Showing papers on "Discrete-time Fourier transform published in 2008"

FFTs on the Rotation Group

Abstract: We discuss an implementation of an efficient algorithm for the numerical computation of Fourier transforms of bandlimited functions defined on the rotation group SO(3). The implementation is freely available on the web. The algorithm described herein uses O(B 4) operations to compute the Fourier coefficients of a function whose Fourier expansion uses only (the O(B 3)) spherical harmonics of degree at most B. This compares very favorably with the direct O(B 6) algorithm derived from a basic quadrature rule on O(B 3) sample points. The efficient Fourier transform also makes possible the efficient calculation of convolution over SO(3) which has been used as the analytic engine for some new approaches to searching 3D databases (Funkhouser et al., ACM Trans. Graph. 83–105, [2003]; Kazhdan et al., Eurographics Symposium in Geometry Processing, pp. 167–175, [2003]). Our implementation is based on the “Separation of Variables” technique (see, e.g., Maslen and Rockmore, Proceedings of the DIMACS Workshop on Groups and Computation, pp. 183–237, [1997]). In conjunction with techniques developed for the efficient computation of orthogonal polynomial expansions (Driscoll et al., SIAM J. Comput. 26(4):1066–1099, [1997]), our fast SO(3) algorithm can be improved to give an algorithm of complexity O(B 3log 2 B), but at a cost in numerical reliability. Numerical and empirical results are presented establishing the empirical stability of the basic algorithm. Examples of applications are presented as well.

A First Course in Fourier Analysis

Abstract: This book provides a meaningful resource for applied mathematics through Fourier analysis. It develops a unified theory of discrete and continuous (univariate) Fourier analysis, the fast Fourier transform, and a powerful elementary theory of generalized functions and shows how these mathematical ideas can be used to study sampling theory, PDEs, probability, diffraction, musical tones, and wavelets. The book contains an unusually complete presentation of the Fourier transform calculus. It uses concepts from calculus to present an elementary theory of generalized functions. FT calculus and generalized functions are then used to study the wave equation, diffusion equation, and diffraction equation. Real-world applications of Fourier analysis are described in the chapter on musical tones. A valuable reference on Fourier analysis for a variety of students and scientific professionals, including mathematicians, physicists, chemists, geologists, electrical engineers, mechanical engineers, and others.

Digital Computation of Linear Canonical Transforms

Abstract: We deal with the problem of efficient and accurate digital computation of the samples of the linear canonical transform (LCT) of a function, from the samples of the original function. Two approaches are presented and compared. The first is based on decomposition of the LCT into chirp multiplication, Fourier transformation, and scaling operations. The second is based on decomposition of the LCT into a fractional Fourier transform followed by scaling and chirp multiplication. Both algorithms take ~ N log N time, where N is the time-bandwidth product of the signals. The only essential deviation from exactness arises from the approximation of a continuous Fourier transform with the discrete Fourier transform. Thus, the algorithms compute LCTs with a performance similar to that of the fast Fourier transform algorithm in computing the Fourier transform, both in terms of speed and accuracy.

Fast Fourier Transform

Abstract: This chapter contains sections titled: Introduction Development of the FFT Algorithm with Radix-2 Decimation-in-Frequency FFT Algorithm with Radix-2 Decimation-in-Time FFT Algorithm with Radix-2 Bit Reversal for Unscrambling Development of the FFT Algorithm with Radix-4 Inverse Fast Fourier Transform Programming Examples References

A Framework for Discrete Integral Transformations I—The Pseudopolar Fourier Transform

Abstract: The Fourier transform of a continuous function, evaluated at frequencies expressed in polar coordinates, is an important conceptual tool for understanding physical continuum phenomena. An analogous tool, suitable for computations on discrete grids, could be very useful; however, no exact analogue exists in the discrete case. In this paper we present the notion of pseudopolar grid (pp grid) and the pseudopolar Fourier transform (ppFT), which evaluates the discrete Fourier transform at points of the pp grid. The pp grid is a type of concentric-squares grid in which the radial density of squares is twice as high as usual. The pp grid consists of equally spaced samples along rays, where different rays are equally spaced in slope rather than angle. We develop a fast algorithm for the ppFT, with the same complexity order as the Cartesian fast Fourier transform; the algorithm is stable, invertible, requires only one-dimensional operations, and uses no approximate interpolations. We prove that the ppFT is invertible and develop two algorithms for its inversion: iterative and direct, both with complexity $O(n^{2}\log{n})$, where $n \times n$ is the size of the reconstructed image. The iterative algorithm applies conjugate gradients to the Gram operator of the ppFT. Since the transform is ill-conditioned, we introduce a preconditioner, which significantly accelerates the convergence. The direct inversion algorithm utilizes the special frequency domain structure of the transform in two steps. First, it resamples the pp grid to a Cartesian frequency grid and then recovers the image from the Cartesian frequency grid.

