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Method Of Splitting Signals By The Paired Transform
Abstract
The mathematical structure of the discrete Fourier transform (DFT) contains a unitary transform, that defines the splitting of the DFT by a certain set of short and separable transforms. This unitary transform is called the paired transform, a wavelet-like transform, which defines the representation (or decomposition) of a discrete signal in the form of a set of independent short signals. Properties of such paired representation are considered and the basis paired functions are described. The paired transform is fast, and many operations over signals can be reduced to processing their short splitting-signals. Examples of decomposition and filtration of signals by splitting-signals are given
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In this paper, a general, efficient, manageable split algorithm to
compute one-dimensional (1-D) unitary transforms, by using the special
partitioning in the frequency domain, is introduced. The partitions
determine fast transformations that split the N-point unitary transform
into a set of Ni-point transforms i=1: n(N1+...N
n=N). Here, we introduce a class of splitting
transformations: the so-called paired transforms. Based on these
transforms, the decompositions of the Fourier transforms of arbitrary
orders are given, and the corresponding algorithms are considered.
Comparative estimates revealing the efficiency of the proposed
algorithms with respect to the known ones are given. In particular, a
proposed method of calculating the 2r-point Fourier transform
requires 2r-1(r-3)+2 multiplications and
2r-1(r+9)-r2-3r-6 additions. In terms of the
paired transforms, the splitting of the 2r-point Hadamard
transform is described. As a result, the proposed algorithm for
computing this transform uses on the average no more than six operations
of additions per sample
A new concept of the A-wavelet transform is introduced, and the representation of the Fourier transform by the A-wavelet transform is described. Such a wavelet transform uses a fully scalable modulated window but not all possible shifts. A geometrical locus of frequency-time points for the A-wavelet transform is derived, and examples are given. The locus is considered "optimal" for the Fourier transform when a signal can be recovered by using only values of its wavelet transform defined on the locus. The inverse Fourier transform is also represented by the A*-wavelet transform defined on specific points in the time-frequency plane. The concept of the A-wavelet transform can be extended for representation of other unitary transforms. Such an example for the Hartley transform is described, and the reconstruction formula is given.
A concept of multipaired unitary transforms is introduced. These
kinds of transforms reveal the mathematical structure of Fourier
transforms and can be considered intermediate unitary transforms when
transferring processed data from the original real space of signals to
the complex or frequency space of their images. Considering paired
transforms, we analyze simultaneously the splitting of the
multidimensional Fourier transform as well as the presentation of the
processed multidimensional signal in the form of the short
one-dimensional (1-D) “signals”, that determine such
splitting. The main properties of the orthogonal system of paired
functions are described, and the matrix decompositions of the Fourier
and Hadamard transforms via the paired transforms are given. The
multiplicative complexity of the two-dimensional (2-D)
2r×2r-point discrete Fourier transform by
the paired transforms is 4r/2(r-7/3)+8/3-12 (r>3), which
shows the maximum splitting of the 5-D Fourier transform into the number
of the short 1-D Fourier transforms. The 2-D paired transforms are not
separable and represent themselves as frequency-time type wavelets for
which two parameters are united: frequency and time. The decomposition
of the signal is performed in a way that is different from the
traditional Haar system of functions