Paulo J. S. G. Ferreira

DNA sequences Toeplitz matrices Shannon sampling Sensor networks and precision agriculture Superoscillations Maxwell equations Real and DFT codes Uncertainty-based and fuzzy information Frames, codes and reconstruction Sorting and the analog median filter Missing samples Nonlinear systems Approximation and coding DFT Miscellaneous articles related to teaching Copyrights

Discrete Fourier transform

The subject of the papers in this section is the discrete Fourier transform.

The first paper describes a set of permutations that commute with the discrete Fourier transform. In other words, let x be a signal, and X its DFT. The permutations described in the paper have the following property: if x is permuted, the DFT of the permuted signal can be obtained by permuting the DFT X in a similar way.

P. J. S. G. Ferreira. A group of permutations that commute with the discrete Fourier transform. IEEE Transactions on Signal Processing, vol. 42, no. 2, pp. 444-445, Feb. 1994.

This paper characterizes a potentially useful set of permutation matrices that commute with the Fourier matrix. The set of all such permutation matrices is a group under matrix multiplication, and every element of the group is its own inverse. The number of these permutations as a function of the Fourier matrix order is studied. It is shown that it is a multiplicative function of the Fourier matrix size N.

How many such permutations exist? Express N as a product of primes,

2^k0 p₁^k1 p₂^k2... p_r^kr

The number of permutations is 2^r, if k₀ is zero or one, or 2^r+1, if k₀ is two, or 2^r+2, if k₀ is greater than or equal to three.

P. J. S. G. Ferreira. A note on the block sum transformation. Mechanical Systems and Signal Processing, vol. 7, no. 2, pp. 191-192, Mar. 1993.

This note shows that the block sum transformation, a method intended as an alternative to the discrete Fourier transform (DFT), is equivalent to an existing algorithm.

Universidade de Aveiro
The home page of Paulo J. S. G. Ferreira
Overview Research Birds Contact	DNA sequences Toeplitz matrices Shannon sampling Sensor networks and precision agriculture Superoscillations Maxwell equations Real and DFT codes Uncertainty-based and fuzzy information Frames, codes and reconstruction Sorting and the analog median filter Missing samples Nonlinear systems Approximation and coding DFT Miscellaneous articles related to teaching Copyrights Discrete Fourier transform The subject of the papers in this section is the discrete Fourier transform. The first paper describes a set of permutations that commute with the discrete Fourier transform. In other words, let x be a signal, and X its DFT. The permutations described in the paper have the following property: if x is permuted, the DFT of the permuted signal can be obtained by permuting the DFT X in a similar way. P. J. S. G. Ferreira. A group of permutations that commute with the discrete Fourier transform. IEEE Transactions on Signal Processing, vol. 42, no. 2, pp. 444-445, Feb. 1994. This paper characterizes a potentially useful set of permutation matrices that commute with the Fourier matrix. The set of all such permutation matrices is a group under matrix multiplication, and every element of the group is its own inverse. The number of these permutations as a function of the Fourier matrix order is studied. It is shown that it is a multiplicative function of the Fourier matrix size N. How many such permutations exist? Express N as a product of primes, 2^k0 p₁^k1 p₂^k2... p_r^kr The number of permutations is 2^r, if k₀ is zero or one, or 2^r+1, if k₀ is two, or 2^r+2, if k₀ is greater than or equal to three. P. J. S. G. Ferreira. A note on the block sum transformation. Mechanical Systems and Signal Processing, vol. 7, no. 2, pp. 191-192, Mar. 1993. This note shows that the block sum transformation, a method intended as an alternative to the discrete Fourier transform (DFT), is equivalent to an existing algorithm.
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