The idea of unified Fibonacci-type topology is used for construction of wide classes of discrete orthogonal transforms, including Rademacher-Fibonacci, Walsh-Fibonacci, Haar-Fibonacci-type transforms, etc. Efficient algorithms for proposed transforms directly related with the generalized Fibonacci topology are derived. The generation of discrete wavelets and wavelet packets based on Fibonacci-type recursions is established.

https://ieeexplore.ieee.org/iel3/4078/12060/00555547.pdf

Discrete orthogonal transforms based on Fibonacci-type recursions

We present a new interconnection topology called generalized Fibonacci cubes, which unifies a wide range of connection topologies such as the Boolean cube (or hypercube), classical Fibonacci cube, etc. We study the properties of generalized Fibonacci corresponding codes, e.g. we find an Zeckendorfs' theorem for the generalized Fibonacci codes. Some basic properties of generalized Fibonacci cubes and graphs, especially, their topological properties, are also established. Finally, the classes of unitary generalized Fibonacci-Walsh and Fibonacci-Haar transforms are proposed for which efficient computation algorithms and corresponding generalized Fibonacci flowcharts are presented.

Generalized Fibonacci cubes and trees for DSP applications

Multidimensional nuclear spectra compression using fast adaptive Fourier-based transforms

Three dimensional spatial signal representation employing transforms is generally performed by applying a 2-D transform to the 2-D slices of the 3-D data or a 3-D transform for the entire data. Transforms such as DCT, Walsh and Haar are frequently considered because of their ease of computation. This paper develops a technique called the multi-transform technique for the efficient representation of 3-D spatial data. The main feature of the technique is that two or more transforms can be applied to represent the signal thus benefiting from the compression qualities of each of the transforms. The signal to noise ratio (SNR) of the reconstructed signal employing the proposed technique is higher than the SNR of the signal employing a single transform alone. Results are presented for a 3-D MRI signal representation using the 2-D and 3-D multi-transform techniques employing the DCT/Walsh combination. Comparisons with using DCT alone are made for both cases. >

Applications of multi-transforms to 3-D spatial signal representation

In this paper we define the Walsh-Fourier bispectral density and establish a statistical methodology for bispectral analysis of integer valued time series. The theoretical properties and results pertaining to the definition and estimation of the Walsh-Fourier bispectral density are obtained. Many of the results we obtain have their analogues in the conventional Fourier frequency domain. We present simulation results of Walsh Fourier bispectral density for linear and bilinear integer valued time series models.

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On Walsh-Fourier Bispectral Analysis

Orthogonal transforms for digital signal processing: N. Ahmed and K.R. Rao. 275 pages, diagrams, 6×9 in. New York, Springer-Verlag Inc. 1975. Price $24.10.

Douglas F. Elliott

Papers

Orthogonal transforms for digital signal processing: N. Ahmed and K.R. Rao. 275 pages, diagrams, 6×9 in. New York, Springer-Verlag Inc. 1975. Price $24.10.