Blind extraction of target signals

Abstract

A system and method allow for extracting target signals from the overall measurement data without knowing how and where these data are collected, the locations and characteristics of target sources as well as those of random background noise sources. In essence, it uses an innovative and advanced signal processing technique to reveal certain critical information from a mixture of data that is hard to obtain otherwise. In particular, it allows for denoising the measured data that have been contaminated by various interfering and background noise, making it possible to extract certain target information that may be otherwise difficult to observe. The only assumption made in this method is that the target signal is incoherent with respect to all interfering signals and background noise. The more information about a target signal is available, the more complete the extracted signal becomes.

Claims

1. A method for extracting a target signal from measured signal including the steps of: a) measuring a target signal and background noise to obtain measurement data; b) using a computer to extract coherent structure from the measurement data using wavelets and corresponding wavelet coefficients; c) comparing the wavelet coefficients of the wavelets with a threshold with the computer; d) eliminating the wavelet coefficients of the wavelets that are less than the threshold with the computer; e) determining approximation coefficients at a selected scale m after said steps a)-d) based upon the wavelet coefficients at scale m; f) determining a wavelet vector at scale m based upon the approximation coefficients; and g) extracting information at scale m based upon the wavelet vector with all wavelet coefficients set to zero except those at the scale m.

2. The method of claim 1 wherein the step of extracting includes setting all wavelet coefficients of the wavelet vector to zero except those at scale m.

3. The method of claim 2 wherein the at least one sensor includes a microphone and the target signal is a person's voice.

4. The method of claim 2 wherein the target signal is sound.

5. The method of claim 2 wherein the target signal is a patient's breathing.

6. A system for extracting a target signal from measured signal including: at least one sensor for measuring a target signal and background noise to obtain measurement data; a computer programmed to extract coherent structure from the measurement data using wavelets and corresponding wavelet coefficients; and the computer further programmed to compare the wavelet coefficients of the wavelets with a threshold and eliminate the wavelet coefficients of the wavelets that are less than the threshold, the computer programmed to determine approximation coefficients at a selected scale m based upon the wavelet coefficients at scale m, the computer programmed to determine a wavelet vector at scale m based upon the approximation coefficients, wherein the computer is further programmed to extract information at scale m based upon the wavelet vector with all wavelet coefficients set to zero except those at the scale m.

7. The system of claim 6 wherein the computer is further programmed set all wavelet coefficients of the wavelet vector to zero except those at scale m.

8. The system of claim 7 wherein the at least one sensor includes a microphone and the target signal is a person's voice.

9. The system claim 7 wherein the at least one sensor includes a microphone and the target signal is sound.

10. The system of claim 7 wherein the target signal is a patient's breathing.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a schematic of the target information extraction system according to one embodiment.

(2) FIG. 2 is a schematic of this new process to extract some target information from measured data.

(3) FIG. 3 is a comparison of the extracted breathing signal of a patient versus the directly measured data. Extraction was done using Eq. (33) without any prior knowledge of the characteristics of both target features and interfering and background noise whatsoever, where and how signals are collected.

DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT

(4) A target information extraction system 10 according to the present invention includes a computer 12 having a processor 14 and storage 16 including memory and mass storage, such as magnetic, electronic and/or optical storage. One or more transducers 20, such as microphones, probes and other sensors, may be used to measure sound pressure (or other physical signals) and send signals to the computer 12 indicating the capture of sound pressure (or other physical signals) at the location of the transducers 20.

(5) The target information extraction system 10 of the present invention uses an algorithm described below to extract information from one or more target sources 30. The algorithm is stored in storage 16 and executed by the processor 14 in the computer 12. The computer 12 is programmed to perform the outlined steps using the algorithms described herein and any necessary functions. The location, number and nature of the sources 30 and background noise sources 32 may be unknown. The transducers 20 or sensors are suitable for the type of target signals being measured, such as microphones, vibration sensors, etc.

(6) FIG. 2 shows the schematic of the system 10 to extract target information from a mixture of data collected in a non-ideal environment. The input to this new process is directly measured data that may have been contaminated by interfering signals and random background noise. The output is the target information. This algorithm can process the time-domain signals collected by any hardware and/or data acquisition system. It has no special requirement for specific hardware required for this algorithm to work.

