Complex and Hypercomplex Discrete Fourier Transforms Based on Matrix Exponential Form of Euler's Formula

We show that the discrete complex, and numerous hypercomplex, Fourier transforms defined and used so far by a number of researchers can be unified into a single framework based on a matrix exponential version of Euler's formula $e^{j\theta}=\cos\theta+j\sin\theta$, and a matrix root of -1 isomorphic to the imaginary root $j$. The transforms thus defined can be computed using standard matrix multiplications and additions with no hypercomplex code, the complex or hypercomplex algebra being represented by the form of the matrix root of -1, so that the matrix multiplications are equivalent to multiplications in the appropriate algebra. We present examples from the complex, quaternion and biquaternion algebras, and from Clifford algebras Cl1,1 and Cl2,0. The significance of this result is both in the theoretical unification, and also in the scope it affords for insight into the structure of the various transforms, since the formulation is such a simple generalization of the classic complex case. It also shows that hypercomplex discrete Fourier transforms may be computed using standard matrix arithmetic packages without the need for a hypercomplex library, which is of importance in providing a reference implementation for verifying implementations based on hypercomplex code.Comment: The paper has been revised since the second version to make some of the reasons for the paper clearer, to include reviews of prior hypercomplex transforms, and to clarify some points in the conclusion

Instantaneous frequency and amplitude of complex signals based on quaternion Fourier transform

The ideas of instantaneous amplitude and phase are well understood for signals with real-valued samples, based on the analytic signal which is a complex signal with one-sided Fourier transform. We extend these ideas to signals with complex-valued samples, using a quaternion-valued equivalent of the analytic signal obtained from a one-sided quaternion Fourier transform which we refer to as the hypercomplex representation of the complex signal. We present the necessary properties of the quaternion Fourier transform, particularly its symmetries in the frequency domain and formulae for convolution and the quaternion Fourier transform of the Hilbert transform. The hypercomplex representation may be interpreted as an ordered pair of complex signals or as a quaternion signal. We discuss its derivation and properties and show that its quaternion Fourier transform is one-sided. It is shown how to derive from the hypercomplex representation a complex envelope and a phase. A classical result in the case of real signals is that an amplitude modulated signal may be analysed into its envelope and carrier using the analytic signal provided that the modulating signal has frequency content not overlapping with that of the carrier. We show that this idea extends to the complex case, provided that the complex signal modulates an orthonormal complex exponential. Orthonormal complex modulation can be represented mathematically by a polar representation of quaternions previously derived by the authors. As in the classical case, there is a restriction of non-overlapping frequency content between the modulating complex signal and the orthonormal complex exponential. We show that, under these conditions, modulation in the time domain is equivalent to a frequency shift in the quaternion Fourier domain. Examples are presented to demonstrate these concepts

Azúcar y nervios: Explanatory models and treatment experiences of Hispanics with diabetes and depression

This study examined the explanatory models of depression, perceived relationships between diabetes and depression, and depression treatment experiences of low-income, Spanish-speaking, Hispanics with diabetes and depression. A purposive sample (n = 19) was selected from participants enrolled in a randomized controlled trial conducted in Los Angeles, California (United States) testing the effectiveness of a health services quality improvement intervention. Four focus groups followed by 10 in-depth semi-structured qualitative interviews were conducted. Data were analyzed using the methodology of coding, consensus, co-occurrence, and comparison, an analytical strategy rooted in grounded theory. Depression was perceived as a serious condition linked to the accumulation of social stressors. Somatic and anxiety-like symptoms and the cultural idiom of nervios were central themes in low-income Hispanics' explanatory models of depression. The perceived reciprocal relationships between diabetes and depression highlighted the multiple pathways by which these two illnesses impact each other and support the integration of diabetes and depression treatments. Concerns about depression treatments included fears about the addictive and harmful properties of antidepressants, worries about taking too many pills, and the stigma attached to taking psychotropic medications. This study provides important insights about the cultural and social dynamics that shape low-income Hispanics' illness and treatment experiences and support the use of patient-centered approaches to reduce the morbidity and mortality associated with diabetes and depression

