ROBUST SIGNAL DETECTION USING CORRENTROPY

STATEMENT AS TO FEDERALLY SPONSORED RESEARCH [0001] The invention was made with U.S. government support under grant number ECS-0601271 awarded by the National Science Foundation. The U.S. government may have certain rights in the invention.

FIELD OF THE INVENTION

[0002] The present invention relates to the field of signal processing, and more particularly, to detecting a low-level periodic signal in a high noise environment.

BACKGROUND

[0003] Detecting low-level information carrying signals in high-level noise is a challenging task. Various signal processing techniques exist for separating noise components from information carrying signals. Performance of the various signal processing techniques may depend on the characteristics of noise and the characteristics of the information carrying signal. As an example, statistical signal processing methods can incorporate noise statistics for modeling the noise source and removing the noise. Signal processing methods can project the noisy signal into multiple subspaces in an attempt to separate the noise components from the information carrying components. Principal Component Analysis is one such method which decomposes a noisy signal into multiple principal components. The information carrying components may be distributed to only a few principal components, depending on the projection. However, PCA is data dependent and the projection may not effectively separate the information carrying signal from the noise signal.

[0004] Accordingly a need exists for a method of robust signal detection that is data independent.

SUMMARY

[0005] One embodiment is a method for detecting a periodic signal in noise. The method can include applying correntropy to a noisy signal to generate a non-linear mapping, and applying a sub-space projection to the non-linear mapping to produce principal components. A non-linear autocorrelation matrix of the input signal can be generated from the non-linear mapping. In practice, a correntropy kernel can be applied to the noisy signal to generate the non-linear autocorrelation matrix. The correntropy kernel maps the input data to a reproducing kernel Hubert Space (RKHS). In one configuration, as a result of the non-linear mapping, the non-linear autocorrelation matrix can be represented as a Gram matrix. A size of the correntropy kernel can also be chosen to generate the Gram matrix and adjust a detection performance. Thereafter, a Temporal Principal Component Analysis (TPCA) can be performed on the Gram matrix. TPCA can include applying an eigen-decomposition to the Gram matrix to produce eigen-vectors of the principal components. In one arrangement, a length of the Gram matrix can be chosen to correspond to a desired number of principal components. The method can further include choosing a principal component having high energy for detecting the periodic signal. To detect the periodic signal, a power spectral density (PSD) of the correntropy derived principal component, and a maximum peak in the PSD can be identified, wherein the maximum peak corresponds to the periodic signal. In one arrangement, the spectral analysis can be performed on a second principal component that does not have a mean bias. More generally, a PSD analysis of any component that contains the signal of interest or any of its parts can be performed. [0006] Another embodiment is a method for Correntropy Principal Component Analysis (CPCA). The method can include applying correntropy to an input data to generate a non-linear autocorrelation matrix, and performing Principal Component Analysis on the non-linear autocorrelation matrix to produce correntropy principal components. Correntropy can project the input data to a linear reproducing kernel Hubert Space (RKHS) to produce the non-linear autocorrelation matrix. More specifically, a correntropy kernel can be applied to the input data to map the input data to a reproducing kernel Hubert Space (RKHS). The correntropy kernel can induce a scalar non-linear mapping that retains a measure of similarity in the input data. The

RKHS can have a same dimension as the input data, and the correntropy kernel is data independent.

[0007] Yet another embodiment is a computer programmable readable storage medium having computer instructions for applying correntropy to an input data to generate a Gram matrix, and performing Principal Component Analysis (PCA) on the Gram matrix to produce correntropy principal components. A Gaussian kernel can be applied to the input data to generate entries of the Gram matrix. In such regard, the Gaussian kernel induces a non-linear mapping in a linear reproducing kernel Hubert Space. The Gram matrix can have an input dimension equal to a number of desired principal components. Principal components having a highest energy can be selected for detecting a periodic signal in the input data.