(7) In general, the coherent structure of a mixture of data can be extracted and written as follows by using the D4 Daubechies wavelets as follows
{x}.sub.N1=2.sup.1/2[C].sub.NN.sup.T{.sup.(1)}.sub.N1,(1)

(8) where {x}.sub.N1 represents a column vector containing the extracted data, [C].sub.NN.sup.T indicates the transpose of the square matrix [C].sub.NN that contains the scaling coefficients, and .sup.(1) is the wavelet vector at scale m=1 after threshold. Note that one can select any threshold. In this case, we choose the universal threshold
.sub.U={square root over (2 ln N)}{circumflex over ()},(2)

(9) where (2 ln N).sup.1/2 is the expected maximum value of a white noise sequence of length N, namely, there are N data points in the sequence, and {circumflex over ()} is a robust estimate of the standard deviation of the noise in the input data under consideration. Typically, {circumflex over ()} is taken at the median of absolute deviation (MAD) of the wavelet coefficients at the smallest scale m=1, divided by 0.6745 to calibrate with the standard deviation of a Gaussian distribution, where {circumflex over ()}=0.6745median[|T.sub.1,imedian(T.sub.1,i)|], T.sub.1,i represents the wavelet coefficients at scale m=1.

(10) In the universal threshold, all wavelet coefficients are compared with .sub.U. The coefficients that are larger than this threshold are deemed to correspond to the coherent parts of target information and kept intact; and those less the threshold .sub.U are deemed incoherent or noisy parts of the input data and are eliminated.

(11) $\begin{array}{cc}{\hat{W}}_i^{\left(1\right)}=\{\begin{array}{cc}0& .Math.{\hat{W}}_i^{\left(1\right)}.Math.<{}_U\\ {\hat{W}}_i^{\left(1\right)}& .Math.{\hat{W}}_i^{\left(1\right)}.Math.{}_U\end{array},& \left(3\right)\end{array}$

(12) Note that the elements of the square matrix [C].sub.NN in Eq. (1) change with the choice of extensions of the input data, depending on the type of Daubechies wavelets are used. When a simple convolution is used to extend the input data, the square matrix [C].sub.NN is expressible as

(13) $\begin{array}{cc}{\left[C\right]}_{NN}={\left[\begin{array}{cccccccccc}{c}_0& c_1& c_2& c_3& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& 0& 0\\ c_3& -c_2& c_1& -c_0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& 0& 0\\ 0& 0& c_0& c_1& c_2& c_3& 0& \mathrm{.Math.}& 0& 0\\ 0& 0& c_3& -c_2& c_1& -c_0& 0& \mathrm{.Math.}& 0& 0\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ 0& 0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ c_2& c_3& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& 0& 0& c_0& c_1\\ c_1& -c_0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& 0& 0& c_3& -c_2\end{array}\right]}_{NN}& \left(4\right)\end{array}$

(14) Because of the constraint conditions for the D4 Daubechies wavelets, we have
c.sub.0.sup.2+c.sub.1.sup.2+c.sub.2.sup.2+c.sub.3.sup.2=3 and c.sub.0c.sub.2+c.sub.1c.sub.3=0.(5)

(15) Therefore, the product of [C].sub.NN.sup.T and [C].sub.NN is a diagonal and unitary matrix multiplied by 2.
[C].sub.NN.sup.T[C].sub.NN=2[I].sub.NN,(6)

(16) where [I].sub.NN indicates an identify square matrix. Accordingly, we can write

(17) $\begin{array}{cc}{\left[C\right]}_{NN}^T={\left[\begin{array}{cccccccccc}{c}_0& c_3& 0& 0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& c_2& c_1\\ c_1& -c_2& 0& 0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& c_3& -c_0\\ c_2& c_1& c_0& c_3& 0& 0& 0& \mathrm{.Math.}& 0& 0\\ c_3& -c_0& c_1& -c_2& 0& 0& 0& \mathrm{.Math.}& 0& 0\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ 0& 0& 0& 0& \mathrm{.Math.}& 0& c_0& c_3& 0& 0\\ 0& 0& 0& 0& \mathrm{.Math.}& 0& c_1& -c_2& 0& 0\\ 0& 0& 0& 0& \mathrm{.Math.}& 0& c_2& c_1& c_0& c_3\\ 0& 0& 0& 0& \mathrm{.Math.}& 0& c_3& -c_0& c_1& -c_2\end{array}\right]}_{NN}.& \left(7\right)\end{array}$

