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Measurement and Adaptive Identification of Nonstationary Acoustic Impulse Responses

Measurement and Adaptive Identification of Nonstationary Acoustic Impulse Responses Hindawi Advances in Acoustics and Vibration Volume 2019, Article ID 4948034, 6 pages https://doi.org/10.1155/2019/4948034 Research Article Measurement and Adaptive Identification of Nonstationary Acoustic Impulse Responses M. Mekarzia Department of Aeronautics, University of Blida, Algeria Correspondence should be addressed to M. Mekarzia; m mekarzia@yahoo.fr Received 7 July 2018; Accepted 19 June 2019; Published 16 July 2019 Academic Editor: Kim M. Liew Copyright ©  M. Mekarzia. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this work, we present a method of measurement of nonstationary acoustic impulse responses identified by the fast version of the Recursive Least Squares algorithm (FRLS), using professional acoustic equipment. is measurement bench realized in a deaf room presents several tests of capability of adaptive algorithm to tracking the nonstationarities of true system to be identified. e tests of tracking capability obtained are stronger compared to what is encountered in real life and can be used in several applications. 1. Introduction In addition, the risk increases considerably when the transducers (LS and MIC) are pulsed. We describe the mea- Acoustic phenomena in an acoustic medium are measured surement method that is better adapted to the requirements from the notion of the acoustic channel (Figure ), which of IR measurement. We can apply this method to the impulse depends on three main elements: response measurement of the individual head []. (i) e shape and acoustic properties of the walls and objects in the room 2. ImpulseResponseofthe IndividualHead (ii) e source with its spectral emission diagram and To measure individual impulse responses of the head (Fig- disposition (LS: loudspeaker) ure ) which allow to obtain a better sound spatial restitution (iii) e receiver with its directivity diagram and its for the noninvasive sensory substitution aid for blind people position (MIC: microphone) by using the auditory pathway for the representation of the frontal visual stage real time, we must take into account the In the approximation of linear acoustics, this channel is a following: linear filter whose input is the signal x (t) and whose output is y (t): (i) Synthesis of echoes to restore the distance (ii) Minimal time analysis between two sounds y(t)= h(t)∗ x(t) () S: it is the source in terms of signal x(t). where h(t) is the impulse response (IR) of the acoustic Ear: it is the receiver in terms of signal y(t) [ ]. channels [, ]. e purpose of the measurement is to find the (IR) 3. Utilized Equipment h(t) which completely characterizes the acoustic channel. In acoustic rooms, these IR have a duration of the order of one e equipment used in the test bench is described as follows: second, and the desired dynamics is of the order of dB. e conventional measurement technique that generates an (i) Two-channel real-time frequency analyzer of type impulse and records the RI does not allow to simultaneously  B & K(B& K= Bruel&Kjaer). We usedthe meet all these requirements [, ]. latter to, on the one hand, generate the excitation  Advances in Acoustics and Vibration Operation time Room Time Shi ft x(t) Acoustic Channel h(t) register x(n) Sampling Signals of clock y(t) x(n) and y(n) Production of x(n) F  : Acoustic channel []. x(t) Results h(n) Acoustic channel y(n) Recording (n) F  : Flowchart of the measurement method []. Direct 4. Principle of Measurement of Reflection Impulse Responses e general flowchart of the measurement method is given in Figure . e main element of this method is the identifica- F  : Example of application [ ]. tion of the impulse response h(t) from the two signals x (t) and y (t) [, ]. signal which is a pseudorandom white noise x (n) of 5. Table of Fast Recursive Leas Squares frequency band equal to  KHz and amplitude  volts Algorithm (FRLS) [9] effective. On the other hand, this analyzer allows the acquisition in real time and in synchronism with the 𝛾 𝛽 𝛼 (i) Proper choice of𝜇 .