Fourier-Based Spectral Analysis with Adaptive Resolution

Fourier-Based Spectral Analysis with Adaptive Resolution Andrey Khilko email: 155tm2@gmail.com Abstract Despite being the most popular methods of data analysis, Fourier-based techniques suffer from the problem of static resolution that is currently believed to be a fundamental limitation of the Fourier Transform. Although alternative solutions overcome this limitation, none provide the simplicity, versatility, and convenience of the Fourier analysis. The lack of convenience often prevents these alternatives from replacing classical spectral methods – even in applications that suffer from the limitation of static resolution. This work demonstrates that, contrary to the generally accepted belief, the Fourier Transform can be generalized to the case of adaptive resolution. The generalized transform provides backward compatibility with classical spectral techniques and introduces minimal computational overhead. Introduction The measurement of spectral characteristics of data involves a fundamental trade-off that is often referred to as the Uncertainty Principle: better resolution in frequency domain results in lower resolution in time domain and vice versa. In existing Fourier-based methods, resolution is generally static: decision about the time-frequency resolution trade-off must be made a priori and cannot be changed in process of measu- rement based on actual characteristics of measured data without prior knowledge of these characteristics. If processing of real time non-stationary signals is required, and their properties are not known in advance, existing Fourier-based analysis cannot provide accurate information about the presence of a particular frequency in the data spectrum and the time interval where the frequency is present. Due to a mismatch between the decided measurement resolution and the actual characteristics of the

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measured data causing undesirable smearing in both, time and frequency domains, the accuracy of such analysis is generally lower than the fundamental limit that is implied by the Uncertainty Principle. A similar problem exists when data contains concurrent spectral components with different uncertainties that cannot be analyzed jointly when resolution is static. Although the time-frequency resolution trade-off is the most common manifestation, the Uncertainty Principle can also manifest itself as spatial resolution vs. frequency resolution. For clarity reasons, only the time-frequency resolution is mentioned below in this paper. In actual application, either manifestation can arise depending on the application nature. A number of alternatives have been proposed to address the problem of static resolution of Fourier-based analysis. The most popular one is probably Multiresolution Analysis (MRA) based on Continuous Wavelet Transform (CWT) or Discrete Wavelet Transform (DWT; a good explanation of wavelets and MRA can be found e.g. in [1]). However, the CWT is computationally expensive and operates different terminology whereas the DWT provides nonlinear frequency grid where resolution is not easily scalable from practical prospective. Therefore in most cases, these alternatives cannot be immediately used as direct replacements of Fourier-based techniques and require new analysis methodology. Because the Fourier Transform (FT) is historically by far the most popular engineering tool for analyzing characteristics of data, a need still exists for an enhanced Fourier-based method capable of working with non-stationary signals. However, the scientific community currently agrees in belief that the FT itself cannot provide this functionality. Therefore, when the FT needs to be used beyond the realm of static resolution, researches seek for application-specific workarounds. In work [2], the FT was used for spectral analysis of speech signals. Because all phonemes1 could not be reliably identified using same time-frequency resolution, the

1 The smallest units of speech that distinguish meaning.

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latter was tuned based on known phonetic structure of speech by adjusting length of sampling window for individual phonemes. While this approach can be applied to speech (known as a highly redundant signal) or sometimes even to a signal of unknown nature [3], it may not work well in situations when important spectral features that require different resolutions are concurrent and cannot be resolved serially. The problem of concurrent spectral components was addressed in [4]. Instead of varying sampling window length for different data fragments, the same data was analyzed multiple times using different static resolutions. One can see however, that this approach introduces serious computational expenses without adequate so

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