(18) In what follows, we consider the D4 Daubechies wavelets and extend the input data by using a simple convolution. Accordingly, the approximate coefficients at scale m=1 contained in .sup.(1) after threshold can be determined by the following matrix equation.

(19) $\begin{array}{cc}{\left\{\begin{array}{c}{\hat{S}}_{1,0}\\ {\hat{S}}_{1,1}\\ {\hat{S}}_{1,2}\\ {\hat{S}}_{1,3}\\ \mathrm{.Math.}\\ \mathrm{.Math.}\\ \mathrm{.Math.}\\ \mathrm{.Math.}\\ {\hat{S}}_{1,\left(\frac{N}{2}-2\right)}\\ {\hat{S}}_{1,\left(\frac{N}{2}-1\right)}\end{array}\right\}}_{\frac{N}{2}1}={\frac{1}{\sqrt{2}}\left[\begin{array}{cccccccccc}{c}_0& c_3& 0& 0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& c_2& c_1\\ c_1& -c_2& 0& 0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& c_3& -c_0\\ c_2& c_1& c_0& c_3& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& 0& 0\\ c_3& -c_0& c_1& -c_2& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& 0& 0\\ 0& 0& c_2& c_1& c_0& c_3& \mathrm{.Math.}& \mathrm{.Math.}& 0& 0\\ 0& 0& c_3& -c_0& c_1& -c_2& \mathrm{.Math.}& \mathrm{.Math.}& 0& 0\\ 0& 0& 0& 0& 0& 0& & \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& & \mathrm{.Math.}& \mathrm{.Math.}\\ 0& 0& 0& 0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& c_0& c_3\\ 0& 0& 0& 0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& c_1& -c_2\end{array}\right]}_{\frac{N}{2}\frac{N}{2}}{\left\{\begin{array}{c}{\hat{S}}_{2,0}\\ {\hat{T}}_{2,0}\\ {\hat{S}}_{2,1}\\ {\hat{T}}_{2,1}\\ \mathrm{.Math.}\\ \mathrm{.Math.}\\ \mathrm{.Math.}\\ \mathrm{.Math.}\\ {\hat{S}}_{2,\left(\frac{N}{4}-1\right)}\\ {\hat{T}}_{2,\left(\frac{N}{4}-1\right)}\end{array}\right\}}_{\frac{N}{2}1}.& \left(8\right)\end{array}$

(20) Note that ({circumflex over (T)}.sub.2,0,{circumflex over (T)}.sub.2,1,{circumflex over (T)}.sub.2,2, . . . ,{circumflex over (T)}.sub.2,(N/4-1)) on the right side of Eq. (8) contain the wavelet coefficients at scale m=2 after threshold, which are known. The approximation coefficients (.sub.2,0,.sub.2,1,.sub.2,2, . . . ,.sub.2,(N/4-1)) are to be determined by the following matrix equation.

