The Fourier Transform and its Applications

doi:10.1049/EP.1965.0268

Journal Article•DOI•

The Fourier Transform and its Applications

01 Jan 1965-Electronics and Power-Vol. 11, Iss: 10, pp 357

About: This article is published in Electronics and Power.The article was published on 1965-01-01. It has received 1534 citations till now. The article focuses on the topics: Fractional Fourier transform & Hartley transform.

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Localisation of the complex spectrum : The S transform

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R. G. Stockwell, Lalu Mansinha, Robert P. Lowe

01 Jan 1996-Journal of Association of Exploration Geophysicists

TL;DR: The S transform as discussed by the authors is an extension to the ideas of the Gabor transform and the Wavelet transform, based on a moving and scalable localising Gaussian window and is shown here to have characteristics that are superior to either of the transforms.

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Abstract: The S transform, an extension to the ideas of the Gabor transform and the Wavelet transform, is based on a moving and scalable localising Gaussian window and is shown here to have characteristics that are superior to either of the transforms. The S transform is fully convertible both forward and inverse from the time domain to the 2-D frequency translation (time) domain and to the familiar Fourier frequency domain. Parallel to the translation (time) axis, the S transform collapses as the Fourier transform. The amplitude frequency-time spectrum and the phase frequency-time spectrum are both useful in defining local spectral characteristics. The superior properties of the S transform are due to the fact that the modulating sinusoids are fixed with respect to the time axis while the localising scalable Gaussian window dilates and translates. As a result, the phase spectrum is absolute in the sense that it is always referred to the origin of the time axis, the fixed reference point. The real and imaginary spectrum can be localised independently with a resolution in time corresponding to the period of the basis functions in question. Changes in the absolute phase ofa constituent frequency can be followed along the time axis and useful information can be extracted. An analysis of a sum of two oppositely progressing chirp signals provides a spectacular example of the power of the S transform. Other examples of the applications of the Stransform to synthetic as well as real data are provided.

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2,323 citations

Journal Article•DOI•

Optical coherence tomography - principles and applications

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Adolf Friedrich Fercher¹, Wolfgang Drexler¹, Christoph K. Hitzenberger¹, Theo Lasser²•Institutions (2)

University of Vienna¹, École Polytechnique Fédérale de Lausanne²

20 Jan 2003-Reports on Progress in Physics

TL;DR: OCT as discussed by the authors synthesises cross-sectional images from a series of laterally adjacent depth-scans, which can be used to assess tissue and cell function and morphology in situ.

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Abstract: There have been three basic approaches to optical tomography since the early 1980s: diffraction tomography, diffuse optical tomography and optical coherence tomography (OCT). Optical techniques are of particular importance in the medical field, because these techniques promise to be safe and cheap and, in addition, offer a therapeutic potential. Advances in OCT technology have made it possible to apply OCT in a wide variety of applications but medical applications are still dominating. Specific advantages of OCT are its high depth and transversal resolution, the fact, that its depth resolution is decoupled from transverse resolution, high probing depth in scattering media, contact-free and non-invasive operation, and the possibility to create various function dependent image contrasting methods. This report presents the principles of OCT and the state of important OCT applications. OCT synthesises cross-sectional images from a series of laterally adjacent depth-scans. At present OCT is used in three different fields of optical imaging, in macroscopic imaging of structures which can be seen by the naked eye or using weak magnifications, in microscopic imaging using magnifications up to the classical limit of microscopic resolution and in endoscopic imaging, using low and medium magnification. First, OCT techniques, like the reflectometry technique and the dual beam technique were based on time-domain low coherence interferometry depth-scans. Later, Fourier-domain techniques have been developed and led to new imaging schemes. Recently developed parallel OCT schemes eliminate the need for lateral scanning and, therefore, dramatically increase the imaging rate. These schemes use CCD cameras and CMOS detector arrays as photodetectors. Video-rate three-dimensional OCT pictures have been obtained. Modifying interference microscopy techniques has led to high-resolution optical coherence microscopy that achieved sub-micrometre resolution. This report is concluded with a short presentation of important OCT applications. Ophthalmology is, due to the transparent ocular structures, still the main field of OCT application. The first commercial instrument too has been introduced for ophthalmic diagnostics (Carl Zeiss Meditec AG). Advances in using near-infrared light, however, opened the path for OCT imaging in strongly scattering tissues. Today, optical in vivo biopsy is one of the most challenging fields of OCT application. High resolution, high penetration depth, and its potential for functional imaging attribute to OCT an optical biopsy quality, which can be used to assess tissue and cell function and morphology in situ. OCT can already clarify the relevant architectural tissue morphology. For many diseases, however, including cancer in its early stages, higher resolution is necessary. New broad-bandwidth light sources, like photonic crystal fibres and superfluorescent fibre sources, and new contrasting techniques, give access to new sample properties and unmatched sensitivity and resolution.

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1,914 citations

Cites background or methods from "The Fourier Transform and its Appli..."

