{ "id": "epnaqs8hgm", "slug": "signal-separation", "title": "Signal separation", "updated": "2026-10-15", "topic_path": [ { "id": "technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/topics/technology" }, { "id": "technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing" }, { "id": "technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai" }, { "id": "technology.computing.ai.algorithms", "label": "Algorithms and computational methods", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.algorithms" }, { "id": "technology.computing.ai.numerical_string_geometry", "label": "Numerical, string, and geometric algorithms", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.numerical_string_geometry" }, { "id": "technology.computing.ai.numerical_string_geometry.fourier_signal_transforms", "label": "Fourier and signal transforms", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.numerical_string_geometry.fourier_signal_transforms" } ], "geo": [ { "id": "geo.nongeo.t1946.technology.computing.ai.numerical_string_geometry.fourier_signal_transforms", "label": "Non-geographic · 1946 to 2000: Fourier and signal transforms", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.numerical_string_geometry.fourier_signal_transforms", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.technology", "label": "Technology and the built world", "api_url": 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"geo.nongeo.t1946.technology.computing.ai.numerical_string_geometry.fourier_signal_transforms", "label": "Fourier and signal transforms", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.numerical_string_geometry.fourier_signal_transforms" } ] }, { "id": "geo.nongeo.t2001.technology.computing.ai.numerical_string_geometry", "label": "Non-geographic · 2001 to 2020: Numerical, string, and geometric algorithms", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.numerical_string_geometry", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t2001", "label": "Non-geographic · 2001 to 2020", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001" }, { "id": "geo.nongeo.t2001.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology" }, { "id": "geo.nongeo.t2001.technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing" }, { "id": "geo.nongeo.t2001.technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai" }, { "id": "geo.nongeo.t2001.technology.computing.ai.algorithms", "label": "Algorithms and computational methods", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.algorithms" }, { "id": "geo.nongeo.t2001.technology.computing.ai.numerical_string_geometry", "label": "Numerical, string, and geometric algorithms", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.numerical_string_geometry" } ] } ], "excerpt": "Signal separation, also called blind source separation (BSS), is a family of signal processing methods that recover individual source signals from observed mixtures without knowing the mixing process.", "snippet": "Signal separation, also called blind source separation (BSS), is a family of signal processing methods that recover individual source signals from observed mixtures without knowing the mixing process.", "node": "technology.computing.ai.numerical_string_geometry.fourier_signal_transforms", "markdown": "# Signal separation\n\nSignal separation, also called blind source separation (BSS), is a family of signal processing and machine learning methods that recover individual source signals from observed mixtures without knowing the mixing process. The standard model treats each sensor output as a linear mixture \\( x = A \\cdot s \\) of statistically independent sources \\( s \\) with an unknown invertible mixing matrix \\( A \\); the method estimates an unmixing matrix \\( W \\approx A^{-1} \\) and returns estimated source waveforms \\( \\hat{s} = W \\cdot x \\), along with an estimate of \\( A \\).