(21) $\begin{array}{cc}{\left\{\begin{array}{c}{\hat{S}}_{2,0}\\ {\hat{S}}_{2,1}\\ {\hat{S}}_{2,2}\\ {\hat{S}}_{2,3}\\ \mathrm{.Math.}\\ \mathrm{.Math.}\\ \mathrm{.Math.}\\ \mathrm{.Math.}\\ {\hat{S}}_{2,\left(\frac{N}{4}-2\right)}\\ {\hat{S}}_{2,\left(\frac{N}{4}-1\right)}\end{array}\right\}}_{\frac{N}{4}1}={\frac{1}{\sqrt{2}}\left[\begin{array}{cccccccccc}{c}_0& c_3& 0& 0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& c_2& c_1\\ c_1& -c_2& 0& 0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& c_3& -c_0\\ c_2& c_1& c_0& c_3& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& 0& 0\\ c_3& -c_0& c_1& -c_2& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& 0& 0\\ 0& 0& c_2& c_1& c_0& c_3& \mathrm{.Math.}& \mathrm{.Math.}& 0& 0\\ 0& 0& c_3& -c_0& c_1& -c_2& \mathrm{.Math.}& \mathrm{.Math.}& 0& 0\\ 0& 0& 0& 0& c_2& c_1& & \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& & \mathrm{.Math.}& \mathrm{.Math.}\\ 0& 0& 0& 0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& c_0& c_3\\ 0& 0& 0& 0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& c_1& -c_2\end{array}\right]}_{\frac{N}{4}\frac{N}{4}}{\left\{\begin{array}{c}{\hat{S}}_{3,0}\\ {\hat{T}}_{3,0}\\ {\hat{S}}_{3,1}\\ {\hat{T}}_{3,1}\\ \mathrm{.Math.}\\ \mathrm{.Math.}\\ \mathrm{.Math.}\\ \mathrm{.Math.}\\ {\hat{S}}_{3,\left(\frac{N}{8}-1\right)}\\ {\hat{T}}_{3,\left(\frac{N}{8}-1\right)}\end{array}\right\}}_{\frac{N}{4}1}& \left(9\right)\end{array}$

(22) Once again, the elements ({circumflex over (T)}.sub.3,0,{circumflex over (T)}.sub.3,1,{circumflex over (T)}.sub.3,2, . . . ,{circumflex over (T)}.sub.3,(N/8-1)) on the right side of Eq. (9) represent the wavelet coefficients at scale m=3 after threshold, which are specified, but (.sub.3,0,.sub.3,1,.sub.3,2, . . . ,.sub.3,(N/8-1)) are to be determined by the matrix equation involving wavelet and approximation coefficients at the next scale after threshold. At scale m=M2, we can solve (.sub.(M-2),0,.sub.(M-2),1,.sub.(M-2),2,.sub.(M-2),3) in terms of ({circumflex over (T)}.sub.(M-1),0,{circumflex over (T)}.sub.(M-1),1) and (.sub.(M-1),0,.sub.(M-1),1) at scale m=M1 after threshold as

(23) $\begin{array}{cc}{\left\{\begin{array}{c}{\hat{S}}_{\left(M-2\right),0}\\ {\hat{S}}_{\left(M-2\right),1}\\ {\hat{S}}_{\left(M-2\right),2}\\ {\hat{S}}_{\left(M-2\right),3}\end{array}\right\}}_{41}={\frac{1}{\sqrt{2}}\left[\begin{array}{cccc}{c}_0& c_3& c_2& c_1\\ c_1& -c_2& c_3& -c_0\\ c_2& c_1& c_0& c_3\\ c_3& -c_0& c_1& -c_2\end{array}\right]}_{44}{\left\{\begin{array}{c}{\hat{S}}_{\left(M-1\right),0}\\ {\hat{T}}_{\left(M-1\right),0}\\ {\hat{S}}_{\left(M-1\right),1}\\ {\hat{T}}_{\left(M-1\right),1}\end{array}\right\}}_{41},& \left(10\right)\end{array}$

(24) where (.sub.(M-1),0,.sub.(M-1),1) on the right side of Eq. (10) are given by

(25) $\begin{array}{cc}{\left\{\begin{array}{c}{\hat{S}}_{\left(M-1\right),0}\\ {\hat{S}}_{\left(M-1\right),1}\end{array}\right\}}_{21}={\frac{1}{\sqrt{2}}\left[\begin{array}{cc}{c}_0+c_2& c_1+c_3\\ c_1+c_3& -c_0-c_2\end{array}\right]}_{22}{\left\{\begin{array}{c}{\hat{S}}_{M,0}\\ {\hat{T}}_{M,0}\end{array}\right\}}_{21},& \left(11\right)\end{array}$

