Blind Components Processing

Blind Components Processing A Novel Approach to Array Signal Processing A Research Orientation

Mahdi Khosravy†, Neeraj Gupta† , Ninoslav Marina† Faramarz Asharif‡ , Mohammad Reza Asharif ‡ and Ishwar K. Sethi⋆ , † University of Information Science and Technology “Sant Paul the Apostle”, Ohrid, Macedonia Emails: {mahdi.khosravy, neeraj.gupta, Rector}@uist.edu.mk ‡ Information Department, Faculty of Engineering University of the Ryukyus, Okinawa, Japan Emails: faramarz [email protected], [email protected] ⋆ Department of Computer Science and Engineering Oakland University in Rochester, Michigan, USA Email: [email protected] Abstract—Blind Components Processing (BCP), a novel approach in processing of data (signal,image, etc.) components, is introduced as well some applications to information communications technology (ICT) are proposed. The newly introduced BCP is with capability of deployment orientation in a wider range of applications. The fundamental of BCP is based on Blind Source Separation (BSS), a methodology which searches for unknown sources of mixtures without a prior knowledge of either the sources or the mixing process. Most of the natural, biomedical as well as industrial observed signals are mixtures of different components while the components and the way they mixed are unknown. If we decompose the signal into its components by BSS, then we can process the components separately without interfering the other components signal/data. Such internal access to signal components leads to extraction of plenty of information as well more efficient processing compared to normal signal processing where in all the structure of the signal is gone under processing and modification. This manuscript besides the introducing blind component processing, proposes some practical application with technical merit.

I.

I NTRODUCTION

The proposed idea of Blind Components Processing (BCP) is individual processing of signal unknown components without any interference with other component of the signal. Every signal, every image and in general every observed phenomena in this world is a mixture of unknown (blind) independent sources as we call them“Blind Components”. As an example, after separation of the light into its components, using a prism, a rainbow of colors appears. A lot of natural events around us are composed of unknown natural sources, while we just observe the result combination of these sources. Theses phenomenon includes a wide range from genomes to multiuser communications signals. Blind Source Separation (BSS)[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] is the mean to extract those original sources and discovers the components of the

fundamental phenomenon involved. BSS does not need a priori knowledge of neither the mixing process nor the sources. Indeed, BSS blindly separates the mixed sources. BSS is now one of the emerging areas in Signal/Image Processing, Information and Communications Technology with a variety of applications such as crosstalk rejection in 3D-stacked inter-chip communication [13], muscle artifact removal in scalp-EEG [14], Automated identification of cardiac signals [15],brian imaging [16], multichannel MAC protocols [17], detection of medial temporal discharges [18], Indoor vehicle pass-by noise applications [19], audio watermarking [20], Audio Recognition Scheme in Multiple SoundSources Environment [21], remote sensing images [22], peak detection [23], Echo Cancellation [24], image mosaic [25], blind MIMO-OFDM system [26], [27], etc. Here, in this manuscript a BSS-based general methodology is illustrated call Blind Component Processing (BCP). BCP enables us for different applications instead of applying the processing methods on whole the data/signal to apply the techniques on a desired component without any interference from/to other component. Note, BCP provides higher efficient processing since first it stop the interference of the other nonobjected components during the processing. Indeed by the separation of the components, the processing is done in the absence of the non-objected components and it results in more efficient processing. Second, since the non-objected components are zero affected by the processing, the processing is with minimum side effects of any non-desired effect. A very clear example of this matter is noise cancellation where in always some parts of signal is removed too along with noise cancellation. The contribution of the paper is as follows. First, a short introduction to BSS is presented in section 2. Then, Section 3 clarifies the idea of Blind Component Processing. Section 4, propose a number of applications for the introduced BCP, and section 5 concludes the

s1 (t)

A

s2 (t)

x1 (t)

y1 (t)

x2 (t)

y2 (t) De-mixing Process

sK (t) Blind Mixing

Blind Sources

Observed Sequences

xM (t)

W

yK (t) Recovered Sources

Fig. 2. The block diagram of BSS problem.

x(t) =  x1 (t)  x2 (t)   .  =  ..  

