Abstract
Based on the generalized discrete Fourier transform, the generalized orthogonal discrete W transform and its fast algorithm are proposed and derived in this paper. The orthogonal discrete W transform proposed by Zhongde Wang has only four types. However, the generalized orthogonal discrete W transform proposed by us has infinite types and subsumes a family of symmetric transforms. The generalized orthogonal discrete W transform is a real-valued orthogonal transform, and the real-valued orthogonal transform of a real sequence has the advantages of simple operation and facilitated transmission and storage. The generalized orthogonal discrete W transforms provide more basis functions with new frequencies and phases and hence lead to more powerful analysis and processing tools for communication, signal processing, and numerical computing.
Highlights
Discrete Dynamics in Nature and Society field of complex numbers (N ≥ 2)
We prove that if ωN is an Nth-order primitive root of unity in the field of complex numbers (N ≥ 2), the row vectors of the transform matrix associated with the kernel function ω(Nl+α)(x+β) are orthogonal to each other, where l and x denote, respectively, the row and column indices of the transform matrix, while the α and β parameters can be any real numbers. e generalized discrete Fourier transform is c(o1n/√st Nr uc)ωte(Ndl+α)(uxs+iβn).g the normalized kernel function
For each of these complex-valued orthogonal transforms, a real-valued orthogonal transform can be obtained by adding the real and imaginary parts of the kernel function of the complex-valued orthogonal transform together to form the kernel function of the real-valued orthogonal transform. Many of these constructed real-valued orthogonal transforms were found to resemble the discrete Hartley transform (DHT) and are collectively called the discrete Hartley-type transforms [17,18,19,20]. ese include the Zhang–Hartley transform and the discrete W transform (DWT). Such transforms have been applied in spectral analysis, data compression, convolution, data security, and so on. ese transforms have been widely used in communication and signal processing [21,22,23,24,25,26,27,28,29,30]
Summary
The base exp(i(2π/N)) of the kernel function can be further extended to any Nth-order primitive root of unity in the field of complex numbers, and (l, x) can be extended to (l + α, x + β), where α and β can be any real numbers. We prove that if ωN is an Nth-order primitive root of unity in the field of complex numbers (N ≥ 2), the row vectors of the transform matrix associated with the kernel function ω(Nl+α)(x+β) are orthogonal to each other, where l and x denote, respectively, the row and column indices of the transform matrix, while the α and β parameters can be any real numbers.
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