In recent work, redressed warped frames have been introduced for the analysis and synthesis of audio signals with nonuniform frequency and time resolutions. In these frames, the allocation of frequency bands or time intervals of the elements of the representation can be uniquely described by means of a warping map. Inverse warping applied after time-frequency sampling provides the key to reduce or eliminate dispersion of the warped frame elements in the conjugate variable, making it possible, e.g., to construct frequency warped frames with synchronous time alignment through frequency. The redressing procedure is however exact only when the analysis and synthesis windows have compact support in the domain where warping is applied. This implies that frequency warped frames cannot have compact support in the time domain. This property is undesirable when online computation is required. Approximations in which the time support is finite are however possible, which lead to small reconstruction errors. In this paper we study the approximation error for compactly supported frequency warped analysis-synthesis elements, providing a few examples and case studies.

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Time and frequency warping provide effective methods for fitting signal representations to desired physical or psychoacoustic characteristics. However, warping in one of the variables, e.g. frequency, disrupts the organization of the representation with respect to the conjugate variable, e.g. time. In recent papers we have considered methods to eliminate or mitigate the dispersion introduced by warping in time frequency representations and Gabor frames. To this purpose, we introduced redressing methods consisting in further warping with respect to the transformed variables. These methods proved not only useful for the visualization of the transform but also to simplify the computation of the transform in terms of shifted precomputed warped elements, without the need for warping in the computation of the transform. In other linear representations, such as time-scale, warping generally modifies the transform operators, making visualization less informative and computation more difficult. Sound signal representations almost invariably need time as one of the coordinates in view of the fact that we normally wish to follow the time evolution of features and characteristics. In this paper we devise methods for the redressing of dispersion introduced by warping in wavelet transforms and in other expansions where time-shift plays a role.

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Time warping is an important paradigm in sound processing, which consists of composing the signal with another function of time called the warping map. This paradigm leads to different points of view in signal processing, fostering the development of new effects or the conception of new implementations of existing ones. While the introduction of time warping in continuous-time signals is in principle not problematic, time warping of discretetime signals is not self-evident. On one hand, if the signal samples were obtained by sampling a bandlimited signal, the warped signal is not necessarily bandlimited: it has a sampling theorem of its own, based on irregular sampling, unless the map is linear. On the other hand, most signals are regularly sampled so that the samples at non-integer multiples of the sampling interval are not known. While the use of interpolation can partly solve the problem it usually introduces artifacts. Moreover, in many sound applications, the computation already involves a phase vocoder. In this paper we introduce new methods and algorithms for time-warping based on warped time-frequency representations. These lead to alternative algorithms for warping for use in sound processing tools and digital audio effects and shed new light in the interaction of time warping with phase vocoders. We also outline the applications of time warping in digital audio effects.

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