Fig. 1. A typical cocktail party problem wherein a listener attempts to focus upon his partner conversation as a single speaker in a scene of other speakers.

paper. II.

B LIND S OURCE S EPARATION

The problem of un-mixing mixtures to their unknown sources is known as blind source separation (BSS). A good and intuitive example is “cocktail party problem (Figure 1). It is the term commonly used to describe the problem experienced in a party by a listener who attempts to focus upon his partner conversation as a single speaker in a scene of other speakers. In such situation, human brain successfully separates the desired talk from a mixture of undesired interfering ones. As well, the statistical DSP approach, BSS, as a blind identification technique can offer an adaptive, intelligent solution to the “cocktail party problem”. Audio signals can be blindly retrieved from the mixture, that is, without a prior knowledge of the audio sources and sensors. When sources are statistically independent, their mixtures are not independent. This is because each source is shared between both mixtures. A BSS strategy is to extract the sources by maximizing a measure of independence[1]. A part from independent maximization approach to BSS, there is a number of other strategies too. One of them is Stone’s BSS[28] based on predictability maximization. The author has established a theory foundation and generalization for Stones BSS in [29]. A. Blind Source Separation Formulations As basic model of BSS, K unknown source signals upon transmission through a medium have been linearly mixed together and mixture signals are collected by M sensors. The source signals are unknown and also no information is available about the mixing process. In mathematical model, the M × 1 vector x of observed signals x(t) = (x1 (t) x2 (t) · · · xM (t))T is multiplication of K × 1 vector s of unknown source signals s(t) = (s1 (t) s2 (t) · · · sK (t))T by unknown mixing matrix A ∈ RM×K :

xM (t)

As(t) + n(t)     n1 (t) s1 (t)  s2 (t)   n2 (t)     A  ...  +  ... 

(1) (2)

nK (t)

sK (t)

where n(t) is a possible additive noise. Given only the observation signals x(t) = (x1 (t) x2 (t) · · · xM (t))T , the solution of BSS problem seeks for the best W ∈ RK×M as un-mixing matrix to extract signals as much as possible close to unknown source signals as follows: y(t)  y1 (t)  y2 (t)   .   ..  

yK (t)

= Wx(t)   x1 (t)  x2 (t)   = W  ... 

(3) (4)

xM (t)

where y(t) is K × 1 vector of extracted signals y(t) = (y1 (t) y2 (t) · · · yK (t))T . Each row of W is a 1 × M un-mixing vector wi related to one of extracted signals. Figure 2 shows the block diagram of a general BSS problem. III.

B LIND C OMPONENT P ROCESSING

This section introduces and clarifies the methodology of Blind Components Processing (BCP). As an example, let x(t) is a acoustic signal, and x1 (t), x2 (t), · · · , xM (t) are different measurement signals of x(t) in M different views. Supposing every chosen ith view Ti [.] is a signal transfer (linear or non-linear) that its reverse transfer Ti−1 [.] is available. Different views of a signal can be different records by different instruments, with different transfer functions. Here, we consider the M observed signals in different views as M mixtures of signal components. Then, by applying blind source separation, we will obtain K component signals as follows;     yˆ1 (t) x1 (t)  yˆ2 (t)   x2 (t)   . =W ˆ  .  (5)  ..   ..  yˆK (t)

xM (t)

ˆ is an approximation to un-mixing matrix where W obtained by BSS and yˆi are approximated components (W ∈ RK×M & M > K). Later we will see that as a great advantage of blind component processing, it is not sensitive to the accuracy of BSS method.

T1 [.]

x1 (t)

T2 [.] x(t)

x2 (t)

yˆ1 (t) Blind

yˆ2 (t)

Source Separation

TM [.] xM (t)

ˆ M ×K yˆK (t) W

P1 [.] P2 [.]