[1](https://arxiv.org/pdf/1404.2986.pdf) • [2](https://www.cs.helsinki.fi/u/ahyvarin/papers/NN00new.pdf) \"Blind\" means very little is known about the mixing and few assumptions are made on the sources; the assumptions that do the work are statistical, typically independence, non-Gaussianity, or sparsity.[3](https://www.ee.columbia.edu/~dpwe/papers/HyvO99-icatut.pdf) BSS names the problem; independent component analysis (ICA), which estimates components that are both statistically independent and non-Gaussian, is perhaps the most widely used method for performing it.[3](https://www.ee.columbia.edu/~dpwe/papers/HyvO99-icatut.pdf) • [4](https://www2.iap.fr/users/cardoso/papers/ProcIEEE.pdf)\n\n| Key fact | Statement | Citation |\n|---|---|---|\n| Model | Observed mixtures are \\( x = A \\cdot s \\) with unknown invertible square \\( A \\); output is \\( W \\approx A^{-1} \\) and \\( \\hat{s} = W \\cdot x \\) | [1](https://arxiv.org/pdf/1404.2986.pdf) |\n| Identifiability | Separation is unique up to scales, signs, and ordering when at most one component is Gaussian | [5](https://www.gipsa-lab.grenoble-inp.fr/~pierre.comon/FichiersPdf/como94-SP.pdf) |\n| Whitening | Reduces free parameters from \\( n^{2} \\) to \\( n(n-1)/2 \\), solving \"half of the problem of ICA\" | [2](https://www.cs.helsinki.fi/u/ahyvarin/papers/NN00new.pdf) |\n| Convergence | FastICA converges cubically with a kurtosis contrast and quadratically with a generic cost | [6](http://users.ics.aalto.fi/zyuan/dissertation/article-3.pdf) |\n| Sensor count | ICA handles determined (sources = sensors) or overdetermined cases; underdetermined cases need sparsity, with fewer than \\( n/2 \\) active sources for uniqueness | [7](https://www.cambridge.org/core/journals/apsipa-transactions-on-signal-and-information-processing/article/review-of-blind-source-separation-methods-two-converging-routes-to-ilrma-originating-from-ica-and-nmf/83175898245678E77B464369563ABB89) • [8](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2008/papers/1569104438.pdf) |\n| Practical size limit | JADE is limited in practice to roughly 40 or 50 sources by available memory | [9](http://www2.iap.fr/users/cardoso/compsep_classic.html) |\n\n## How it works\n\nThe key theoretical result is identifiability through non-Gaussianity: if a vector has independent components of which at most one is Gaussian, the decomposition is unique up to diagonal scaling and permutation.[5](https://www.gipsa-lab.grenoble-inp.fr/~pierre.comon/FichiersPdf/como94-SP.pdf) The intuition comes from the central limit theorem: a mixture of independent non-Gaussian signals is closer to Gaussian than any single source, so a direction that maximizes non-Gaussianity of \\( w^{T} \\cdot x \\) tends to isolate one source.[2](https://www.cs.helsinki.fi/u/ahyvarin/papers/NN00new.pdf) • [10](https://web.mit.edu/~gari/teaching/6.555/lectures/ch15_bss.pdf)\n\nSecond-order statistics are not enough. Cancelling the \\( n(n-1)/2 \\) second-order cross moments supplies only half the equations needed for the unknowns, which is why principal component analysis (PCA) and factor analysis, which only decorrelate, cannot separate the signals.[11](https://www.gipsa-lab.grenoble-inp.fr/~pierre.comon/FichiersPdf/ComonJH-SP91.pdf) • [2](https://www.cs.helsinki.fi/u/ahyvarin/papers/NN00new.pdf) [Mutual information](https://www.edgepedia.org/mutual-information) involves statistics of all orders except for jointly Gaussian variables, so independence carries strictly more information than uncorrelatedness.[12](https://papers.cnl.salk.edu/PDFs/An%20Information-Maximization%20Approach%20to%20Blind%20Separation%20and%20Blind%20Deconvolution%201995-3631.pdf) For Gaussian sources the problem is ill posed: after whitening, any rotation yields a new set of independent sources, so infinitely many solutions exist.