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Simple matrix-theory proof of the discrete dyadic convolution theorem

Simple matrix-theory proof of the discrete dyadic convolution theorem

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© The Institution of Electrical Engineers
Published

A matrix-theory proof of the (periodic) discrete dyadic convolution theorem based on the determination of the eigen-values and eigenvectors of the dyadic convolution matrix is given. The eigenvalues are shown to be equal to the discrete Walsh transform of the dyadic convolution system's impulse response, and the matrix of the eigenvectors corresponds to the discrete Walsh-transform matrix.

Inspec keywords: Walsh functions; eigenvalues and eigenfunctions; transforms; matrix algebra

Other keywords: Walsh functions; eigenvalues and eigenfunctions; transforms; discrete dyadic convolution theorem; matrix algebra

Subjects: Algebra; Algebra

References

    1. 1)
      • Pratt, W.K.: `Linear and non-linear filtering in the Walsh domain', Proceedings of the symposium on applications of Walsh functions, April 1971, p. 166–170.
    2. 2)
      • H. Kremer . On the representation of Walsh functions and fast Walsh transformations. Angew. Inf. , 7 - 20
    3. 3)
      • Pichler, F.: `On state-space description of linear dyadic invariant systems', Proceedings of the symposium on applications of Walsh functions, April 1971, p. 166–170.
    4. 4)
      • B.R. Hunt . A matrix theory proof of the discrete convolution theorem. IEEE Trans. , 285 - 288
    5. 5)
      • B.L.N. Kennett . A note on the finite Walsh transform. IEEE Trans. , 489 - 491
    6. 6)
      • F. Ayres . (1962) , Theory and problems of matrices.
    7. 7)
      • S.C. Gupta . (1966) , Transform and state variable methods in linear systems.
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