| Budget Amount *help |
¥3,380,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥180,000)
Fiscal Year 2007: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥1,600,000 (Direct Cost: ¥1,600,000)
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| Research Abstract |
Digital signal processing is the theory of matrices related to unitary matrices. The goal of this research is a basic study to investigate the possibility for extending the field of complex numbers on which the existing many methods of signal processings are based to the Clifford algebra, the fields of quaternion and hyper complex numbers. We investigated precisely discrete Fourier Transform (DFT) from the point of view for symmetry groups, we found a relation between 2-dimesional DFT and permutations. From the point of view we investigated to extend the results to general unitary matrices which have intrinsically symmetries with more than 2 order. The extension of DFT to quaternions and Clifford algebra is done by replaced the imaginary units by three units vectors of quaternions, but this extension is intrinsically related to a permutation with order 2. Therefore, the extension along this idea cannot be applied to general unitary matrices which have symmetries with more than 2 order.
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In this research, we investigated basically and theoretically to extend complex numbers to quaternions and Clifford algebra for unitary matrices of which eigenvalues are of finite orders., and we found the possibility of theoretical extension. When the unitary matrices have the symmetry of e order for one-dimensional signals, applying these unitary matrices to two-dimensional signals, we have a symmetry of e^2 order. But without reducing the symmetry of e^2 order, using e^2 volume elements of Clifford algebra Cl_n, it is indicated that we may not be able to embed unitary matrices into C4-matrices so that we can keep the symmetry of e^2 order. Using a subset B of which elements are disjoint bivectors each other, and the commutative subalgebra R_<2n,o>^+, it yields the reduction of the symmetry of e^2 order, but this embedding to Cl_n,-matrices still keeps about a half of e^2 symmetries. When the symmetries of two-dimensional signals are of more than 2 order, we can investigate the symmetries by Kronecker product for Cl_n-matrices, and but, for more than three-dimensional signals, we have to use tensor product for embedding unitary matrices to C4-matrices. This problem is still open. We shall study the problems. Less
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