Real-time time-varying frequency warping via short-time Laguerre transform
In this paper we address the problem of the real-time implementation of time-varying frequency warping. Frequency warping based on a one-parameter family of one-to-one warping maps can be realized by means of the Laguerre transform and implemented in a non-causal structure. This structure is not directly suited for real-time implementation since each output sample is formed by combining all of the input samples. Similarly, the recently proposed time-varying Laguerre transform has the same drawback. Furthermore, long frequency dependent delays destroy the time organization or macrostructure of the sound event. Recently, the author has introduced the Short-Time Laguerre Transform for the approximate real-time implementation of frequency warping. In this transform the short-time spectrum rather than the overall frequency spectrum is frequency warped. The input is subdivided into frames that are tapered by a suitably selected window. By careful design, the output frames correspond to warped versions of the input frames modulated by a stretched version of the window. It is then possible to overlap-add these frames without introducing audible distortion. The overlap-add technique can be generalized to time-varying warping. However, several issues concerning the design of the window and the selection of the overlap parameters need to be addressed. In this paper we discuss solutions for the overlap of the frames when the Laguerre parameter is kept constant but distinct in each frame and solutions for the computation of full time-varying frequency warping when the Laguerre parameter is changing within each frame.