By J. H Poynting and J. J. Thomson
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Additional info for A text-book of physics: Sound (Fourth Edition)
The method programmed by Maher (1989) for use in the SNDAN analysis/synthesis package (Beauchamp, 1993) convolves a Hamming-windowed sinc function with the input signal, and the upsampled result is linearly interpolated. For the DFT requirement, the substitution m ← m + n − N /2 is made in Eq. 17c), resulting in fs = c˜k (n) = e jπ k P(1−2n/N ) P N N −1 w (N /2 − m)s(m + n − N /2)e− j2π k Pm/N . 19a) m=0 With P = 1, the summation of Eq. 19a) is in the correct form for the DFT. However, with P > 1, Eq.
Another distinct method uses the cepstrum, which seeks to determine the periodicity of the DFT magnitude by taking its log and then applying a second FFT. This has been used extensively in speech applications (Noll, 1967) but seldom for music [for an exception, see Chen (2001)]. All of these methods rely on the selection of a minimum or maximum of a function. Determining which of several maxima or minima correspond to the correct fundamental frequency turns out to be the biggest problem in making these methods reliable.
27) becomes 8 θk (0). 29b) H where i is the frame number, running from i = 0 to i = I − 1 (there are I frames). This results in I equations with I + 2 unknowns. 29d) H and a similar equation obtains for i = I − 1. Therefore, there are now I equations with I unknowns, and these can be readily solved by a process of elimination. 8 f k1 = Analysis and Synthesis of Musical Instrument Sounds 23 The same process can be applied to interpolation of the amplitude values. The quadratic interpolation method works very well.
A text-book of physics: Sound (Fourth Edition) by J. H Poynting and J. J. Thomson