BRIEF DESCRIPTION OF THE DRAWINGS

[0008] Various features of the system are set forth with particularity in the appended claims. The embodiments herein, can be understood by reference to the following description, taken in conjunction with the accompanying drawings, in which: [0009] FIG. 1 is a schematic diagram of a processor employing correntropy in accordance with one embodiment of the invention;

[00010] FIG. 2 is a method for detecting a periodic signal in noise using correntropy in accordance with one embodiment of the invention;

[00011] FIG. 3 is more detailed method for correntropy in accordance with one embodiment of the invention;

[00012] FIG. 4 is a method for Correntropy Principal Component Analysis (CPCA) in accordance with the invention;

[00013] FIG. 5 is a non-linear mapping of correntropy in accordance with the invention;

[00014] FIG. 6 is a correntropy power spectrum in accordance with the invention; [00015] FIG 7 is an exemplary illustration for signal detecting using correntropy based on a power spectral density (PSD) of the correntropy in accordance with the invention; [00016] FIG. 8 is a table presenting PCA performance comparisons; and [00017] FIG. 9 is a table identifying effects of kernel size on TPCA performance.

DETAILED DESCRIPTION

[00018] Broadly stated, embodiments of the invention are directed to robust signal detection using correntropy. Referring to FIG. 1 , an exemplary communication system 100 is shown. The exemplary communication system can include a transmitter 190 that transmits a periodic signal (e.g. sine wave) and a receiver having a processor 1 10 that receives the periodic signal. During transmission, the periodic signal may be corrupted by noise such as channel noise or environmental noise. The signal to noise ratio (SNR) of the periodic signal may be very low compared to the noise components as a result of the transmission or the communication environment. The processor 1 10 can monitor and detect a faint periodic signal 141 in the noisy signal 101 using methods of correntropy herein presented.

[00019] The noisy signal 101 may be a communication signal having a periodic carrier component, a medical signal having a fundamental frequency, a voice signal having a fundamental pitch, a surveillance signal having a periodic beacon signal, a radar signal having a periodic component, or any other periodic signal corrupted by noise. Broadly stated, the processor 1 10 can non-linearly project the noisy signal 101 (signal 141 plus noise) to a subspace to increase a signal to noise ratio (120), and then detect the periodic signal 141 in a subspace projection that has a higher signal to noise ratio (130). The non-linear projection allows the processor 100 to more effectively detect the periodic signal 141 in the noisy signal 101.

[00020] Referring to FIG. 2, a method 200 for detecting a faint periodic signal in a noisy signal is shown. The method 200 can be practiced with more or less than the number of steps shown. Moreover, the method 200 is not limited to the order shown in FIG. 2. Reference will be also made to FIGS 3-6 when discussing method 200. The method 200 can start in state 201 wherein the processor 100 receives the noisy signal 101 , having a low level periodic signal 141.

[00021] At step 210 the processor 100 can apply correntropy on the noisy signal to generate a non-linear mapping. Briefly, correntropy can be defined as a generalized similarity measure between two variables x and Y defined by

V _σ {X,Y) =ε[κ _σ {X-Y)\ (1 )

where ^ _σ ( ^{χ ~χ} i) is the Gaussian kernel that provides a non-linear mapping,

The Gaussian kernel can be substituted by any other positive definite function. When detecting a low-level periodic signal in a noisy signal, the similarity measure of EQ (1 ) evaluates the similarity for multiple samples of the noisy signal. The noise may be Gaussian or non-Gaussian. More specifically, the X and Y variables correspond to time delayed samples of the noisy signal and are used in EQ (1 ) to evaluate a correlation in a non-linear domain. In practice, a joint PDF for the variables X and Y is unknown and only a finite number of data { ( ^X P JJ ) are available, leading to the sample estimator of correntropy

[00022] At step 220, the processor 100 can apply a sub-space projection to the nonlinear mapping to produce principal components. Briefly, the processor 100 applies a subspace projection to the non-linear mapping to decompose the noisy signal 101 into principal components (e.g. sub-spaces). As a result of the non-linear mapping, one of the principal components contains a majority of the periodic signal 141 energy at a high signal to noise ratio. When the noise in the noisy signal 101 is large and uncorrelated, the noise power is divided equally among the principal components, while the periodic signal 141 is concentrated in primarily only one of the principal components. [00023] At step 230, the processor 100 can choose a principal component having a majority of overall signal energy. As previously noted, the noise power is divided equally by the principal components as a result of the sub-space projection, while the periodic component 141 is concentrated in one of the principal component directions. Hence, signal to noise ratio is increased and detection of the signal may be accomplished with

more robust performance. Accordingly, the processor 100 can more accurately detect the periodic component in the principal components with the higher overall energy, which corresponds to the principal component with the higher signal to noise ratio. At step 241 , the method 200 can end.