[13](https://www.isip.uni-luebeck.de/fileadmin/files/publications/Condurache15_Tutorial_BSS_ICA.pdf) Estimates carry inherent indeterminacies: the signs and scales of components and their ordering are not determined.[14](https://royalsocietypublishing.org/doi/10.1098/rsta.2011.0534)\n\n## How it is done\n\nA typical workflow has three stages. First, centering: subtract the mean of each observed signal; the source means can be recovered afterwards as \\( A^{-1} \\cdot m \\). Second, whitening by eigenvalue decomposition of the covariance matrix, which transforms the mixing problem into an orthogonal one and reduces the free parameters from \\( n^{2} \\) to \\( n(n-1)/2 \\).[3](https://www.ee.columbia.edu/~dpwe/papers/HyvO99-icatut.pdf) • [2](https://www.cs.helsinki.fi/u/ahyvarin/papers/NN00new.pdf) Third, contrast optimization: a measure of non-Gaussianity or independence, such as negentropy, kurtosis, or mutual information, is maximized or minimized over the remaining rotation.[10](https://web.mit.edu/~gari/teaching/6.555/lectures/ch15_bss.pdf) • [13](https://www.isip.uni-luebeck.de/fileadmin/files/publications/Condurache15_Tutorial_BSS_ICA.pdf)\n\nFastICA performs this optimization by a fixed-point iteration on a negentropy approximation, using nonlinearities such as \\( g_{1}(u) = \\tanh(a_{1} \\cdot u) \\) with \\( 1 \\le a_{1} \\le 2 \\) or \\( g_{2}(u) = u\\exp(-u^{2}/2) \\), and can also be derived as an approximative Newton method; it has no step-size parameters.[2](https://www.cs.helsinki.fi/u/ahyvarin/papers/NN00new.pdf) • [3](https://www.ee.columbia.edu/~dpwe/papers/HyvO99-icatut.pdf) Components are extracted either one after another (deflation) or simultaneously (symmetric); the optimization landscape has \\( 2n \\) local maxima, two per component corresponding to the sign ambiguity.[3](https://www.ee.columbia.edu/~dpwe/papers/HyvO99-icatut.pdf) Symmetric FastICA has local quadratic convergence with a generic cost function and cubic convergence for the kurtosis contrast, and it outperformed popular gradient-descent ICA methods in convergence speed by a clear margin because it is an approximative Newton method.[6](http://users.ics.aalto.fi/zyuan/dissertation/article-3.pdf) Because local minima exist and entropy is hard to estimate from finite data, practitioners validate results with multiple correlation measures or bootstrap procedures.[1](https://arxiv.org/pdf/1404.2986.pdf) [Reference](https://www.edgepedia.org/reference) implementations expose these choices directly: scikit-learn's FastICA defaults to the parallel algorithm, unit-variance whitening, the logcosh nonlinearity, at most 200 iterations, and a tolerance of \\( 10^{-4} \\).[15](https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html?highlight=fastica) If the algorithm has not converged after 2000 iterations it is often cycling through a loop of values, and growing the iteration count usually does not help.[16](https://journal.r-project.org/articles/RJ-2018-046/RJ-2018-046.pdf)\n\n## Origin\n\nThe blind separation problem was stated in a two-part 1991 issue of Signal Processing: Christian Jutten and Jeanny Herault described an adaptive algorithm based on a neuromimetic architecture,[17](https://doi.org/10.1016/0165-1684%2891%2990079-x) and Pierre Comon, Christian Jutten, and Jeanny Herault formalized the problem statement of recovering statistically independent sources from observed mixtures assuming only linearity and independence.[18](https://doi.org/10.1016/0165-1684%2891%2990080-3) • [5](https://www.gipsa-lab.grenoble-inp.fr/~pierre.comon/FichiersPdf/como94-SP.pdf) • [19](https://research.ics.aalto.fi/ica/book/intro.pdf)\n\nComon's 1994 paper gave ICA its formal definition and a practical algorithm executable in polynomial time without exhaustive search, even with non-Gaussian noise.