Terhardt, E. (1985). Fourier transformation of time signals - Conceptual revision. Acustica 57, 242-256

With regard to spectral representation of audio-signals such as speech and music, Fourier transformation is reconsidered in the framework of a simple concept of parametric signal representation. In the formula which specifies the complex coefficients of a Fourier series, the integration interval T is regarded as an analysis interval instead of the signal period. Thereby that formula gets the status of a basic and universal transformation which specifies the so-called Fourier-amplitude spectrum of that portion of a time signal which falls into the analysis interval. The length T of the analysis interval, and its position on the time scale, denoted by its terminating instant t, are parameters of that transformation. While for the purpose of spectrum analysis both a finite and an infinite analysis interval T may be appropriate, for application in linear systems theory T must begin at t = -infinity and may end at a finite instant t, such that T is infinite. With a finite T, the Fourier transformation is discrete in the sense that the spectrum of a continuous signal is significant only at discrete frequencies. With an infinite T, it bears information at any frequency and thus is continuous. When the spectrum is normalized to 2/T, the so-called amplitude-density spectrum is achieved which also is either discrete or continuous, depending on whether T is finite or infinite. When causality of signals and systems is presumed, the analysis interval's terminating instant t does in no case need to be infinite. One consequence thereof is, that with one-sided signals the Fourier integral's limits both are finite, such that no convergence problems do exist. Another consequence is, that with one-sided signals which are analytically specified in any observation interval of time, the integral's upper limit can be ignored and a reduced Fourier transform (RFT) is achieved which fulfills the same purposes as Laplace transform. Finally, the transformation parameter t can be made continuously gliding, such that in every instant a spectrum is obtained which pertains to another parameter t, yet fully meets the requirements of linear systems theory. That type of transformation, which thereby gets theoretical justification, is called Fourier-t-transformation (FTT). It appears most adequate for reflecting essential characteristics of "natural" spectrum analyzers such as the human ear. An efficient method for digital computation of FTT, particularly for audio applications, is outlined.

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