Introduction
This webpage provides resources and experimental results for the accompanying research paper
P. Balazs, M. Dörfler, N. Holighaus, F. Jaillet and G. Velasco, “Theory, Implementation and Application of Nonstationary Gabor Frames," Preprint
Abstract. Signal analysis with classical Gabor frames leads to a fixed time-frequency resolution over the whole time-frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time-frequency resolution changing over time or frequency. We describe the construction of the resulting nonstation- ary Gabor frames and give the explicit formula for the canonical dual frame for a particular case, the painless case. We show that wavelet transforms, constant-Q transforms and more general filter banks may be modeled in the framework of nonstationary Gabor frames. Further, we present the results in the finite-dimensional case, which provides a method for implementing the above-mentioned transforms with perfect reconstruction. Finally, we elaborate on two applications of nonstationary Gabor frames in audio signal processing, namely a method for automatic adaptation to transients and an algorithm for an invertible constant-Q transform.
In the future, material accompanying related papers by the authors will be linked here:
Constructing an invertible constant-Q transform with Nonstationary Gabor frames
Download the NSGT toolbox here.
Nonstationary Gabor Toolbox V0.01
The onset detection routines use LTFAT's Gabor transform implementation, to use them LTFAT 0.97 or higher has to installed. LTFAT is an open-source MATLAB/octave toolbox for linear time-frequency analysis and available for download here: http://ltfat.sourceforge.net/
Experiments on nonstationary Gabor transforms
Time-side nonstationary Gabor transform
1. Onset-based scale frames
2. Sparse reconstruction experiments
Frequency-side nonstationary Gabor transform
1. Examples of constant-Q nonstationary Gabor transform spectrograms
Onset-based scale frames
Scale frames are nonstationary Gabor frames built from a single window prototype and its dilations. In the following examples, an onset detection function was used to adapt the signals at transient positions. The scale frames in all examples were built from a Hann window, using 8 scales with the shortest window being 192 samples long
Results and sound files for the remaining signals can be found here (For a detailed explanation of the onset detection parameters, see S. Dixon, “Onset detection revisited,” in Proceedings of DAFx-06, September 2006, pp. 133–137.):
|
spectrogram comparison |
sound file |
threshold |
range |
factor |
onset shift |
|
0.3 |
8 |
2 |
2 |
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|
0.3 |
8 |
2 |
2 |
||
|
0.2 |
8 |
2 |
2 |
||
|
0.3 |
8 |
3 |
2 |
||
|
0.2 |
6 |
2 |
2 |
||
|
0.3 |
8 |
1 |
2 |
To reproduce the plots above, run this script in MATLAB in the folder you saved the test signals and specify the signal name as parameter. (Requires LTFAT)
Sparse reconstruction experiments
We applied the sparse reconstruction procedure described in the Section 4.1.2 of the paper to our 7 test signals. The results are listed below. Additionally, we provide several sparse reconstructions using approximately a specified number of coefficients, using tight analysis and synthesis systems. The samples illustrate that the artifacts introduced are similar for all the transform procedures. As to be expected, a non-structured sparsity approach will, from a perceptive point of view, not yield a good reconstruction for any of them.
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Glockenspiel sparse reconstructions (by number of coefficients) |
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30000 coefficients |
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15000 coefficients |
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Glockenspiel sparse reconstructions (by RMS) |
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|
Transform |
1% |
2% |
5% |
10% |
Coefficients |
|
NSGT (192, 8) |
25366 |
14138 |
7142 |
3979 |
439488 |
|
GT (768, 1280) |
44274 |
28248 |
14655 |
7786 |
441600 |
|
GT (1536, 2560) |
48914 |
32475 |
16186 |
7582 |
448000 |
|
GT (3072, 5120) |
58048 |
39382 |
19386 |
7441 |
460800 |
Results for the remaining test signals: Ligeti, Hancock, Tufrial (results and sound examples), Nuriel, Zophiel, Rimmon (results only).
Examples of constant-Q nonstationary Gabor transform (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.
The following are additional signals as well as the corresponding CQ-NSGT spectrogram:
|
CQ-NSGT spectrogram |
sound file |
min frequency |
Bins per octave |
|
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 |
||
|
130 Hz |
48 |
||
|
130 Hz |
48 |
To produce plots comparing the presented CQ-NSGTs with ordinary Gabor transforms, run this script in MATLAB in the folder you saved the test signals and specify the signal name as parameter. (Requires LTFAT)