[20](https://doi.org/10.1016/0165-1684%2894%2990029-9) Bell and Sejnowski's 1995 paper introduced the Infomax learning rule, stochastic gradient ascent on the output entropy of a sigmoidal network, which performs separation by reducing redundancy between outputs.[12](https://papers.cnl.salk.edu/PDFs/An%20Information-Maximization%20Approach%20to%20Blind%20Separation%20and%20Blind%20Deconvolution%201995-3631.pdf) J. F. Cardoso and A. Souloumiac published the JADE algorithm, based on joint diagonalization of cumulant matrices, in 1993;[21](https://doi.org/10.1049/ip-f-2.1993.0054) Cardoso and B. H. Laheld published equivariant adaptive source separation via the relative gradient in 1996;[22](https://doi.org/10.1109/78.553476) and Aapo Hyvärinen and Erkki Oja published the FastICA fixed-point algorithm in 1997.[23](https://doi.org/10.1162/neco.1997.9.7.1483) As a precursor, objective functions such as kurtosis and standardized negative Shannon entropy had been proposed in geophysical blind deconvolution in the late 1970s.[24](https://jasoncantarella.com/downloads/bse_to_bss.pdf)\n\n## Variants\n\nThe number of sensors relative to sources determines which variant applies. ICA and [independent vector analysis](https://www.edgepedia.org/independent-vector-analysis) (IVA) require determined or overdetermined cases; underdetermined cases, with more sources than sensors, are handled by clustering, for example with Gaussian mixture models, followed by time-frequency masking.[7](https://www.cambridge.org/core/journals/apsipa-transactions-on-signal-and-information-processing/article/review-of-blind-source-separation-methods-two-converging-routes-to-ilrma-originating-from-ica-and-nmf/83175898245678E77B464369563ABB89) In the underdetermined case the mixing matrix is not square and not invertible, so knowing it does not directly recover the sources; uniqueness of a sparse solution requires the number of active sources \\( k \\) to satisfy \\( k < n/2 \\).[8](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2008/papers/1569104438.pdf)\n\nAlgorithm families differ in contrast and update scheme. JADE is an off-line method with no parameter tuning but limited to roughly 40 or 50 sources by memory; EASI uses stochastic relative gradient updates, an idea that appears identically in the literature as the natural gradient.[9](http://www2.iap.fr/users/cardoso/compsep_classic.html) RobustICA uses an exact line search on kurtosis and avoids prewhitening; EFICA is a FastICA variant designed to attain the Cramér–Rao lower bound.[25](https://webusers.i3s.unice.fr/~zarzoso/biblio/ica07.pdf) • [26](https://doi.org/10.1109/tnn.2006.875991) Sparse component analysis assumes sources have sparse representations in a possibly overcomplete dictionary such as wavelets, extending maximum a posteriori separation to more sources than mixtures; speech is usually sparser in the time-frequency or time-scale domains than in time.[27](https://www.bcl.hamilton.ie/~barak/papers/sparse-ica-99a.pdf) • [8](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2008/papers/1569104438.pdf) [Non-negative matrix factorization](https://www.edgepedia.org/non-negative-matrix-factorization) (NMF) imposes non-negativity and low-rank structure instead of full independence. For convolutive mixtures, applying ICA per frequency bin leaves the ordering of source estimates arbitrary, the permutation problem; IVA, which exploits higher-order frequency dependencies across bins, and ILRMA, which combines IVA with multichannel NMF, solve it through shared source models, and TRINICON treats convolutive BSS in a unified optimization framework.