### Short-Term Fourier Analysis, Modification, and Resynthesis

The Fourier duality of the overlap-add and filter-bank-summation short-time Fourier transform (discussed in Chapter 9) appeared in the late 1970s [7,9]. This unification of downsampled filter-banks and FFT processors spawned considerable literature in STFT processing [158,8,219,192,98,191]. While the phase vocoder is normally regarded as a fixed bandpass filter bank. The STFT, in contrast, is usually regarded as a time-ordered sequence of overlapping FFTs (the ``overlap-add'' interpretation). Generally speaking, sound reconstruction by STFT during this period was nonparametric. A relatively exotic example was signal reconstruction from STFT magnitude data (*magnitude-only reconstruction*) [103,192,219,20].

In the speech-modeling world, parametric sinusoidal modeling of the STFT apparently originated in the context of the magnitude-only reconstruction problem [221]. Since the phase vocoder was in use for measuring amplitude and frequency envelopes for additive synthesis no later than 1977,

^{G.12}it is natural to expect that parametric ``inverse FFT synthesis'' from sinusoidal parameters would have begun by that time. Instead, however, traditional banks of sinusoidal (and more general wavetable) oscillators remained in wide use for many more years. Inverse FFT synthesis of sound was apparently first published in 1980 [35]. Thus, parametric reductions of STFT data (in the form of instantaneous amplitude and frequency envelopes of vocoder filter-channel data) were in use in the 1970s, but we were not yet resynthesizing sound by STFT using spectral buffers synthesized from parameters.

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Sinusoidal Modeling Systems

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Further Reading on Vocoders