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

Ultrasonic Continuous Wave Spirometer

There exists a problem of accurately performing spirographic measurements under physical stress situations. Existing systems, which use mechanical structures in the measurement process, have response times that are too slow, or are too bulky to be considered portable. The proposed system solves these problems and has a number of attractive characteristics. The system uses relatively inexpensive solid state electronic components which implies a minimal of mechanical parts; portability; and a linear, fast response time. The system presented in this thesis determines the velocity and temperature fluctuations of the human breath by measuring the difference and sum of the transit times for two continuous sound waves travelling in opposite directions along the air path. The information about the transit times is contained in the phase differencces of the two sound waves across the path. A phase-locked loop is used to keep the differences across the parth constant, irrespective of air - and sound - velocity variations. Therefore, the phase information is converted to frequency variations in the phase-locked loop

Criterial Noise Effects on Rule-Based Category Learning: The Impact of Delayed Feedback

Variability in the representation of the decision criterion is assumed in many category learning models yet few studies have directly examined its impact. On each trial, criterial noise should result in drift in the criterion and will negatively impact categorization accuracy, particularly in rule-based categorization tasks where learning depends upon the maintenance and manipulation of decision criteria. The results of three experiments test this hypothesis and examine the impact of working memory on slowing the drift rate. Experiment 1 examined the effect of drift by inserting a 5 s delay between the categorization response and the delivery of corrective feedback, and working memory demand was manipulated by varying the number of decision criteria to be learned. Delayed feedback adversely affected performance, but only when working memory demand was high. Experiment 2 built upon a classic finding in the absolute identification literature and demonstrated that distributing the criteria across multiple dimensions decreases the impact of drift during the delay. Experiment 3 confirmed that the effect of drift during the delay is moderated by working memory. These results provide important insights into the interplay between criterial noise and working memory as well as providing important constraints for models of rule-based category learning

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

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

Cavity-enhanced direct frequency comb spectroscopy

Cavity-enhanced direct frequency comb spectroscopy combines broad spectral bandwidth, high spectral resolution, precise frequency calibration, and ultrahigh detection sensitivity, all in one experimental platform based on an optical frequency comb interacting with a high-finesse optical cavity. Precise control of the optical frequency comb allows highly efficient, coherent coupling of individual comb components with corresponding resonant modes of the high-finesse cavity. The long cavity lifetime dramatically enhances the effective interaction between the light field and intracavity matter, increasing the sensitivity for measurement of optical losses by a factor that is on the order of the cavity finesse. The use of low-dispersion mirrors permits almost the entire spectral bandwidth of the frequency comb to be employed for detection, covering a range of ~10% of the actual optical frequency. The light transmitted from the cavity is spectrally resolved to provide a multitude of detection channels with spectral resolutions ranging from a several gigahertz to hundreds of kilohertz. In this review we will discuss the principle of cavity-enhanced direct frequency comb spectroscopy and the various implementations of such systems. In particular, we discuss several types of UV, optical, and IR frequency comb sources and optical cavity designs that can be used for specific spectroscopic applications. We present several cavity-comb coupling methods to take advantage of the broad spectral bandwidth and narrow spectral components of a frequency comb. Finally, we present a series of experimental measurements on trace gas detections, human breath analysis, and characterization of cold molecular beams.Comment: 36 pages, 27 figure

Genomic Dissection of Bipolar Disorder and Schizophrenia, Including 28 Subphenotypes

publisher: Elsevier articletitle: Genomic Dissection of Bipolar Disorder and Schizophrenia, Including 28 Subphenotypes journaltitle: Cell articlelink: https://doi.org/10.1016/j.cell.2018.05.046 content_type: article copyright: © 2018 Elsevier Inc

Quaternion involutions and anti-involutions

An involution or anti-involution is a self-inverse linear mapping. In this paper we study quaternion involutions and anti-involutions. We review formal axioms for such involutions and anti-involutions. We present two mappings, one a quaternion involution and one an anti-involution, and a geometric interpretation of each as reflections. We present results on the composition of these mappings and show that the quaternion conjugate may be expressed using three mutually perpendicular anti-involutions. Finally, we show that projection of a vector or quaternion can be expressed concisely using three mutually perpendicular anti-involutions. © 2007 Elsevier Ltd. All rights reserved