PK [.]

yˆ1P (t)

xCP 1 (t)

yˆ2P (t)

Reverse

xCP 2 (t)

Mixing Process

ˆ −1 W

P (t) yˆK

xCP M (t)

T1−1 [.] T2−1 [.]

Σ

xCP (t)

−1 TM [.]

Components Information

Fig. 3. Block diagram of blind component processing.

By having the approximated components, we can process each component yˆi individually and separately by different processing approaches Pi [.]; yˆiP (t) = Pi [ˆ yi (t)]

(6)

The acquired Pi [.] depends on the signal information that we are looking for to detect, extract, cancel, or etc. At this stage the processed components can be remixed ˆ −1 in order to remake the x signal views but with by W internally process components;  CP   P  x1 (t) yˆ1 (t) xCP   yˆ2P (t)  (t) 2    ˆ −1  (7)  ..  = W  ..   .   .  xCP M (t)

M X

T −1 [xCP i (t)]

BCP has a number of advantages as follows;

•

•

IV.

BCP

APPLICATION CANDIDATES

This section proposes some applications wherein BCP can efficiently plays role. The role of BCP is not limited to these applications, but also a wide range of array processing is covered by BCP application. The BCP applications illustrated in this sections are just a way to clarify its capability. A. Harmonic Noise Cancellation at Components Level

(8)

i=1

•

Forth, the ambiguities inherent to BSS like permutation alignment and scaling ambiguity are not problem anymore. Because of deploying the reverse of the same un-mixing matrix for remixing the components after their process, the permutation and scaling of the components will be returned back to the initial sort of the array. Indeed, every component with the same scale and permutation will return to the mixture but it would be individually processed.

Figure 3 shows the general block diagram of blind component processing.

P yˆK (t)

where the indexes .P and .CP respectively indicate “Processed” and “Component Processed”. Applying the reverse transfer of each view T −1 [.] to each component processed view xCP i (t), gives us a component processed version of x as T −1 [xCP i (t)]. We consider their average as final “Blind Component Processed” x; xCP =

•

First and the most, the processing does not affect the non-objective components of the signal. In this way we have the minimum side effects of any processing and modifications applied to the signal. Second, as well as not the other components being affected by the processing, their presence does not interference the processing efficiency. Indeed, in the absence of the other components, the processing is more efficient. Third, BCP is not sensitive to the accuracy of the deployed BSS methodology. If the BSS methods approximately works well, component processing will sufficiently works. It is because any obtained approximate to un-mixing matrix ˆ to be deployed, its reverse W ˆ −1 will be W used for remixing the components. Therefore the initial array will be recovered after even not a perfect BSS applications on the array. However, a more efficient BSS result in more clear advantages of BCP.

In many applications, harmonic noise always degrades the quality of desired signal. Harmonic noise is made of unwanted periodic interference signals. Here are some examples; •

Power generator interference in hybrid cars: In RF, antenna picks up desired signal with harmonic interference on the same band. For example in hybrid cars, the power generator (power control unit PCU) for charging the battery, produces harmonic signal which is near to AM-RF radio band (500-1600 kHz).

•

Engine speed oscillations for hybrid vehicles: The periodic fuel combustion in cylinders in conventional internal combustion engines and oscillating masses result in pulsating engine torque. In hybrid powertrains, an electrical motor is proposed to control such pulsations [30].

•

Power line 50-60 Hz disturbance: Another well-known harmonic noise is power line frequencies disturbances in the processing of the electrocardiogram (ECG) [31].

•

Electronic devices harmonic noise: Harmonic noise produced by electronic devices corrupts

x1 (t) x2 (t)

Blind

yˆ2 (t)

Source Separation

xM (t)

ˆ M ×K yˆK (t) W

yˆ1periodic (t) Identifying harmonic noise components & setting them to zero.

