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In 1929, Fisher proposed a test for periodicity based on the largest periodogram ordinate. If the true frequency lies between two consecutive Fourier frequencies and the signal to noise ratio is low, the test may conclude that there is no periodicity. This loss of power was noted by Whittle in 1952, as well as the necessary assumption that the noise be white. Whittle and subsequent authors suggested remedies for the white noise assumption. This paper proposes simple tests, based on the Fourier coefficients, that is, the Fourier transforms at the Fourier frequencies, that have good power properties at all frequencies.

This article is concerned with determining whether two independent time series have been generated by underlying stochastic processes with the same spectral shape. There are many methods that do so using the periodogram. Alternative approaches test for the equality of a finite number of autocovariances or autocorrelations. Non-parametric methods usually have low power when compared with parametric methods. The parametric approach we introduce fits autoregressions to the two time series and tests whether the model parameters are equal using a likelihood ratio test. The test performs well when the time series are from autoregressions. However, problems arise when this is not the case. A modification to the test is proposed, which fits fixed order autoregressions. Simulations show that the modified test performs well even when the two time series are not from autoregressive processes. The parametric approach is shown to outperform non-parametric alternatives in a power study.

Heart rate variability (HRV) and baroreceptor sensitivity (BRS) quantify autonomic variability in heart pacing and the autonomic response to blood pressure changes respectively. By necessity, the signals used to calculate HRV and BRS (systolic blood pressure (SBP) and RR interval) have one data point every cardiac cycle. Due to inherent variability in heart rate, these are non-uniformly sampled data. A number of calculation methods exist that adjust for non-uniform sampled signals. This study compared frequency domain methods of HRV and BRS calculation to ascertain whether more complex methods resulted in different results to simpler methods. Wistar rats (n=10), and rats with induced diabetes (n=8) were anesthetized and SBP and RR interval measured for a period of approximately 5 minutes. This data were analyzed using the sequence technique (for BRS), fast Fourier transform (FFT), non-uniform discrete Fourier transform (NDFT) and an extended Lomb-Scargle Periodogram (LSP). There were small but significant differences in NDFT from LSP technique for both BRS in the low frequency range (p=0.005) and HRV in the high frequency range (p=0.001). The NDFT technique was also significantly different to FFT technique for BRS in the low frequency range (p=0.023). All other methods were not statistically different. However, all techniques showed the same results comparing diabetic to control rats. This study shows more complex methods that correct for the non-uniformity of the sampling have significant differences but those differences are small to the point of not altering findings associated with HRV or BRS.

Kay has proposed a technique for estimating the frequency of a complex sinusoid in additive noise, the real and imaginary parts of which are independent and normally distributed with means zero and the same variance. For fixed sample size the estimator achieves the Cramer–Rao lower bound for unbiased estimators of the frequency in the limit as the signal to noise ratio approaches infinity. It has been noted by Lovell and Williamson, however, that the estimator is biased. It is shown in this paper that under Kay's assumptions the estimator is not consistent (for fixed signal to noise ratio the estimator converges almost surely, as the sample size N increases, to a number that is not the true frequency, no matter how large the signal to noise ratio). A class of distributions for the additive noise is proposed under which the technique is strongly consistent and has the correct order of asymptotic variance, namely N−3, for the case where there is some a priori knowledge concerning the range of the frequency. For this class a normal central limit theorem is developed.

In Quinn & Fernandes (1991), an iterative technique was presented for the estimation of the frequency of a sinusoid in the presence of noise. The technique is more robust to poor initial estimates than is the periodogram maximiser. A generalisation to the case of multiple sinusoids is presented in this paper, and it is shown that the original estimator may be interpreted as the maximiser of a smoothed version of the periodogram.

Estimating the coefficients of a noisy polynomial phase signal is important in fields including radar, biology and radio communications. One approach attempts to perform polynomial regression on the phase of the signal. This is complicated by the fact that the phase is wrapped modulo 2π and must be unwrapped before regression can be performed. In this paper, we consider an estimator that performs phase unwrapping in a least squares manner. We call this the least squares unwrapping (LSU) estimator. The LSU estimator can be computed in a reasonable amount of time for data sets of moderate size using existing general purpose algorithms from algebraic number theory. Under mild conditions on the distribution of the noise we describe the asymptotic properties of this estimator, showing that it is strongly consistent and asymptotically normally distributed. A key feature is that the LSU estimator is accurate over a far wider range of parameters than many popular existing estimators. Monte-Carlo simulations support our theoretical results and demonstrate the excellent statistical performance of the LSU estimator when compared with existing state-of-the-art estimators. Refereed/Peer-reviewed