On the Use of a Lower Sampling Rate for Broken Rotor Bar Detection With DTFT and AR-Based Spectrum Methods

Abstract: Broken rotor bars in an induction motor create asymmetries and result in abnormal amplitude of the sidebands around the fundamental supply frequency and its harmonics. Motor current signature analysis (MCSA) techniques are applied to inspect the spectrum amplitudes at the broken rotor bar specific frequencies for abnormality and to decide about broken rotor bar fault detection and diagnosis. In this paper, we have demonstrated with experimental results that the use of a lower sampling rate with a digital notch filter is feasible for MCSA in broken rotor bar detection with discrete-time Fourier transform and autoregressive-based spectrum methods. The use of the lower sampling rate does not affect the performance of the fault detection, while requiring much less computation and low cost in implementation, which would make it easier to implement in embedded systems for motor condition monitoring.

Optical image encryption based on the multiple-parameter fractional Fourier transform

Abstract: A novel image encryption algorithm is proposed based on the multiple-parameter fractional Fourier transform, which is a generalized fractional Fourier transform, without the use of phase keys. The image is encrypted simply by performing a multiple-parameter fractional Fourier transform with four keys. Optical implementation is suggested. The method has been compared with existing methods and shows superior robustness to blind decryption.

Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing

Abstract: Preface. Acknowledgments. 1. Vector Spaces, Signals, and Images. 1.1 Overview. 1.2 Some common image processing problems. 1.3 Signals and images. 1.4 Vector space models for signals and images. 1.5 Basic wave forms the analog case. 1.6 Sampling and aliasing. 1.7 Basic wave forms the discrete case. 1.8 Inner product spaces and orthogonality. 1.9 Signal and image digitization. 1.10 Infinitedimensional inner product spaces. 1.11 Matlab project. Exercises. 2. The Discrete Fourier Transform. 2.1 Overview. 2.2 The time domain and frequency domain. 2.3 A motivational example. 2.4 The onedimensional DFT. 2.5 Properties of the DFT. 2.6 The fast Fourier transform. 2.7 The twodimensional DFT. 2.8 Matlab project. Exercises. 3. The discrete cosine transform. 3.1 Motivation for the DCT: compression. 3.2 Initial examples thresholding. 3.3 The discrete cosine transform. 3.4 Properties of the DCT. 3.5 The twodimensional DCT. 3.6 Block transforms. 3.7 JPEG compression. 3.8 Matlab project. Exercises. 4. Convolution and filtering. 4.1 Overview. 4.2 Onedimensional convolution. 4.3 Convolution theorem and filtering. 4.4 2D convolution filtering images. 4.5 Infinite and biinfinite signal models. 4.6 Matlab project. Exercises. 5. Windowing and Localization. 5.1 Overview: Nonlocality of the DFT. 5.2 Localization via windowing. 5.3 Matlab project. Exercises. 6. Filter banks. 6.1 Overview. 6.2 The Haar filter bank. 6.3 The general onstage twochannel filter bank. 6.4 Multistage filter banks. 6.5 Filter banks for finite length signals. 6.6 The 2D discrete wavelet transform and JPEG 2000. 6.7 Filter design. 6.8 Matlab project. 6.9 Alternate Matlab project. Exercises. 7. Wavelets. 7.1 Overview. 7.2 The Haar Basis. 7.3 Haar wavelets versus the Haar filter bank. 7.4 Orthogonal wavelets. 7.5 Biorthogonal wavelets. 7.6 Matlab Project. Exercises. References. Solutions. Index.