(26) where (.sub.M,0,{circumflex over (T)}.sub.M,0) on the right side of Eq. (11) are the approximation and wavelet coefficients at scale m=M after threshold, respectively, which are known. Substituting these quantities into the wavelet vector at scale m=1 after threshold, we obtain:
.sup.(1)=(.sub.1,0,{circumflex over (T)}.sub.1,0,.sub.1,1,{circumflex over (T)}.sub.1,1,.sub.1,2,{circumflex over (T)}.sub.1,2, . . . ,.sub.1,(N/2-1),{circumflex over (T)}.sub.1,(N/2-1)).(12)

(27) The coherent structure of the input data can now be extracted by Eq. (1). It is important to understand that in general the noise embedded in the input data cannot be completely eliminated, and the extracted parts may be imperfect. This is because no prior knowledge about the test environment and conditions, and background noise are available. Suppression of noise in the input data is accomplished through threshold, which either sets the wavelet coefficients to zero or leave them as they are. Also, threshold tends to reduce the amplitudes of the extracted information. These will affect the quality of the extracted information. Thus, the more information about the target signal can be incorporated in the process, the higher the quality of the extracted information becomes.

(28) Equation (1) enables us to suppress noise imbedded in the input data. However, there can be useful information in these sanitized data at each individual scale. Detailed formulations for extracting this useful information are derived at each scale. These are described as follows.

(29) Suppose that we want to extract specific information at scale m from the sanitized data given by Eq. (1). Note that each scale describes a different frequency band. The higher the scale value m is, the higher the frequency band is. The formulations for extracting specific information from each scale m can be expressed as
{x.sup.(m)}.sub.N1=2.sup.1/2[C].sub.NN.sup.T{W.sup.(m)}.sub.N1,(13)

(30) where m=1, 2, . . . , M1. The wavelet vector W.sup.(m) is specified with all wavelet coefficients set to zero, except those at the scale m.

(31) For example, to extract target information at scale m=1, we can start from the fully decomposed wavelet vector after threshold
W.sup.M=(.sub.M,0,{circumflex over (T)}.sub.M,0,{circumflex over (T)}.sub.(M-1),0,{circumflex over (T)}.sub.(M-1),1, . . . ,{circumflex over (T)}.sub.2,0,{circumflex over (T)}.sub.2,1,{circumflex over (T)}.sub.2,2, . . . ,{circumflex over (T)}.sub.2,(N/4-1),{circumflex over (T)}.sub.1,0,{circumflex over (T)}.sub.1,1,{circumflex over (T)}.sub.1,2, . . . ,{circumflex over (T)}.sub.1,(N/2-1)).(14)

(32) Next, we set all coefficients set to zero, except the wavelet coefficients at the scale m=1.
W.sup.(M)=(0,0, . . . ,{circumflex over (T)}.sub.1,0,{circumflex over (T)}.sub.1,1,{circumflex over (T)}.sub.1,2, . . . ,{circumflex over (T)}.sub.1,(N/2-1)).(15)

(33) The approximation coefficients from scales m=M1 down to m=1 can be determined by,

(34) $\begin{array}{cc}{\hat{S}}_{m-1,n}=\frac{1}{\sqrt{2}}\underset{k=0}{\overset{N_k-1}{.Math.}}\left(c_{n-2k}{\hat{S}}_{m,k}+b_{n-2k}{\hat{T}}_{m,k}\right),& \left(16\right)\end{array}$

(35) where b.sub.k=(1).sup.kc.sub.3-k, and the sum of all the coefficients b.sub.k is zero. Once the approximation coefficients are specified, the wavelet vector at scale m=1 is established. Using the universal threshold (3), we can write
W.sup.(1)=({tilde over (S)}.sub.1,0,{tilde over (T)}.sub.1,0,{tilde over (S)}.sub.1,1,{tilde over (T)}.sub.1,1, . . . ,{tilde over (S)}.sub.1,(N/2-1),{tilde over (T)}.sub.1,(M/2-1)).(17)

(36) The extracted information at scale m=1 can then be written as
{x.sup.(1)}.sub.N1=2.sup.1/2[C].sub.NN.sup.T{W.sup.(1)}.sub.N1.(18)

