Introduction
This webpage provides resources and experimental results for the accompanying research paper
M. Dörfler, N. Holighaus, T. Grill and G. Velasco, “ Constructing an Invertible Constant-Q Transform with Nonstationary Gabor Frames," to appear in Proceedings of the 14th International Conference on Digital Audio Effects (DAFx 11), Paris, France, 2011
Abstract. An efficient and perfectly invertible signal transform featuring a constant-Q frequency resolution is presented. The proposed approach is based on the idea of the recently introduced nonstationary Gabor frames. Exploiting the properties of the operator corresponding to a family of analysis atoms, this approach overcomes the problems of the classical implementations of constant-Q transforms, in particular, computational intensity and lack of invertibility. Perfect reconstruction is guaranteed by using an easy to calculate dual system in the synthesis step and computation time is kept low by applying FFT-based processing. The proposed method is applied to real-life signals and evaluated in comparison to a related approach, recently introduced specifically for audio signals.
More material on nonstationary Gabor transforms:
Theory and Implementation of nonstationary Gabor Frames
Download the NSGT toolbox here.
Nonstationary Gabor Toolbox V0.01
Experiments on constant-Q nonstationary Gabor transform (CQ-NSGT):
1. Examples of CQ-NSGT spectrograms
2. Masking experiments
3. Transposition experiments
Examples of CQ-NSGT spectrograms
The frequency-side version of the nonstationary Gabor transform allows one to construct a perfectly invertible constant-Q transform. Below are examples of CQ-NSGT spectrograms of the Glockenspiel signal. The minimum frequency for both representations is set set at 200 Hz, while the bins per octave for the two representations are set at 12 and 48, respectively.
For comparision, below are spectrograms of the Glockenspiel signal: a standard Gabor spectrogram obtained by using a Hann window of length 1024 samples with a hop-size of 512, and a constant-Q transform spectrogram (from the original formulation of J. Brown) with minimum frequency set at 200 Hz and 48 bins per octave.
The following are additional signals as well as the corresponding CQ-NSGT spectrogram:
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CQ-NSGT spectrogram |
sound file |
min frequency |
Bins per octave |
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130 Hz |
48 |
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130 Hz |
48 |
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130 Hz |
48 |
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130 Hz |
48 |
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130 Hz |
48 |
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130 Hz |
48 |
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130 Hz |
48 |
Masking experiment
The perfect reconstruction property of CQ-NSGT can be used to cut out components from a signal by directly modifying the time-frequency coefficients. The example shows a mask for extracting a portion from the Glockenspiel signal, along with the masked spectrogram, as well as the spectrograms of the synthesized, processed signal and remainder.
Results
of the masking experiment: Original
signal, Processed
signal, Remainder
Transposition experiment
It is possible to use the invertibility of the CQ-NSGT to easily transpose a complete harmonic structure by translation along the frequency bins. We illustrate this by transposing a chord, played on a piano. Note that the onset has been damped since transient events have to be handled separately to avoid phasing effects.
Sound
files: Original signal, 2
semitone transpose, 5
semitone transpose
