Signal separation

<h2 id="applications">Applications</h2>

<h3>Cocktail party problem</h3>
<p class="note">See also: <a href="/facts/Cocktail_party_effect/qnGLrtpr">Cocktail party effect</a></p>
<p>At a cocktail party, there is a group of people talking at the same time. You have multiple microphones picking up mixed signals, but you want to isolate the speech of a single person. BSS can be used to separate the individual sources by using mixed signals. In the presence of noise, dedicated optimization criteria need to be used.
</p>
<h3>Image processing</h3>

<p>Figure 2 shows the basic concept of BSS.  The individual source signals are shown as well as the mixed signals which are received signals. BSS is used to separate the mixed signals with only knowing mixed signals and nothing about original signal or how they were mixed.  The separated signals are only approximations of the source signals.  The separated images, were separated using <a href="/facts/Python_(programming_language)/YbuGqofa">Python</a> and the <a href="/facts/Shogun_(toolbox)/46cERwRK">Shogun toolbox</a> using Joint Approximation Diagonalization of Eigen-matrices (<a href="/facts/Joint_Approximation_Diagonalization_of_Eigen-matrices/SpKC2yne">JADE</a>) algorithm which is based on <a href="/facts/Independent_component_analysis/gsdVaDFR">independent component analysis</a>, ICA.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>   This toolbox method can be used with multi-dimensions but for an easy visual aspect images(2-D) were used.
</p>
<h3>Medical imaging</h3>
<p>One of the practical applications being researched in this area is <a href="/facts/Medical_imaging/pL6JyJEp">medical imaging</a> of the brain with <a href="/facts/Magnetoencephalography/epvUfx7A">magnetoencephalography</a> (MEG). This kind of imaging involves careful measurements of <a href="/facts/Magnetic_field/Ta8Ev588">magnetic fields</a> outside the head which yield an accurate 3D-picture of the interior of the head. However, external sources of <a href="/facts/Electromagnetic_field/MacraS8y">electromagnetic fields</a>, such as a wristwatch on the subject's arm, will significantly degrade the accuracy of the measurement. Applying source separation techniques on the measured signals can help remove undesired artifacts from the signal.
</p>
<h3>EEG</h3>
<p>In <a href="/facts/Electroencephalogram/XGvpcBl5">electroencephalogram</a> (EEG) and <a href="/facts/Magnetoencephalography/epvUfx7A">magnetoencephalography</a> (MEG), the interference from muscle activity masks the desired signal from brain activity. BSS, however, can be used to separate the two so an accurate representation of brain activity may be achieved.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h3>Music</h3>
<p>Another application is the separation of <a href="/facts/Music/hDVP0xsQ">musical</a> signals. For a stereo mix of relatively simple signals it is now possible to make a fairly accurate separation, although some <a href="/facts/Sonic_artifact/RYTxFVnj">artifacts</a> remain.
</p>
<h3>Others</h3>
<p>Other applications:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<ul><li>Communications</li>
<li>Stock Prediction</li>
<li>Seismic Monitoring</li>
<li>Text Document Analysis</li></ul>
<h2 id="mathematical-representation">Mathematical representation</h2>

<p>The set of individual source signals, 
  
    
      
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    {\displaystyle s(t)=(s_{1}(t),\dots ,s_{n}(t))^{T}}
  
, is 'mixed' using a matrix, 
  
    
      
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, to produce a set of 'mixed' signals, 
  
    
      
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    {\displaystyle x(t)=(x_{1}(t),\dots ,x_{m}(t))^{T}}
  
, as follows.  Usually, 
  
    
      
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 is equal to 
  
    
      
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    {\displaystyle m}
  
. If 
  
    
      
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, then the system of equations is overdetermined and thus can be unmixed using a conventional linear method. If 
  
    
      
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    {\displaystyle n>m}
  
, the system is underdetermined and a non-linear method must be employed to recover the unmixed signals. The signals themselves can be multidimensional.
</p><p>
  