[7](https://www.cambridge.org/core/journals/apsipa-transactions-on-signal-and-information-processing/article/review-of-blind-source-separation-methods-two-converging-routes-to-ilrma-originating-from-ica-and-nmf/83175898245678E77B464369563ABB89) • [28](https://doi.org/10.1109/tasl.2006.872618) • [29](https://www.mdpi.com/1424-8220/23/1/493) • [30](https://doi.org/10.1007/1-4020-7769-6_10) Blind signal extraction, recovering only some non-Gaussian components, is useful when sensor counts exceed 120 in EEG or MEG and only some components are wanted.[24](https://jasoncantarella.com/downloads/bse_to_bss.pdf)\n\n## Applications\n\nTypical applications named in the method literature are removing artifacts from brain signal recordings, finding hidden factors in financial time series, and reducing noise in natural images.[2](https://www.cs.helsinki.fi/u/ahyvarin/papers/NN00new.pdf) ICA filtering has also been applied to medical signals including EEG, MEG, and MRI, to biological assays such as microarrays, and to audio and photographic image analysis.[1](https://arxiv.org/pdf/1404.2986.pdf) JADE has been used in mobile telephony, airport radar, and biomedical signals including ECG and multi-electrode neural recordings.[9](http://www2.iap.fr/users/cardoso/compsep_classic.html) In fMRI, reliability comparisons found that Infomax always presented better reliability than other non-deterministic algorithms (FastICA, EVD, and COMBI), and among non-deterministic algorithms only FastICA showed good spatial consistency with Infomax results.[31](https://pmc.ncbi.nlm.nih.gov/articles/PMC9236259/)\n\n## Limitations and alternatives\n\nClassical ICA fails for Gaussian sources, in underdetermined and single-channel scenarios, and under strong statistical assumptions that may not hold in real acoustics; kurtosis is disproportionately sensitive to distribution tails and outliers, motivating negentropy-based contrasts, and cumulant-based indexes tolerate additive Gaussian noise only approximately at finite samples and fail at too-low signal-to-noise ratios.[13](https://www.isip.uni-luebeck.de/fileadmin/files/publications/Condurache15_Tutorial_BSS_ICA.pdf) • [32](https://arxiv.org/html/2508.10830) • [33](https://www.cs.helsinki.fi/u/ahyvarin/papers/IMPS01.pdf) • [24](https://jasoncantarella.com/downloads/bse_to_bss.pdf) Small samples cause non-convergence, and extracting a Gaussian component early can make the expected error diverge.[16](https://journal.r-project.org/articles/RJ-2018-046/RJ-2018-046.pdf) Compared with PCA and factor analysis, ICA is not restricted to an orthogonal basis and, unlike ordinary factor analysis, determines the factor rotation uniquely because its latent factors are non-Gaussian; PCA is equivalent to ICA for Gaussian data.[1](https://arxiv.org/pdf/1404.2986.pdf) • [33](https://www.cs.helsinki.fi/u/ahyvarin/papers/IMPS01.pdf)\n\n[Deep learning](https://www.edgepedia.org/deep-learning) has largely displaced classical BSS in audio. The mainstream speech separation pipeline is an encoder, a separator, audio estimation by masking or direct prediction, and a decoder.[32](https://arxiv.org/html/2508.10830) Conv-TasNet replaced the STFT with a learned convolutional encoder-decoder and a temporal convolutional network trained with SI-SNR loss under permutation invariant training, significantly surpassing ideal time-frequency magnitude masks in SI-SNRi and SDRi with a smaller model; earlier time-domain classical methods such as ICA and NMF had not been comparable in scalability.[34](https://arxiv.org/pdf/1809.07454) [Deep clustering](https://www.edgepedia.org/deep-clustering) introduced discriminative embeddings for time-frequency segmentation,[35](https://doi.org/10.48550/arxiv.1508.04306) and the field has since shifted from time-frequency masking to complex spectrum and time-domain waveform estimation with dual- or multi-path architectures, plus hybrid DNN mask estimation with beamforming.