Showing papers on "Fractional Fourier transform published in 1998"
Book•
01 Jan 1998
TL;DR: In this paper, the Discrete Fourier Transform and Numerical Computations (DFT) were used for time-frequency analysis in periodic signals and periodical signals.
Abstract: Signals and Systems.- Periodic Signals.- The Discrete Fourier Transform and Numerical Computations.- The Lebesgue Integral.- Spaces.- Convolution and the Fourier Transform of Functions.- Analog Filters.- Distributions.- Convolution and the Fourier Transform of Distributions.- Filters and Distributions.- Sampling and Discrete Filters.- Current Trends: Time-Frequency Analysis.- References.
263 citations
Book•
01 Jan 1998
TL;DR: This chapter discusses the Fourier Transform and its applications to Discrete-Time Signal Systems, as well as some of the techniques used to design and implement these systems in the real-time world.
Abstract: BACKGROUND B1 Complex Numbers B2 Sinusoids B3 Sketching Signals B4 Cramer's Rule B5 Partial Fraction Expansion B6 Vectors and Matrices B7 Miscellaneous CHAPTER 1 INTRODUCTION TO SIGNALS AND SYSTEMS 11 Size of a Signal 12 Classification of Signals 13 Some Useful Signal Operations 14 Some Useful Signal Models 15 Even and Odd Functions 16 Systems 17 Classification of Systems 18 System Model: Input-Output Description CHAPTER 2 TIME-DOMAIN ANALYSIS OF CONTINUOUS-TIME SYSTEMS 21 Introduction 22 System Response to Internal Conditions: Zero-Input Response 23 The Unit Impulse Response h(t) 24 System Response to External Input: Zero-State Response 25 Classical Solution of Differential Equations 26 System Stability 27 Intuitive Insights into System Behavior 28 Appendix 21: Determining the Impulse Response CHAPTER 3 SIGNAL REPRESENTATION BY FOURIER SERIES 31 Signals and Vectors 32 Signal Comparison: Correlation 33 Signal Representation by Orthogonal Signal Set 34 Trigonometric Fourier Series 35 Exponential Fourier Series 36 Numerical Computation of D[n 37 LTIC System response to Periodic Inputs 38 Appendix CHAPTER 4 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER TRANSFORM 41 Aperiodic Signal Representation by Fourier Integral 42 Transform of Some Useful Functions 43 Some Properties of the Fourier Transform 44 Signal Transmission through LTIC Systems 45 Ideal and Practical Filters 46 Signal Energy 47 Application to Communications: Amplitude Modulation 48 Angle Modulation 49 Data Truncation: Window Functions CHAPTER 5 SAMPLING 51 The Sampling Theorem 52 Numerical Computation of Fourier Transform: The Discrete Fourier Transform (DFT) 53 The Fast Fourier Transform (FFT) 54 Appendix 51 CHAPTER 6 CONTINUOUS-TIME SYSTEM ANALYSIS USING THE LAPLACE TRANSFORM 61 The Laplace Transform 62 Some Properties of the Laplace Transform 63 Solution of Differential and Integro-Differential Equations 64 Analysis of Electrical Networks: The Transformed Network 65 Block Diagrams 66 System Realization 67 Application to Feedback and Controls 68 The Bilateral Laplace Transform 69 Appendix 61: Second Canonical Realization CHAPTER 7 FREQUENCY RESPONSE AND ANALOG FILTERS 71 Frequency Response of an LTIC System 72 Bode Plots 73 Control System Design Using Frequency Response 74 Filter Design by Placement of Poles and Zeros of H(s) 75 Butterworth Filters 76 Chebyshev Filters 77 Frequency Transformations 78 Filters to Satisfy Distortionless Transmission Conditions CHAPTER 8 DISCRETE-TIME SIGNALS AND SYSTEMS 81 Introduction 82 Some Useful Discrete-Time Signal Models 83 Sampling Continuous-Time Sinusoids and Aliasing 84 Useful Signal Operations 85 Examples of Discrete-Time Systems CHAPTER 9 TIME-DOMAIN ANALYSIS OF DISCRETE-TIME SYSTEMS 91 Discrete-Time System Equations 92 System Response to Internal Conditions: Zero-Input Response 93 Unit Impulse Response h[k] 94 System Response to External Input: Zero-State Response 95 Classical Solution of Linear Difference Equations 96 System Stability 97 Appendix 91: Determining Impulse Response CHAPTER 10 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS 101 Periodic Signal Representation by Discrete-Time Fourier Series 102 Aperiodic Signal Representation by Fourier Integral 103 Properties of DTFT 104 DTFT Connection with the Continuous-Time Fourier Transform 105 Discrete-Time Linear System Analysis by DTFT 106 Signal Processing Using DFT and FFT 107 Generalization of DTFT to the Z-Transform CHAPTER 11 DISCRETE-TIME SYSTEM ANALYSIS USING THE Z-TRANSFORM 111 The Z-Transform 112 Some Properties of the