A Review of the Frequency Estimation and Tracking Problems

This report presents a concise review of some frequency estimation and frequency tracking problems. In particular, the report focusses on aspects of these problems which have been addressed by members of the Frequency Tracking and Estimation project of the Centre for Robust and Adaptive Systems. The report is divided into four parts: problem specification and discussion, associated problems, frequency estimation algorithms and frequency tracking algorithms. Part I begins with a definition of the various frequency estimation and tracking problems. Practical examples of where each problem may arise are given. A comparison is made between the frequency estimation and tracking problems. In Part II, block frequency estimation algorithms, fast block frequency estimation algorithms and notch filtering techniques for frequency estimation are dealt with. Frequency tracking algorithms are examined in Part III. Part IV of this report examines various problems associated with frequency estimation. Associated problems include Cramer-Rao lower bounds, theoretical algorithm performance, frequency resolution, use of the analytic signal and model order selection

Nonlinear adaptive estimation with application to sinusoidal identification

Parameter estimation of a sinusoidal signal in real-time is encountered in applications in numerous areas of engineering. Parameters of interest are usually amplitude, frequency and phase wherein frequency tracking is the fundamental task in sinusoidal estimation. This thesis deals with the problem of identifying a signal that comprises n (n ≥ 1) harmonics from a measurement possibly affected by structured and unstructured disturbances. The structured perturbations are modeled as a time-polynomial so as to represent, for example, bias and drift phenomena typically present in applications, whereas the unstructured disturbances are characterized as bounded perturbation. Several approaches upon different theoretical tools are presented in this thesis, and classified into two main categories: asymptotic and non-asymptotic methodologies, depending on the qualitative characteristics of the convergence behavior over time. The first part of the thesis is devoted to the asymptotic estimators, which typically consist in a pre-filtering module for generating a number of auxiliary signals, independent of the structured perturbations. These auxiliary signals can be used either directly or indirectly to estimate—in an adaptive way—the frequency, the amplitude and the phase of the sinusoidal signals. More specifically, the direct approach is based on a simple gradient method, which ensures Input-to-State Stability of the estimation error with respect to the bounded-unstructured disturbances. The indirect method exploits a specific adaptive observer scheme equipped with a switching criterion allowing to properly address in a stable way the poor excitation scenarios. It is shown that the adaptive observer method can be applied for estimating multi-frequencies through an augmented but unified framework, which is a crucial advantage with respect to direct approaches. The estimators’ stability properties are also analyzed by Input-to-State-Stability (ISS) arguments. In the second part we present a non-asymptotic estimation methodology characterized by a distinctive feature that permits finite-time convergence of the estimates. Resorting to the Volterra integral operators with suitably designed kernels, the measured signal is processed, yielding a set of auxiliary signals, in which the influence of the unknown initial conditions is annihilated. A sliding mode-based adaptation law, fed by the aforementioned auxiliary signals, is proposed for deadbeat estimation of the frequency and amplitude, which are dealt with in a step-by-step manner. The worst case behavior of the proposed algorithm in the presence of bounded perturbation is studied by ISS tools. The practical characteristics of all estimation techniques are evaluated and compared with other existing techniques by extensive simulations and experimental trials.Open Acces

A comparison of frequency estimation techniques for high-dynamic trajectories

A comparison is presented for four different estimation techniques applied to the problem of continuously estimating the parameters of a sinusoidal Global Positioning System (GPS) signal, observed in the presence of additive noise, under extremely high-dynamic conditions. Frequency estimates are emphasized, although phase and/or frequency rate are also estimated by some of the algorithms. These parameters are related to the velocity, position, and acceleration of the maneuvering transmitter. Estimated performance at low carrier-to-noise ratios and high dynamics is investigated for the purpose of determining the useful operating range of an approximate Maximum Likelihood (ML) estimator, an Extended Kalman Filter (EKF), a Cross-Product Automatic Frequency Control (CPAFC) loop, and a digital phase-locked loop (PPL). Numerical simulations are used to evaluate performance while tracking a common trajectory exhibiting high dynamics

A parallel prefiltering approach for the identification of a biased sinusoidal signal: theory and experiments

The problem of estimating the amplitude, frequency, and phase of an unknown sinusoidal signal from a noisy-biased measurement is addressed in this paper by a family of parallel prefiltering schemes. The proposed methodology consists in using a pair of linear filters of specified order to generate a suitable number of auxiliary signals that are used to estimate\u2014in an adaptive way\u2014the frequency, the amplitude, and the phase of the sinusoid. Increasing the order of the prefilters improves the noise immunity of the estimator, at the cost of an increase of the computational complexity. Among the whole family of estimators realizable by varying the order of the filters, the simple parallel prefilters of orders 2 C 2 and 3 C 3 are discussed in detail, being the most attractive from the implementability point of view. The behavior of the two algorithms with respect to bounded external disturbances is characterized by input-to-state stability arguments. Finally, the effectiveness of the proposed technique is shown both by comparative numerical simulations and by a real experiment addressing the estimation of the frequency of the electrical mains from a noisy voltage measurement