𝜇 .𝜇 𝑎𝑛𝑑𝜆 excitation x(n) of the response signal of the room y(n) [, ]. (ii) Variables available at the moment t: (ii) A loudspeaker of type B & K that transforms the excitation signal x(n) into a sound pressure that will 𝑎 ,𝑏 ,∁ ,𝛾 ,𝛼 ,𝛽 ,𝐻 () 𝑁,−𝑡 1 𝑁,−𝑡 1 𝑁,−𝑡 1 𝑁,−𝑡 1 𝑁,−𝑡 1 𝑁,−𝑡 1 𝑁,−𝑡 1 be broadcast in the room [, ]. (iii) A rotating arm microphone of type  B & K (iii) New information:𝑥 𝑎𝑛𝑑𝑦 that converts acoustic pressures into electrical signals 𝑡 𝑡 usable on the spectrum analyzer. (iv) Modeling of𝑥 𝑎𝑛𝑑𝑥 : 𝑡 𝑡−𝑁 e microphone-speaker pair defines the acoustic channel of the room [, ]. 𝑒 =𝑥 −𝑎 𝑋 𝑁,𝑡 𝑡 𝑁,−𝑡 1 𝑁,−𝑡 1 (iv) A precision integrating modular sound meter of type .𝛼 =𝜆𝛼 +𝛾 e B & K that measures the level of ambient noise in N,t N,t−1 N,t−1 N,t the room during measurement [, ]. 𝜆𝛼 𝑁,−𝑡 1 𝛾 = 𝛾 𝑁+1,𝑡 𝑁,−𝑡 1 e measurement of the impulse response of the 𝑁,𝑡 acoustic channel is carried out in two steps: 0 1 𝑁,𝑡 𝑐 =[ ]− [ ] 𝑁+1,𝑡 (i) Synchronous acquisition of the signals x(n) and y(n) 𝜆𝛼 𝑐 −𝛼 𝑁,−𝑡 1 𝑁,−𝑡 1 𝑁,−𝑡 1 with the equipment described above. 𝑎 =𝑎 −𝑒 𝛾 𝑐 𝑁,𝑡 𝑁,−𝑡 1 𝑁,𝑡 𝑁,−𝑡 1 𝑁,−𝑡 1 (ii) Identification in deferred time of the impulse re- sponse. 𝑟 =𝑥 −𝑏 𝑋 𝑁,𝑡 𝑡−𝑁 𝑁,𝑡 𝑁,−𝑡 1 Advances in Acoustics and Vibration e excitation signal x(n) of the loudspeaker (of type ) is a stationary pink noise generated by the spectrum Analyzer analyzer (of type  ) whose sampling frequency is fs = kHz. e two signals x(n) and y(n) are separated and stored on disk in separate files for further processing. e nonstation- arity of the acoustic channel results in nonstationarity in the signal y (n) picked up by the microphone. To check for non- Microphone Loudspeaker stationarities caused by moving a person into the room, we have identified the signal y(n) from the signal x(n) using the fast recursive least squares algorithm (RLS) with exponential oblivion factor𝜆 . An oblivion factor other than  allows the algorithm to continue tracking the nonstationarities of the acoustic channel. e identification error is given by 𝜀 (𝑛 )=𝑦 (𝑛 )−𝐻 𝑥 (𝑛 ) () F  : Room configuration during measure experiments [ ]. where the adaptation of H is carried out by RLS algorithm with an exponential oblivion factor𝜆 = . and a size 𝑁+1 𝜉 =𝑟 +𝜆𝛽 𝑐 𝑁,𝑡 𝑁,𝑡 𝑁,−𝑡 1 𝑁+1,𝑡 of the impulse response equal to  points allows the algorithm to track the nonstationarities of the acoustic 𝑟 =𝑟 +𝜇 𝜉 𝑁,𝑡 𝑁,𝑡 𝑁,𝑡 channel. e performance criterion used is the time evolution 𝑟 =𝑟 +𝜇 𝜉 𝑁,𝑡 𝑁,𝑡 𝑁,𝑡 energy of the normalized error(𝑦(𝑛)− 𝑦(𝑛))̂ by the energy of the signal. 𝑏 𝑏 𝑟 =𝑟 +𝜇 𝜉 𝑁,𝑡 𝑁,𝑡 𝑁,𝑡 is criterion is given by 𝑐 −𝑏 𝑁,−𝑡 1 𝑁,−𝑡 1 2 2 𝑁+1 𝐽 (𝑛 )=10.log ⟨⟨𝜀 ⟩/⟨𝑦 (𝑛 )⟩⟩ ( [ ]=𝑐 −𝑐 [ ] 10 𝑁 𝑁+1,𝑡 𝑁+1,𝑡 0 1 where <> is a short-term time average over a number of −𝑏 𝑏 =𝑏 −𝑟 𝛾 𝑐 𝑁,𝑡 𝑁,−𝑡 1 𝑁,𝑡 𝑁,𝑡 𝑁,𝑡 consecutive samples (, ...) [ ]. isperformance indexiswidely used inadaptive cancel- −𝛽 lation of acoustic echo []. 