Z-Transform 113 Z-Transform Solution of Linear Difference Equations 114 System Realization 115 Connection Between the Laplace and the Z-Transform 116 Sampled-Data (Hybrid) Systems 117 The Bilateral Z-Transform CHAPTER 12 FREQUENCY RESPONSE AND DIGITAL FILTERS 121 Frequency Response of Discrete-Time Systems 122 Frequency Response From Pole-Zero Location 123 Digital Filters 124 Filter Design Criteria 125 Recursive Filter Design: The Impulse Invariance Method 126 Recursive Filter Design: The Bilinear Transformation Method 127 Nonrecursive Filters 128 Nonrecursive Filter Design CHAPTER 13 STATE-SPACE ANALYSIS 131 Introduction 132 Systematic Procedure for Determining State Equations 133 Solution of State Equations 134 Linear Transformation of State Vector 135 Controllability and Observability 136 State-Space Analysis of Discrete-Time Systems ANSWERS TO SELECTED PROBLEMS SUPPLEMENTARY READING INDEX Each chapter ends with a Summary
255 citations
TL;DR: The quantum algorithms of Deutsch, Simon and Shor are described in a way which highlights their dependence on the Fourier transform and an efficient quantum factoring algorithm based on a general formalism of Kitaev is described.
Abstract: The quantum algorithms of Deutsch, Simon and Shor are described in a way which highlights their dependence on the Fourier transform. The general construction of the Fourier transform on an Abelian group is outlined and this provides a unified way of understanding the efficacy of the algorithms. Finally we describe an efficient quantum factoring algorithm based on a general formalism of Kitaev and contrast its structure to the ingredients of Shor9 algorithm.
205 citations
TL;DR: A new convolution structure for the FRFT is introduced that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters.
Abstract: The fractional Fourier transform (FRFT), which is a generalization of the Fourier transform, has many applications in several areas, including signal processing and optics. Almeida (see ibid., vol.4, p.15-17, 1997) and Mendlovic et al. (see Appl. Opt., vol.34, p.303-9, 1995) derived fractional Fourier transforms of a product and of a convolution of two functions. Unfortunately, their convolution formulas do not generalize very well the classical result for the Fourier transform, which states that the Fourier transform of the convolution of two functions is the product of their Fourier transforms. This paper introduces a new convolution structure for the FRFT that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters.
194 citations
TL;DR: The generalized Hilbert transform has similar properties to those of the ordinary Hilbert transform, but it lacks the semigroup property of the fractional Fourier transform.
Abstract: The analytic part of a signal f(t) is obtained by suppressing the negative frequency content of f, or in other words, by suppressing the negative portion of the Fourier transform, f/spl circ/, of f. In the time domain, the construction of the analytic part is based on the Hilbert transform f/spl circ/ of f(t). We generalize the definition of the Hilbert transform in order to obtain the analytic part of a signal that is associated with its fractional Fourier transform, i.e., that part of the signal f(t) that is obtained by suppressing the negative frequency content of the fractional Fourier transform of f(t). We also show that the generalized Hilbert transform has similar properties to those of the ordinary Hilbert transform, but it lacks the semigroup property of the fractional Fourier transform.
108 citations
TL;DR: In this paper, the eigenvalues and eigenvectors of the discrete Fourier and Hartley transform matrices are investigated, and the results of the eigendecomposition of the transform matrix are used to define DFRHT and DFRFT.
Abstract: This paper is concerned with the definitions of the discrete fractional Hartley transform (DFRHT) and the discrete fractional Fourier transform (DFRFT). First, the eigenvalues and eigenvectors of the discrete Fourier and Hartley transform matrices are investigated. Then, the results of the eigendecompositions of the transform matrices are used to define DFRHT and DFRFT. Also, an important relationship between DFRHT and DFRFT is described, and numerical examples are illustrated to demonstrate that the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT. Finally, a filtering technique in the fractional Fourier transform domain is applied to remove chirp interference.