[36](https://merl.com/publications/docs/TR2025-036.pdf)\n\nSince 2023, unsupervised and generative approaches have grown quickly. UNSSOR trains separation networks directly on over-determined multi-microphone mixtures by constraining filtered per-speaker estimates to sum to each observed mixture, avoiding the over-separation problems of the mixture-of-mixtures (MixIT) paradigm.[37](https://papers.nips.cc/paper_files/paper/2023/file/6b44765c9201730a27f7931afb4d7434-Paper-Conference.pdf) • [32](https://arxiv.org/html/2508.10830) Diffusion-prior methods treat separation as an inverse problem: ArrayDPS uses a single-speaker speech diffusion model plus microphone mixtures and is comparable to supervised methods in SDR,[38](https://proceedings.mlr.press/v267/xu25f.html) and ZeroSep separates mixtures zero-shot with a pre-trained text-guided audio diffusion model via latent inversion and conditioned denoising.[39](https://proceedings.neurips.cc/paper_files/paper/2025/file/a24a75ef009ee73b160653c16b18f00e-Paper-Conference.pdf) Unified models now cover multiple tasks: USE infers source count and acoustic clues automatically, reporting a 1.4 dB SDR improvement in separation and 86% target-sound-extraction accuracy,[40](https://ojs.aaai.org/index.php/AAAI/article/view/40635) and the task-aware unified source separation (TUSS) model of Kohei Saijo and colleagues accepts a variable number of learnable prompts and outputs the corresponding number of separated sources across speech, sound, music, and cinematic audio tasks, using a permutation-invariant loss for source ordering.[41](https://doi.org/10.48550/arxiv.2410.23987)\n\n## References\n\n1. [A Tutorial on Independent Component Analysis (Shlens, 2014)](https://arxiv.org/pdf/1404.2986.pdf)\n2. [Independent Component Analysis: Algorithms and Applications (Hyvärinen & Oja, Neural Networks, 2000)](https://www.cs.helsinki.fi/u/ahyvarin/papers/NN00new.pdf)\n3. [Independent Component Analysis: A Tutorial (Hyvärinen & Oja, 1999)](https://www.ee.columbia.edu/~dpwe/papers/HyvO99-icatut.pdf)\n4. [Blind signal separation: statistical principles (Cardoso, Proceedings of the IEEE)](https://www2.iap.fr/users/cardoso/papers/ProcIEEE.pdf)\n5. [Independent component analysis, a new concept? (P. Comon, Signal Processing 36 (1994) 287-314)](https://www.gipsa-lab.grenoble-inp.fr/~pierre.comon/FichiersPdf/como94-SP.pdf)\n6. [Erkki Oja and Zhijian Yuan. The FastICA Algorithm revisited: convergence](http://users.ics.aalto.fi/zyuan/dissertation/article-3.pdf)\n7. [A review of blind source separation methods: two converging routes to ILRMA originating from ICA and NMF (APSIPA Transactions)](https://www.cambridge.org/core/journals/apsipa-transactions-on-signal-and-information-processing/article/review-of-blind-source-separation-methods-two-converging-routes-to-ilrma-originating-from-ica-and-nmf/83175898245678E77B464369563ABB89)\n8. [Estimating the mixing matrix in underdetermined sparse component analysis (SCA) using consecutive ICA (EUSIPCO 2008)](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2008/papers/1569104438.pdf)\n9. [Blind source separation and Independent component analysis (Jean-François Cardoso's research page)](http://www2.iap.fr/users/cardoso/compsep_classic.html)\n10. [Blind source separation and ICA (MIT 6.555 lecture notes, ch. 15)](https://web.mit.edu/~gari/teaching/6.555/lectures/ch15_bss.pdf)\n11. [Blind separation of sources, Part II: Problems statement (Comon, Jutten & Herault, Signal Processing 24 (1991) 11-21)](https://www.gipsa-lab.grenoble-inp.fr/~pierre.comon/FichiersPdf/ComonJH-SP91.pdf)\n12. [An Information-Maximization Approach to Blind Separation and Blind Deconvolution (Bell & Sejnowski, Neural Computation 1995)](https://papers.cnl.salk.edu/PDFs/An%20Information-Maximization%20Approach%20to%20Blind%20Separation%20and%20Blind%20Deconvolution%201995-3631.pdf)\n13. [A Tutorial on Blind Source Separation using ICA (Condurache, 