High-dynamic GPS tracking

The results of comparing four different frequency estimation schemes in the presence of high dynamics and low carrier-to-noise ratios are given. The comparison is based on measured data from a hardware demonstration. The tested algorithms include a digital phase-locked loop, a cross-product automatic frequency tracking loop, and extended Kalman filter, and finally, a fast Fourier transformation-aided cross-product frequency tracking loop. The tracking algorithms are compared on their frequency error performance and their ability to maintain lock during severe maneuvers at various carrier-to-noise ratios. The measured results are shown to agree with simulation results carried out and reported previously

Performance evaluation of a linear predictor frequency estimator for mobile flat fading wireless channels

A well known frequency estimation algorithm using the linear prediction method is analyzed for flat fading wireless channels. The estimator outputs are statistically analyzed and its jitter performances are compared with the non-fading case and the Cramer-Rao bound. We provide a closed form solution for the distribution and the variance of the frequency estimates under fading conditions by making valid assumptions. We also verify the theoretical model using simulations. Analysis shows that the variance of the estimates for flat fading channels reaches a threshold point and increasing the transmit power does not necessarily improve the performances any further

A new transfer function analyser

This thesis investigates the concept and design of a portable on-line transfer function analyser (TFA). It is eminently suitable for the identification of plants and other controlled feedback systems in which normal operating records are available. A point by point representation in the frequency domain, requiring a maximum of three records, allows Nyquist plots to be carried out, either visually or by plotter facilities. The basic theory relies heavily upon statistical concepts whereby, least squares estimates of the transfer function are obtained from a combination of heterodyning, exact filtering and adaptive loops. The resultant output, on both channels (real and imaginary), is the culmination of the solution to two linear differential equations with stochastic coefficients, so mechanised when the adaptive loops reach a stable equilibrium. Throughout, emphasis is placed upon the electronics combining the best of analog and digital techniques, in order that six parallel paths may be analysed in similar mode. This is especially true of the heterodyning and filtering operations. Practical shortcomings of the instrument noted by comparing estimates with those from the best currently available commercial apparatus, operating on deterministic signals. Examples of a feedback loop, subjected to both deterministic and random stimuli, with and without the presence of extraneous noise sources, are used to illustrate the ease and simplicity by which the instrument can be used in place of complex computing schemes, which tend, in consequence, to be solely of local academic interest. The practical features of the thesis have led to the submission of four papers to the technical press. Two of them deal, exclusively, with a capacitor ratio commutated filter – not apparently described in publications to date - which is also the subject of a proposed patent application in conjunction with NRDC. It is intended, in the near future, to submit the complete instrument as the basis for a second patent proposal

Technique for Measurement of Weld Resistance for AC Resistance Spot Welding via Instantaneous Phasor Measurement

The resistance measurement in the resistance spot welding (RSW), is an ongoing research topic. The high current flow during the welding process induces an electromagnetic field in the wires which are attached to the electrodes to measure tip voltage. This results an additional voltage drop which is proportional to the derivative of current. Also the presence of silicon controlled rectifier (SCR) in the welding power supply generates harmonics in both supply voltage and current. These issues together complicate the methods for resistance estimation. A set of simultaneous linear equations is derived for the on-line measurement of dynamic resistance and induced voltage constant by using the dynamic circuit analysis of weld setup. This can be solved to determine the weld resistance using instantaneous phasors measurements for the 1st, 3rd and 5th harmonics of current and measured voltage signals. The instantaneous phasor measurements for these desired harmonics are obtained by employing the following proposed method. In this thesis, a new method for the measurement of instantaneous phasor is proposed for the narrow band signals. The proposed algorithm is based on the internal model principle (IMP) defined for the cancellation of a sinusoidal disturbance signal. The IMP has two states, exhibiting the properties of being sinusoidal and orthogonal. The instantaneous values of IMP states are defined as real and imaginary components of a complex signal at each time instant. The instantaneous measurements of envelope and phase of a sinusoidal signal are determined from instantaneous values of complex signal by using arithmetic properties of complex numbers. In case of signal comprising of sum of sinusoids of different frequencies, the approach for obtaining instantaneous phasor for each sinusoidal component is presented by connecting multiple internal models in the parallel and open-loop configuration. The instantaneous phasor measurement of fundamental frequency signal is not only advantageous in detecting faults like short circuiting, harmonic distortion and frequency variations but it can also be applied to protect power system from these faults. In this work, the applicability of the proposed instantaneous phasor measurement algorithm is analyzed for scenarios of power disturbances due to the the harmonic distortion and decaying DC offset. The results are discussed and compared with few existing methods

Estimation of FM synthesis parameters from sampled sounds

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Architecture, 1988.Includes bibliographical references.by Thomas M. Sullivan.M.S