105 citations
TL;DR: A novel generic tool for data compression and filtering: the generalized Karhunen-Loeve (GKL) transform, which minimizes a distance between any given reference and a transformation of some given data where the transform has a predetermined maximum possible rank.
Abstract: We present a novel generic tool for data compression and filtering: the generalized Karhunen-Loeve (GKL) transform. The GKL transform minimizes a distance between any given reference and a transformation of some given data where the transform has a predetermined maximum possible rank. The GKL transform is also a generalization of the relative Karhunen-Loeve (RKL) transform by Yamashita and Ogawa (see IEEE Trans. Signal Processing, vol.44, p.661-72, Mar. 1996) where the latter assumes that the given data consist of the given reference (signal) and an independent noise. This letter provides a very simple and yet complete description of the GKL transform and shows useful engineering insights into the GKL transform.
99 citations
TL;DR: This paper develops a 2D DFRFT which can preserve the rotation properties and provide similar results to continuous FRFT.
98 citations
TL;DR: A general definition of the fractional Fourier transform (FRT) for all signal classes and the multiplicity (which is intrinsic in a fractional operator) is clearly developed.
Abstract: The paper investigates the possibility for giving a general definition of the fractional Fourier transform (FRT) for all signal classes [one-dimensional (1-D) and multidimensional, continuous and discrete, periodic and aperiodic]. Since the definition is based on the eigenfunctions of the ordinary Fourier transform (FT), the preliminary conditions is that the signal domain/periodicity be the same as the FT domain/periodicity. Within these classes, a general FRT definition is formulated, and the FRT properties are established. In addition, the multiplicity (which is intrinsic in a fractional operator) is clearly developed. The general definition is checked in the case in which the FRT is presently available and, moreover, to establish the FRT in new classes of signals.
93 citations
TL;DR: In this paper, a new design approach for diffractive phase elements (DPE) that implement beam shaping in the fractional Fourier transform (FRFT) domain is presented.
Abstract: A new design approach for the diffractive phase elements (DPE’s) that implement beam shaping in the fractional Fourier transform (FRFT) domain is presented. The new algorithm can successfully achieve the design of DPE’s for beam shaping in both unitary and nonunitary transform systems. The unitarity transform condition of the FRFT domain is discussed. Modeling designs of the DPE’s are carried out for several fractional orders and different parameters of the beam for converting a Gaussian profile into a uniform beam. Our approach can realize beam shaping well for a nonunitary transform system in the FRFT domain.
88 citations
TL;DR: This work allows the fractional Fourier transform orders to be specified independently for the two dimensions but also allow the input and output scale parameters and the residual spherical phase factors to be controlled.
Abstract: We provide a general treatment of optical two-dimensional fractional Fourier transforming systems. We not only allow the fractional Fourier transform orders to be specified independently for the two dimensions but also allow the input and output scale parameters and the residual spherical phase factors to be controlled. We further discuss systems that do not allow all these parameters to be controlled at the same time but are simpler and employ a fewer number of lenses. The variety of systems discussed and the design equations provided should be useful in practical applications for which an optical fractional Fourier transforming stage is to be employed.
TL;DR: A one- dimensional fractional Hilbert transform acting on a one-dimensional rectangle function is analyzed and it is shown how it produces an output image that is selectively edge enhanced.
Abstract: The Hilbert transform is of interest for image-processing applications because it forms an image that is edge enhanced relative to an input object. Recently a fractional Hilbert transform was introduced that can select which edges are enhanced and to what degree the edge enhancement occurs. Although experimental results of this selective edge enhancement were presented, there was no explanation of this phenomenon. We analyze a one-dimensional fractional Hilbert transform acting on a one-dimensional rectangle function and show how it produces an output image that is selectively edge enhanced.
Book•
01 Jun 1998
TL;DR: This work presentsrepresentations of fully complex functions on real-time spatial light modulators on the basis of the fractional Fourier transform to optical pattern recognition, and describes the properties and applications of bacteriorhodopsin Q.
