{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T20:20:42Z","timestamp":1776889242967,"version":"3.51.2"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,3,12]],"date-time":"2025-03-12T00:00:00Z","timestamp":1741737600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,3,12]],"date-time":"2025-03-12T00:00:00Z","timestamp":1741737600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100007143","name":"S\u00e4hk\u00f6tekniikan Korkeakoulu, Aalto-yliopisto","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100007143","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J AUDIO SPEECH MUSIC PROC."],"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>A common bane of artificial reverberation algorithms is spectral coloration in the synthesized sound, typically manifesting as metallic ringing, leading to a degradation in the perceived sound quality. In delay network methods, coloration is more pronounced when fewer delay lines are used. This paper presents an optimization framework in which a tiny differentiable feedback delay network, with as few as four delay lines, is used to learn a set of parameters to iteratively reduce coloration. The parameters under optimization include the feedback matrix, as well as the input and output gains. The optimization objective is twofold: to maximize spectral flatness through a spectral loss while maintaining temporal density by penalizing sparseness in the parameter values. A favorable narrow distribution of modal excitation is achieved while maintaining the desired impulse response density. In a subjective assessment, the new method proves effective in reducing perceptual coloration of late reverberation. Compared to the author\u2019s previous work, which serves as the baseline and utilizes a sparsity loss in the time domain, the proposed method achieves computational savings while maintaining performance. The effectiveness of this work is demonstrated through two application scenarios where smooth-sounding synthetic room impulse responses are obtained via the introduction of attenuation filters and an optimizable scattering feedback matrix.<\/jats:p>","DOI":"10.1186\/s13636-025-00401-w","type":"journal-article","created":{"date-parts":[[2025,3,12]],"date-time":"2025-03-12T01:48:41Z","timestamp":1741744121000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Optimizing tiny colorless feedback delay networks"],"prefix":"10.1186","volume":"2025","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-8094-4333","authenticated-orcid":false,"given":"Gloria","family":"Dal Santo","sequence":"first","affiliation":[]},{"given":"Karolina","family":"Prawda","sequence":"additional","affiliation":[]},{"given":"Sebastian J.","family":"Schlecht","sequence":"additional","affiliation":[]},{"given":"Vesa","family":"V\u00e4lim\u00e4ki","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,3,12]]},"reference":[{"issue":"5","key":"401_CR1","doi-asserted-by":"publisher","first-page":"1421","DOI":"10.1109\/TASL.2012.2189567","volume":"20","author":"V V\u00e4lim\u00e4ki","year":"2012","unstructured":"V. V\u00e4lim\u00e4ki, J.D. Parker, L. Savioja, J.O. Smith, J.S. Abel, Fifty years of artificial reverberation. IEEE Trans. Audio Speech Lang. Process. 20(5), 1421\u20131448 (2012)","journal-title":"IEEE Trans. Audio Speech Lang. Process."},{"key":"401_CR2","doi-asserted-by":"crossref","unstructured":"M.R. Schroeder, B.F. Logan, \u201cColorless\u201d artificial reverberation. IRE Trans. Audio 6, 209\u2013214 (1961)","DOI":"10.1109\/TAU.1961.1166351"},{"key":"401_CR3","unstructured":"J.M. Jot, A. Chaigne, in Proceedings of the Audio Engineering Society Convention 90, Digital delay networks for designing artificial reverberators (Paris, France, 1991)"},{"key":"401_CR4","unstructured":"S.J. Schlecht, Feedback delay networks in artificial reverberation and reverberation enhancement (Ph.D. thesis, Friedrich-Alexander-Universitaet Erlangen-Nuernberg, Germany, 2018)"},{"issue":"4","key":"401_CR5","doi-asserted-by":"publisher","first-page":"186","DOI":"10.17743\/jaes.2022.0053","volume":"71","author":"C Kirsch","year":"2023","unstructured":"C. Kirsch, T. Wendt, S. Van De Par, H. Hu, S.D. Ewert, Computationally-efficient simulation of late reverberation for inhomogeneous boundary conditions and coupled rooms. J. Audio Eng. Soc. 71(4), 186\u2013201 (2023)","journal-title":"J. Audio Eng. Soc."