2015)](https://www.isip.uni-luebeck.de/fileadmin/files/publications/Condurache15_Tutorial_BSS_ICA.pdf)\n14. [Independent component analysis: recent advances (Hyvärinen, Phil. Trans. R. Soc. A, 2013)](https://royalsocietypublishing.org/doi/10.1098/rsta.2011.0534)\n15. [scikit-learn FastICA documentation](https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html?highlight=fastica)\n16. [fICA: FastICA Algorithms and Their Improvements (R Journal)](https://journal.r-project.org/articles/RJ-2018-046/RJ-2018-046.pdf)\n17. [Blind separation of sources, part I: An adaptive algorithm based on neuromimetic architecture (Signal Processing, 1991)](https://doi.org/10.1016/0165-1684%2891%2990079-x)\n18. [Blind separation of sources, part II: Problems statement (Signal Processing, 1991)](https://doi.org/10.1016/0165-1684%2891%2990080-3)\n19. [Independent Component Analysis (Hyvärinen, Karhunen & Oja, book introduction chapter)](https://research.ics.aalto.fi/ica/book/intro.pdf)\n20. [Independent component analysis, A new concept? (Signal Processing, 1994)](https://doi.org/10.1016/0165-1684%2894%2990029-9)\n21. [J.F. Cardoso, A. Souloumiac (1993). Blind beamforming for non-gaussian signals. IEE Proceedings F Radar and Signal Processing.](https://doi.org/10.1049/ip-f-2.1993.0054)\n22. [J.-F. Cardoso, B.H. Laheld (1996). Equivariant adaptive source separation. 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[Hershey, John R. and colleagues (2015). Deep clustering: Discriminative embeddings for segmentation and separation. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1508.04306)\n36. [30+ Years of Source Separation Research: Achievements and Future Challenges (Araki, Ito, Haeb-Umbach, Wichern, Wang, Mitsufuji; MERL TR2025-036, March 2025)](https://merl.com/publications/docs/TR2025-036.pdf)\n37. [UNSSOR: Unsupervised Neural Speech Separation by Leveraging Over-determined Training Mixtures (NeurIPS 2023)](https://papers.nips.cc/paper_files/paper/2023/file/6b44765c9201730a27f7931afb4d7434-Paper-Conference.pdf)\n38. [ArrayDPS: Unsupervised Blind Speech Separation with a Diffusion Prior (ICML 2025, PMLR v267)](https://proceedings.mlr.press/v267/xu25f.html)\n39. [Separate Anything in Audio with Zero Training (ZeroSep, NeurIPS 2025)](https://proceedings.neurips.cc/paper_files/paper/2025/file/a24a75ef009ee73b160653c16b18f00e-Paper-Conference.pdf)\n40. [USE: A Unified Model for Universal Sound Separation and Extraction (AAAI-26)](https://ojs.aaai.org/index.php/AAAI/article/view/40635)\n41. [Saijo, Kohei and colleagues (2024). Task-Aware Unified Source Separation. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.2410.23987)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Algorithms and computational methods › Numerical, string, and geometric algorithms › Fourier and signal transforms*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://www.ee.columbia.edu/~dpwe/papers/HyvO99-icatut.pdf", "https://web.mit.edu/~gari/teaching/6.555/lectures/ch15_bss.pdf" ], "url": "https://www.edgepedia.org/signal-separation", "markdown_url": "https://www.edgepedia.org/signal-separation.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Signal separation\", Edgepedia (EdgeChat), https://www.edgepedia.org/signal-separation. Edgepedia Community License 1.0.", "credit_md": "\"[Signal separation](https://www.edgepedia.org/signal-separation)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/signal-separation](https://www.edgepedia.org/signal-separation). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Signal separation\", Edgepedia (EdgeChat), https://www.edgepedia.org/signal-separation. Edgepedia Community License 1.0.", "speakable": "Signal separation, also called blind source separation, is a family of signal processing methods that recover individual source signals from observed mixtures without knowing the mixing process." }