signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...

, signal subspace methods are empirical linear methods for

dimensionality reduction Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally ...

and

noise reduction Noise reduction is the process of removing noise from a signal. Noise reduction techniques exist for audio and images. Noise reduction algorithms may distort the signal to some degree. Noise rejection is the ability of a circuit to isolate an u ...

. These approaches have attracted significant interest and investigation recently in the context of speech enhancement, speech modeling, and speech classification research. The signal subspace is also used in

radio direction finding Direction finding (DF), radio direction finding (RDF), or radiogoniometry is the use of radio waves to determine the direction to a radio source. The source may be a cooperating radio transmitter or may be an inadvertent source, a natural ...

using the MUSIC (algorithm). Essentially the methods represent the application of a

principal components analysis Principal component analysis (PCA) is a Linear map, linear dimensionality reduction technique with applications in exploratory data analysis, visualization and Data Preprocessing, data preprocessing. The data is linear map, linearly transformed ...

(PCA) approach to ensembles of observed time-series obtained by sampling, for example sampling an

audio Audio most commonly refers to sound, as it is transmitted in signal form. It may also refer to: Sound *Audio signal, an electrical representation of sound *Audio frequency, a frequency in the audio spectrum *Digital audio, representation of sound ...

signal. Such samples can be viewed as vectors in a high-

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

over the

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s. PCA is used to identify a set of orthogonal

basis vector In mathematics, a set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as ...

s (basis signals) which capture as much as possible of the energy in the ensemble of observed samples. The vector space spanned by the basis vectors identified by the analysis is then the ''signal subspace''. The underlying assumption is that information in speech signals is almost completely contained in a small

linear subspace In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping''); * linearity of a ''polynomial''. An example of a li ...

of the overall space of possible sample vectors, whereas additive noise is typically distributed through the larger space isotropically (for example when it is

white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used with this or similar meanings in many scientific and technical disciplines, i ...

). By projecting a sample on a signal subspace, that is, keeping only the component of the sample that is in the ''signal subspace'' defined by linear combinations of the first few most energized basis vectors, and throwing away the rest of the sample, which is in the remainder of the space orthogonal to this subspace, a certain amount of noise filtering is then obtained. Signal subspace noise-reduction can be compared to Wiener filter methods. There are two main differences: * The basis signals used in Wiener filtering are usually harmonic sine waves, into which a signal can be decomposed by

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

. In contrast, the basis signals used to construct the signal subspace are identified empirically, and may for example be

chirp A chirp is a signal in which the frequency increases (''up-chirp'') or decreases (''down-chirp'') with time. In some sources, the term ''chirp'' is used interchangeably with sweep signal. It is commonly applied to sonar, radar, and laser syste ...

s, or particular characteristic shapes of transients after particular triggering events, rather than pure sinusoids. * The Wiener filter grades smoothly between linear components that are dominated by signal, and linear components that are dominated by noise. The noise components are filtered out, but not quite completely; the signal components are retained, but not quite completely; and there is a transition zone which is partly accepted. In contrast, the signal subspace approach represents a sharp cut-off: an orthogonal component either lies within the signal subspace, in which case it is 100% accepted, or orthogonal to it, in which case it is 100% rejected. This reduction in dimensionality, abstracting the signal into a much shorter vector, can be a particularly desired feature of the method. In the simplest case signal subspace methods assume white noise, but extensions of the approach to colored noise removal and the evaluation of the subspace-based speech enhancement for robust

speech recognition Speech recognition is an interdisciplinary subfield of computer science and computational linguistics that develops methodologies and technologies that enable the recognition and translation of spoken language into text by computers. It is also ...

have also been reported.

References

* {{DEFAULTSORT:Signal Subspace Signal processing Noise reduction