Fast computation of the ambiguity function and the Wigner distribution on arbitrary line segments

By using the fractional Fourier transformation of the time-domain signals, closed-form expressions for the projections of their auto or cross ambiguity functions are derived. Based on a similar formulation for the projections of the auto and cross Wigner distributions and the well known two-dimensional (2-D) Fourier transformation relationship between the ambiguity and Wigner domains, closed-form expressions are obtained for the slices of both the Wigner distribution and the ambiguity function. By using discretization of the obtained analytical expressions, efficient algorithms are proposed to compute uniformly spaced samples of the Wigner distribution and the ambiguity function located on arbitrary line segments. With repeated use of the proposed algorithms, samples in the Wigner or ambiguity domains can be computed on non-Cartesian sampling grids, such as polar grids

Efficient computation of the Ambiguity Function and the Wigner Distribution on arbitrary line segments

Efficient algorithms are proposed for the computation of Wigner distribution and ambiguity function samples on arbitrary line segments based on the relationship of Wigner distribution and ambiguity function with the fractional Fourier transformation. The proposed algorithms make use of an efficient computation algorithm of fractional Fourier transformation to compute N uniformly spaced samples O(N log N) flops. The ability of obtaining samples on arbitrary line segments provides significant freedom in the shape of the grids used in the Wigner distribution or in ambiguity function computations

Time-frequency component analyzer

Cataloged from PDF version of article.In this thesis, a new algorithm, time–frequency component analyzer (TFCA), is proposed to analyze composite signals, whose components have compact time–frequency supports. Examples of this type of signals include biological, acoustic, seismic, speech, radar and sonar signals. By conducting its time–frequency analysis in an adaptively chosen warped fractional domain the method provides time–frequency distributions which are as sharp as the Wigner distribution, while suppressing the undesirable interference terms present in the Wigner distribution. Being almost fully automated, TFCA does not require any a priori information on the analyzed signal. By making use of recently developed fast Wigner slice computation algorithm, directionally smoothed Wigner distribution algorithm and fractional domain incision algorithm in the warped fractional domain, the method provides an overall time-frequency representation of the composite signals. It also provides time–frequency representations corresponding to the individual signal components constituting the composite signal. Since, TFCA based analysis enables the extraction of the identified components from the composite signals, it allows detailed post processing of the extracted signal components and their corresponding time–frequency distributions, as well.Özdemir, Ahmet KemalPh.D

Exact equations for smoothed Wigner transforms and homogenization of wave propagation

The Wigner Transform (WT) has been extensively used in the formulation of phase-space models for a variety of wave propagation problems including high-frequency limits, nonlinear and random waves. It is well known that the WT features counterintuitive 'interference terms', which often make computation impractical. In this connection, we propose the use of the smoothed Wigner Transform (SWT), and derive new, exact equations for it, covering a broad class of wave propagation problems. Equations for spectrograms are included as a special case. The 'taming' of the interference terms by the SWT is illustrated, and an asymptotic model for the Schroedinger equation is constructed and numerically verified.Comment: 16 pages, 8 figure

High resolution time frequency representation with significantly reduced cross-terms

A novel algorithm is proposed for efficiently smoothing the slices of the Wigner distribution by exploiting the recently developed relation between the Radon transform of the ambiguity function and the fractional Fourier transformation. The main advantage of the new algorithm is its ability to suppress cross-term interference on chirp-like auto-components without any detrimental effect to the auto-components. For a signal with N samples, the computational complexity of the algorithm is O(N log N) flops for each smoothed slice of the Wigner distribution

A new time-frequency analysis technique for neuroelectric signals

Cataloged from PDF version of article.In the presence of external stimuli, the functioning brain emits neuroelectrical signals which can be recorded as the Event Related Potential (ERP) signals. To understand the brain cognitive functions, ERP signals have been the subject matter of many applications in the field of cognitive psychophysiology. Due to the non–stationary nature of the ERP signals, commonly used time or frequency analysis techniques fail to capture the time–frequency domain localized nature of the ERP signal components. In this study, the newly developed Time–Frequency Component Analyzer (TFCA) approach is adapted to the ERP signal analysis. The results obtained on the actual ERP signals show that the TFCA does not have a precedent in resolution and extraction of uncontaminated individual ERP signal components. Furthermore, unlike the existing ERP analysis techniques, the TFCA based analysis technique can reliably measures the subject dependent variations in the ERP signals, which iiiopens up new possibilities in the clinical studies. Thus, TFCA serves as an ideal tool for studying the intricate machinery of the human brain.Tüfekçi, D. İlhanM.S

The multiresolution Fourier transform : a general purpose tool for image analysis

The extraction of meaningful features from an image forms an important area of image analysis. It enables the task of understanding visual information to be implemented in a coherent and well defined manner. However, although many of the traditional approaches to feature extraction have proved to be successful in specific areas, recent work has suggested that they do not provide sufficient generality when dealing with complex analysis problems such as those presented by natural images. This thesis considers the problem of deriving an image description which could form the basis of a more general approach to feature extraction. It is argued that an essential property of such a description is that it should have locality in both the spatial domain and in some classification space over a range of scales. Using the 2-d Fourier domain as a classification space, a number of image transforms that might provide the required description are investigated. These include combined representations such as a 2-d version of the short-time Fourier transform (STFT), and multiscale or pyramid representations such as the wavelet transform. However, it is shown that these are limited in their ability to provide sufficient locality in both domains and as such do not fulfill the requirement for generality. To overcome this limitation, an alternative approach is proposed in the form of the multiresolution Fourier transform (MFT). This has a hierarchical structure in which the outermost levels are the image and its discrete Fourier transform (DFT), whilst the intermediate levels are combined representations in space and spatial frequency. These levels are defined to be optimal in terms of locality and their resolution is such that within the transform as a whole there is a uniform variation in resolution between the spatial domain and the spatial frequency domain. This ensures that locality is provided in both domains over a range of scales. The MFT is also invertible and amenable to efficient computation via familiar signal processing techniques. Examples and experiments illustrating its properties are presented. The problem of extracting local image features such as lines and edges is then considered. A multiresolution image model based on these features is defined and it is shown that the MET provides an effective tool for estimating its parameters.. The model is also suitable for representing curves and a curve extraction algorithm is described. The results presented for synthetic and natural images compare favourably with existing methods. Furthermore, when coupled with the previous work in this area, they demonstrate that the MFT has the potential to provide a basis for the solution of general image analysis problems