A noniterative method for the reconstruction of the short-time fourier transform (STFT) phase from the magnitude is presented. The method is based on the direct relationship between the partial derivatives of the phase and the logarithm of the magnitude of the un-sampled STFT with respect to the Gaussian window. Although the theory holds in the continuous setting only, the experiments show that the algorithm performs well even in the discretized setting (discrete Gabor transform) with low redundancy using the sampled Gaussian window, the truncated Gaussian window and even other compactly supported windows such as the Hann window. Due to the noniterative nature, the algorithm is very fast and it is suitable for long audio signals. Moreover, solutions of iterative phase reconstruction algorithms can be improved considerably by initializing them with the phase estimate provided by the present algorithm. We present an extensive comparison with the state-of-the-art algorithms in a reproducible manner.

The discrete wavelet transform module is a recent addition to the Large Time-Frequency Analysis Toolbox (LTFAT). It provides implementations of various generalizations of Mallat's well-known algorithm (iterated filterbank) such that completely general filterbank trees, dual-tree complex wavelet transforms, and wavelet packets can be computed. The resulting transforms can be equivalently represented as filterbanks and analyzed as filterbank frames using fast algorithms.

Envelope representations such as the auditory or traditional spectrogram can be defined by the set of envelopes from the outputs of a filterbank. Common envelope extraction methods discard information regarding the fast fluctuations, or phase, of the signal. Thus, it is difficult to invert, or reconstruct a time-domain signal from, an arbitrary envelope representation. To address this problem, a general optimization approach in the time domain is proposed here, which iteratively minimizes the distance between a target envelope representation and that of a reconstructed time-domain signal. Two implementations of this framework are presented for auditory spectrograms, where the filterbank is based on the behavior of the basilar membrane and envelope extraction is modeled on the response of inner hair cells. One implementation is direct while the other is a two-stage approach that is computationally simpler. While both can accurately invert an auditory spectrogram, the two-stage approach performs better on time-domain metrics. The same framework is applied to traditional spectrograms based on the magnitude of the short-time Fourier transform. Inspired by human perception of loudness, a modification to the framework is proposed, which leads to a more accurate inversion of traditional spectrograms.

This paper describes a method for obtaining a perceptually motivated and perfectly invertible time-frequency representation of a sound signal. Based on frame theory and the recent non-stationary Gabor transform, a linear representation with resolution evolving across frequency is formulated and implemented as a non-uniform filterbank. To match the human auditory time-frequency resolution, the transform uses Gaussian windows equidistantly spaced on the psychoacoustic “ERB” frequency scale. Additionally, the transform features adaptable resolution and redundancy. Simulations showed that perfect reconstruction can be achieved using fast iterative methods and preconditioning even using one filter per ERB and a very low redundancy (1.08). Comparison with a linear gammatone filterbank showed that the ERBlet approximates well the auditory time-frequency resolution.