Abstract: Contributors Preface 1. Pattern recognition with optics Francis T. S. Yu and Don A. Gregory 2. Hybrid neural networks for nonlinear pattern recognition Taiwei Lu 3. Wavelets, optics, and pattern recognition Yao Li and Yunglong Sheng 4. Applications of the fractional Fourier transform to optical pattern recognition David Mendlovic, Zeev Zalesky and Haldum M. Oxaktas 5. Optical implementation of mathematical morphology Tien-Hsin Chao 6. Nonlinear optical correlators with improved discrimination capability for object location and recognition Leonid P. Yaroslavsky 7. Distortion-invariant quadratic filters Gregory Gheen 8. Composite filter synthesis as applied to pattern recognition Shizhou Yin and Guowen Lu 9. Iterative procedures in electro-optical pattern recognition Joseph Shamir 10. Optoelectronic hybrid system for three-dimensional object pattern recognition Guoguang Mu, Mingzhe Lu and Ying Sun 11. Applications of photrefractive devices in optical pattern recognition Ziangyang Yang 12. Optical pattern recognition with microlasers Eung-Gi Paek 13. Optical properties and applications of bacteriorhodopsin Q. Wang Song and Yu-He Zhang 14. Liquid-crystal spatial light modulators Aris Tanone and Suganda Jutamulia 15. Representations of fully complex functions on real-time spatial light modulators Robert W. Cohn and Laurence G. Hassbrook Index.
Book•
30 Nov 1998TL;DR: In this article, the NDFT was used to construct a 1-D and 2-D antenna pattern synthesis with Prescribed Nulls, and the Dual-Tone Multi-Frequency Signal Decoding (DTMSD) was proposed.
Abstract: 1. Introduction. 2. The Nonuniform Discrete Fourier Transform. 3. 1-D Fir Filter Design Using the NDFT. 4. 2-D Fir Filter Design Using the NDFT. 5. Antenna Pattern Synthesis with Prescribed Nulls. 6. Dual-Tone Multi-Frequency Signal Decoding. 7. Conclusions. References. Index.
TL;DR: By using the Fourier representation and the fast Fourier transform, one generation of the infinite population simple genetic algorithm can be computed in time O(cl log2 3), where c is arity of the alphabet and l is the string length.
Abstract: This paper is the first part of a two-part series. It proves a number of direct relationships between the Fourier transform and the simple genetic algorithm. (For a binary representation, the Walsh transform is the Fourier transform.) The results are of a theoretical nature and are based on the analysis of mutation and crossover. The Fourier transform of the mixing matrix is shown to be sparse. An explicit formula is given for the spectrum of the differential of the mixing transformation. By using the Fourier representation and the fast Fourier transform, one generation of the infinite population simple genetic algorithm can be computed in time O(cl log2 3), where c is arity of the alphabet and l is the string length. This is in contrast to the time of O(c3l) for the algorithm as represented in the standard basis. There are two orthogonal decompositions of population space that are invariant under mixing. The sequel to this paper will apply the basic theoretical results obtained here to inverse problems and asymptotic behavior.
TL;DR: In this article, the authors extend the fractional Fourier transform to different spaces of generalized functions using two different techniques, one analytic and the other algebraic, which makes the transform of a convolution of two functions almost equal to the product of their transform.
Abstract: In recent years the fractional Fourier transform (FRFT), which is a generalization of the Fourier transform, has been the focus of many research papers because of its application in several areas, including signal processing and optics. In this paper, we extend the fractional Fourier transform to different spaces of generalized functions using two different techniques, one analytic and the other algebraic. The algebraic approach requires the introduction of a new convolution operation for the fractional Fourier transform that makes the transform of a convolution of two functions almost equal to the product of their transform.
14 Oct 1998
TL;DR: The Fourier transform is a very useful tool for signal studies as mentioned in this paper. Nevertheless there are many problems in using it; but these problems are very well known and correctly explained in literature.
Abstract: The Fourier transform is a very useful tool for signal studies. Nevertheless there are many problems in using it; but these problems are very well known and correctly explained in literature. Wavelets are not usual in power network analysis. However, they are easy to use and give good results; the edge effects are transient and the computation time may be reasonable. Prony analysis is only found in a few papers about power networks. There are few high-performance decomposition programs. The best ones remain sensitive to noise. They require a long observation time with many samples. But, when the analysis succeed, this method is the most powerful to explain what happens in a power network transient. This paper explains as simply as possible the wavelet and the Prony analyses and shows, qualitatively, their performances and their limits.