...Furthermore, multiple scattered light decisively influences OCT probing depth and signal-to-noise ratio (SNR). In the end equations (2.9) and (2.14) are based on Maxwell’s equations. There are other treatments of scattering based on Maxwell’s equations. For example, Brodsky et al (2000) use a path-integral analysis of the vector Helmholtz equation. A reasonable agreement has been obtained with experiments on the maximum LCI signal as a function of particle size and concentration by Thurber et al (2000). Furthermore, the properties of multiple scattered light have been analysed analytically using the extended Huygens–Fresnel formulation of beam propagation in the turbulent atmosphere developed by Yura (1979) and adapted to the OCT problem by Schmitt and Knüttel (1997) and by Thrane et al (2000). These studies have revealed that LCI and OCT probing depths are not only dependent on absorption and scattering coefficients but also on scattering anisotropy, aperture of the sample lens, and, in particular, on the sample distribution between lens and coherence gate....
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...A corresponding technique using a space-coherence gate has recently been investigated by Rosen and Takeda (2000). These authors suggested that the spatial spectrum of the beam illuminating the object be varied by spatial masks in order to use the longitudinal component of the spatial coherence as a coherence gate for depth ranging....
[...]
...Furthermore, multiple scattered light decisively influences OCT probing depth and signal-to-noise ratio (SNR). In the end equations (2.9) and (2.14) are based on Maxwell’s equations. There are other treatments of scattering based on Maxwell’s equations. For example, Brodsky et al (2000) use a path-integral analysis of the vector Helmholtz equation. A reasonable agreement has been obtained with experiments on the maximum LCI signal as a function of particle size and concentration by Thurber et al (2000). Furthermore, the properties of multiple scattered light have been analysed analytically using the extended Huygens–Fresnel formulation of beam propagation in the turbulent atmosphere developed by Yura (1979) and adapted to the OCT problem by Schmitt and Knüttel (1997) and by Thrane et al (2000)....
[...]
...Furthermore, multiple scattered light decisively influences OCT probing depth and signal-to-noise ratio (SNR). In the end equations (2.9) and (2.14) are based on Maxwell’s equations. There are other treatments of scattering based on Maxwell’s equations. For example, Brodsky et al (2000) use a path-integral analysis of the vector Helmholtz equation....
[...]
...A similar plot, employing parameters for OCT of scattering tissues can be found in Rollins and Izatt (1999). Unbalanced versus balanced operation in OCT has been analysed by Podoleanu (2000), who showed that the balanced OCT configuration, besides the attenuation of vibration-induced noise in the depth-scanner and the avoidance of back-reflections to the optical source, has signal-to-noise advantages in particular for fast OCT systems....
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Journal Article•DOI•

Crosscorrelation properties of pseudorandom and related sequences

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Dilip V. Sarwate¹, Michael B. Pursley¹•Institutions (1)

University of Illinois at Urbana–Champaign¹

01 May 1980

TL;DR: This paper presents a survey of recent results and provides several new results on the periodic and aperiodic crosscorrelation functions for pairs of m-sequences and for Pair of related (but not maximal-length) binary shift register sequences.

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Abstract: Binary maximal-length linear feedback shift register sequences (m-sequences) have been successfully employed in communications, navigation, and related systems over the past several years. For the early applications, m-sequences were used primarily because of their excellent periodic autocorrelation properties. For many of the recent systems applications, however, the crosscorrelation properties of such sequences are at least as important as the autocorrelation properties, and the system performance depends upon the aperiodic correlation in addition to the periodic correlation. This paper presents a survey of recent results and provides several new results on the periodic and aperiodic crosscorrelation functions for pairs of m-sequences and for pairs of related (but not maximal-length) binary shift register sequences. Also included are several recent results on correlation for complex-valued sequences as well as identities relating the crosscorrelation functions to autocorrelation functions. Examples of problems in spread-spectrum communications are employed to motivate the choice of correlation parameters that are considered in the paper.

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1,686 citations

Journal Article•DOI•

Retinotopic organization in human visual cortex and the spatial precision of functional MRI.

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Stephen A. Engel¹, Gary H. Glover¹, Brian A. Wandell¹•Institutions (1)

Stanford University¹

01 Mar 1997-Cerebral Cortex

TL;DR: This work identified the borders between several retinotopically organized visual areas in the posterior occipital lobe and estimated the spatial resolution of the fMRI signal and found that signal amplitude falls to 60% at a spatial frequency of 1 cycle per 9 mm of visual cortex.

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Abstract: A method of using functional magnetic resonance imaging (fMRI) to measure retinotopic organization within human cortex is described. The method is based on a visual stimulus that creates a traveling wave of neural activity within retinotopically organized visual areas. We measured the fMRI signal caused by this stimulus in visual cortex and represented the results on images of the flattened cortical sheet. We used the method to locate visual areas and to evaluate the spatial precision of fMRI. Specifically, we: (i) identified the borders between several retinotopically organized visual areas in the posterior occipital lobe; (ii) measured the function relating cortical position to visual field eccentricity within area V1; (iii) localized activity to within 1.1 mm of visual cortex; and (iv) estimated the spatial resolution of the fMRI signal and found that signal amplitude falls to 60% at a spatial frequency of 1 cycle per 9 mm of visual cortex. This spatial resolution is consistent with a linespread whose full width at half maximum spreads across 3.5 mm of visual cortex. In a series of experiments, we measured the retinotopic organization of human cortical area V1 and identified the locations of other nearby retinotopically organized visual areas. We also used the retinotopic organization of human primary visual cortex to measure the spatial localization and spatial resolution that can be obtained from functional magnetic resonance imaging (fMRI) of human visual cortex. Human primary visual cortex (area V1) is located in the

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1,585 citations

Journal Article•DOI•

The Rapid Calculation of Potential Anomalies

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Robert L. Parker¹•Institutions (1)

University of California, San Diego¹

01 Mar 1973-Geophysical Journal International

TL;DR: In this paper, it is shown how a series of Fourier transforms can be used to calculate the magnetic or gravitational anomaly caused by an uneven, non-uniform layer of material.

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Abstract: Summary It is shown how a series of Fourier transforms can be used to calculate the magnetic or gravitational anomaly caused by an uneven, non-uniform layer of material. Modern methods for finding Fourier transforms numerically are very fast and make this approach attractive in situations where large quantities of observations are available.

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1,365 citations

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The Fourier Transform and its Applications

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