},{"issue":"5","key":"401_CR6","doi-asserted-by":"publisher","first-page":"229","DOI":"10.17743\/jaes.2022.0074","volume":"71","author":"J Herre","year":"2023","unstructured":"J. Herre, S. Disch, MPEG-I immersive audio \u2014 reference model for the virtual\/augmented reality audio standard. J. Audio Eng. Soc. 71(5), 229\u2013240 (2023)","journal-title":"J. Audio Eng. Soc."},{"issue":"1","key":"401_CR7","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1109\/89.554269","volume":"5","author":"D Rocchesso","year":"1997","unstructured":"D. Rocchesso, J.O. Smith, Circulant and elliptic feedback delay networks for artificial reverberation. IEEE Trans. Speech Audio Process. 5(1), 51\u201363 (1997)","journal-title":"IEEE Trans. Speech Audio Process."},{"issue":"6","key":"401_CR8","doi-asserted-by":"publisher","first-page":"1554","DOI":"10.1109\/TSP.2016.2637323","volume":"65","author":"SJ Schlecht","year":"2016","unstructured":"S.J. Schlecht, E.A. Habets, On lossless feedback delay networks. IEEE Trans. Signal Proc. 65(6), 1554\u20131564 (2016)","journal-title":"IEEE Trans. Signal Proc."},{"issue":"2","key":"401_CR9","doi-asserted-by":"publisher","first-page":"374","DOI":"10.1109\/TASLP.2016.2635027","volume":"25","author":"SJ Schlecht","year":"2016","unstructured":"S.J. Schlecht, E.A. Habets, Feedback delay networks: Echo density and mixing time. IEEE\/ACM Trans. Audio Speech Lang. Process. 25(2), 374\u2013383 (2016)","journal-title":"IEEE\/ACM Trans. Audio Speech Lang. Process."},{"key":"401_CR10","doi-asserted-by":"crossref","unstructured":"J. Heldmann, S.J. Schlecht, in Proceedings of the 24th International Conference on Digital Audio Effects (DAFx20in21), The role of modal excitation in colorless reverberation (Online, 2021), pp. 206\u2013213","DOI":"10.23919\/DAFx51585.2021.9768217"},{"issue":"20","key":"401_CR11","doi-asserted-by":"publisher","first-page":"5340","DOI":"10.1109\/TSP.2019.2937286","volume":"67","author":"SJ Schlecht","year":"2019","unstructured":"S.J. Schlecht, E.A. Habets, Modal decomposition of feedback delay networks. IEEE Trans. Signal Proc. 67(20), 5340\u20135351 (2019)","journal-title":"IEEE Trans. Signal Proc."},{"key":"401_CR12","unstructured":"G.\u00a0Dal Santo, K.\u00a0Prawda, S.\u00a0Schlecht, V.\u00a0V\u00e4lim\u00e4ki, in Proceedings of the 26th International Conference on Digital Audio Effects (DAFx23), Differentiable feedback delay network for colorless reverberation (Copenhagen, Denmark, 2023), pp. 244\u2013251"},{"issue":"1","key":"401_CR13","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1186\/s13636-024-00371-5","volume":"2024","author":"AI Mezza","year":"2024","unstructured":"A.I. Mezza, R. Giampiccolo, E. De Sena, A. Bernardini, Data-driven room acoustic modeling via differentiable feedback delay networks with learnable delay lines. EURASIP J Audio Speech Music Process. 2024(1), 51 (2024)","journal-title":"EURASIP J Audio Speech Music Process."},{"key":"401_CR14","doi-asserted-by":"crossref","unstructured":"A.I. Mezza, R. Giampiccolo, A. Bernardini, in Proceedings of the 27th International Conference on Digital Audio Effects (DAFx24), Modeling the frequency-dependent sound energy decay of acoustic environments with differentiable feedback delay networks (Guildford, UK, 2024), pp. 238\u2013245","DOI":"10.1186\/s13636-024-00371-5"},{"key":"401_CR15","unstructured":"R. Giampiccolo, A.I. Mezza, A. Bernardini, in Proceedings of the 27th International Conference on Digital Audio Effects (DAFx24), Differentiable MIMO feedback delay networks for multichannel room impulse response modeling (Guildford, UK, 2024), pp. 278\u2013285"},{"key":"401_CR16","unstructured":"M. Karjalainen, H. J\u00e4rvel\u00e4inen, in Proceedings of the 111th Audio Engineering Society Convention, More about this reverberation science: Perceptually good late reverberation (New