Signal separation

The classical example of a source separation problem is the cocktail party problem, where a number of people are talking simultaneously in a room (for example, at a cocktail party), and a listener is trying to follow one of the discussions.

This problem is in general highly underdetermined, but useful solutions can be derived under a surprising variety of conditions.

Much of the early literature in this field focuses on the separation of temporal signals such as audio.

However, blind signal separation is now routinely performed on multidimensional data, such as images and tensors, which may involve no time dimension whatsoever.

Some of the more successful approaches are principal components analysis and independent component analysis, which work well when there are no delays or echoes present; that is, the problem is simplified a great deal.

In human perception this ability is commonly referred to as auditory scene analysis or the cocktail party effect.

You have multiple microphones picking up mixed signals, but you want to isolate the speech of a single person.

BSS can be used to separate the individual sources by using mixed signals.

The separated images, were separated using Python and the Shogun toolbox using Joint Approximation Diagonalization of Eigen-matrices (JADE) algorithm which is based on independent component analysis, ICA.

[1] This toolbox method can be used with multi-dimensions but for an easy visual aspect images(2-D) were used.

One of the practical applications being researched in this area is medical imaging of the brain with magnetoencephalography (MEG).

However, external sources of electromagnetic fields, such as a wristwatch on the subject's arm, will significantly degrade the accuracy of the measurement.

BSS, however, can be used to separate the two so an accurate representation of brain activity may be achieved.

For a stereo mix of relatively simple signals it is now possible to make a fairly accurate separation, although some artifacts remain.

, then the system of equations is overdetermined and thus can be unmixed using a conventional linear method.

, the system is underdetermined and a non-linear method must be employed to recover the unmixed signals.

Since the chief difficulty of the problem is its underdetermination, methods for blind source separation generally seek to narrow the set of possible solutions in a way that is unlikely to exclude the desired solution.

In one approach, exemplified by principal and independent component analysis, one seeks source signals that are minimally correlated or maximally independent in a probabilistic or information-theoretic sense.

A second approach, exemplified by nonnegative matrix factorization, is to impose structural constraints on the source signals.

These structural constraints may be derived from a generative model of the signal, but are more commonly heuristics justified by good empirical performance.

This approach can be particularly effective if one requires not the whole signal, but merely its most salient features.