Blind signal separation concepts involved
1, the blind signal separation means is observed from the plurality of mixed signals to the original signal analysis is not observed. Typically observed mixed signal output from a plurality of sensors, the independence and the output signal (sensor linearly independent ). "Blind" Blind Signal word emphasized two points: 1) does not know the original signal; 2) the method does not know the mixed signal. In most studies, only we discuss linear mixed model, when the hybrid model is non-linear, are generally unable to recover the source signals from the mixed data, unless there is further a priori knowledge of the signal and hybrid models Keziliyong.
2, the algorithm has made the following assumptions:
- Having
independent sources
and
independent observations
, and observations source has the following relationship
meaning equation is m n-dimensional source signal obtained by mixing the vector of observed data. Wherein
, the output of which the respective sensor elements is obtained;
,
is a
coefficient matrix, which indicates the mixing elements of the signal, the original problem becomes known
and
independence on demand
- Assuming as blind signal separation problem formulation is: the mixing matrix A and the unknown source signals, only the separation matrix W is determined from the observed data vector x (t), such that the transformed output y (t) = Wx ( t) is the source signal vector s (t) is estimated copy or, W is a
coefficient matrix, the problem becomes how to make an estimate of the matrix W.
3, the algorithm involves basic assumptions:
- Each source signals
are zero-mean signal, real random variables, statistical independence between the signals. If the source signal
, the
probability density is:
- The number of source signal
. Mixing matrix
matrix. It assumed
- Signal source allows only a Gaussian distribution, when more than a Gaussian distribution, the source signal becomes inseparable.
4, there are two of uncertainty or ambiguity blind signal separation: order and uncertainty signal complex amplitude (amplitude and the initial phase) separated uncertainty mainly for blind signal separation of non-hybrid matrix A. complete identification.
5, the definition of two matrices M 1 and N equal is called nature, and recorded as 
N, such that if there is a matrix G 
, where G is a generalized switching matrix, and which elements have unit norm. The definition of a blind signal separation problem can be described as: the nature of the mixing matrix A only the identification matrix is equal to / or recover the source signals from the sensor output x (t).
6, [1] proved separability blind signal: for each element independently of each other, and only one Gaussian signal vector s, if the y = Cs (where C is an arbitrary invertible matrix) mutually independent elements , then y is a copy of s.
7, signal components si (t), normalized Kurtosis is defined 
, for a Gaussian signal, the normalized kurtosis is equal to zero if. 
> 0, called si (t) is a super-Gaussian signal; if 
<0, then si (t) is the sub-Gaussian signal.
8, 2 is defined so that A = A (x) is a batch mixing matrix A of the estimator. If for any invertible matrix M has a constant A (Mx) = MA (x ) is called estimators A like change, and 
referred to changes in conditions and other conditions change another expression: Let W (t) is a blind source separation algorithm obtained separating matrix, and C (t) = W (t ) a is separated from the mixed synthesis system, the algorithm and the like is changed, if C (t) satisfies
, Where H (C (t) s (t)) is the matrix product C (t) s (t) is a function matrix, which is independent of the mixing matrix A and the separation matrix W (t).
9, assume that the estimated source signal s (t) = A- 1X ( t), where A = A (x) is the mixing matrix A variation estimator and the like, it is easy to prove 
, i.e., if the signal separation algorithm like variability , the signal separation performance of the hybrid matrix (i.e., a signal transmission channel) is completely independent of the algorithm, depends only on the original signal, this performance as homogeneous properties.
10, the blind signal separation there are many algorithms that can be divided into the following three categories (the latter two methods equivalent):
- After the signal is converted, so that dependencies between different signal components (de2pendency) is minimized. Such methods referred to as independent component analysis, in 1994, proposed by Comon.
- Output transfer function of the nonlinear transform, so that the output distribution comprises a limited hypercube; then will force the output entropy maximization as uniformly dispersed in the hypercube Such methods called entropy maximization method, Yes. Bell and Sejnowski proposed in 1995.
- Nonlinear principal component analysis is a generalization of linear principal component analysis method, proposed by Oja and Karhumen and others in 1994.
Independent component analysis
1, independent component analysis (ICA) basic purpose is to determine the linear transformation matrix W, such that the converted output components yi (t) are statistically independent as possible.
2, Comparative function definition 3 output vector y is referred to as 
, defined as y probability density distribution of the set of mapped to a real-valued function of the operator 
, and the mapping function 
satisfies the following conditions: (1) if the vector y elements yi changing the arrangement position, the function 
remains unchanged, that is constant for all the switching matrix P 
; (2) if the element yi y change the "scale" function 
remains the same, i.e., all invertible diagonal matrix D constancy 
.
3, several typical independent component analysis:
- Stochastic gradient algorithm: instantaneous or stochastic gradient type instead of
the true gradient, i.e., to obtain stochastic gradient algorithm proposed by Bell and Sejnoeski is 
where η (t) is the learning rate or steps are major drawback of this algorithm: convergence rate slow, as it involves the simultaneous separation matrix W (t) inversion, once the W (t) number of conditions deteriorate during the update process, the algorithm may diverge. - Natural gradient algorithm: stochastic gradient 5H (z; W) in formula (16) 5W with natural gradient 5H (z; W) after the place 5WWTW, to obtain natural gradient algorithm is as follows: W (t + 1) = W (t ) + η (t) (I - <(y (t)) yT (t)) W (t) (17) where, the nonlinear transform function <i (yi) = αi (κi3, κi4) y2i + βi (κi3, κi4) y3i (18) where κi3 = E {y3i, k} and κi4 E {y4i, k} = - 3 represent yi skewness and kurtosis, and ai (κi3, κi4) = -12κi3 + 94κi3κi4 (19 a) βi (κi3, κi4) = -16κi4 + 32 (κi3) 2 + 34 (κi4) 2 (19 b) skewness and kurtosis updated with the following formula: κi3, k + 1 = κi3, k - u · T · (κi3, k- y3i, k) (20) κi4, k + 1 = κi4, k- u · T · (κi4, k- y4i, k + 3) (21) is the first natural gradient algorithm Cichocki et al in 1994 [13], and later Amari et al [2, 3, 48] theoretically proved its effectiveness (3) EASI algorithm: in 1996, Cardoso and Laheld [9]
Nonlinear principal component analysis
【1】Cao X R ,Liu R W. A general approach to blind source separation [J ] .IEEE Trans. Signal Processing ,1996 ,44 :562 - 571.