[00024] FIG. 3 shows a more detailed approach to the method steps 200 of FIG. 2. At step 21 1 , the processor 100 can generate a non-linear autocorrelation matrix as a result of the non-linear mapping. The non-linear autocorrelation matrix is generated from a non-linear mapping of the correntropy analysis of EQ 1. The sample estimator of correntropy shown in EQ 3 uses a correntropy kernel (e.g. Gaussian kernel), K. The correntropy kernel, K, induces a scalar nonlinear mapping V which maps the noisy signal, where [ X ₁ is the noisy signal data set, as while preserving the similarity measure, as shown below:

m _x ®-η _x (i+t)]=V(ij+t)=ε[κ(x(i)-x(i+t))l

0≤t ≤N-l ⁽⁴⁾

where x(i) - x(i + 1) is the similarity measure of EQ (1 ) that is non-linearly mapped to a multiplication of η _x (i)η(i + t) _x as a result of the correntropy kernel, K.

The term η _x (i)η(i + t) _x represents the non-linear autocorrelation of the noisy signal.

[00025] Based on EQ (4), the non-linear autocorrelation matrix (e.g. left side of EQ 4, "non-linear mapping") can be generated by applying correntropy analysis (e.g. right side of EQ 4, "similarity measure") to the noisy signal 141. That is, the non-linear autocorrelation matrix is generated by applying correntropy to the noisy signal 141 on a sample-by-sample basis over time. Moreover, when m _η denotes the mean of the transformed data, then the square of rn _η is an asymptotic estimate of the information potential of the original data as N — > ∞ , i.e.

[00026] At step 221 , the processor 100 can perform Temporal Principal Component Analysis (TPCA) on the non-linear autocorrelation matrix to generate the principal components. TPCA is a method of sub-space projection that decomposes a signal into principal components (See step 220 FIG 2). Recall, the non-linear autocorrelation matrix is generated by a non-linear mapping of the correntropy kernel, K, as shown in EQ (4). In such regard, the principal components can be directly generated from correntropy. Moreover, the kernel, K, is data independent. Furthermore, the size of the correntropy kernel can be selected so as to project the data in a principal component having the same dimension as the noisy data. The size can also be selected to generate the size of the non-autocorrelation matrix. One unique arrangement of the non-linear autocorrelation matrix based on the kernel size allows for a data independent transformation using TCPA. That is, when the non-linear autocorrelation matrix is arranged as a Gram matrix the principal components can be directly generated from the input data (e.g. noisy signal 101 ).

[00027] In conventional PCA, an autocorrelation matrix is used to generate principal components. In the current embodiment, the autocorrelation matrix is replaced with the Gram matrix for TPCA, wherein the Gram matrix is generated as a result of the correntropy kernel. In the foregoing, a brief description of TPCA with regard to the autocorrelation matrix and Gram matrix is provided. Suppose the noisy signal 141 is

{x(i),i=\2,-,N+L—l} _anc | j _S mapped as a trajectory of N points in the reconstruction space of dimension L. With the data matrix

[00028] Principal Component Analysis estimates the eigen-filters and principal components (PC). An autocorrelation matrix and Gram matrix of the noisy signal 141 are denoted as R and K respectively, and written as

where r(k) = E[x(i)x(i + k)] \ _s the autocorrelation function of X. When ν and L are large, EQ (7) and EQ (8) are good approximations of the autocorrelation. In the following derivation, L is adjustable, and can be set appropriately to the signal detection application. That is, L can be set as a function of the input data dimension. Assuming L < N , by Singular Value Decomposition (SVD)

X = UDV (9)

where U, V are two orthonormal matrices and D is a pseudo-diagonal Lx N matrix with singular values (Vλ ' V^ '-V^ I as its entries. In PCA the singular values are the eigen values of the eigen vectors. Therefore,

