulation Curve, Q= Alg: Simulation Curve, Q= 0 0.5 0.4 0.3 0. 0. 0 0. 0. 0.3 0.4 0.5 Offset From Bin Center, Fig.. Plot of the ratio of the variance of the Alg and Alg to the asymptotic CRB as a function of, the frequency offset from the in center. Simulation curves for and iterations are shown. The theoretical curve is also displayed. 0000 runs were averaged at a signal to noise ratio of 0dB. Standard Deviation of Frequency Estimates (Hz 0 0 0 0 0 3 0 4 Cramer Rao Bound Curve Alg: Simulation Curve, Q= Alg: Simulation Curve, Q= 0 5 0 8 6 4 0 8 6 4 0 Signal to oise Ratio (db Fig.. Plot of the standard deviation of the frequency estimation error for Alg and Alg as a function of the signal to noise ratio. The Cramer Rao Bound curve is also shown. 0000 runs at each signal to noise ratio were averaged. Q =, the simulation and theoretical results agree closely. For Alg, the oundary etween regions and is clearly visile. The plot also shows that after the second iteration, the performance of oth algorithms is uniform over the entire interval. The ratio of the variance of oth estimators is, as expected, very close to the theoretical value of.047. Fig. presents the simulation results of the noise performance of oth algorithms as a function of the signal to noise ratio. The CRB curve is shown for the purpose of comparison. Both algorithms exhiit almost identical performances that are on the CRB curve. The threshold effect that is characteristic of the ML estimator and which results from the coarse estimation stage, is visile. V. COCLUSIO In this paper we have proposed and analyzed two new estimators for the frequency of a complex exponential in additive noise. The estimators consist of a coarse search followed y a fine search algorithm. The coarse search is implemented using the standard Maximum Bin Search. Two new fine search algorithms have een proposed and their asymptotic performances derived. The estimator were implemented iteratively and the resulting procedure shown to converge to the true signal frequency. The estimation variance of the iterative algorithm was also shown to converge asymptotically to its minimum value. This results in an improvement in the performances of the estimators when implemented iteratively. The numer of iterations required for convergence was found to e for oth algorithms. Hence the iterative estimator has a computational load of the same order as the FFT. Finally, the theoretical results were verified y simulations. ACKOWLEDGEMETS The authors appreciate the constructive comments of the anonymous reviewers. REFERECES ] B. G. Quinn, Estimating frequency y interpolation using Fourier coefficients, IEEE Transactions on Signal Processing, vol. 4, no. 5, pp. 64 8, 994. ] B. 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