High resolution 4-D spectroscopy with sparse concentric shell sampling and FFT-CLEAN

Abstract: Recent efforts to reduce the measurement time for multidimensional NMR experiments have fostered the development of a variety of new procedures for sampling and data processing. We recently described concentric ring sampling for 3-D NMR experiments, which is superior to radial sampling as input for processing by a multidimensional discrete Fourier transform. Here, we report the extension of this approach to 4-D spectroscopy as Randomized Concentric Shell Sampling (RCSS), where sampling points for the indirect dimensions are positioned on concentric shells, and where random rotations in the angular space are used to avoid coherent artifacts. With simulations, we show that RCSS produces a very low level of artifacts, even with a very limited number of sampling points. The RCSS sampling patterns can be adapted to fine rectangular grids to permit use of the Fast Fourier Transform in data processing, without an apparent increase in the artifact level. These artifacts can be further reduced to the noise level using the iterative CLEAN algorithm developed in radioastronomy. We demonstrate these methods on the high resolution 4-D HCCH-TOCSY spectrum of protein G’s B1 domain, using only 1.2% of the sampling that would be needed conventionally for this resolution. The use of a multidimensional FFT instead of the slow DFT for initial data processing and for subsequent CLEAN significantly reduces the calculation time, yielding an artifact level that is on par with the level of the true spectral noise.

A Framework for Discrete Integral Transformations II—The 2D Discrete Radon Transform

Abstract: Although naturally at the heart of many fundamental physical computations, and potentially useful in many important image processing tasks, the Radon transform lacks a coherent discrete definition for two-dimensional (2D) discrete images which is algebraically exact, invertible, and rapidly computable. We define a notion of 2D discrete Radon transforms for 2D discrete images, which is based on summation along lines of absolute slope less than 1. Values at nongrid locations are defined using trigonometric interpolation on a zero-padded grid. Our definition is shown to be geometrically faithful: the summation avoids wrap-around effects. Our proposal uses a special collection of lines in $\mathbb{R}^{2}$ for which the transform is rapidly computable and invertible. We describe a fast algorithm using $O(N\log{N})$ operations, where $N =n^{2}$ is the number of pixels in the image. The fast algorithm exploits a discrete projection-slice theorem, which associates the discrete Radon transform with the pseudopolar Fourier transform [A. Averbuch et al., SIAM J. Sci. Comput., 30 (2008), pp. 764-784]. Our definition for discrete images converges to a natural continuous counterpart with increasing refinement.

Discrete Sums for the Rapid Determination of Exponential Decay Constants

Abstract: Several computational methods are presented for the rapid extraction of decay time constants from discrete exponential data. Two methods are found to be comparably fast and highly accurate. They are corrected successive integration and a method involving the Fourier transform (FT) of the data and the application of an expression that does not assume continuous data. FT methods in the literature are found to introduce significant systematic error owing to the assumption that data are continuous. Corrected successive integration methods in the literature are correct, but we offer a more direct way of applying them which we call linear regression of the sum. We recommend the use of the latter over FT-based methods, as the FT methods are more affected by noise in the original data.

Uncertainty relation for the discrete Fourier transform.

Abstract: We derive an uncertainty relation for two unitary operators which obey a commutation relation of the form UV=e(i phi) VU Its most important application is to constrain how much a quantum state can be localized simultaneously in two mutually unbiased bases related by a discrete fourier transform It provides an uncertainty relation which smoothly interpolates between the well-known cases of the Pauli operators in two dimensions and the continuous variables position and momentum This work also provides an uncertainty relation for modular variables, and could find applications in signal processing In the finite dimensional case the minimum uncertainty states, discrete analogues of coherent and squeezed states, are minimum energy solutions of Harper's equation, a discrete version of the harmonic oscillator equation

Discrete and continuous fourier transforms : analysis, applications and fast algorithms

