A note on `Notch Fourier transform´ [by Y. Tadokodo and K. Abe]

In the above-titled paper (see ibid., vol.ASSP-35, p.1282, Sept. 1987), Tadokodo and Abe presented a clever method to compute the Fourier coefficients at arbitrary frequencies. But their proof of (16) is incomplete, since in their proof the number of sampled data N is assumed to be power of two (i.e., N=2^q). However, N=2r, where r is the number of frequency components to be analyzed, is not a necessary power of two even in the discrete Fourier transform (DFT) case. The note gives a complete proof on their (16) without any restrictions on N