York, NY, USA, 2001)"},{"key":"401_CR17","unstructured":"J.M. Jot, in Proceedings of the Audio Engineering Society Convention 139, Proportional parametric equalizers\u2014Application to digital reverberation and environmental audio processing (New York, NY, USA, 2015)"},{"key":"401_CR18","unstructured":"S.J. Schlecht, E.A. Habets, in Proceedings of the 20th International Conference on Digital Audio Effects (DAFx17), Accurate reverberation time control in feedback delay networks (Edinburgh, 2017), pp. 337\u2013344"},{"issue":"12","key":"401_CR19","doi-asserted-by":"publisher","first-page":"278","DOI":"10.1049\/el:19760215","volume":"11","author":"MA Gerzon","year":"1976","unstructured":"M.A. Gerzon, Unitary (energy-preserving) multichannel networks with feedback. Electron. Lett. 11(12), 278\u2013279 (1976)","journal-title":"Electron. Lett."},{"issue":"1","key":"401_CR20","doi-asserted-by":"publisher","first-page":"52","DOI":"10.2307\/3680358","volume":"6","author":"J Stautner","year":"1982","unstructured":"J. Stautner, M. Puckette, Designing multi-channel reverberators. Comput. Music. J. 6(1), 52\u201365 (1982)","journal-title":"Comput. Music. J."},{"key":"401_CR21","doi-asserted-by":"crossref","unstructured":"J.A. Moorer, About this reverberation business. Comput. Music. J. 3(2), 13\u201328 (1979)","DOI":"10.2307\/3680280"},{"key":"401_CR22","unstructured":"M. Lezcano-Casado, D. Mart\u00ednez-Rubio, in Proceedings of the International Conference on Machine Learning, Cheap orthogonal constraints in neural networks: A simple parametrization of the orthogonal and unitary group (Long Beach, CA, USA, 2019), pp. 3794\u20133803"},{"key":"401_CR23","doi-asserted-by":"crossref","unstructured":"H.\u00a0Kuttruff, Room Acoustics, 5th edn. (CRC Press, 2009)","DOI":"10.3397\/1.3455049"},{"issue":"4","key":"401_CR24","first-page":"273","volume":"8","author":"H Kuttruff","year":"1958","unstructured":"H. Kuttruff, Eigenschaften und Auswertung von Nachhallkurven. Acta Acustica U. Acustica 8(4), 273\u2013280 (1958)","journal-title":"Acta Acustica U. Acustica"},{"key":"401_CR25","unstructured":"J.S. Abel, P.\u00a0Huang, in Proceedings of the Audio Engineering Society Convention 121, A simple, robust measure of reverberation echo density (Audio Engineering Society, 2006)"},{"key":"401_CR26","unstructured":"A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, A. Lerer, in 31st Neural Information Processing Systems Automatic differentiation Workshop (Long Beach, CA, USA, 2017)"},{"key":"401_CR27","unstructured":"PyTorch complex numbers. https:\/\/pytorch.org\/docs\/stable\/complex_numbers.html. Accessed 23 Aug 2024"},{"key":"401_CR28","doi-asserted-by":"crossref","unstructured":"J.M. Jot, in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 2. An analysis\/synthesis approach to real-time artificial reverberation (San Francisco, CA, USA, 1992), pp. 221\u2013224","DOI":"10.1109\/ICASSP.1992.226080"},{"key":"401_CR29","unstructured":"K. Prawda, S.J. Schlecht, V. V\u00e4lim\u00e4ki, in Proceedings of the 22nd International Conference on Digital Audio Effects, Improved reverberation time control for feedback delay networks (Birmingham, UK, 2019), pp. 1\u20137"},{"key":"401_CR30","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1109\/LSP.2024.3352510","volume":"31","author":"V V\u00e4lim\u00e4ki","year":"2024","unstructured":"V. V\u00e4lim\u00e4ki, K. Prawda, S.J. Schlecht, Two-stage attenuation filter for artificial reverberation. IEEE Signal Proc. Lett. 31, 391\u2013395 (2024)","journal-title":"IEEE Signal Proc. Lett."},{"key":"401_CR31","doi-asserted-by":"crossref","unstructured":"V. V\u00e4lim\u00e4ki, J.D. Reiss, All about audio equalization: Solutions and frontiers. Appl. Sci. 6(5), 25\u201370 (2016)","DOI":"10.3390\/app6050129"},{"key":"401_CR32","unstructured":"J. Liski, V. V\u00e4lim\u00e4ki, in Proceedings of the 20th International Conference on Digital Audio Effects, The quest for the best graphic equalizer (Edinburgh, UK, 2017), pp. 95\u2013102"},{"key":"401_CR33","unstructured":"G.\u00a0Dal\u00a0Santo, B.\u00a0Alary, K.\u00a0Prawda, S.\u00a0Schlecht, V.