The Estimation and Tracking of Frequency

Abstract

Many electronic and acoustic signals can be modelled as sums of sinusoids and noise. However, the amplitudes, phases and frequencies of the sinusoids are often unknown and must be estimated in order to characterise the periodicity or near-periodicity of a signal and consequently to identify its source. This 2001 book presents and analyses several practical techniques used for such estimation. The problem of tracking slow frequency changes over time of a very noisy sinusoid is also considered. Rigorous analyses are presented via asymptotic or large sample theory, together with physical insight. The book focuses on achieving extremely accurate estimates when the signal to noise ratio is low but the sample size is large. Each chapter begins with a detailed overview, and many applications are given. Matlab code for the estimation techniques is also included. The book will thus serve as an excellent introduction and reference for researchers analysing such signals.

Adaptive tests for periodic signal detection with applications to laser vibrometry

“…The results summarized in Table 4.3 show that, at such a signal to noise ratio, our test is powerful. The classical tests used in periodic signal detection, such as those developed in the books of Brockwell and Davis [4] or Quinn and Hannan [18], essentially use a parametric approach. As noticed in [6] on page 298, they are particularly adapted to the detection of periodic signals having a small number of harmonics such as sinusoids or trigonometric polynomials with small degrees, which is not the case of the signals of our interest.…”

Numerical questions in ODE boundary value problems

“…However, the unstable case corresponding to h = 0.1 does show more influence of instability than the case h = .01 . The small denominator in (12) suggests the likely explanation.…”

“…A higher rate (O n −3/2 ) is theoretically possible in maximum likelihood estimates in the frequency estimation problem, but direct maximization is not the way to obtain these quantities [12].…”

“…Consider the trapezoidal rule denominator in (12). This suggests large and spurious amplification is likely in case h = 0.1 ; but this is not evident in the stable computation.…”

On frequency tracking in harmonic acoustic signals

“…Frequency and harmonics tracking of a periodic signal in noise is a well studied problem; for an overview, see [1]. One approach includes methods which assume that the frequency is approximately constant over a certain interval (batch).…”