{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T03:02:16Z","timestamp":1771297336290,"version":"3.50.1"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,4,15]],"date-time":"2020-04-15T00:00:00Z","timestamp":1586908800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,4,15]],"date-time":"2020-04-15T00:00:00Z","timestamp":1586908800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J AUDIO SPEECH MUSIC PROC."],"published-print":{"date-parts":[[2020,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we introduce a quadratic approach for single-channel noise reduction. The desired signal magnitude is estimated by applying a linear filter to a modified version of the observations\u2019 vector. The modified version is constructed from a Kronecker product of the observations\u2019 vector with its complex conjugate. The estimated signal magnitude is multiplied by a complex exponential whose phase is obtained using a conventional linear filtering approach. We focus on the linear and quadratic maximum signal-to-noise ratio (SNR) filters and demonstrate that the quadratic filter is superior in terms of subband SNR gains. In addition, in the context of speech enhancement, we show that the quadratic filter is ideally preferable in terms of perceptual evaluation of speech quality (PESQ) and short-time objective intelligibility (STOI) scores. The advantages, compared to the conventional linear filtering approach, are particularly significant for low input SNRs, at the expanse of a higher computational complexity. The results are verified in practical scenarios with nonstationary noise and in comparison to well-known speech enhancement methods. We demonstrate that the quadratic maximum SNR filter may be superior, depending on the nonstationary noise type.<\/jats:p>","DOI":"10.1186\/s13636-020-00174-4","type":"journal-article","created":{"date-parts":[[2020,4,15]],"date-time":"2020-04-15T12:02:37Z","timestamp":1586952157000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Quadratic approach for single-channel noise reduction"],"prefix":"10.1186","volume":"2020","author":[{"given":"Gal","family":"Itzhak","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jacob","family":"Benesty","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Israel","family":"Cohen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,4,15]]},"reference":[{"issue":"2","key":"174_CR1","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1109\/TASSP.1980.1163394","volume":"28","author":"R. McAulay","year":"1980","unstructured":"R. McAulay, M. Malpass, Speech enhancement using a soft-decision noise suppression filter. IEEE Trans. Acoust. Speech Sig. Proc.28(2), 137\u2013145 (1980).","journal-title":"IEEE Trans. Acoust. Speech Sig. Proc."},{"issue":"6","key":"174_CR2","doi-asserted-by":"publisher","first-page":"1109","DOI":"10.1109\/TASSP.1984.1164453","volume":"32","author":"Y. Ephraim","year":"1984","unstructured":"Y. Ephraim, D. Malah, Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator. IEEE Trans. Acoust. Speech Sig. Proc.32(6), 1109\u20131121 (1984).","journal-title":"IEEE Trans. Acoust. Speech Sig. Proc."},{"issue":"2","key":"174_CR3","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1109\/TASSP.1985.1164550","volume":"33","author":"Y. Ephraim","year":"1985","unstructured":"Y. Ephraim, D. Malah, Speech enhancement using a minimum mean-square error log-spectral amplitude estimator. IEEE Trans. Acoust. Speech Sig. Proc.33(2), 443\u2013445 (1985).","journal-title":"IEEE Trans. Acoust. Speech Sig. Proc."},{"key":"174_CR4","volume-title":"Spectral Enhancement Methods","year":"2008","unstructured":"I. Cohen, S. Gannot, Spectral Enhancement Methods. (J. Benesty, M. Sondhi, Y. Huang, eds.) (Springer, Berlin, Heidelberg, 2008)."},{"key":"174_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-23250-3","volume-title":"Speech Enhancement in the STFT Domain","author":"J. Benesty","year":"2012","unstructured":"J. Benesty, J. Chen, E. Habets, Speech Enhancement in the STFT Domain (Springer-Verlag Berlin Heidelberg, Berlin, 2012)."},{"key":"174_CR6","volume-title":"Fundamentals of Signal Enhancement and Array Signal Processing","author":"J. Benesty","year":"2018","unstructured":"J. Benesty, I. Cohen, J. Chen, Fundamentals of Signal Enhancement and Array Signal Processing (Wiley-IEEE Press, Singapore, 2018)."},{"issue":"4","key":"174_CR7","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1109\/97.1001645","volume":"9","author":"I. Cohen","year":"2002","unstructured":"I. Cohen, Optimal speech enhancement under signal presence uncertainty using log-spectral amplitude estimator. IEEE Sig. Proc. Lett.9(4), 113\u2013116 (2002).","journal-title":"IEEE Sig. Proc. Lett."