A note on 2-D Hartley transform

Geophysics ◽  
1994 ◽  
Vol 59 (7) ◽  
pp. 1150-1155 ◽  
Author(s):  
N. L. Mohan ◽  
L. Anand Babu
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Seismic Signal ◽  
Clear Definition ◽  
Hartley Transform ◽  
The Fourier Transform ◽  
Definition Of ◽  
Seismic Signal Processing

In recent years the application of the Hartley transform, originally introduced by Hartley (1942), has gained importance in seismic signal processing and interpretation (Saatcilar et al., 1990, 1992). The Hartley transform is similar to the Fourier transform but is computationally much faster than even the fast Fourier transform (Bracewell, 1983; Bracewell et al., 1986; Sorensen et al., 1985; Pei and Wu, 1985; Duhamel and Vetterli, 1987; Zhou, 1992). Surprisingly, we have not seen a clear definition of the 2-D Hartley transform in the published literature.

Discrete Time, Discrete Frequency

2021 ◽  
pp. 106-155
Author(s):  
Victor Lazzarini
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Discrete Time ◽  
Time Domain ◽  
Continuous Time ◽  
Time Varying ◽  
Discrete Frequency ◽  
The Fourier Transform ◽  
Definition Of ◽  
The Time Domain

This chapter is dedicated to exploring a form of the Fourier transform that can be applied to digital waveforms, the discrete Fourier transform (DFT). The theory is introduced and discussed as a modification to the continuous-time transform, alongside the concept of windowing in the time domain. The fast Fourier transform is explored as an efficient algorithm for the computation of the DFT. The operation of discrete-time convolution is presented as a straight application of the DFT in musical signal processing. The chapter closes with a detailed look at time-varying convolution, which extends the principles developed earlier. The conclusion expands the definition of spectrum once more.

2-D Hartley transforms

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 262-267 ◽  
Author(s):  
N. Sundararajan
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Digital Signal Processing ◽  
Digital Signal ◽  
Physical Significance ◽  
The Real ◽  
Hartley Transform ◽  
Real Domain ◽  
Hartley Transforms ◽  
The Fourier Transform

Two different versions of kernels associated with the 2-D Hartley transforms are investigated in relation to their Fourier counterparts. This newly emerging tool for digital signal processing is an alternate means of analyzing a given function in terms of sinusoids and is an offshoot of Fourier transform. Being a real‐valued function and fully equivalent to the Fourier transform, the Hartley transform is more efficient and economical than its progenitor. Hartley and Fourier pairs of complete orthogonal transforms comprise mathematical twins having definite physical significance. The direct and inverse Hartley transforms possess the same kernel, unlike the Fourier transform, and hence have the dual distinction of being both self reciprocal and having the convenient property of occupying the real domain. Some of the properties of the Hartley transform differ marginally from those of the Fourier transform.

scholarly journals Two Dimensional Short Time Hartley Transforms

2016 ◽  
Vol 21 (1) ◽  
pp. 41 ◽  
Author(s):  
Narasimman Sundararajan ◽  
A. Ebrahimi ◽  
Nannappa Vasudha
Keyword(s):  
Fourier Transform ◽  
Seismic Data ◽  
Statistical Properties ◽  
Two Dimensional ◽  
Short Time Fourier Transform ◽  
Hartley Transform ◽  
Hartley Transforms ◽  
The Fourier Transform ◽  
Definition Of ◽  
Short Time

The Hartley transform, as in the case of the Fourier transform, is not suitably applicable to non-stationary representations of signals whose statistical properties change as a function of time. Hence, different versions of 2-D short time Hartley transforms (STHT) are given in comparison with the short time Fourier transform (STFT). Although the two different versions of STHT defined here with their inverses are equally applicable, one of them is mathematically incorrect/incompatible due to the incorrect definition of the 2-D Hartley transform in literature. These definitions of STHTs can easily be extended to multi-dimensions. Computations of the STFT and the two versions of STHTs are illustrated based on 32 channels (traces) of synthetic seismic data consisting of 256 samples in each trace. Salient features of STHTs are incorporated. 

Fast Fourier Transform – Calculation and Interpretation

2008 ◽  
Vol 3 (4) ◽  
pp. 74-86
Author(s):  
Boris A. Knyazev ◽  
Valeriy S. Cherkasskij
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Discrete Fourier Transform ◽  
Signal Theory ◽  
Correct Interpretation ◽  
The Fourier Transform ◽  
Application Programs ◽  
Physics Problems

The article is intended to the students, who make their first steps in the application of the Fourier transform to physics problems. We examine several elementary examples from the signal theory and classic optics to show relation between continuous and discrete Fourier transform. Recipes for correct interpretation of the results of FDFT (Fast Discrete Fourier Transform) obtained with the commonly used application programs (Matlab, Mathcad, Mathematica) are given.

The whole can be very much less than the sum of its parts

2019 ◽  
Vol 103 (556) ◽  
pp. 117-127
Author(s):  
Peter Shiu
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Mr Imaging ◽  
Fast Fourier Transform ◽  
Data Compression ◽  
Mobile Phones ◽  
Processing Data ◽  
Speed Up ◽  
Indispensable Tool ◽  
Computing Machines

This Article is on the discrete Fourier transform (DFT) and the fast Fourier transform (FFT). As we shall see, FFT is a slight misnomer, causing confusion to beginners. The idiosyncratic title will be clarified in §4.Computing machines are highly efficient nowadays, and much of the efficiency is based on the use of the FFT to speed up calculations in ultrahigh precision arithmetic. The algorithm is now an indispensable tool for solving problems that involve a large amount of computation, resulting in many useful and important applications: for example, in signal processing, data compression and photo-images in general, and WiFi, mobile phones, CT scanners and MR imaging in particular.

The Spectra of Filters

2021 ◽  
pp. 204-268
Author(s):  
Victor Lazzarini
Keyword(s):  
Fourier Transform ◽  
Impulse Response ◽  
Discrete Time ◽  
Filter Design ◽  
Finite Impulse Response ◽  
Infinite Impulse Response ◽  
Main Body ◽  
Analysis Tool ◽  
The Fourier Transform ◽  
Definition Of

This chapter now turns to the discussion of filters, which extend the notion of spectrum beyond signals into the processes themselves. A gentle introduction to the concept of delaying signals, aided by yet another variant of the Fourier transform, the discrete-time Fourier transform, allows the operation of filters to be dissected. Another analysis tool, in the form of the z-transform, is brought to the fore as a complex-valued version of the discrete-time Fourier transform. A study of the characteristics of filters, introducing the notion of zeros and poles, as well as finite impulse response (FIR) and infinite impulse response (IIR) forms, composes the main body of the text. This is complemented by a discussion of filter design and applications, including ideas related to time-varying filters. The chapter conclusion expands once more the definition of spectrum.

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