(37) In a similar manner, we can extract information at scale m by setting all other coefficients of the fully decomposed wavelet vector to zero after threshold, except the wavelet coefficients at the scale m.
W.sup.(M)=(0,0, . . . ,{circumflex over (T)}.sub.m,0,{circumflex over (T)}.sub.m,1,{circumflex over (T)}.sub.m,2, . . . , 0,0, . . . )(19)

(38) Using Eq. (16) and the universal threshold (3), we can determine the wavelet vector at scale m as
W.sup.(m)=({tilde over (S)}.sub.1,0,{tilde over (T)}.sub.1,0,{tilde over (S)}.sub.1,1,{tilde over (T)}.sub.1,1. . . ,{tilde over (S)}.sub.1,(N/2-1),{tilde over (T)}.sub.1,(N/2-1)).(20)

(39) Substitute W.sup.(m) into Eq. (13) leads to the extracted information at scale m.

(40) To summarize, the new process enables one to sanitize the directly measured data and extract specific information at any scale m by using, say, the D4 Daubechies wavelets. These are depicted by Eqs. (13) to (20). Detailed steps involved in these procedures are as follows:

(41) 1. Extend any set of measured data {x}.sub.N1 using a simple convolution, where N=2.sup.M.

(42) 2. Set the approximation coefficients S.sub.0,i=x.sub.0,i, here x.sub.0,i are the given set of input data with i=0, 1, 2, . . . , N.

(43) 3. Calculate the scaling and wavelet coefficients S.sub.m,n and T.sub.m,n using the following equations

(44) $\begin{array}{cc}{S}_{m+1,n}=\frac{1}{\sqrt{2}}\underset{k=0}{\overset{N_k-1}{.Math.}}{c}_k{S}_{m,2n+k}\mathrm{or}{S}_{m+1,n}=\frac{1}{\sqrt{2}}\underset{k=0}{\overset{N_k-1}{.Math.}}{c}_{k-2n}{S}_{m,k}& \left(21\right)\\ T_{m+1,n}=\frac{1}{\sqrt{2}}\underset{k=0}{\overset{N_k-1}{.Math.}}{b}_k{S}_{m,2n+k}\mathrm{or}{T}_{m+1,n}=\frac{1}{\sqrt{2}}\underset{k=0}{\overset{N_k-1}{.Math.}}{b}_{k-2n}{S}_{m,k}& \left(22\right)\end{array}$

(45) 4. Once the approximation and wavelet coefficients at all scales are determined, the signal is said to be fully decomposed.
W.sup.(M)=(S.sub.M,0,T.sub.M,0,T.sub.(M-1),0,T.sub.(M-1),1, . . . ,T.sub.2,0,T.sub.2,1, . . . ,T.sub.2,(N/4-1),T.sub.1,0,T.sub.1,1,T.sub.1,2, . . . ,T.sub.1,(N/2-1)).(23)

(46) 5. Apply the universal threshold .sub.U to W.sup.(M) at scale m=M

(47) 0$\begin{array}{cc}{W}_i^{\left(M\right)}=\{\begin{array}{cc}0& .Math.W_i.Math.<{}_U\\ W_i& .Math.W_i.Math.{}_U\end{array},& \left(24\right)\end{array}$

(48) where the threshold is given by
.sub.U={square root over (2 ln N)}{circumflex over ()},(25)

(49) where (2 ln N).sup.1/2 is the expected maximum value of the white noise sequence with length N and {circumflex over ()} is a robust estimate of the standard deviation of the noise in the signal under consideration. Typically {circumflex over ()} is taken at the median of absolute deviation (MAD) of the wavelet coefficients at the smallest scale m=1 divided by 0.6745 to calibrate with the standard deviation of Gaussian distribution, where {circumflex over ()}=0.6745median[|T.sub.1,imedian(T.sub.1,i)|].

