Power spectrum (PS) estimation methods have a long history. At their beginning the word ''spectrum''was introduced by Newton in Opticks, i.e. in his studies of decomposition of the white light into a band of light colours that was published in 1706. It is known that the classical approaches of spectral analysis include the ordinary discrete Fourier transform (DFT) as well as its variants, which are typically based on smoothing the DFT spectral estimate or windowing the data. The DFT methods are very robust because they do not make any a priori assumptions on the spectral form. On the other hand, the spectral resolution of DFT techniques is frequently poor and depends on the number of samples to be processed. In total, the power spectrum estimation methods are divided into nonparametric, parametric, and subspace methods. The nonparametric and parametric methods have been investigated by many researches, but the analysis of their performance is difficult and fewer results are available, especially, when the signal to be processed is short. Subspace methods generate PS estimates for a signal based on the eigenanalysis of the correlation matrix. We analyze eleven most important power spectrum estimation techniques and algorithms, beginning with the nonparametric methods, based on the periodogram, and concluding with the modern parametric method, based on the AR linear models. A new approach is presented that improves the frequency estimates of the short-length signal in a noisy frame. In such a case, the Burg algorithm with the extrapolation technique is used and investigated by simulation on a computer as well. The frequency modulated (FM) signal estimation in a noisy environment is important for many commercial and military applications. A new forward-backward approach grounded on basic functions is proposed for time-varying frequency estimation of the frequency-modulated signal in a noisy frame. In many cases time series in econometrics, biometrics, techniques, and other applications have missing data. Restoration of missing data is the estimation of the lost samples of a signal using the known samples at the neighbouring segments. We consider a problem of restoration of a missing data segment in a relatively short data sequence on the assumption that the data can be modeled as a finite order AR process contaminated with the additive Gaussian noise. The restoration must be done so that the restored data fit the assumed model as well as possible in the least squares (LS) sense. The recursive DFT and DCT (discrete-time cosine transform) algorithms are worked out and they are analysed in the sense of the burden of ordinary arithmetical operations necessary for their calculations in wireless sensor networks. The parallel DFT (PDFT) procedure is proposed here that noticeably decreases the amount of operations needed to calculate DFT, and, on the contrary, increases the speed of DFT estimation. We analysed here the total number of arithmetic operations of two-dimensional DFT, too. The recursive algorithms are worked out that let us to solve PS problem more effectively. A problem to recognize the non-aliased realization in the set of multifold downsampled realizations is solved here, too. At last, there is proposed an approach, how efficiently to determine some statistical characteristics of downsampled realizations.

Schlagwörter: signal; spectrum; estimation; decimation; digital processing

Mathematik, Informatik