    
      
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    {\displaystyle x(t)=A\cdot s(t)}

</p><p>The above equation is effectively 'inverted' as follows. Blind source separation separates the set of mixed signals, 
  
    
      
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    {\displaystyle x(t)}
  
, through the determination of an 'unmixing' matrix, 
  
    
      
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    {\displaystyle B=[B_{ij}]\in \mathbb {R} ^{n\times m}}
  
, to 'recover' an approximation of the original signals, 
  
    
      
        y
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    {\displaystyle y(t)=(y_{1}(t),\dots ,y_{n}(t))^{T}}
  
.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p><p>
  
    
      
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    {\displaystyle y(t)=B\cdot x(t)}

</p>
<h2 id="approaches">Approaches</h2>
<p>Since the chief difficulty of the problem is its underdetermination, methods for blind source separation generally seek to narrow the set of possible solutions in a way that is unlikely to exclude the desired solution. In one approach, exemplified by <a href="/facts/Principal_components_analysis/pCbVTqdR">principal</a> and <a href="/facts/Independent_components_analysis/gsdVaDFR">independent</a> component analysis, one seeks source signals that are minimally <a href="/facts/Correlation/egkluAEm">correlated</a> or maximally <a href="/facts/Independence_(probability)/NUzQtnUL">independent</a> in a probabilistic or <a href="/facts/Information_theory/qNLH3U5P">information-theoretic</a> sense. A second approach, exemplified by <a href="/facts/Nonnegative_matrix_factorization/DIBLH8B0">nonnegative matrix factorization</a>, is to impose structural constraints on the source signals. These structural constraints may be derived from a generative model of the signal, but are more commonly heuristics justified by good empirical performance. A common theme in the second approach is to impose some kind of low-complexity constraint on the signal, such as <a href="/facts/Sparsity/FFtN0XjR">sparsity</a> in some <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a> for the signal space. This approach can be particularly effective if one requires not the whole signal, but merely its most salient features.
</p>
<h3>Methods</h3>
<p>There are different methods of blind signal separation:
</p>
<ul><li><a href="/facts/Principal_components_analysis/pCbVTqdR">Principal components analysis</a></li>
<li><a href="/facts/Singular_value_decomposition/8t3v6wk3">Singular value decomposition</a></li>
<li><a href="/facts/Independent_component_analysis/gsdVaDFR">Independent component analysis</a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> <a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a></li>
<li><a href="/facts/Dependent_component_analysis/jp5afPwx">Dependent component analysis</a></li>
<li><a href="/facts/Non-negative_matrix_factorization/DIBLH8B0">Non-negative matrix factorization</a></li>
<li>Low-complexity coding and decoding</li>
<li><a href="/facts/Stationary_subspace_analysis/GEQmjCMk">Stationary subspace analysis</a></li>
<li><a href="/facts/Common_spatial_pattern/vAtdPJNw">Common spatial pattern</a></li>
<li><a href="/facts/Canonical_correlation_analysis/LuX8FltG">Canonical correlation analysis</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Adaptive_filtering/SzNepmBJ">Adaptive filtering</a></li>
<li><a href="/facts/Celemony_Software/AtDEpNj3">Celemony Software#Direct Note Access</a></li>
<li><a href="/facts/Colin_Cherry/4rUHq61R">Colin Cherry</a></li>
<li><a href="/facts/Deconvolution/Q5ECQykB">Deconvolution</a></li>
<li><a href="/facts/Factorial_code/5eQdusgX">Factorial codes</a></li>
<li><a href="/facts/Infomax_principle/6JCHoxdF">Infomax principle</a></li>
<li><a href="/facts/Segmentation_(image_processing)/jofAhbxa">Segmentation (image processing)</a></li>
<li><a href="/facts/Speech_segmentation/UCoL3rfQ">Speech segmentation</a></li></ul>

<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Source separation.

