A method of extraction of nonstationary sinusoids
Introduction
An algorithm which is capable of extraction of sinusoids and estimation of their parameters in real-time finds applications in diverse areas of engineering. Parameters of interest are usually amplitude, total and constant phase and frequency of sinusoidal signals [12]. Examples of its general applications are frequency estimation [2], [18], estimation of time-varying biomedical signals [1], active noise and vibration control [6], and sinusoidal disturbance rejection [3].
There have been numerous attempts to design an algorithm to extract a single sinusoidal signal out of a multi-component nonstationary input signal. Adaptive notch filtering is an example of one such techniques [17]. An adaptive notch filter with a very sharp notch whose center frequency adaptively tracks that of the desired component of the input signal has been the subject of an active field of research. An ideal frequency domain analysis tool such as discrete Fourier transform (DFT) loses its effectiveness when the frequency of the input signal happens to vary with time [8]. Conventional methods of signal analysis, therefore, have long been left aside when dealing with nonstationary signals. Time-frequency domain signal processing tools are being developed to tackle such problems, an example of which is wavelet transform [1]. Alternative methods of sinusoidal signal extraction and analysis have been proposed. One is referred to [14], [10] for a new class of signal processing algorithms, development of which does not follow conventional methods.
This paper presents the development and mathematical properties of a signal processing algorithm capable of extracting and estimating a specified single sinusoidal component of its input signal and tracking variations of the amplitude, phase, and frequency of such a sinusoid over time. This novel algorithm is found to have numerous applications in diverse areas of electrical engineering ranging from biomedical [19], [20] to mechanical engineering [21], [22], [23].
A set of nonlinear differential equations governs the dynamics of the algorithm. Gradient descent method has been used to minimize the least-squares error between the input signal and the desired sinusoidal signal. Section 2 presents the detailed procedure of derivation of the governing differential equations. Since the error is not a quadratic function of the parameters to be estimated, the gradient descent method, by itself, does not guarantee the convergence and stability of the dynamical system. Poincaré map theorem has been employed to prove the mathematical properties of the algorithm. Section 3 summarizes the proofs of the existence, uniqueness and stability of a periodic orbit of the dynamical system governing the proposed algorithm. Mathematically predicted behavior of the algorithm is confirmed by the use of computer simulations. Section 4 presents results of such simulations. To verify the functionality of the proposed algorithm in practical scenarios, it has been implemented on a digital signal processor (DSP) platform. Section 5 presents the laboratory verification of the performance of the proposed algorithm. In Section 7, the proposed method is compared with two of the reportedly most promising existing methods.
Section snippets
Proposed algorithm
Let u(t) represent a voltage or current signal. This function is usually continuous and almost periodic. A sinusoidal component of this function, , is of interest where A is the amplitude and φ(t) represents the total phase of this component. Where the frequency is fixed, φ(t) term may be expressed as ωt+δ in which ω is the frequency (in rad/s) and δ is the constant phase. Ideally, parameters A, ω, and δ are fixed quantities; but, in practice, this assumption does not hold true. In
Mathematical properties of the algorithm
The expression for the error function (10) may be used in , , to yield a more explicit form. Presence of sine and cosine terms in the expressions suggests the framing of the equations in spherical coordinate system. If the explicit form of the governing differential equations of the algorithm is framed in spherical coordinates by replacing , and by r, θ and φ, the set of differential equations becomes
Let u(t)=uo(t
Computer simulations
The signal processing algorithm described by the dynamical system , , has a very simple structure. It can be easily implemented in any scientific programming language or schematic design environment. Where a first-order approximation for derivatives is assumed, the discretized form of the equations can be written aswhere Ts is the sampling time and n is the
Laboratory verification
Fig. 10 shows a snapshot of real-time performance of the algorithm implemented on a Texas InstrumentsTM TMS320C6711 floating point DSP. The algorithm, embedded in a C code, was translated into DSP assembly language using the integrated development environment of the DSP and was subsequently downloaded into the DSP platform. The input signal is an intentionally distorted sinusoid oscillating at sampled at with a resolution. The values of parameters were chosen as μ1=50, μ
Applications of the proposed algorithm
The proposed algorithm is a technique for the analysis of nonstationary signals and, as such, finds applications in diverse areas of science and engineering. It provides a means of estimation of parameters (amplitude, phase, and frequency) of a sinusoid of time-varying characteristics, and is thus a fundamental tool for amplitude/phase/frequency estimation. Moreover, the algorithm provides an estimate of the sinusoidal component of interest itself which is synchronous to the input signal. In
Comparison of the proposed algorithm with existing methods
In this section, the proposed method of extraction of nonstationary sinusoids is compared with two of the reportedly most promising methods that are of a similar kind. Extended Kalman filtering is chosen as representing an example of the conventional methods with reported success in the literature. The method of Regalia et al. [10], [14], which has recently attracted the attention of researchers, is another example of the successful methods reported so far. In each case, the existing method is
Conclusions
A method of extraction of nonstationary sinusoids and estimation of their parameters is presented. A nonlinear set of differential equations governs the dynamics of the algorithm. The dynamical system is shown to possess a unique asymptotically stable periodic orbit. In signal processing terms, the algorithm locks into a more or less specified sinusoidal component of its input signal and follows its variations over time.
The structure of the algorithm is very simple; it can be easily implemented
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