# Blind Signal Processing

Within the wide Digital Signal Processing (DSP) world, a relevant role is played by Adaptive DSP, devoted to those solutions where a structure changing its characteristics in response to specific stimuli and according to certain rules occur. In this field a relevant distinction has to be made, that is between supervised and unsupervised methods employed to carry out the adaptation of the structure chosen for solving the task under study. Usually, the latter seems to be more realistic but more difficult, and sometimes impractical if we do not receive any help from the learning environment. Along this direction, “Blind Signal Processing (BSP) is now one of the hottest and emerging areas in Signal Processing with solid theoretical foundations and many potential applications”, in several areas like biomedical engineering, speech enhancement, communication systems, data mining, and so on. Furthermore, BSP “techniques principally do not use any training data and do not assume any a-priori knowledge about parameters of convolutive, filtering and mixing systems”. Many researchers are involved in this field and lots of scientific contributions have been proposed in the last 10-15 years. Nevertheless the research activity is very alive and new topics and algorithms are continuously investigated resulting in more and more florid literature.

One of the most important BSP problems is **Blind Source Separation (BSS)**, which can be encountered in various areas, such as acoustics, radio, medical signal and image processing. To effectively describe it, let us consider a situation in which some physical sources, located in different positions, are emitting some signals. For example, the sources could be different people speaking in the same room (“cocktail party problem”) – in which case the signals are speech signals – or different areas of the brain, emitting electrical signals. Furthermore suppose that there are some physical sensors used to record the underlying signals (e.g. microphones, or other proper receivers). Now, each sensor is not able to record just a single source signal, but it “captures” a combination (or mixture) of all the sources present in the environment, and each source signal is weighted by a different coefficient, depending upon the distance from the emitter. Given the previous picture, the blind source separation problem can be stated as follows: given only the amplitudes of all the mixtures at every time instant, recover the original source signals when the mixing model is unknown.

A3LAB has put many efforts over the past years in this field, and the following related sub-topics can be taken here into consideration:

**Instantaneous BSS**. If we neglect the time delays and suppose there are no echoes or reverberations, the generic mixture value at the certain time instant will be a linear combination (with real coefficients) of the source amplitudes at the same time instant; therefore a matrix can be built that expresses the analytic dependence of all the mixture signals on the source signals. An interesting related case study is represented by the underdetermined BSS problem, that it occurs when the number of sources is greater than the number of observables: significant contributions have been proposed by the A3LAB guys in their research work.**Nonlinear BSS**. A more challenging but complex situation w.r.t. the previous one is obtained considering nonlinear mixing models. Indeed it can be shown that for these kinds of model, also under the only assumption of statistically independent source signals, the original signals can be recovered only up to a nonlinear distortion, which is not allowable in most practical applications. For this reason only particular kinds of nonlinearities must be used. A particularly interesting nonlinear mixing model is the Post Non Linear (PNL) model, that can be used in all real situations where a system with a linear transmission channel and with sensors introducing nonlinear memoryless distortions (e.g.: saturation-like distortions) has to be modeled. The quadratic PNL BSS problem is separable under the only assumption of statistically independent source signals. A3LAB has successfully faced the underdetermined PNL mixing model with delays is studied, exploiting Gaussianization based approaches also in presence of delays in the mixing models.**Multichannel Blind Deconvolution (MBD)**. This is the problem of recovering the sources mixed by a convolutive system into a certain number of mixture signals. The deconvolution procedure has to be carried out without knowing the sources and the system. This also referred as Convolutive BSS problem. Several solutions have appeared in the literature so far (like those based on Independent component Analysis or Second-orde Statistics) and A3LAB work has been mainly oriented to apply such techniques to audio problems in realistic scenarios.**Natural Gradient based learning**. Gradient descent is a widely-used approach for unconstrained optimization, related to the problem under study. Due to the frequently slow convergence of gradient-descent, a relevant task is the improvement of the convergence speed of corresponding algorithms. The proposal of substituting the standard gradient with a new one, namely the natural gradient, goes towards this direction. This approach has been effectively applied to BSS and MBD problems in the literature, with significant element of innovations provided by the A3LAB group itself.

## Related pubblications

- Pignotti , A. , Marcozzi , D. , Cifani , S. , S. , Squartini , Piazza , F. (2009), "A Blind Source Separation Based Approach for Speech Enhancement in Noisy and Reverberant Environment.", Cross-Modal Analysis of Speech, Gestures, Gaze and Facial Expressions :356-367.
- Bastari , A. , Squartini , S. , Piazza , F. (2007), "Discrete Stockwell Transform and Reduced Redundancy Versions from Frame Theory Viewpoint", Circuits and Systems, 2007. ISCAS 2007. IEEE International Symposium on :2319-2322.
- Squartini , S. , Cecchi , S. , Moretti , E. , Piazza , F. (2007), "Overcomplete Blind Separation of Speech Sources in the Post Nonlinear Case Through Extended Gaussianization", Fundamentals of Verbal and Nonverbal Communication and the Biometric Issue:208.
- Tummarello , G. , Squartini , S. , Piazza , F. (2007), "An MPEG-7 Architecture with a Blind Signal Processing Based Front-End for Spoken Document Retrieval", Language:195--207.
- Squartini , S. , Arcangeli , A. , Piazza , F. (2007), "Stability Analysis of Natural Gradient Learning Rules in Complete ICA: A Unifying Perspective", Signal Processing Letters, IEEE,14:54-57.
- Squartini , S. , Piazza , F. , Theis , F.J. (2006), "New Riemannian metrics for speeding-up the convergence of over- and underdetermined ICA", Circuits and Systems, 2006. ISCAS 2006. Proceedings. 2006 IEEE International Symposium on :4 pp.-.
- Squartini , S. , Bastari , A. , Piazza , F. (2006), "A practical Approach Based on Gaussianization for Post-Nonlinear Underdetermined BSS", Communications, Circuits and Systems Proceedings, 2006 International Conference on ,1:528-532.
- Principi , E. , Squartini , S. , Piazza , F. (2005), "An ICA based approach for blind deconvolution of three-dimensional signals", Circuits and Systems, 2005. ISCAS 2005. IEEE International Symposium on : 5714-5717 Vol. 6.
- Squartini , S. , Piazza , F. , Shawker , A. (2005), "New Riemannian metrics for improvement of convergence speed in ICA based learning algorithms", Circuits and Systems, 2005. ISCAS 2005. IEEE International Symposium on : 3603-3606 Vol. 4.
- Bastari , A. , Squartini , S. , Piazza , F. (2005), "Underdetermined Blind Separation of Speech Signals with Delays in Different Time-Frequency Domains", Nonlinear speech modeling and applications: advanced lectures and revised selected papers:136.
- Pomponi , E. , Squartini , S. , Piazza , F. (2005), "Signal Sparsity Enhancement Through Wavelet Transforms in Underdetermined BSS", Lecture notes in computer science,3445:384.
- Tomassoni , M. , Squartini , S. , Piazza , F. (2005), "An alternative natural gradient approach for multichannel blind deconvolution", Circuits and Systems, 2005. ISCAS 2005. IEEE International Symposium on : 5742-5745 Vol. 6.
- Arcangeli , A. , Squartini , S. , Piazza , F. (), "An alternative natural gradient approach for ICA based learning algorithms in blind source separation", EUSIPCO 2004 .