Abstract: Preface Fundamentals, Analysis, and Applications Analytical and Graphical Representation of Function Contents Time and Frequency Contents of a Function The Frequency-Domain Plots as Graphical Tools Identifying the Cosine and Sine Modes Using Complex Exponential Modes Using Cosine Modes with Phase or Time Shifts Periodicity and Commensurate Frequencies Review of Results and Techniques Expressing Single Component Signals General Form of a Sinusoid in Signal Application Fourier Series: A Topic to Come Terminology Sampling and Reconstruction of Functions-Part I DFT and Band-Limited Periodic Signal Frequencies Aliased by Sampling Connection: Anti-Aliasing Filter Alternate Notations and Formulas Sampling Period and Alternate Forms of DFT Sample Size and Alternate Forms of DFT The Fourier Series Formal Expansions Time-Limited Functions Even and Odd Functions Half-Range Expansions Fourier Series Using Complex Exponential Modes Complex-Valued Functions Fourier Series in Other Variables Truncated Fourier Series and Least Squares Orthogonal Projections and Fourier Series Convergence of the Fourier Series Accounting for Aliased Frequencies in DFT DFT and Sampled Signals Deriving the DFT and IDFT Formulas Direct Conversion between Alternate Forms DFT of Concatenated Sample Sequences DFT Coefficients of a Commensurate Sum Frequency Distortion by Leakage The Effects of Zero Padding Computing DFT Defining Formulas Per Se Sampling and Reconstruction of Functions-Part II Sampling Nonperiodic Band-Limited Functions Deriving the Fourier Transform Pair The Sine and Cosine Frequency Contents Tabulating Two Sets of Fundamental Formulas Connections with Time/Frequency Restrictions Fourier Transform Properties Alternate Form of the Fourier Transform Computing the Fourier Transform Computing the Fourier Coefficients Sampling and Reconstruction of Functions-Part III Impulse Functions and Their Properties Generating the Fourier Transform Pairs Convolution and Fourier Transform Periodic Convolution and Fourier Series Convolution with the Impulse Function Impulse Train as a Generalized Function Impulse Sampling of Continuous-Time Signals Nyquist Sampling Rate Rediscovered Sampling Theorem for Band-Limited Signal Sampling of Band-Pass Signals The Fourier Transform of a Sequence Deriving the Fourier Transform of a Sequence Properties of the Fourier Transform of a Sequence Generating the Fourier Transform Pairs Duality in Connection with the Fourier Series The Fourier Transform of a Periodic Sequence The DFT Interpretation The Discrete Fourier Transform of a Windowed Sequence A Rectangular Window of Infinite Width A Rectangular Window of Appropriate Finite Width Frequency Distortion by Improper Truncation Windowing a General Nonperiodic Sequence Frequency-Domain Properties of Windows Applications of the Windowed DFT Discrete Convolution and the DFT Linear Discrete Convolution Periodic Discrete Convolution The Chirp Fourier Transform Applications of the DFT in Digital Filtering and Filters The Background Application-Oriented Terminology Revisit Gibbs Phenomenon from the Filtering Viewpoint Experimenting with Digital Filtering and Filter Design Fast Algorithms Index Mapping and Mixed-Radix FFTs Algebraic DFT versus FFT-Computed DFT The Role of Index Mapping The Recursive Equation Approach Other Forms by Alternate Index Splitting Kronecker Product Factorization and FFTs Reformulating the Two-Factor Mixed-Radix FFT From Two-Factor to Multifactor Mixed-Radix FFT Other Forms by Alternate Index Splitting Factorization Results by Alternate Expansion Unordered FFT for Scrambled Input Utilities of the Kronecker Product Factorization The Family of Prime Factor FFT Algorithms Connecting the Relevant Ideas Deriving the Two-Factor PFA Matrix Formulation of the Two-Factor PFA Matrix Formulation of the Multifactor PFA Number Theory and Index Mapping by Permutations The In-Place and In-Order PFA Efficient Implementation of the PFA On Computing the DFT of Large Prime Length Performance of FFT for Prime N Fast Algorithm I: Approximating the FFT Fast Algorithm II: Using Bluestein's FFT Bibliography Index

Application of the wavelet transform and the enhanced Fourier spectrum in the impact echo test

