Editing Speech processing (section)

== Techniques ==

=== Dynamic time warping ===
{{Main|Dynamic time warping}}Dynamic time warping (DTW) is an [[algorithm]] for measuring similarity between two [[Time series|temporal sequences]], which may vary in speed. In general, DTW is a method that calculates an [[Optimal matching|optimal match]] between two given sequences (e.g. time series) with certain restriction and rules. The optimal match is denoted by the match that satisfies all the restrictions and the rules and that has the minimal cost, where the cost is computed as the sum of absolute differences, for each matched pair of indices, between their values.{{citation needed|date=December 2018}}

=== Hidden Markov models ===
{{Main|Hidden Markov model}}A hidden Markov model can be represented as the simplest [[dynamic Bayesian network]]. The goal of the algorithm is to estimate a hidden variable x(t) given a list of observations y(t). By applying the [[Markov property]], the [[conditional probability distribution]] of the hidden variable ''x''(''t'') at time ''t'', given the values of the hidden variable ''x'' at all times, depends ''only'' on the value of the hidden variable ''x''(''t'' − 1). Similarly, the value of the observed variable ''y''(''t'') only depends on the value of the hidden variable ''x''(''t'') (both at time ''t'').{{citation needed|date=December 2018}}

=== Artificial neural networks ===
{{Main|Artificial neural network}}An artificial neural network (ANN) is based on a collection of connected units or nodes called [[artificial neuron]]s, which loosely model the [[neuron]]s in a biological [[brain]]. Each connection, like the [[synapse]]s in a biological [[brain]], can transmit a signal from one artificial neuron to another. An artificial neuron that receives a signal can process it and then signal additional artificial neurons connected to it. In common ANN implementations, the signal at a connection between artificial neurons is a [[real number]], and the output of each artificial neuron is computed by some non-linear function of the sum of its inputs.{{citation needed|date=December 2018}}

===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| url-access = subscription}}</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| url-access = subscription}}</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| url-access = subscription}}</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" />