K = X ⁷ X = VEfDV ¹ (1 1 )

[00029] From EQ (10) and EQ (1 1 ), the columns of U and V are eigenvectors of R and K respectively. Rewriting (9) as

U ^r X = DV ^r (12) or equivalently,

U ₁ ⁷ X = ^V ₁ ⁷ , i = l, 2, ..., L _{(1 3} )

[00030] Here U ₁ and V ₁ are the i-th columns of U and V respectively. This equation reveals that the data (X) projected onto the i-th eigenvector ( U ₁ ) of R (e.g. left side of

EQ 13) is exactly the scaled [JX ₁ ) i-th eigenvector [ V ₁ ) of K (e.g. right side of EQ 13). EQ (13) indicates that the principal components can be obtained directly from the Gram matrix (e.g. V ₁ ) without dependence on the input data (X). In contrast, the autocorrelation matrix (e.g. U ₁ ) requires the input data (X) to be projected for generating the principal components, thus establishing a dependence on the input data. Accordingly, it can be noted from EQ (13), that the principal components can be obtained by 1 ) eigen-decomposing the autocorrelation matrix and then projecting the data, or by 2) eigen-decomposing the Gram matrix directly. Notably, the latter method of using the Gram matrix is a data independent approach. [00031] Moreover, as shown in EQ (6), there exists a scalar nonlinear mapping η{-) which maps the signal as [ ⁷ J _x Q)J = l,2,...,N+L—ϊ\ while preserving the similarity measure

E[^(O - ⁷ J _x U)] = E[κtx(ϋ - *( J))] (14)

[00032] As shown in EQ (14), correntropy includes evaluating a similarity measure of time shifted input data x(i) - x(i + t) , and applying a Gaussian kernel, K, to the similarity measure to generate entries of the non-linear autocorrelation matrix ⁷ J _x (O ¹ J(J) _x - Notably, the autocorrelation function of ⁷ J _x (O is given by the correntropy function of x. It should be noted that EQ (14) provides a correntropy extension to temporal PCA. More specifically, the correntropy extension to TPCA replaces autocorrelation entries of EQ(8) with correntropy entries from EQ (14). Furthermore, the principal components can be obtained by eigen-decomposition of the Gram matrix, K, directly in a data independent manner.

[00033] Referring to FIG. 4, a method 300 for correntropy principal component analysis (CPCA) is shown. The method 300 can be practiced with more or less than the number of steps shown. Moreover, the method 300 is not limited to the order shown in FIG. 4. The method 300 can start in state wherein the processor 100 receives the noisy signal 101 having a low level periodic signal 141 corrupted by noise. [00034] At step 310, the processor 100 (FIG 1 ) can perform a correntropy analysis on the noisy signal 141 to generate a Gram matrix in accordance with the method steps of FIG. 2. In particular, the processor 100 can apply a correntropy kernel (e.g. Gaussian kernel) to the input data to generate the Gram matrix as shown in step 31 1. Step 31 1 is a specific implementation of correntropy analysis that corresponds to method step 210 of FIG. 2 for generating a non-linear mapping. Briefly referring to FIG. 5, method step 31 1 can include applying a Gaussian kernel to map the input data to a reproducing kernel Hubert Space (RKHS). As shown in EQ (13), the non-linear autocorrelation matrix is generated by applying a Gaussian kernel to the input data. The Gaussian kernel produces a non-linear mapping of the input data.

[00035] Returning back to FIG. 4, at step 320, the processor 100 can perform Principal Component Analysis (PCA) on the Gram matrix to produce correntropy principal components. In particular, the processor 100 can eigen-decompose the Gram matrix to produce eigen vectors of the principal components as shown in step 321. Step 321 is a specific implementation of applying a subspace projection that corresponds to method step 220 of FIG. 2. The non-linear mapping of applying the correntropy kernels implicitly projects non-linearly the input data to a reproducing kernel Hubert space