Abstract: The objective of this study is to develop a reliable and effective method to analyze the signal of the impact echo test. The impact echo test is a nondestructive testing technique for civil structures. In the test, the surface response of the target structure due to an impact is measured. Then, the Fourier transform is adopted to transform the signal from the time domain to the frequency domain. Owing to the multiple reflections induced by cracks, voids, or other interfaces, peaks will form in the Fourier spectrum. The frequencies of the peaks can then be used to determine the size of the structure or the location of the defect. Several difficulties are encountered when applying the Fourier transform to impact echo data. Because the impact echo data are non-stationary and contains multiple reflections, ripples and multiple peaks appear in the Fourier spectrum, which may mislead the follow-up diagnosis. Furthermore, the existence of the high-energy surface wave and structural vibrations often complicates the spectrum and makes the data interpretation even more difficult. To overcome these difficulties, this research adopts the wavelet transform in the analysis of impact echo data. Theoretically, the wavelet transform can avoid ripple and multiple-peak phenomena. Furthermore, the frequency range and time span of surface wave can be observed in the wavelet scalogram. However, the spectral resolution of the wavelet marginal spectrum is inferior to that of the Fourier transform. Therefore, two approaches are proposed in this paper. One is to combine the Fourier spectrum and the wavelet marginal spectrum to determine the precise location of the echo peak. The other is to take the product of the two spectra to establish the enhanced Fourier spectrum. As such, the interference in the Fourier spectrum is suppressed while the peak is enhanced. Numerical and experimental tests were performed to verify the effectiveness and reliability of the proposed approaches.

Modified Fourier-Hankel method based on analysis of errors in Abel inversion using Fourier transform techniques

Abstract: Errors in discrete Abel inversion methods using Fourier transform techniques have been analyzed. The Fourier expansion method is very accurate but sensitive to noise. The Fourier-Hankel method has a significant systematic negative deviation, which increases with the radius; inversion error of the method can be reduced by adjusting the value of a factor. With a decrease of the factor both methods show a noise filtering property. Based on the analysis, a modified Fourier-Hankel method that is accurate, computationally efficient, and has the ability to filter noise in the inversion process is proposed for applying to experimental data.

Research progress on discretization of fractional Fourier transform

Abstract: As the fractional Fourier transform has attracted a considerable amount of attention in the area of optics and signal processing, the discretization of the fractional Fourier transform becomes vital for the application of the fractional Fourier transform. Since the discretization of the fractional Fourier transform cannot be obtained by directly sampling in time domain and the fractional Fourier domain, the discretization of the fractional Fourier transform has been investigated recently. A summary of discretizations of the fractional Fourier transform developed in the last nearly two decades is presented in this paper. The discretizations include sampling in the fractional Fourier domain, discrete-time fractional Fourier transform, fractional Fourier series, discrete fractional Fourier transform (including 3 main types: linear combination-type; sampling-type; and eigen decomposition-type), and other discrete fractional signal transform. It is hoped to offer a doorstep for the readers who are interested in the fractional Fourier transform.

On the Security of a Cryptosystem Based on Multiple-Parameters Discrete Fractional Fourier Transform

Abstract: Pei and Hsue (IEEE Signal Processing Letters, Vol. 13, No. 6, pp. 329-332, June 2006) proposed an encryption scheme based on multiple-parameter discrete fractional Fourier transform. We show that all the building blocks of this scheme are linear, and hence, breaking this scheme, using a known plaintext attack, is equivalent to solving a set of linear equations.

Method for CMF Signal Processing Based on the Recursive DTFT Algorithm With Negative Frequency Contribution

Abstract: There is a long convergence stage when using the sliding Goertzel algorithm to measure the phase difference between signals of a Coriolis mass flowmeter (CMF) because the contribution of negative frequency is neglected in the algorithm. A novel method for CMF signal processing is proposed based on the recursive discrete-time Fourier transform (DTFT) algorithm with negative frequency contribution. First, an adaptive lattice notch filter is applied to filter the sensor output signals of the CMF and calculate the frequency. Then, a new method based on the recursive DTFT algorithm with negative frequency contribution is introduced to calculate the real-time phase difference between two enhanced signals. With the frequency and the phase difference obtained, the time interval of the two signals is calculated, and then, the mass flowrate is derived. The method is validated in experiments using CMF signals acquired for different flowrates. Simulation and experimental results show that the convergence stage of both the phase difference and time interval calculations has been largely shortened with higher accuracy of the CMF, as compared with the existing method based on the sliding Goertzel algorithm.