TL;DR: The classical Wiener filter can be implemented in O(n) time for space-varying degradations and nonstationary processes, however, the optimal linear estimate requires O(N 2 ) time for implementation as discussed by the authors.
Abstract: The classical Wiener filter, which can be implemented in O(N log N) time, is suited best for space-invariant degradation models and space-invariant signal and noise characteristics. For space-varying degradations and nonstationary processes, however, the optimal linear estimate requires O(N2) time for implementation. Optimal filtering in fractional Fourier domains permits reduction of the error compared with ordinary Fourier domain Wiener filtering for certain types of degradation and noise while requiring only O(N log N) implementation time. The amount of reduction in error depends on the signal and noise statistics as well as on the degradation model. The largest improvements are typically obtained for chirplike degradations and noise, but other types of degradation and noise may also benefit substantially from the method (e.g., nonconstant velocity motion blur and degradation by inhomegeneous atmospheric turbulence). In any event, these reductions are achieved at no additional cost.
TL;DR: The fundamentals of Fourier analysis are reviewed with emphasis on the analysis of transient signals, and the human saccade is considered to illustrate the pitfalls and advantages of various Fourier analyses.
TL;DR: In this paper, a numerical algorithm based on a single fast Fourier transform is proposed, which shows better precision and calculation efficiency than those of previously published algorithms, and if specific conditions are met, the numerical calculations of two successive fractional Fourier transforms produce results that are similar to the analytical solution.
Abstract: A numerical algorithm based on a single fast Fourier transform is proposed. Its precision and calculation efficiency show better performance than those of previously published algorithms. It is also shown that if specific conditions are met, the numerical calculations of two successive fractional Fourier transforms produce results that are similar to the analytical solution.
TL;DR: A recursive algorithm to implement phase retrieval from two intensities in the fractional Fourier transform domain is proposed that can significantly simplify computational manipulations and does not need an initial phase estimate compared with conventional iterative algorithms.
Abstract: We first discuss the discrete fractional Fourier transform and present some essential properties. We then propose a recursive algorithm to implement phase retrieval from two intensities in the fractional Fourier transform domain. This approach can significantly simplify computational manipulations and does not need an initial phase estimate compared with conventional iterative algorithms. Simulation results show that this approach can successfully recover the phase from two intensities.
TL;DR: In this paper, the inverse loop transform of a regular Borel measure on the moduli space is derived from the Riesz-Markov theorem, which states that every positive linear functional on the space of continuous functions thereon qualifies as the loop transform.
Abstract: The loop transform in quantum gauge field theory can be recognized as the Fourier transform (or characteristic functional) of a measure on the space of generalized connections modulo gauge transformations. Since this space is a compact Hausdorff space, conversely, we know from the Riesz–Markov theorem that every positive linear functional on the space of continuous functions thereon qualifies as the loop transform of a regular Borel measure on the moduli space. In the present article we show how one can compute the finite joint distributions of a given characteristic functional, that is, we derive the inverse loop transform.
TL;DR: The generalized Calderon reproducing formula involving wavelet measure is established for functions f ∈ Lp(ℝn) in this article, which gives rise to the continuous decomposition of f into wavelets, and enables one to obtain inversion formulae for generalized windowed X-ray transforms, the Radon transform, and k-plane transforms.
Abstract: The generalized Calderon reproducing formula involving “wavelet measure” is established for functions f ∈ Lp(ℝn). The special choice of the wavelet measure in the reproducing formula gives rise to the continuous decomposition of f into wavelets, and enables one to obtain inversion formulae for generalized windowed X-ray transforms, the Radon transform, and k-plane transforms. The admissibility conditions for the wavelet measure μ are presented in terms of μ itself and in terms of the Fourier transform of μ.
TL;DR: This correspondence proposes a systematic method, based on the structure of the FRT, which not only provides unambiguous extensions of ordinary operations but permits writing the applicable expressions simply by inspection.
Abstract: The fractional Fourier transform (FRT) permits a variety of associated fractional operations. This correspondence proposes a systematic method, based on the structure of the FRT, which not only provides unambiguous extensions of ordinary operations but permits writing the applicable expressions simply by inspection. The approach also exposes the possible paths for implementing such operations.