On harmonic analysis of vector-valued signals

A vector‐valued signal in N dimensions is a signal whose value at any time instant is an N‐dimensional vector, that is, an element of urn:x-wiley:mma:media:mma3938:mma3938-math-0001. The sum of an arbitrary number of such signals of the same frequency is shown to trace an ellipse in N‐dimensional space, that is, to be confined to a plane. The parameters of the ellipse (major and minor axes, represented by N‐dimensional vectors; and phase) are obtained algebraically in terms of the directions of oscillation of the constituent signals, and their phases. It is shown that the major axis of the ellipse can always be determined algebraically. That is, a vector, whose value can be computed algebraically (without decisions or comparisons of magnitude) from parameters of the constituent signals, always represents the major axis of the ellipse. The ramifications of this result for the processing and Fourier analysis of signals with vector values or samples are discussed, with reference to the definition of Fourier transforms, particularly discrete Fourier transforms, such as have been defined in several hypercomplex algebras, including Clifford algebras. The treatment in the paper, however, is entirely based on signals with values in urn:x-wiley:mma:media:mma3938:mma3938-math-0002. Although the paper is written in terms of vector signals (which are taken to include images and volumetric images), the analysis clearly also applies to a superposition of simple harmonic motions in N dimensions

Excitonic Photoluminescence in Semiconductor Quantum Wells: Plasma versus Excitons

Time-resolved photoluminescence spectra after nonresonant excitation show a distinct 1s resonance, independent of the existence of bound excitons. A microscopic analysis identifies excitonic and electron-hole plasma contributions. For low temperatures and low densities the excitonic emission is extremely sensitive to even minute optically active exciton populations making it possible to extract a phase diagram for incoherent excitonic populations.Comment: 9 pages, 4 figure

Integrating user-centred design in the development of a silent speech interface based on permanent magnetic articulography

Abstract: A new wearable silent speech interface (SSI) based on Permanent Magnetic Articulography (PMA) was developed with the involvement of end users in the design process. Hence, desirable features such as appearance, port-ability, ease of use and light weight were integrated into the prototype. The aim of this paper is to address the challenges faced and the design considerations addressed during the development. Evaluation on both hardware and speech recognition performances are presented here. The new prototype shows a com-parable performance with its predecessor in terms of speech recognition accuracy (i.e. ~95% of word accuracy and ~75% of sequence accuracy), but significantly improved appearance, portability and hardware features in terms of min-iaturization and cost

On harmonic analysis of vector-valued signals

A vector‐valued signal in N dimensions is a signal whose value at any time instant is an N‐dimensional vector, that is, an element of urn:x-wiley:mma:media:mma3938:mma3938-math-0001. The sum of an arbitrary number of such signals of the same frequency is shown to trace an ellipse in N‐dimensional space, that is, to be confined to a plane. The parameters of the ellipse (major and minor axes, represented by N‐dimensional vectors; and phase) are obtained algebraically in terms of the directions of oscillation of the constituent signals, and their phases. It is shown that the major axis of the ellipse can always be determined algebraically. That is, a vector, whose value can be computed algebraically (without decisions or comparisons of magnitude) from parameters of the constituent signals, always represents the major axis of the ellipse. The ramifications of this result for the processing and Fourier analysis of signals with vector values or samples are discussed, with reference to the definition of Fourier transforms, particularly discrete Fourier transforms, such as have been defined in several hypercomplex algebras, including Clifford algebras. The treatment in the paper, however, is entirely based on signals with values in urn:x-wiley:mma:media:mma3938:mma3938-math-0002. Although the paper is written in terms of vector signals (which are taken to include images and volumetric images), the analysis clearly also applies to a superposition of simple harmonic motions in N dimensions

Fundamental representations and algebraic properties of biquaternions or complexified quaternions

The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner and outer products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically

Is Pressure Stressful? The Impact of Pressure on the Stress Response and Category Learning

We examine the basic question of whether pressure is stressful. We propose that when examining the role of stress or pressure in cognitive performance it is important to consider the type of pressure, the stress response, and the aspect of cognition assessed. In Experiment 1, outcome pressure was not experienced as stressful but did lead to impaired performance on a rule-based (RB) category learning task and not a more procedural information-integration (II) task. In Experiment 2, the addition of monitoring pressure resulted in a modest stress response to combined pressure and impairment on both tasks. Across experiments, higher stress appraisals were associated with decreased performance on the RB, but not the II, task. In turn, higher stress-reactivity (heart rate) was associated with enhanced performance on the II, but not the RB, task. This work represents an initial step towards integrating the stress-cognition and pressure-cognition literatures and suggests that integrating these fields may require consideration of the type of pressure, the stress-response, and the cognitive system mediating performance