HAF1

yˆ2periodic (t) HAF2 periodic yˆK (t)

HAFK

Periodic Components

yˆ1non-periodic(t) Harmonic Adaptive Filtering of the recovered components

yˆ2non-periodic(t)

non-periodic yˆK (t)

Non-periodic Components

xˆ1 (t)

+ Reverse

+

xˆ2 (t)

Mixing Process

ˆ −1 W +

xˆM (t)

Harmonic noise canceled

Multi-sensory records

yˆ1 (t)

Fig. 4. Block diagram of BCP based Harmonic Noise Cancellation.

image signal that is sent to earth from a space discovery shuttle robots [32]. Here, we propose the newly introduced “Blind Component Processing” with internally employed “Harmonic Adaptive Filters” (HAF) to extract different periodic components by adaptively synthesizing them via following the components. Harmonic Adaptive Filtering (HAF) is a blind method to synthesis and separate the periodic part of the signal out of the scope of this paper, but under work by the author [33]. Let’s assume having at least two different synchronized records of the signal contaminated by harmonic noise. For example in the case of harmonic noise of power generators in hybrid cars, we can deploy two receiver antennas for AM-FM radios in different distance and view angles. In order to apply BCP, we directly apply BSS to the sensors records array as x1 (t), x2 (t), · · · , xM (t). Due to separation by BSS demixing matrix W, we have the estimated blind components, y1 (t), y2 (t), · · · , yK (t). At this stage periodic components extraction is done. To aim this goal, the “Harmonic Adaptive Filter” decomposes each compoperiodic nent ayi (t) into a periodic component yi (t) and a non-periodic non-periodic component yi (t). Therefore, we have two sets of components at this stage. One set contains the periodic components and the other set contains the non-periodic components. Among the periodic components, some of them belong to Harmonic noise and the others belong to original content of the signal. At this stage by having a prior knowledge of harmonic noise possible contents, the periodic components of the harmonic noise are detected and set to zero in remixing back the signal components. The remixed signal components in the absence of periodic components of harmonic noise will shape a harmonic noise canceled signal. Figure 5 shows block diagram of harmonic noise cancellation by harmonic adaptive filtering at components level.

It may be asked that “Is not possible to apply the process of harmonic adaptive noise detection directly over the signal and not going through the BCP steps?” The answer is “Yes”. It is possible. But since it might be overlap between frequency contents of harmonic noise and frequency content of the signal, signal content would be partially lost by directly applying “HAF”. It is expected that BCP based harmonic adaptive filtering would conclude higher efficiency. B. Stock pricing model aided by BCP While emerging markets suffer by huge frequent changes and well developed markets bubble burst effect, the resultant heavy tails violate the assumption of normality of stock data for both types of market. By assuming a linear time invariance behavior for stock options, the heavy tails can be removed at blind component level by deploying BCP. The stock with heavy tails removed in components would matches better with available stock pricing models because of their higher normality. To aim this goal, we consider the following assumptions. •

Assumption 1: Stock options as linear time invariant systems. Stock options behavior is considered as linear time invariant systems.

The assumption of time invariance can be fair enough for a short period of time while the properties and conditions of economical system does not change. As an example, the working quality of staff, the quality of products, the demand for a brand in a market and etcetera can not be a subject of sudden change. As well the assumption of linearity is subjected to the volume of changes put in system. Normally the response of economical systems to reasonable changes is linear. If all the cost imposed to an economical systems becomes twice, we can expect the outputs of the system to be twice more expensive.

Stock as an LTI system

Shock

Stock shock response

Fig. 5. Stock response to shock as decaying sparse response in an LTI system.

=( Total stock response to shock

)+( Normal stock price

) Separated shock response

Fig. 6. Total stock price as superposition of normal stock price and response to huge sudden change (heavy tail).

•

Assumption 2: The same shock response with different amplitude and direction. Since all economic components belong to the same economical system, their responses to a shock (like huge economical changes or bubble burst) are considered the same for all of them but with different magnitudes and phases (direction).