TL;DR: A new, efficient method for 3-D radon inversion, i.e., reconstruction of the image from the radial derivative of the 3- D radon transform, called direct Fourier inversion (DFI) is presented, based directly on the 3D Fourier slice theorem.
Abstract: The radial derivative of the three-dimensional (3-D) radon transform of an object is an important intermediate result in many analytically exact cone-beam reconstruction algorithms. The authors briefly review Grangeat's (1991) approach for calculating radon derivative data from cone-beam projections and then present a new, efficient method for 3-D radon inversion, i.e., reconstruction of the image from the radial derivative of the 3-D radon transform, called direct Fourier inversion (DFI). The method is based directly on the 3-D Fourier slice theorem. From the 3-D radon derivative data, which is assumed to be sampled on a spherical grid, the 3-D Fourier transform of the object is calculated by performing fast Fourier transforms (FFTs) along radial lines in the radon space. Then, an interpolation is performed from the spherical to a Cartesian grid using a 3-D gridding step in the frequency domain. Finally, this 3-D Fourier transform is transformed back to the spatial domain via 3-D inverse FFT. The algorithm is computationally efficient with complexity in the order of N/sup 3/ log N. The authors have done reconstructions of simulated 3-D radon derivative data assuming sampling conditions and image quality requirements similar to those in medical computed tomography (CT).
TL;DR: A version of the resolution of the identity and some properties of FRWPT connected with those of FRFT and WPT are shown.
Abstract: We introduce the concept of the Fractional Wave Packet Transform(FRWPT), based on the idea of the Fractional Fourier Transform(FRFT) and Wave Packet Transform(WPT). We show a version of the resolution of the identity and some properties of FRWPT connected with those of FRFT and WPT.
TL;DR: It has been shown that transformation from a rectangular to a quincunx lattice can be associated with fractional Fourier transformation, and a Gaussian function, which plays an important role as a window function in Gabor theory, is an eigenfunction of fractional fourier transformation.
Abstract: Transformations of Gabor lattices have been associated with operations on the window functions that arise in Gabor theory. In particular, it has been shown that transformation from a rectangular to a quincunx lattice can be associated with fractional Fourier transformation. Since a Gaussian function, which plays an important role as a window function in Gabor theory, is an eigenfunction of fractional Fourier transformation, this transformation has a clear advantage over other operations that are used to transform a rectangular lattice into a quincunx lattice.
TL;DR: Some basic properties of the FRHT such as Parseval's theorem and its optical implementation are discussed qualitatively and the integral representation of a fractional Hankel transform (FRHT) is derived from the fractional Fourier transform.
Abstract: We derive the integral representation of a fractional Hankel transform (FRHT) from a fractional Fourier transform. Some basic properties of the FRHT such as Parseval's theorem and its optical implementation are discussed qualitatively.
01 Nov 1998
TL;DR: A preprocessing of the received signal by an FRFT-based excision scheme prior to demodulation is proposed, which often improves the bit error performance of the receiver when compared to the case where there is no preprocessing.
Abstract: We demonstrate the use of the fractional Fourier transform (FRFT) in excising broadband, linear FM (chirp) type interferences in spread spectrum communication systems. This method is predicated on the fact that the FRFT perfectly localizes a linear FM signal as an impulse when the angle parameter of the transform matches the sweep rate (chirp rate) of the linear chirp signal. Therefore, a transform domain thresholding can often eliminate linear-FM type interferences without severely affecting the desired part of the received signal. Thus we propose a preprocessing of the received signal by an FRFT-based excision scheme prior to demodulation. The simulations demonstrate that this technique often improves the bit error performance of the receiver when compared to the case where there is no preprocessing.
TL;DR: In this article, the wavelet transform is described from the perspective of a Fourier transform and an adaptive window is presented that may be optimally tailored to suit one's needs and hence, possibly, the scaling functions and the wavelets.
Abstract: The wavelet transform is described from the perspective of a Fourier transform. The relationships among the Fourier transform, the Gabor (1946) transform (windowed Fourier transform), and the wavelet transform are described. The differences are also outlined, to bring out the characteristics of the wavelet transform. The limitations of the wavelets in localizing responses in various domains are also delineated. Finally, an adaptive window is presented that may be optimally tailored to suit one's needs, and hence, possibly, the scaling functions and the wavelets.