International Audio Laboratories Erlangen

Vandermonde Tools

Introduction

This software package provides implementations of the most central functions required for using the Vandermonde transform, including Vandermonde factorization of symmetric Toeplitz matrices. For an introduction about the Vandermonde transform, see [1], implementation details are available in [2,3,4,5] and some applications can be found in [6,7].

Contents

The package contains the following functions

• Vandermonde factorization using FFT [2,5] or the approach from Boley [3].
• Forward and backward transforms, that is, multiplication of a vector with the Vandermonde matrix, its inverse, Hermitian or inverse Hermitian [3].
• Leja-ordering for numerical stabilization (pre-conditioning) of the matrix inversions [3,4,5].

The sub-functions include

• Refinement of polynomial roots with the Durand-Kerner and Aberth-Ehrlich methods [2].
• Adaptation of the Levinson-Durbin algorithm for the case where the samples of the right hand side vector are all ones.

Read the README.TXT for usage and syntax.

Download

The MATLAB implementation package is available here: Download

References

1. Bäckström, Tom
   Vandermonde Factorization of Toeplitz Matrices and Applications in Filtering and Warping
   IEEE Transactions on Signal Processing, 61(24): 6257–6263, 2013. Bibtex

   @article{backstrom2013vandermonde,
   title={Vandermonde Factorization of Toeplitz Matrices and Applications in Filtering and Warping},
   author={B{\"a}ckstr{\"o}m, Tom},
   journal={IEEE Transactions on Signal Processing},
   volume={61},
   number={24},
   pages={6257--6263},
   year={2013},
   publisher={IEEE}
   }

2. Bäckström, T. and Fischer~Pedersen, C.
   Superfast Vandermonde Factorization of Autocorrelation Matrices of Finite Length Signals
   submitted to IEEE Signal Processing Letters, 2014. Bibtex

   @article{backstrom2014superfast,
   title={Superfast Vandermonde Factorization of Autocorrelation
   Matrices of Finite Length Signals},
   author={Bäckström, T. and Fischer~Pedersen, C.},
   journal={submitted to IEEE Signal Processing Letters},
   year=2014,
   }

3. T. Bäckström, J. Fischer, and D. Boley
   Implementation and Evaluation of the Vandermonde Transform
   In Proc EUSIPCO 2014, 2014. Bibtex

   @inproceedings{backstrom2014implementation,
   AUTHOR={T. Bäckström and J. Fischer and D. Boley},
   TITLE={Implementation and Evaluation of the {Vandermonde} Transform},
   BOOKTITLE={Proc EUSIPCO 2014},
   ADDRESS="Lisbon, Portugal",
   MONTH=sep,
   YEAR=2014,
   KEYWORDS="time-frequency transforms, decorrelation, Vandermonde matrix, Toeplitz
   matrix, speech and audio processing",
   ABSTRACT="We have recently presented a time-frequency transform, the Vandermonde
   transform, which simultaneously decorrelates the input signal and provides
   a frequency-domain representation thereof. While the discrete Fourier
   transform also provides a frequency-domain representation, its
   decorrelating performance is only approximate. Although this is good enough
   for many applications, in applications which require relatively short
   windows, such as speech and audio processing, the penalty due to the
   approximate decorrelation limits overall performance. This paper focuses on
   the practical implementation of the Vandermonde transform. Our purpose is
   to provide a short review of state-of-the-art methods, a step-by-step
   instructions for implementation and an evaluation of the performance."
   }

4. Pedersen, C. F.
   Leja ordering LSFs for accurate estimation of predictor coefficients
   In International Conference on Spoken Language Processing (ICSLP), 2011. Bibtex

   @inproceedings{pedersen2011leja,
   author = {Pedersen, C. F.},
   title = {Leja ordering {LSFs} for accurate estimation of predictor coefficients},
   booktitle = {International Conference on Spoken Language Processing (ICSLP)},
   year = {2011},
   address = {Florence, Italy},
   publisher = {{ISCA}}
   }

5. Bäckström, T., Fischer~Pedersen, C., Fischer, J., and Pietrzyk, G.
   Finding line spectral frequencies using the Fast Fourier Transform
   In Proc. ICASSP, 2015. Bibtex

   @inproceedings{backstrom2015fftlsf,
   author = {Bäckström, T. and Fischer~Pedersen, C. and Fischer, J. and Pietrzyk, G.},
   title = {Finding line spectral frequencies using the Fast Fourier Transform},
   booktitle = {Proc. ICASSP},
   year = {2015},
   month = apr,
   }

6. Bäckström, Tom and Helmrich, Christian R.
   Decorrelated Innovative Codebooks for ACELP Using Factorization of Autocorrelation Matrix
   In Proc Interspeech, 2014. Bibtex

   @inproceedings{backstrom2014decorrelated,
   title={Decorrelated Innovative Codebooks for {ACELP} Using Factorization of Autocorrelation Matrix},
   author={B{\"a}ckstr{\"o}m, Tom and Helmrich, Christian R.},
   booktitle={Proc Interspeech},
   year={2014}
   }

7. Pedersen, Christian Fischer and Bäckström, Tom
   Sparse Time-Frequency Representation of Speech by the Vandermonde Transform
   In Proc Interspeech, 2014. Bibtex

   @inproceedings{pedersen2014sparse,
   title={Sparse Time-Frequency Representation of Speech by the {Vandermonde} Transform},
   author={Pedersen, Christian Fischer and B{\"a}ckstr{\"o}m, Tom},
   booktitle={Proc Interspeech},
   year={2014}
   }

Contact

For more information contact Tom Bäckström.