(RKHS), which is a linear space. Temporal PCA is performed in the RKHS for distributing noise among principal components. Because RKHS is a linear space, measures of similarity such as distance in the input data space are preserved. Moreover, using a correntropy kernel to non-linearly map the input data to a RKHS allows for the RKHS to have a same dimension as the input data. Furthermore, the correntropy kernel is independent of the input data as shown by EQ (13). [00036] At step 330, upon determining the principal components, the processor 101 can identify the principal component having the highest energy. Step 360 is a specific implementation for choosing a subspace that corresponds to method step 230 of FIG. 2. Recall, TCPA equally distributes the noise energy among the principal components while preserving the periodic signal in across one, or only a few, principal components. At step 370, the processor 101 can perform a a power spectral density (PSD) of the correntropy on a high energy principal component. As an example, a Fast Fourier Transform (FFT) may be applied to the principal component in a digital domain to produce a spectrum. As another example, a bank of band-pass filters can be applied to the principal component in the analog domain to produce a spectrum. At step 380, the processor 101 can identify a maximum peak in the spectral analysis. [00037] For example, FIG. 6 shows an exemplary FFT of a principal component. The FFT reveals a peak 385 that corresponds to a frequency of the periodic signal 141. A threshold detector can be employed to identify peaks in the spectrum. For example, any frequency components that exceed a threshold 386 can be considered a candidate for the periodic signal 141. Notably, the peak is enhanced (e.g. magnitude) due to a higher signal to noise ratio of the principal component as a result of TCPA. Moreover, the principal component was generated directly from the noisy signal 101 using correntropy in a data independent manner.

[00038] The method of correntropy TPCA can be applied to various applications such as communications, surveillance, and medicine. For instance, the periodic signal can be a communication signal, a surveillance signal such as an image, or a monitoring signal which may not even have a time structure (multidimensional random variables). As one example, correntropy TPCA can be used to detect a number of users in an allocated frequency band of a communication signal prior to assigning new users to the allocated

frequency band. As a practical example, correntropy TPCA (CTPCA) can be applied to a sinusoidal signal corrupted with impulsive noise for illustrative purposes

x(m) = sin(2π fm) + A - z(m) (15)

for m = l, 2,..., N + L-l . z(jή) is a white noise process drawn from the PDF

^(n) = o.8xN(o,o.i) +o.ixN(4,o.i) +o.ixN(-4,o.i) (16)

In the example, N = 256 , / = o.3 and 3ν data is generated to estimate N point autocorrelation and correntropy functions. In TPCA, a larger subspace (larger L) produces a higher the Signal Noise Ratio (SNR). For a fair comparison, an N-by-N Gram matrix is eigen-decompose for both conventional TPCA and correntropy TPCA. Results of eigen-decomposing an L-by-L autocorrelation matrix and then projecting the data are also presented for comparison. For each case, 1000 Monte Carlo trials with different noise realizations are run to evaluate the improvement of CTPCA over TPCA. For A = 5 , a probability of detecting the sinusoidal signal successfully as the largest peak in the spectrum is 100% for CTPCA, compared with 15% for N-by-N autocorrelation. The kernel size is set to σ = 1 in CTPCA throughout the simulation. In this particular application of finding sinusoids in noise, the kernel size can be scanned until the best line spectrum is obtained.

[00039] In CTPCA, the transformed data may not be centered in the feature space and the mean can introduce a large DC component that is picked as the first principal component. This can be shown as follows. We denote C as the correntropy matrix,

C _c as the centered correntropy matrix, ^m _η as the mean of the transformed data, i _N as an N-by-1 column vector with all entries equal to 1 and l _NxN as an N-by-N matrix with all entries equal to 1. Thus,

C _c = C - m^l _N - V _N (17)

[00040] If the eigenvector I _N is normalized to a unit norm, the corresponding eigenvalue is Nm* which can be very large if N is large as in this example. Therefore, the first principal component of the correntropy matrix in this example is always a DC component. Accordingly, a second principal component can be used in place of the first principal component. Another way to center the correntropy matrix is to estimate the square of the mean in (17) directly by the Information Potential (IP) as shown in EQ (4) to EQ (6).