An adaptive resolution computationally efficient short-time Fourier transform

Abstract: The short-time Fourier transform (STFT) is a classical tool, used for characterizing the time varying signals. The limitation of the STFT is its fixed time-frequency resolution. Thus, an enhanced version of the STFT, which is based on the cross-level sampling, is devised. It can adapt the sampling frequency and the window function length by following the input signal local characteristics. Therefore, it provides an adaptive resolution time-frequency representation of the input signal. The computational complexity of the proposed STFT is deduced and compared to the classical one. The results show a significant gain of the computational efficiency and hence of the processing power.

Schemes for quantum key distribution with higher-order alphabets using single-photon fractional Fourier optics

Abstract: We propose generalized quantum key distribution schemes using spatially encoded $d$-dimensional qudits based on fractional Fourier transform operations. We determine the necessary conditions on the orders of the transforms which ensure a shared secret random key string and briefly discuss the transmission rate and a possible encoding procedure. We also show that the fractional Fourier transform can be used to analyze more general eavesdropping strategies, including an intermediate-basis attack. The error rate and information gain for the intercept-resend and intermediate-basis attacks are briefly analyzed for a particular example. Effects of atmospheric turbulence in a free-space transmission are considered.

Fourier transform and middle convolution for irregular D-modules

Abstract: SBlock and HEsnault constructed the local Fourier transform for D-modules We present a different approach to the local Fourier transform, which makes its properties almost tautological We apply the local Fourier transform to compute the local version of Katz's middle convolution

A simple phase unwrapping approach based on filtering by windowed Fourier transform: A note on the threshold selection

Abstract: A simple phase unwrapping approach based on windowed Fourier filtering was proposed recently [K. Qian et al. A simple phase unwrapping approach based on filtering by windowed Fourier transform. Opt Laser Technol 2005;37:458–62]. The windowed Fourier filtering algorithm is an essential ingredient that suppresses the noise effectively and makes the phase unwrapping trivial. This paper adds a note on the threshold selection in the windowed Fourier filtering algorithm. A large interval can be selected as the threshold to obtain almost optimal filtering results. Once the selected threshold is suitable, it is almost optimal. This makes the threshold selection in the windowed Fourier filtering algorithm extremely easy.

The Radon transform on SO(3): a Fourier slice theorem and numerical inversion *

Abstract: The inversion of the one-dimensional Radon transform on the rotation group SO(3) is an ill-posed inverse problem which applies to x-ray tomography with polycrystalline materials. This paper presents a novel approach to the numerical inversion of the one-dimensional Radon transform on SO(3). Based on a Fourier slice theorem the discrete inverse Radon transform of a function sampled on the product space of two two-dimensional spheres is determined as the solution of a minimization problem, which is iteratively solved using fast Fourier techniques for and SO(3). The favorable complexity and stability of the algorithm based on these techniques has been confirmed with numerical tests.