},{"key":"174_CR8","doi-asserted-by":"crossref","unstructured":"P. J. Wolfe, J. S. Godsill, Efficient alternatives to the Ephraim and Malah suppression rule for audio signal enhancement. EURASIP J. Adv. Sig. Proc.2003(10) (2003).","DOI":"10.1155\/S1110865703304111"},{"key":"174_CR9","doi-asserted-by":"publisher","first-page":"845","DOI":"10.1109\/TSA.2005.851927","volume":"13","author":"R. Martin","year":"2005","unstructured":"R. Martin, Speech enhancement based on minimum mean-square error estimation and supergaussian priors. IEEE Trans. Speech Audio Proc.13:, 845\u2013856 (2005). https:\/\/doi.org\/10.1109\/TSA.2005.851927.","journal-title":"IEEE Trans. Speech Audio Proc."},{"issue":"3","key":"174_CR10","doi-asserted-by":"publisher","first-page":"336","DOI":"10.1016\/j.specom.2005.02.011","volume":"47","author":"I. Cohen","year":"2005","unstructured":"I. Cohen, Speech enhancement using super-gaussian speech models and noncausal a priori snr estimation. Speech Commun.47(3), 336\u2013350 (2005).","journal-title":"Speech Commun."},{"key":"174_CR11","doi-asserted-by":"crossref","unstructured":"R. C. Hendriks, H. Richard, J. Jensen, Log-Spectral Magnitude MMSE Estimators Under Super-Gaussian Densities, (2009).","DOI":"10.21437\/Interspeech.2009-411"},{"key":"174_CR12","unstructured":"R. Martin, in Proceedings of the 27th IEEE International Conference Acoustics Speech Signal Processing, ICASSP-02, 1. Speech enhancement using MMSE short time spectral estimation with gamma distributed speech priors, (2002), pp. 253\u2013256."},{"issue":"6","key":"174_CR13","doi-asserted-by":"publisher","first-page":"1741","DOI":"10.1109\/TASL.2007.899233","volume":"15","author":"J. S. Erkelens","year":"2007","unstructured":"J. S. Erkelens, R. C. Hendriks, R. Heusdens, J. Jensen, Minimum mean-square error estimation of discrete fourier coefficients with generalized gamma priors. IEEE Trans. Audio Speech Lang. Proc.15(6), 1741\u20131752 (2007). https:\/\/doi.org\/10.1109\/TASL.2007.899233.","journal-title":"IEEE Trans. Audio Speech Lang. Proc."},{"key":"174_CR14","unstructured":"R. R. Martin, C. Breithaupt, in Proceedings of the 8th International Workshop on Acoustic Echo and Noise Control (IWAENC). Speech enhancement in the DFT domain using Laplacian speech priors, (2003), pp. 87\u201390."},{"issue":"4","key":"174_CR15","doi-asserted-by":"publisher","first-page":"698","DOI":"10.1016\/j.sigpro.2005.06.005","volume":"86","author":"I. Cohen","year":"2006","unstructured":"I. Cohen, Speech spectral modeling and enhancement based on autoregressive conditional heteroscedasticity models. Sig. Proc.86(4), 698\u2013709 (2006).","journal-title":"Sig. Proc."},{"issue":"11","key":"174_CR16","doi-asserted-by":"publisher","first-page":"2403","DOI":"10.1016\/S0165-1684(01)00128-1","volume":"81","author":"I. Cohen","year":"2001","unstructured":"I. Cohen, B. Berdugo, Speech enhancement for non-stationary noise environments. Sig. Proc.81(11), 2403\u20132418 (2001).","journal-title":"Sig. Proc."},{"issue":"5","key":"174_CR17","doi-asserted-by":"publisher","first-page":"870","DOI":"10.1109\/TSA.2005.851940","volume":"13","author":"I. Cohen","year":"2005","unstructured":"I. Cohen, Relaxed statistical model for speech enhancement and a priori SNR estimation. IEEE Trans. Speech Audio Proc.13(5), 870\u2013881 (2005).","journal-title":"IEEE Trans. Speech Audio Proc."},{"issue":"4","key":"174_CR18","doi-asserted-by":"publisher","first-page":"1256","DOI":"10.1109\/TASL.2011.2174226","volume":"20","author":"Y. A. Huang","year":"2012","unstructured":"Y. A. Huang, J. Benesty, A multi-frame approach to the frequency-domain single-channel noise reduction problem. IEEE Trans Audio Speech Lang. Proc.20(4), 1256\u20131269 (2012).","journal-title":"IEEE Trans Audio Speech Lang. Proc."},{"issue":"12","key":"174_CR19","doi-asserted-by":"publisher","first-page":"2034","DOI":"10.1109\/TASLP.2014.2360643","volume":"22","author":"G. Huang","year":"2014","unstructured":"G. Huang, J. Benesty, T. Long, J. Chen, A family of maximum SNR filters for noise reduction. IEEE\/ACM Trans. Audio Speech Lang. Proc.22(12), 2034\u20132047 (2014).","journal-title":"IEEE\/ACM Trans. Audio Speech Lang. Proc."},{"key":"174_CR20","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/j.specom.2019.10.001","volume":"114","author":"G. Itzhak","year":"2019","unstructured":"G. Itzhak, J. Benesty, I. Cohen, Nonlinear kronecker product filtering for multichannel noise reduction. Speech Commun.114:, 49\u201359 (2019).","journal-title":"Speech Commun."