C _C = C -\ _NXN - CIN - C λ _NXN IN +\ _NXN - C λ _NXN IN ² ₍ i ₈ )

[00041] FIG. 7 shows a spectral analysis using CTPCA for the sinusoidal signal EQ (15) corrupted with impulsive noise of EQ (16). Subplot 410 shows the observation spectrum 41 1 which is the sinusoidal signal of EQ (15) corrupted with noise, and the "clean" spectrum 412 of EQ (15). Notably, the clean spectrum 412 shows a strong line spectrum at 0.3 Hz which corresponds to the fundamental frequency of the sinusoidal signal. In contrast, the observation spectrum 41 1 is corrupted with high amplitude noise and essentially buries the line spectrum component of the sinusoidal signal. That is, the observation spectrum does not noticeably reveal a presence of the sinusoidal component 412.

[00042] Subplot 420 shows an FFT of the first principal component of CTPA. A small amplitude line spectrum occurs at the frequency of the sinusoidal signal. However, the magnitude of the line spectrum is not significantly higher than the average spectral magnitude, and is not suitable for robust detection.

[00043] Subplot 430 shows an FFT of the second principal component of CTPA (i.e. centering CPCA). Notably, the line spectrum of the sinusoidal signal is more pronounced than the line spectrum in Subplot 420, and provides for more robust detection.

[00044] FIG. 8 shows a comparison of PCA methods: Correntropy TPCA (CTPCA), Centering CTPCA, PCA by NxN Gram R, and PCA by L-by-L autocorrelation matrix with various lengths. Results of the simulation indicate that the method of CTPCA and Centering CTPCA outperforms the other evaluated methods.

[00045] FIG. 9 presents a table showing the effects of kernel size on correntropy CTPCA. Results of the simulation indicate that a kernel size of 1 provides the highest detection performance.

[00046] Detailed embodiments of the present method and system have been disclosed herein. However, it is to be understood that the disclosed embodiments are merely exemplary, and that the invention can be embodied in various forms. Therefore, specific structural and functional details disclosed herein are not to be interpreted as limiting, but merely as a basis for the claims and as a representative basis for teaching one skilled in the art to variously employ the embodiments of the present invention in virtually any appropriately detailed structure. Further, the terms and phrases used herein are not intended to be limiting but rather to provide an understandable description of the embodiment herein.

[00047] Where applicable, the present embodiments of the invention can be realized in hardware, software or a combination of hardware and software. Any kind of computer system or other apparatus adapted for carrying out the methods described herein are suitable. A typical combination of hardware and software can be a mobile communications device with a computer program that, when being loaded and executed, can control the mobile communications device such that it carries out the methods described herein. Portions of the present method and system may also be embedded in a computer program product, which comprises all the features enabling the implementation of the methods described herein and which when loaded in a computer system, is able to carry out these methods.

[00048] The term "processing" can be defined as number of suitable processors, controllers, units, or the like that carry out a pre-programmed or programmed set of instructions. The terms "program," "software application," and the like as used herein, are defined as a sequence of instructions designed for execution on a computer system. A program, computer program, or software application may include a subroutine, a function, a procedure, an object method, an object implementation, an executable application, a source code, an object code, a shared library/dynamic load library and/or other sequence of instructions designed for execution on a computer system.

[00049] For example, the above-discussed embodiments may be implemented using software modules which perform certain tasks. The software modules discussed herein may include script, batch, or other executable files. The software modules may be stored on a machine-readable or computer-readable storage medium such as a disk drive. Storage devices used for storing software modules in accordance with an embodiment of the invention may be magnetic floppy disks, hard disks, or optical discs such as CD-ROMs or CD-Rs, for example. A storage device used for storing firmware or hardware modules in accordance with an embodiment of the invention may also include a semiconductor-based memory, which may be permanently, removably or remotely coupled to a microprocessor/memory system. Thus, the modules may be stored within a computer system memory to configure the computer system to perform the functions of the module. Other new and various types of computer-readable storage media may be used to store the modules discussed herein.

[00050] While the preferred embodiments of the invention have been illustrated and described, it will be clear that the embodiments of the invention are not so limited. Numerous modifications, changes, variations, substitutions and equivalents will occur to those skilled in the art without departing from the spirit and scope of the present embodiments of the invention as defined by the appended claims.