(50) 6. Use the universal threshold to determine the wavelet coefficients at all scales.
({circumflex over (T)}.sub.M,0,{circumflex over (T)}.sub.(M-1),0,{circumflex over (T)}.sub.(M-1),1, . . . ,{circumflex over (T)}.sub.2,0,{circumflex over (T)}.sub.2,1,{circumflex over (T)}.sub.2,2, . . . ,{circumflex over (T)}.sub.2,(N/4-1),{circumflex over (T)}.sub.1,0,{circumflex over (T)}.sub.1,1,{circumflex over (T)}.sub.1,2, . . . ,{circumflex over (T)}.sub.1,(N/2-1)).(26)

(51) 7. Use the following equation to specify the approximation coefficients from scales m=M1 to m=1 after threshold (.sub.m,0, .sub.m,1, .sub.1,2, . . . , .sub.m(N/2.sub.m.sub.-2), S.sub.1,(N/2.sub.m.sub.-1)).

(52) $\begin{array}{cc}{\hat{S}}_{m-1,n}=\frac{1}{\sqrt{2}}\underset{k=0}{\overset{3}{.Math.}}\left(c_{n-2k}{\hat{S}}_{m,k}+b_{n-2k}{\hat{T}}_{m,k}\right).& \left(27\right)\end{array}$

(53) 8. Determine the wavelet vector at the scale m=1 after threshold using the following equation.
.sup.(1)=(.sub.1,0,{circumflex over (T)}.sub.1,0,.sub.1,1,{circumflex over (T)}.sub.1,1,.sub.1,2,{circumflex over (T)}.sub.1,2, . . . ,.sub.1,(N/2-1),{circumflex over (T)}.sub.1,(N/2-1)).(28)

(54) 9. Use Eq. (4) to set up the square matrix for the scaling coefficients [C].sub.NN, and Eq. (7) to find its inversion.

(55) 10. The original mixture of data can now be sanitized by using the following equation
{x}.sub.N1=2.sup.1/2[C].sub.NN.sup.T{.sup.(1)}.sub.N1,(29)

(56) 11. To extract specific information at any scale m, we set all wavelet coefficients to zero in the fully decomposed wavelet vector after threshold, except those at the scale m.
W.sup.(m)=(0,0, . . . ,{circumflex over (T)}.sub.m,0,{circumflex over (T)}.sub.m,1,{circumflex over (T)}.sub.m,2, . . . ,0,0, . . . ).(30)

(57) 12. Use Eq. (27) to determine the approximation coefficients from scales m=M1 to m=1.

(58) 13. Establish the wavelet vector at scale m and apply the universal threshold (3) to get,
W.sup.(m)=({tilde over (S)}.sub.1,0,{tilde over (T)}.sub.1,0,{tilde over (S)}.sub.1,1,{tilde over (T)}.sub.1,1, . . . ,{tilde over (S)}.sub.1,(N/2-1),{tilde over (T)}.sub.1,(N/2-1)).(32)

(59) 14. The extracted information at scale m can now be written as
{x.sup.(m)}.sub.N1=2.sup.1/2[C].sub.NN.sup.T{W.sup.(m)}.sub.N1.(33)

(60) Extraction of specific information can be very effective when some prior knowledge is incorporated in the process. For example, suppose that information over a specific frequency band is desired. Then we can focus on the scale that corresponds to this frequency band and extract the target information. If no information regarding the target information is available, it becomes a completely blind extraction. Under this condition, one can still use the steps outlined above to extract coherent structures from the measured data or simply sanitize the input data.

(61) FIG. 3 illustrates an example of a complete blind extraction of coherent structures of mixed signals without any prior knowledge of how and where signals are measured, and what the characteristics of target information, and interfering signals and random background noise are.

(62) These data were collected by a sensor taped on an asthma patient to detect early signs asthma attack. The black line in FIG. 3 represents the overall measured data and the white line is the extracted signal. Since the amplitudes of breathing signals are so low compared to those of overall signals, they are completely masked by the interfering signals and random background noise, including the doctor's voice. If the recorded signals are played back, there is no way of noticing the patient's breathing, which makes it impossible to monitor an early asthma attack. However, by following the steps 1 to 14 as outlined above, we are able to sanitize the data and extract patient's breathing signal. In other words, all interfering signals and random background noise including doctor's voice are successfully suppressed.

(63) In accordance with the provisions of the patent statutes and jurisprudence, exemplary configurations described above are considered to represent a preferred embodiment of the invention. However, it should be noted that the invention can be practiced other than as specifically illustrated and described without departing from its spirit or scope.