𝛽 =𝜆𝛽 +𝛾 ( ) 𝑁,𝑡 𝑁,−𝑡 1 𝑁,𝑡 𝑁,𝑡 7. Simulation Results and Comments ( ) e results obtained in the case of a nonstationary acoustic (v) Filtering of𝑦 : channelare showninFigures , , and . e size ofthe 𝜀 =𝑦 −𝐻 𝑋 impulse response of the acoustic channel is N = . e dis- 𝑁,𝑡 𝑡 𝑁,𝑡 𝑁,−𝑡 1 () tance “d” between the microphone and the loudspeaker, the 𝐻 =𝐻 −𝜀 𝛾 𝑐 𝑁,𝑡 𝑁,−𝑡 1 𝑁,𝑡 𝑁,𝑡 𝑁,𝑡 nature of the excitation signal, and the type of nonstationarity are also indicated. In these figures, the nonstationarities of the acoustic 6. Description of the Different channel appear as an increase in the value of the criterion J(n). Measurements Taken In the stationary case, the J(n) criterion regularly Our measurements were taken in a room  meters long,  decreases towards a minimum value. Here the signal x(n) is meters wide, and . meters high. e sound system during stationary; therefore the increase of the criterion J(n) is due the measurements consists of a rotating arm on which are to an increase of the energy of the error𝜖 (𝑛). is increase mounted the microphone and a speaker that radiates in the in error is due to the inability of the adaptive algorithm direction of the microphone (Figure ). to perfectly track nonstationarities in the acoustic channel. e acoustic changes in the room or nonstationary acous- erefore, the visible lobes on the temporal evolutions of tic channel coupling are caused randomly by the movement of J(n) are the nonstationarities caused by the moving person a person between the microphone and the speaker. Depend- in the room. e analysis of these results shows that the ing on the speed of motion of the person, the acoustic changes notion of the temporal rapidity or temporal slowness of the areconsidered over timeas follows: Slow, Medium,and Fast. nonstationarity of the experimenter is not the same at the e movements of the person between the microphone and scale of the adaptive identification algorithm. is rapid or the speaker introduce nonstationarities of acoustic channel slow mobility of the experimenter results in an increase or that can be considered strong compared to those encountered decrease in the number of energy lobes in the evolution of in real-life situations. the criterion J(n). For the adaptive algorithm, all these results Amp Amp Amp Amp Amp Amp Advances in Acoustics and Vibration dB dB 0.96 −0.09 −1.16 −1.09 2 2 −2.52 −2.09 10 10 0.01 0.96 1.92 2.87 3.83 0.01 0.96 1.92 2.87 3.83 samp samp (a) (b) dB 0.28 −0.77 −1.81 0.01 0.96 1.92 2.87 3.83 samp (c) F  : (a) Test of the tracking capacity algorithm with slow variations. (b) Test of the tracking capacity algorithm with medium variations. (c) Test of the tracking capacity algorithm with rapid variations []. dB dB 0.15 0.07 −0.99 −1.10 −2.04 −2.35 0.01 0.96 1.92 2.87 3.83 0.01 0.96 1.92 2.87 3.83 samp samp (a) (b) dB 0.06 −1.24 −2.54 10 0.01 0.96 1.92 2.87 3.83 samp (c) : Nonstationary tests. (a) Near in time with slow variations. (b) With stationary final state with medium variations. (c) Near to an initial state with rapid variations []. canbe considered asfast asthe algorithm cannot perfectly (i) e tracking capacity close to an initial state: the follow the nonstationarities in the acoustic channel slowly or signals do not leaveenough timefor thealgorithm to rapidly variable at the scale of the experimenter. erefore, converge from an initial state. one can say that an adaptive algorithm is more successful (ii) e ability of tracking from a permanent state: the in nonstationary situations when it gives the lowest maxima signals do not leaveenough timefor thealgorithm to possible in J(n) (lobe maxima). reach an acceptable steady state. For the use of the measurements carried out in the tests of the tracking capacity of adaptive algorithms we propose (iii) Nonstationarities close in time: the signals explain the following signals according to the type of the desired that the lobes of nonstationarities caused by the tests: person are close to each other. Amp Amp Amp Advances in Acoustics and Vibration dB dB 10 10 0.68 0.35 −0.34 −0.64 −1.62 10 −1.36 10 0.01 0.96 1.92 2.87 3.83 0.01 0.96 1.92 2.87 3.83 samp samp (a) (b) dB 0.46 −0.77 −2.00 10 0.01 0.96 1.92 2.87 3.83 samp (c) F  : Nonstationary tests. (a) Near to an initial state with slow variations. (b) With stationary permanent state with medium variations. (c) Near in time with rapid variations []. (iv) Nonstationarities with a stationary final state: this in time. We have obtained in this case a better tracking type of signals makes it possible to give an estimate (Figure ). of the dynamics of the nonstationarity with respect to (iii) e third experiment with the following measure- the stationary state. ment parameters: In all the manipulations done in the laboratory, we used the same two-channel frequency analyzer (of type  B&K), as (a) A loudspeaker of type MICDIS well as thesameacoustic medium, butwetested several pairs (b) A distance of m of microphones and loudspeakers. We realized a bench of (b) x(n) that is a pink noise sequence measurement of nonstationary acoustic impulse responses. We quote some practical impulse responses obtained with In this nonstationary acoustic impulse response measure- their own parameters: ment test, we have not given the algorithm enough time to converge. We have created two nonstationary lobes (tests (a) (i) e first experiment with the following measurement and (c)). e signal of test (b) gives the algorithm sufficient parameters: time to reach an acceptable steady state (Figure ). (a) A loudspeaker of type  B & K (iv) e fourth experiment with the following measure- (b) A distance of .m ment parameters: (c) x(n) that is a pink noise sequence (a) A loudspeaker of type MICDIS Convergence curves in this realistic context clearly show (b) A distance of .m that we have given the algorithm time to converge and we (c) x(n) that is a pink noise sequence have caused nonstationarity during the asymptotic phase. For this casewehavegood convergence and better tracking In this measurement test we have several lobes of system (Figure ). nonstationarities, which give the researcher the choice of nonstationarities to test their identification algorithms. On (ii) e second experiment with the following measure- the other hand, these system nonstationarities may allow him ment parameters: to see the utility of a professional speaker and consumer speakers that poorly represent the low frequencies (Figure ). (a) A loudspeaker of type  B & K (b) A distance of m (c)𝑥 (𝑛 ) that is a pink noise sequence 8. Conclusion In this experiment, we have two nonstationary lobes e analysis of these results shows that the notion of time which aim to test the efficiency and robustness of the speed or time slowness of the experimenter nonstationarity identification algorithms to track nonstationary systems close is not the same at the scale of the adaptive identification Amp Amp Amp Advances in Acoustics and Vibration dB dB 0.45 0.30 −0.75 −0.69 2 2 −1.69 10 −1.95 10 0.01 0.96 1.92 2.87 3.83 0.01 0.96 1.92 2.87 3.83 samp samp (a) (b) dB 0.47 −0.70 −1.86 10 0.01 0.96 1.92 2.87 3.83 samp (c) F  : Nonstationary tests. (a) Far in time with slow variations. (b) Near to an initial state with medium variations. (c) Near in time with rapid variations []. algorithm. is rapid or slow mobility of the experimenter [ ] “Condenser microphone and microphone preamplifier for acoustic measurement,” in Data Handbook,Bruel ¨ & Kajaer, leads to an increase or a decrease in the number of energy lobes in the evolution of the J(n) criterion. As far as the [] Sound Level Meter Type 2131 for Acoustic Measurement,Bruel ¨ & adaptive algorithm is concerned, all these results can be Kajaer, . considered as rapid since the algorithm cannot perfectly [] M. Mekarzia and M. Guerti, “Measurement and identification of follow the nonstationarity in the acoustic channel slowly or an acoustic impulse responses,” Journal Building Acoustics,vol. rapidly variable at the experimenter