We examine the problem of estimating the periods of interleaved periodic point processes. We are particularly interested in the case where times of arrival (TOAs) are either measured with noise or not measured at all. This can arise in communications surveillance, where communications signals of different bauds may lie within the same surveillance bandwidth, and likewise in Electronic Surveillance (ES), where pulses or scans from different radars are observed together. In [1], the authors developed a general asymptotic theory for the Bartlett point-process periodogram estimator of the period of a single periodic process. In this paper, we extend the model to multiple periodic processes, each with a distinct period. The TOAs are observed unlabelled and in time order, i.e., they are interleaved. The largest local maximizers of the periodogram are shown to be good estimators of the unknown periods, asymptotically, and central limit theorems are proved. Simulations highlight a number of practical problems, and some problems with outstanding solutions are suggested.

We consider least squares estimators of carrier phase and amplitude from a noisy communications signal. We focus on signaling constellations that have symbols evenly distributed on the complex unit circle, i.e., M-ary phase shift keying. We show, under reasonably mild conditions on the distribution of the noise, that the least squares estimator of carrier phase is strongly consistent and asymptotically normally distributed. However, the amplitude estimator is not consistent, but converges to a positive real number that is a function of the true carrier amplitude, the noise distribution and the size of the constellation. The results of Monte Carlo simulations are provided and these corroborate the theoretical results. Refereed/Peer-reviewed

A parametric form for the received ‘‘instantaneous frequency’’ of a pure tone is derived for the case of relative straight line motion between source and receiver. Ferguson and Quinn [J. Acoust. Soc. Am. 96, 821–827 (1994)] have recently described an algorithm which, given a sequence of high‐accuracy frequency estimators, produces estimators of the closest distance between the source and receiver, the relative speed, the rest frequency, and the time at closest approach. In this paper, an improved algorithm which uses amplitude estimators as well is introduced, and the statistical behavior of the estimators derived. The technique is automatic and will work even if only a section of the Doppler track is available.

The dominant feature in the acoustic spectrum of a propeller‐driven aircraft is the spectral line corresponding to the propeller blade rate that is equal to the product of the propeller shaft rotation rate and the number of blades on the propeller. The frequency of this line, when measured by a stationary observer on the ground, changes with time due to the acoustical Doppler effect. In this paper, the short‐time Fourier transform and the Wigner–Ville distribution are used to estimate the propeller blade rate at short time intervals for a turbo‐prop aircraft flying at a constant altitude and speed over an acoustic sensor located just above ground level. The temporal variation in the observed blade rate is then used to estimate the speed and altitude of the aircraft, together with the source (or rest) frequency of the blade rate. Finally, the estimated values for these parameters are compared with the actual values recorded onboard the aircraft during each of the eighteen transits formed by pairing each element of a set of speeds: 150, 200, and 250 kn, with each element of a set of aircraft altitudes: 250, 500, 750, 1000, 1250, and 1500 ft.

A hidden Markov model (HMM) technique for the estimation of the shape of a towed array is presented. It is assumed that there is a far-field source radiating sound containing possibly weak spectral lines. The technique uses either the Fourier coefficients at a given frequency computed from a single time block or the maximal eigenvector of a sample spectral covariance matrix. The technique is illustrated using several simulations. The results of these simulations indicate that the HMM technique yields shape and bearing estimates more accurate than those provided by a maximum-likelihood array shape estimation technique. >

Single frequency estimation is a long-studied problem with application domains including radar, sonar, telecommunications, astronomy and medicine. One method of estimation, called phase unwrapping, attempts to estimate the frequency by performing linear regression on the phase of the received signal. This procedure is complicated by the fact that the received phase is ‘wrapped’ modulo ???? and therefore must be ‘unwrapped’ before the regression can be performed. In this paper, we propose an estimator that performs phase unwrapping in the least squares sense. The estimator is shown to be strongly consistent and its asymptotic distribution is derived.We then show that the problem of computing the least squares phase unwrapping is related to a problem in algorithmic number theory known as the nearest lattice point problem. We derive a polynomial time algorithm that computes the least squares estimator. The results of various simulations are described for different values of sample size and SNR.