Set-Class Similarity, Voice Leading, and the Fourier Transform

Abstract: In this article, I consider two ways to model distance (or inverse similarity) between chord types, one based on voice leading and the other on shared interval content. My goal is to provide a contrapuntal reinterpretation of Ian Quinn’s work, which uses the Fourier transform to quantify similarity of interval content. The first section of the article shows how to find the minimal voice leading between chord types or set-classes. The second uses voice leading to approximate the results of Quinn’s Fourier-based method. The third section explains how this is possible, while the fourth argues that voice leading is somewhat more flexible than the Fourier transform. I conclude with a few thoughts about realism and relativism in music theory. twentieth-century music often moves flexibly between contrasting harmonic regions: in the music of Stravinsky, Messiaen, Shostakovich, Ligeti, Crumb, and John Adams, we find diatonic passages alternating with moments of intense chromaticism, sometimes mediated by nondiatonic scales such as the whole-tone and octatonic. In some cases, the music moves continuously from one world to another, making it hard to identify precise bound aries between them. Yet we may still have the sense that a particular passage, melody, or scale is, for instance, fairly diatonic, more-or-less octatonic, or less diatonic than whole-tone. A challenge for music theory is to formalize these intuitions by proposing quantitative methods for locating musical objects along the spectrum of contemporary harmonic possibilities. One approach to this problem uses voice leading: from this point of view, to say that two set-classes are similar is to say that any set of the first type can be transformed into one of the second without moving its notes very far. Thus, the acoustic scale is similar to the diatonic because we can transform one into the other by a single-semitone shift; for example, the acoustic scale {C, D, E, F≥, G, A, B≤} can be made diatonic by the single-semitone displacement F≥ → F or B≤ → B. Similarly, when we judge the minor seventh chord Thanks to Rachel Hall, Justin Hoffman, Ian Quinn, Joe Straus, and in particular Clifton Callender, whose investigations into continuous Fourier transforms deeply influenced my thinking. Callender pursued his approach despite strenuous objections on my part, for which I am both appropriately grateful and duly chastened. 252 J O U r n A L o f M U S I C T h E O r Y Dmitri Tymoczko Voice Leading and the Fourier Transform to be very similar to the dominant seventh, we are saying that we can relate them by a single-semitone shift. This conception of similarity dates back to John roeder’s work in the mid-1980s (1984, 1987) and has been developed more recently by Thomas robinson (2006), Joe Straus (2007), and Clifton Callender, Ian Quinn, and myself (2008). The approach is consistent with the thought that composers, sitting at a piano keyboard, would judge chords to be similar when they can be linked by small physical motions. Another approach uses intervallic content: from this point of view, to say that set-classes are similar is to say that they contain similar collections of intervals. (That the two methods are different is shown by “Z-related” or “nontrivially homometric” sets, which contain the same intervals but are nonidentical according to voice leading.) In a fascinating pair of papers, Quinn has demonstrated that the Fourier transform can be used to quantify this approach.1 Essentially, for any number n from 1 to 6, and every pitch class p in a chord, the Fourier transform assigns a two-dimensional vector whose components are Vp,n 5 (cos 2ppn/12, sin 2ppn/12). (1) Adding these vectors together, for one particular n and all the pitch classes p in the chord, produces a composite vector representing the chord as a whole— its “nth Fourier component.” The length (or “magnitude”) of this vector, Quinn astutely observes, reveals something about the chord’s harmonic character: in particular, chords saturated with (12/n)-semitone intervals, or intervals approximately equal to 12/n, tend to score highly on this index of chord quality.2 The Fourier transform thus seems to capture the intuitive sense that chords can be more or less diminished-seventh-like, perfect-fifthy, or wholetonish. It also seems to offer a distinctive approach to set-class similarity: from this point of view, two set-classes can be considered “similar” when their Fourier magnitudes are approximately equal—a situation that obtains when the chords have approximately the same intervals. The interesting question is how these two conceptions relate. In recent years, a number of theorists have tried to reinterpret Quinn’s Fourier magnitudes using voice-leading distances. robinson (2006), for example, pointed out that there is a strong anticorrelation between the magnitude of a chord’s first Fourier component and the size of the minimal voice leading to the nearest chromatic cluster. (See also Straus 2007, which echoes robinson’s point.) however, neither robinson nor Straus found an analogous interpretation of the other Fourier components. In an interesting article in this issue (see pages 219–49), Justin hoffman extends this work, interpreting Fourier components in light of unusual “voice-leading lattices” in which voices move by distances other than one semitone. But despite this intriguing idea, the 1 See Quinn 2006 and 2007. Quinn’s use of the Fourier transform develops ideas in Lewin 1959 and 2001 and Vuza 1993. 2 These magnitudes are the same for transpositionally or inversionally related chords, so it is reasonable to speak of a set-class’s Fourier magnitudes. Dmitri Tymoczko Voice Leading and the Fourier Transform 253 relation between Fourier analysis and more traditional conceptions of voice leading remains obscure. The purpose of this article is to describe a general connection between the two approaches: it turns out that the magnitude of a chord’s nth Fourier component is approximately inversely related to the size of the minimal voice leading to the nearest subset of any perfectly even n-note chord.3 For instance, a chord’s first Fourier component is approximately inversely related to the size of the minimal voice leading to any transposition of {0}; the second Fourier component is approximately inversely related to the size of the minimal voice leading to any transposition of either {0} or {0, 6}; the third component is approximately inversely related to the size of the minimal voice leading to any transposition of either {0}, {0, 4}, or {0, 4, 8}, and so on. Interestingly, however, we can see this connection clearly only when we model chords as multisets in continuous pitch-class space, following the approach of Callender, Quinn, and Tymoczko (2008). (This in fact may be one reason why previous theorists did not notice the relationship.) When we do adopt this perspective, we see that there is a deep relationship between two seemingly very different conceptions of set-class similarity, one grounded in voice leading, the other in interval content. Furthermore, this realization allows us to generalize some of the features of Quinn’s approach, using related methods that transcend some of the limitations of the Fourier transform proper. I. Voice leading and set-class similarity Let me begin by describing the voice-leading approach to set-class similarity (or inverse distance), reviewing along the way some basic definitions. Much of what follows is drawn from (or implicit in) earlier essays, including Tymoczko 2006 and 2008 and Callender, Quinn, and Tymoczko 2008; readers who want to explore these ideas further are hereby referred to these more in-depth discussions. We can label pitch classes using real numbers (not just integers) in the range [0, 12), with C as 0.4 here the octave has size 12, and familiar twelvetone equal-tempered semitones have size 1. This system provides labels for every conceivable pitch class and does not limit us to any particular scale; thus, the number 4.5 refers to “E quarter-tone sharp,” halfway between the twelvetone equal-tempered pitch classes E and F. A voice leading between pitch-class sets corresponds to a phrase like “the C major triad moves to E major by moving C down to B, holding E fixed, and shifting G up by semitone to G≥.” We can notate this more efficiently by writing 3 By “perfectly even n-note chord” I mean the chord that exactly divides the octave into n equally sized pieces, not necessarily lying in any familiar scale. For example, the perfectly even eight-note chord is {0, 1.5, 3, 4.5, 6, 7.5, 9, 10.5}. 4 The notation [x, y) indicates a range that includes the lower bound x but not the upper bound y. Similarly (x, y) includes neither upper nor lower bounds, while [x, y] includes both. 254 J O U r n A L o f M U S I C T h E O r Y Dmitri Tymoczko Voice Leading and the Fourier Transform (C, E, G) 1, 0, 1 (B, E, G≥), indicating that C moves to B by one descending semitone, E moves to E by zero semitones, and G moves to G≥ by one ascending semitone. The order in which voices are listed is not important; thus, (C, E, G) 1, 0, 1 (B, E, G≥) is the same as (E, G, C) 0, 1, 1 (E, G≥, B). The numbers above the arrows represent paths in pitch-class space, or directed distances such as “up two semitones,” “down seven semitones,” “up thirteen semitones,” and so on. When the paths all lie in the range (–6, 6] I eliminate them; thus, a notation like (C, E, G) → (B, E, G≥) indicates that each voice moves to its destination along the shortest possible route, with the arbitrary convention being that tritones ascend. Formally, voice leadings between pitch-class sets can be modeled as multisets of ordered pairs, in which the first element is a pitch class and the second a real number representing a path in pitch-class space. Voice leadings are bijective when they associate each element of one chord with precisely one element of the other. however, it matters whether w

