Editing Speech processing (section)

===Phase-aware processing===
Phase is usually supposed to be random uniform variable and thus useless. This is due wrapping of phase:<ref name="limits">{{Cite journal| doi = 10.1109/TASLP.2015.2430820| issn = 2329-9290| volume = 23| issue = 8| pages = 1283–1294| last1 = Mowlaee| first1 = Pejman| last2 = Kulmer| first2 = Josef| title = Phase Estimation in Single-Channel Speech Enhancement: Limits-Potential| journal = IEEE/ACM Transactions on Audio, Speech, and Language Processing|access-date= 2017-12-03| date = August 2015| s2cid = 13058142| url = https://ieeexplore.ieee.org/document/7103305}}</ref> result of [[arctangent]] function is not continuous due to periodical jumps on <math>2 \pi</math>. After phase unwrapping (see,<ref>{{Cite book| publisher = Wiley| isbn = 978-1-119-23882-9| last1 = Mowlaee| first1 = Pejman| last2 = Kulmer| first2 = Josef| last3 = Stahl| first3 = Johannes| last4 = Mayer| first4 = Florian| title = Single channel phase-aware signal processing in speech communication: theory and practice| location = Chichester| date = 2017}}</ref> Chapter 2.3; [[Instantaneous phase and frequency]]), it can be expressed as:<ref name="limits" /><ref name="vonMises">{{Cite conference| publisher = IEEE| pages = 5063–5067| last1 = Kulmer| first1 = Josef| last2 = Mowlaee| first2 = Pejman| title = Harmonic phase estimation in single-channel speech enhancement using von Mises distribution and prior SNR|book-title= Acoustics, Speech and Signal Processing (ICASSP), 2015 IEEE International Conference on| date = April 2015}}</ref>
<math>\phi(h,l) = \phi_{lin}(h,l) + \Psi(h,l)</math>, where <math>\phi_{lin}(h,l) = \omega_0(l') {}_\Delta t</math> is linear phase (<math>{}_\Delta t</math> is temporal shift at each frame of analysis), <math>\Psi(h,l)</math> is phase contribution of the vocal tract and phase source.<ref name="vonMises" />
Obtained phase estimations can be used for noise reduction: temporal smoothing of instantaneous phase <ref>{{Cite journal| doi = 10.1109/LSP.2014.2365040| issn = 1070-9908| volume = 22| issue = 5| pages = 598–602| last1 = Kulmer| first1 = Josef| last2 = Mowlaee| first2 = Pejman| title = Phase Estimation in Single Channel Speech Enhancement Using Phase Decomposition| journal = IEEE Signal Processing Letters|access-date= 2017-12-03| date = May 2015| bibcode = 2015ISPL...22..598K| s2cid = 15503015| url = https://ieeexplore.ieee.org/document/6936313}}</ref> and its derivatives by time ([[Instantaneous phase and frequency|instantaneous frequency]]) and frequency ([[Group delay and phase delay|group delay]]),<ref name="Advances">{{Cite journal| doi = 10.1016/j.specom.2016.04.002| issn = 0167-6393| volume = 81| pages = 1–29| last1 = Mowlaee| first1 = Pejman| last2 = Saeidi| first2 = Rahim| last3 = Stylianou| first3 = Yannis| title = Advances in phase-aware signal processing in speech communication| journal = Speech Communication|access-date= 2017-12-03| date = July 2016| s2cid = 17409161| url = http://linkinghub.elsevier.com/retrieve/pii/S0167639316300784}}</ref> smoothing of phase across frequency.<ref name="Advances" /> Joined amplitude and phase estimators can recover speech more accurately basing on assumption of von Mises distribution of phase.<ref name="vonMises" />