\u00a0V\u00e4lim\u00e4ki, in Proceedings of the 27th International Conference on Digital Audio Effects (DAFx24), RIR2FDN: An improved room impulse response analysis and synthesis (University of Surrey, 2024), pp. 230\u2013237"},{"key":"401_CR34","doi-asserted-by":"crossref","unstructured":"G.\u00a0G\u00f6tz, R.\u00a0Falc\u00f3n\u00a0P\u00e9rez, S.J. Schlecht, V.\u00a0Pulkki, Neural network for multi-exponential sound energy decay analysis. J. Acoust. Soc. Am. 152(2), 942\u2013953 (2022)","DOI":"10.1121\/10.0013416"},{"key":"401_CR35","doi-asserted-by":"crossref","unstructured":"S.J. Schlecht, E.A. Habets, in Proceedings of the IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA), Dense reverberation with delay feedback matrices (New Paltz, NY, USA, 2019), pp. 150\u2013154","DOI":"10.1109\/WASPAA.2019.8937284"},{"key":"401_CR36","doi-asserted-by":"publisher","first-page":"1915","DOI":"10.1109\/TASLP.2020.3001395","volume":"28","author":"SJ Schlecht","year":"2020","unstructured":"S.J. Schlecht, E.A. Habets, Scattering in feedback delay networks. IEEE\/ACM Trans Audio Speech Lang. Process. 28, 1915\u20131924 (2020)","journal-title":"IEEE\/ACM Trans Audio Speech Lang. Process."},{"key":"401_CR37","unstructured":"D.P. Kingma, J.\u00a0Ba, Adam: A method for stochastic optimization (2014). arXiv:1412.6980"},{"issue":"4","key":"401_CR38","doi-asserted-by":"publisher","first-page":"1130","DOI":"10.1016\/j.laa.2013.02.024","volume":"439","author":"DA Bini","year":"2013","unstructured":"D.A. Bini, V. Noferini, Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method. Linear Algebra Appl. 439(4), 1130\u20131149 (2013)","journal-title":"Linear Algebra Appl."},{"key":"401_CR39","unstructured":"G.\u00a0Dal\u00a0Santo, G.M. De\u00a0Bortoli, K.\u00a0Prawda, S.J. Schlecht, V.\u00a0V\u00e4lim\u00e4ki, FLAMO: An open-source library for frequency-domain differentiable audio processing (2024). arXiv:2409.08723"},{"key":"401_CR40","unstructured":"International Telecommunication Union, Method for the subjective assessment of intermediate quality level of audio systems. Recommendation ITU-R BS.1534\u20133 (2015)"},{"key":"401_CR41","doi-asserted-by":"crossref","unstructured":"M. Schoeffler, S. Bartoschek, F.R. St\u00f6ter, M. Roess, S. Westphal, B. Edler, J. Herre, WebMUSHRA\u2014a comprehensive framework for web-based listening tests. J. Open Res. Softw. 6(1), 1\u201334 (2018)","DOI":"10.5334\/jors.187"},{"issue":"3\/4","key":"401_CR42","doi-asserted-by":"publisher","first-page":"591","DOI":"10.2307\/2333709","volume":"52","author":"SS Shapiro","year":"1965","unstructured":"S.S. Shapiro, M.B. Wilk, An analysis of variance test for normality (complete samples). Biometrika 52(3\/4), 591\u2013611 (1965)","journal-title":"Biometrika"},{"issue":"6","key":"401_CR43","doi-asserted-by":"publisher","first-page":"80","DOI":"10.2307\/3001968","volume":"1","author":"F Wilcoxon","year":"1945","unstructured":"F. Wilcoxon, Individual comparisons by ranking methods. Biom. Bull. 1(6), 80\u201383 (1945)","journal-title":"Biom. Bull."}],"container-title":["EURASIP Journal on Audio, Speech, and Music Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s13636-025-00401-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1186\/s13636-025-00401-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s13636-025-00401-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,12]],"date-time":"2025-03-12T01:48:50Z","timestamp":1741744130000},"score":1,"resource":{"primary":{"URL":"https:\/\/asmp-eurasipjournals.springeropen.com\/articles\/10.1186\/s13636-025-00401-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,12]]},"references-count":43,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["401"],"URL":"https:\/\/doi.org\/10.1186\/s13636-025-00401-w","relation":{},"ISSN":["1687-4722"],"issn-type":[{"value":"1687-4722","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,12]]},"assertion":[{"value":"5 February 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 February 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 March 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"13"}}