Signal subspace
Encyclopedia
In signal processing
, signal subspace methods are empirical linear methods for dimensionality reduction and noise reduction
. These approaches have attracted significant interest and investigation recently in the context of speech enhancement, speech modeling, and speech classification research.
Essentially the methods represent the application of a principal components analysis
(PCA) approach to ensembles of observed time-series obtained by sampling
, for example sampling an audio
signal. Such samples can be viewed as vectors in a high-dimension
al vector space
over the real number
s. PCA is used to identify a set of orthogonal basis vectors (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
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
).
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:
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 have also been reported.
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, signal subspace methods are empirical linear methods for dimensionality reduction and noise reduction
Noise reduction
Noise reduction is the process of removing noise from a signal.All recording devices, both analogue or digital, have traits which make them susceptible to noise...
. These approaches have attracted significant interest and investigation recently in the context of speech enhancement, speech modeling, and speech classification research.
Essentially the methods represent the application of a principal components analysis
Principal components analysis
Principal component analysis is a mathematical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of uncorrelated variables called principal components. The number of principal components is less than or equal to...
(PCA) approach to ensembles of observed time-series obtained by sampling
Sampling (signal processing)
In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave to a sequence of samples ....
, for example sampling an audio
Sound
Sound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...
signal. Such samples can be viewed as vectors in a high-dimension
Dimension
In physics and mathematics, the dimension of a 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 because only one coordinate is needed to specify a point on it...
al vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. PCA is used to identify a set of orthogonal basis vectors (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
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
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
White noise
White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
).
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
Wiener filter
In signal processing, the Wiener filter is a filter proposed by Norbert Wiener during the 1940s and published in 1949. Its purpose is to reduce the amount of noise present in a signal by comparison with an estimation of the desired noiseless signal. The discrete-time equivalent of Wiener's work was...
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 transformFourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
. In contrast, the basis signals used to construct the signal subspace are identified empirically, and may for example be chirpChirpA chirp is a signal in which the frequency increases or decreases with time. In some sources, the term chirp is used interchangeably with sweep signal. It is commonly used in sonar and radar, but has other applications, such as in spread spectrum communications...
s, or particular characteristic shapes of transients after particular triggering events, rather than pure sinusoids. - The Wiener filter grades smoothlySmooth functionIn mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
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 have also been reported.