MRI Reconstruction Using Discrete Fourier Transform: A tutorial

Abstract: The use of Inverse Discrete Fourier Transform (IDFT) implemented in the form of Inverse Fourier Transform (IFFT) is one of the standard method of reconstructing Magnetic Resonance Imaging (MRI) from uniformly sampled K-space data. In this tutorial, three of the major problems associated with the use of IFFT in MRI reconstruction are highlighted. The tutorial also gives brief introduction to MRI physics; MRI system from instrumentation point of view; K-space signal and the process of IDFT and IFFT for One and two dimensional (1D and 2D) data.

Multi-channel sampling theorems for band-limited signals with fractional Fourier transform

Abstract: Multi-channel sampling for band-limited signals is fundamental in the theory of multi-channel parallel A/D environment and multiplexing wireless communication environment. As the fractional Fourier transform has been found wide applications in signal processing fields, it is necessary to consider the multi-channel sampling theorem based on the fractional Fourier transform. In this paper, the multi-channel sampling theorem for the fractional band-limited signal is firstly proposed, which is the generalization of the well-known sampling theorem for the fractional Fourier transform. Since the periodic nonuniformly sampled signal in the fractional Fourier domain has valuable applications, the reconstruction expression for the periodic nonuniformly sampled signal has been then obtained by using the derived multi-channel sampling theorem and the specific space-shifting and phase-shifting properties of the fractional Fourier transform. Moreover, by designing different fractional Fourier filters, we can obtain reconstruction methods for other sampling strategies.