Wikimedia Commons has media related to Blind signal separation.

<ul><li><a href="http://www.cis.hut.fi/projects/ica/">Explanation of Independent Component Analysis (ICA)</a></li>
<li><a href="http://www.volker-koch.com/diss/">A tutorial-style dissertation by Volker Koch that introduces message-passing on factor graphs to decompose EMG signals</a></li>
<li><a href="https://web.archive.org/web/20061128090454/http://visl.technion.ac.il/demos/bss/">Blind source separation flash presentation</a></li>
<li><a href="http://apps.dtic.mil/dtic/tr/fulltext/u2/a455940.pdf">Removing electroencephalographic artifacts by blind source separation</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Kevin Hughes “Blind Source Separation on Images with Shogun” http://shogun-toolbox.org/static/notebook/current/bss_image.html <a href="http://shogun-toolbox.org/static/notebook/current/bss_image.html" target="_blank">http://shogun-toolbox.org/static/notebook/current/bss_image.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Aapo Hyvarinen, Juha Karhunen, and Erkki Oja. “Independent Component Analysis” https://www.cs.helsinki.fi/u/ahyvarin/papers/bookfinal_ICA.pdf pp. 147–148, pp. 410–411, pp. 441–442, p. 448 <a href="https://www.cs.helsinki.fi/u/ahyvarin/papers/bookfinal_ICA.pdf" target="_blank">https://www.cs.helsinki.fi/u/ahyvarin/papers/bookfinal_ICA.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Congedo, Marco; Gouy-Pailler, Cedric; Jutten, Christian (December 2008). "On the blind source separation of human electroencephalogram by approximate joint diagonalization of second order statistics". Clinical Neurophysiology. 119 (12): 2677–2686. arXiv:0812.0494. doi:10.1016/j.clinph.2008.09.007. PMID 18993114. S2CID 5835843. <a href="https://hal.archives-ouvertes.fr/hal-00343628" target="_blank">https://hal.archives-ouvertes.fr/hal-00343628</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Aapo Hyvarinen, Juha Karhunen, and Erkki Oja. “Independent Component Analysis” https://www.cs.helsinki.fi/u/ahyvarin/papers/bookfinal_ICA.pdf pp. 147–148, pp. 410–411, pp. 441–442, p. 448 <a href="https://www.cs.helsinki.fi/u/ahyvarin/papers/bookfinal_ICA.pdf" target="_blank">https://www.cs.helsinki.fi/u/ahyvarin/papers/bookfinal_ICA.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Jean-Francois Cardoso “Blind Signal Separation: statistical Principles” http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.462.9738&rep=rep1&type=pdf <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.462.9738&rep=rep1&type=pdf" target="_blank">http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.462.9738&rep=rep1&type=pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Rui Li, Hongwei Li, and Fasong Wang. “Dependent Component Analysis: Concepts and Main Algorithms” http://www.jcomputers.us/vol5/jcp0504-13.pdf <a href="http://www.jcomputers.us/vol5/jcp0504-13.pdf" target="_blank">http://www.jcomputers.us/vol5/jcp0504-13.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Aapo Hyvarinen, Juha Karhunen, and Erkki Oja. “Independent Component Analysis” https://www.cs.helsinki.fi/u/ahyvarin/papers/bookfinal_ICA.pdf pp. 147–148, pp. 410–411, pp. 441–442, p. 448 <a href="https://www.cs.helsinki.fi/u/ahyvarin/papers/bookfinal_ICA.pdf" target="_blank">https://www.cs.helsinki.fi/u/ahyvarin/papers/bookfinal_ICA.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>P. Comon and C. Jutten (editors). “Handbook of Blind Source Separation, Independent Component Analysis and Applications” Academic Press, ISBN 978-2-296-12827-9 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Shlens, Jonathon. "A tutorial on independent component analysis." arXiv:1404.2986 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
</ol>

Signal separation open-in-new

Signal separation