Erratum to "On discrete Gauss-Hermite functions and eigenvectors of the discrete Fourier transform" [Signal Processing 88 (11) (2008) 2738-2746]

1. Page 2741, caption of Fig. 2: three occurrences of ‘‘commutator’’ instead of ‘‘commutor’’, i.e. the caption should read Fig. 2. Eigenvalue decomposition of the commutor for the CDFT: (a) sorted eigenvalues of the commutor T1 for N 1⁄4 128. (b) Selected symmetric and skew-symmetric eigenvectors of the commutor for k 1⁄4 0,1, 2, 3. 2. Page 2743, fourth sentence after Eq. (14): ‘‘commutator’’ instead of ‘‘commutor’’, i.e. the sentence should read In addition the eigenvalue spectrum on the commutor exhibits deviations from the eigenvalue spectrum seen in the case of the CDFT at two regions as seen in Fig. 4(a) for the choice of a 1⁄4 a1 1⁄4 0, due to the eigenvalue degeneracy. 3. Page 2743, paragraph following Eq. (19): three occurrences of ‘‘commutator’’ instead of ‘‘commutor’’, i.e. the paragraph should read commutes with the DFT matrix. The eigenvalues of the commutor are exactly the same as those seen for the CDFT as seen in Fig. 4(c). The eigenvectors of this commutor are K-symmetric or K-anti-symmetric with K 1⁄4 W as described in the Appendix, where we outline an algorithm to construct K-symmetric and K-anti-symmetric eigenvectors from the even and odd eigen-subspaces of a generalized K-symmetric matrix. The symmetric involution matrix K used in the formulation of the K-symmetric matrix framework for computing the eigenvectors of the commutor can be any cyclic permutation matrix that commutes with the DFT and does not need to be W [17]. 4. Page 2744, caption of Fig. 4: two occurrences of ‘‘commutator’’ instead of ‘‘commutor’’, i.e. the caption should read Fig. 4. Eigenvalue spectrum of the commutor T̃1 for the DFT: (a) symmetric difference of the eigenvalue spectrum of the commutor depicting eigenvalue degeneracy, (b) symmetric difference of the eigenvalue spectrum for N 1⁄4 128 using a2 for the asymmetric vector, (c) the corresponding, symmetric difference spectrum when just Eq. (6) is satisfied. 5. Page 2745, caption of Fig. 5: ‘‘commutator’’ instead of ‘‘commutor’’, i.e. the caption should read Fig. 5. Eigenvalue decomposition of T̃2: symmetric difference of the eigenvalue spectrum of the generalized commutor for N 1⁄4 128. The observed symmetric difference of two for a significant portion of the spectrum is indicative of the uniform integer spacing between eigenvalues—a unique feature of the proposed approach.