The article focuses on the refinement of the Cramer-Rao lower boundary for the single-tone signal amplitude for estimation methods using window functions. The Cramer-Rao lower boundary allows finding the minimum variance of the signal parameter estimate. There are formulas for variances estimations of the amplitude, frequency, and phase of harmonics of harmonic signals, but the known methods for their finding, including algorithms based on the maximum likelihood method, show results above this boundary. The increased variance of the estimate occurs due to the application of window functions on the original signal. The estimation of the accuracy of the parameters harmonics in case using the window functions usually done with a numerical simulation. In the article, the authors derive a formula for the minimum variance for amplitudes of harmonics in the case of the using the window functions. This derivation allows us to understand the mathematical meaning of the broadening the Cramer-Rao lower boundary when using window functions, and the resulting formula brings a faster and more accurate estimation of an amplitude accuration, in comparison with numerical simulation. According to the results of the experiment the calculations by the proposed formula and the numerical experiment data are the same.
Статья посвящена уточнению границы Крамера-Рао амплитуды однотонального сигнала для методов ее оценки с использованием оконных функций. Граница Крамера-Рао позволяет найти минимальную дисперсию оценки параметра сигнала. Для гармонических сигналов известны формулы, оценивающие дисперсии амплитуды, частоты и фазы гармоник, однако, известные методы их оценки показывают результаты выше этой границы. В том числе не достигает границы Крамера-Рао и метод корреляционного анализа, который находит гармоники по методу максимального правдоподобия. Увеличение дисперсии оценки происходит из-за наложения на исходный сигнал оконных функций. Для определения точности оценки при выборе оконной функции обычно использовалось численное моделирование методов оценки гармоник. В статье выводится формула для оценки минимальной дисперсии амплитуды гармоник для методов, использующих оконные функции. Этот вывод позволяет понять математический смысл увеличения границы Крамера-Рао при использовании оконных функции, а полученная формула позволяет быстрее и точнее, по сравнению с численным моделированием, оценить точность определения амплитуды. Результаты экспериментальных исследований показали, что расчеты по предложенной формуле совпадают с результатами, полученными на основе численных экспериментов.
Discrete Fourier Transform (DFT) allows you to determine the discrete spectrum of a signal. Due to the presence of its high-speed implementation, called Fast Fourier Transform (FFT), this transform is widely used in digital signal processing (DSP). Most DSP tasks that deal with analogy signal and spectrum adapt the DFT to find the signal spectrum between harmonics. One of the most commonly used ways of such adaptation – the use of window functions. The analysis of standard window functions (Kaiser, etc.) showed that their direct application to solving the problem of estimating the parameters (frequency, amplitude, and phase) of nonharmonic sinusoidal components of signals leads both to the need for additional corrections of the estimation results and to additional errors in determining the phase of the signal. The paper proposes a method that allows building a window function without the indicated drawbacks based on standard window functions. The essence of the method is to transform the standard window function so that its spectrum does not contain imaginary components, and the amplitude of the fundamental harmonic would be equal to 1. The results of modeling the proposed method on the example of the Kaiser window showed that the phase estimate of the nonharmonic components of the spectrum using the obtained window function, in contrast to the estimate using standard window functions, is not displaced.
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