The magnitude of the huge changes effect depends on the stock type and its values. The decrement or increment directions depend on the stock type too and the way it is positively or negatively impacted. Considering the two above assumptions, the individual response of stock price to huge sudden changes (heavy tails) can be considered as impulse response of a linear system. Such response is linearly added to the sequence of normal stock price. Heavy tails effect similar to impulse response of a linear systems is assumed as a decaying sparse sequence as it is shown in Figure 5. Considering Assumption 1, the total stock response to shock can be expressed as superposition of normal stock price and the shock response. However, as it is shown in Figure, the sequences of stock response to shock and normal stock price sequence are unknown. The observation is total stock price. Blind source separation (BSS) is deployed to catch the components of stock price sequences and extract the shock effect from stock. BSS works over multi-variable data sequences, and fortunately the stock data of a market is multivariable. First, we apply BSS over multi variate stock data. We are looking for detection and extraction of shock effect among the blind components. Considering assumption 1 and 2 together, we consider each stock sequence as the following linear combination of normal stock data and the shock effect; xi = si + βi h th

(9)

where xi is i stock price sequence. si is its normal stock price sequence without shock effect and βi h is the shock response of ith stock system based on Assumption 1 and Assumption 2. βi is the coefficient of the shock

effect for ith stock sequence. Based on Assumption 2, h has been considered the same for all stock sequences of the same market. All the components and parameters on right side of Equation (9) are unknown(blind). The Equation. 9 can be rewritten as follows for all stock multivariate data;      x1 (t) s1 1 0 ... 0 β1 CP β2   ..  x2 (t) 0 1 . . . 0  .  (10)  .  = . . . ..   .   .. .. . . ... .  sM  . h 0 0 . . . 1 βM (t) xM (t) It is summarized in vector notation as x = As

(11)

where x is the vector of all M stock price sequences. A is a M × (M + 1) mixing matrix with unknown βi parameters. A mixes normal stocks with shock response h where all indicated in vector s. Equation 11 with blind parameters on the right side matches well with Equation 2 as the basic equation of BSS problem. However, A in equation 10 is more specified than just an unknown mixing matrix. In order to obtain A as M × (M + 1) matrix with a diagonal M ×M on its left side and last column of βi values on its right side, we need an over-complete approach of BSS. Since, we aim to separate just h as heavy tails effect from stock data, and we are not looking for recovering si components, a non-over-complete BSS can efficiently perform. h has a sparse nature and it brings more sparsity and less normality to stock price data, the BSS approaches which separates the components based on sparsity maximization or normality minimization can separates h than other components in a better way. After separation of components by BSS, the components are remixed back by the inverse of A in the absence of h. The remixed components without h are expected to have higher normality and better match with classical models of stock price based on random walk. The block diagram of removal of shock effect at blind component processing level has been shown in Figure 7. It is a kind of blind

yˆ1 (t)

x2 (t)

Blind

yˆ2 (t)

Source Separation

xM (t)

ˆ M ×K yˆK (t) W

Replacing the highest sparse component with zero sequence.

x1 (t)

yˆ1P (t) yˆ2P (t)

xCP 1 (t) Reverse

xCP 2 (t)

Mixing Process P yˆK (t)

ˆ −1 W

xCP M (t)

Fig. 7. Removal of stock shock effect at blind components level.

component processing wherein the middle processing block is finding the component with highest sparsity and replacing it with a zero sequence before remixing back the components. It is expected that the remixed stock data with removed shock effect by blind component processing have a higher normality than initial stock price data. After modeling the stock price by classical methods over shocked removed stock, the shock effect can be added to the modeled data at components level by keeping the mixing and de-mixing matrices from former blind component processing stage. V.

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C ONCLUSION

The proposed approach of Blind Component Processing enables us to internally access the signal blind components. Such an internal access to the signal components brings us the ability of processing and extracting signal information without interference of other components. As well, the processing can be componentobjective which means processing is done just over the object component and the other component are secured of processing side effects. A more efficient deployed blind source separation will result in a more efficient blind component processing. However, blind component processing is not sensitive to BSS efficiency since the inverse of the same un-mixing matrix is used. Blind component processing makes a new horizon not only for the proposed applications at this articles (efficient harmonic noise cancellation and stock price modeling), but also for all fields of digital array signal processing. R EFERENCES [1]

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