Asymptotic analysis of a fast algorithm for efficient multiple frequency estimation

Abstract

Based on an asymptotic analysis of the contraction mapping (CM) method of Li and Kedem (IEEE Trans. Inform. Theory, vol. 39, pp. 989-998, 1993), a bandwidth shrinkage rule is proposed for fast and accurate estimation of the frequencies of multiple sinusoids from noisy measurements. The CM frequency estimates are defined as the fixed-points of a contractive mapping formed by the lag-one autocorrelation coefficient calculated from the output of a parametric filter applied to the observed time series. With judiciously chosen bandwidth parameters according to the asymptotic analysis, the algorithm is shown to be able to accommodate possibly poor initial values of precision O(n −1/3) and converge to a final estimate whose accuracy is arbitrarily close to O(n −3/2), the optimal error rate for frequency estimation under the Gaussian assumption. The total computational complexity of the algorithm is shown to be O(n log n), which is comparable to that of n-point FFT. A novelty in the asymptotic analysis is that it accommodates closely-spaced frequencies by allowing not only the filter bandwidth but also the frequency separation to be functions of the sample size n. This enables an assessment of the accuracy of the frequency estimates for given bandwidths and initial values in situations where some or all of the frequencies are close to each other.

Statistical analysis of business cycle fluctuations in Poland before and after the crisis

“…However, in order to provide a comparison to other methods used in the literature, Fig. 4 depicts the outcome obtained using the parametric procedure proposed by Li and Song (2002). 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 2 3 4 5 6 7 8 9 10 , cycle length: years / months last observation:…”

“…3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 2 3 4 5 6 7 8 9 10 , cycle length: years / months last observation: Dec 2014 Figure 4. Analysis of the Polish industrial production index (y-o-y growth rates) using the approach of Li and Song (2002) Source: own calculations. CM frequency estimates ranging from 0.05 to 0.35 are depicted on vertical axes, with increasing number of iterations on horizontal axes.…”

“…Differences arising from amplitude or phase shift parameters are not of primary importance in the paper. Moreover, we compare the results with outcome of the method presented in Li and Song (2002).…”

“…Notice that the mean function of the time series {X t : t∈Z} is almost periodic. For details concerning the estimation procedure see Li and Song (2002).…”