},{"key":"174_CR21","doi-asserted-by":"publisher","DOI":"10.1201\/b14529","volume-title":"Speech Enhancement: Theory and Practice, 2nd edn.","author":"P. C. Loizou","year":"2013","unstructured":"P. C. Loizou, Speech Enhancement: Theory and Practice, 2nd edn. (CRC Press, Inc., Boca Raton, 2013)."},{"key":"174_CR22","volume-title":"Noise Reduction in Speech Processing, 1st edn.","author":"J. Benesty","year":"2009","unstructured":"J. Benesty, J. Chen, Y. Huang, I. Cohen, Noise Reduction in Speech Processing, 1st edn. (Springer-Verlag Berlin Heidelberg, Berlin, 2009)."},{"issue":"6","key":"174_CR23","doi-asserted-by":"publisher","first-page":"1109","DOI":"10.1109\/TASL.2009.2020415","volume":"17","author":"J. Benesty","year":"2009","unstructured":"J. Benesty, J. Chen, Y. A. Huang, Noise reduction algorithms in a generalized transform domain. IEEE Trans. Audio Speech Lang. Proc.17(6), 1109\u20131123 (2009).","journal-title":"IEEE Trans. Audio Speech Lang. Proc."},{"key":"174_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/b98818","volume-title":"Matrix Algebra from a Statistician\u2019s Perspective, 1st edn.","author":"D. A. Harville","year":"1997","unstructured":"D. A. Harville, Matrix Algebra from a Statistician\u2019s Perspective, 1st edn. (Springer-Verlag New York, New York, 1997)."},{"key":"174_CR25","volume-title":"Matrix Computations, 3rd edn.","author":"G. H. Golub","year":"1996","unstructured":"G. H. Golub, C. F. V. Loan, Matrix Computations, 3rd edn. (Baltimore, Maryland: The Johns Hopkins University Press, Baltimore, 1996)."},{"key":"174_CR26","volume-title":"Kendall\u2019s Advanced Theory of Statistics, Volume1: Distribution Theory, 6th edn.","author":"A. Stuart","year":"2010","unstructured":"A. Stuart, K. Ord, Kendall\u2019s Advanced Theory of Statistics, Volume1: Distribution Theory, 6th edn. (Wiley, New York, 2010)."},{"key":"174_CR27","unstructured":"DARPA TIMIT acoustic phonetic continuous speech corpus CDROM. NIST (1993)."},{"key":"174_CR28","unstructured":"A. W. Rix, J. G. Beerends, M. P. Hollier, A. P. Hekstra, in 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings, 2. Perceptual evaluation of speech quality (PESQ)-a new method for speech quality assessment of telephone networks and codecs, (2001), pp. 749\u20137522."},{"issue":"7","key":"174_CR29","doi-asserted-by":"publisher","first-page":"2125","DOI":"10.1109\/TASL.2011.2114881","volume":"19","author":"C. H. Taal","year":"2011","unstructured":"C. H. Taal, R. C. Hendriks, R. Heusdens, J. Jensen, An algorithm for intelligibility prediction of time\u2013frequency weighted noisy speech. IEEE Trans. Audio Speech. Lang. Proc.19(7), 2125\u20132136 (2011). https:\/\/doi.org\/10.1109\/TASL.2011.2114881.","journal-title":"IEEE Trans. Audio Speech. Lang. Proc."},{"key":"174_CR30","doi-asserted-by":"crossref","unstructured":"K. Paliwal, K. W\u00f3jcicki, B. Schwerin, Single-channel speech enhancement using spectral subtraction in the short-time modulation domain. Speech Commun., 450\u2013475 (2010).","DOI":"10.1016\/j.specom.2010.02.004"},{"issue":"9","key":"174_CR31","doi-asserted-by":"publisher","first-page":"1355","DOI":"10.1109\/TASLP.2014.2329633","volume":"22","author":"A. Schasse","year":"2014","unstructured":"A. Schasse, R. Martin, Estimation of subband speech correlations for noise reduction via MVDR processing. IEEE\/ACM Trans. Speech Lang. Proc.22(9), 1355\u20131365 (2014).","journal-title":"IEEE\/ACM Trans. Speech Lang. Proc."}],"container-title":["EURASIP Journal on Audio, Speech, and Music Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s13636-020-00174-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1186\/s13636-020-00174-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s13636-020-00174-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,21]],"date-time":"2022-10-21T10:18:35Z","timestamp":1666347515000},"score":1,"resource":{"primary":{"URL":"https:\/\/asmp-eurasipjournals.springeropen.com\/articles\/10.1186\/s13636-020-00174-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,15]]},"references-count":31,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,12]]}},"alternative-id":["174"],"URL":"https:\/\/doi.org\/10.1186\/s13636-020-00174-4","relation":{},"ISSN":["1687-4722"],"issn-type":[{"value":"1687-4722","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,15]]},"assertion":[{"value":"27 September 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 March 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 April 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"7"}}