scale. ,no. , pp. –,. erefore, one can say that an adaptive algorithm tracks [] M. Arezki and A. Benallal, “Fast adaptive filtering algorithm better nonstationarities when it gives the lowest maximums for acoustic noise cancellation,” in Proceedings of the World possible in J(n) (lobe maxima). Congress on Engineering 2012,vol.,London,UK,. [] M. Mekarzia, “LATSI speech team of the institute of electronics Data Availability of the university of blida,” - e data used to support the findings of this study are available from the corresponding author upon request. Conflicts of Interest e author declares that he has no conflicts of interest. References [] M. Mekarzia, “Measure and identification of acoustic impulse responses by NLMS-DC,” . [] M. Mekarzia, Measure And Identification of the Acoustic Impulse Responses, Institute of Aeronautics and Space Studies University of Blida, . [ ] M. Mekarzia, Measure and Identification of the acoustic Impulse responses, esis of doctorate February, polythechnic school El Harrach Algerie, . [] Analyzer of the frequencies real time  / instruction, Analyzer of the frequencies real time 2123/2133 instruction,vol. , Bruel ¨ & Kajaer, . [] Loudspeaker 4224 Data,Bruel ¨ & Kajaer, . 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Measurement and Adaptive Identification of Nonstationary Acoustic Impulse Responses

Advances in Acoustics and Vibration , Volume 2019 – Jul 16, 2019

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Copyright © 2019 M. Mekarzia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Advances in Acoustics and Vibration Volume 2019, Article ID 4948034, 6 pages https://doi.org/10.1155/2019/4948034 Research Article Measurement and Adaptive Identification of Nonstationary Acoustic Impulse Responses M. Mekarzia Department of Aeronautics, University of Blida, Algeria Correspondence should be addressed to M. Mekarzia; m mekarzia@yahoo.fr Received 7 July 2018; Accepted 19 June 2019; Published 16 July 2019 Academic Editor: Kim M. Liew Copyright ©  M. Mekarzia. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this work, we present a method of measurement of nonstationary acoustic impulse responses identified by the fast version of the Recursive Least Squares algorithm (FRLS), using professional acoustic equipment. is measurement bench realized in a deaf room presents several tests of capability of adaptive algorithm to tracking the nonstationarities of true system to be identified. e tests of tracking capability obtained are stronger compared to what is encountered in real life and can be used in several applications. 1. Introduction In addition, the risk increases considerably when the transducers (LS and MIC) are pulsed. We describe the mea- Acoustic phenomena in an acoustic medium are measured surement method that is better adapted to the requirements from the notion of the acoustic channel (Figure ), which of IR measurement. We can apply this method to the impulse depends on three main elements: response measurement of the individual head []. (i) e shape and acoustic properties of the walls and objects in the room 2. ImpulseResponseofthe IndividualHead (ii) e source with its spectral emission diagram and To measure individual impulse responses of the head (Fig- disposition (LS: loudspeaker) ure ) which allow to obtain a better sound spatial restitution (iii) e receiver with its directivity diagram and its for the noninvasive sensory substitution aid for blind people position (MIC: microphone) by using the auditory pathway for the representation of the frontal visual stage real time, we must take into account the In the approximation of linear acoustics, this channel is a following: linear filter whose input is the signal x (t) and whose output is y (t): (i) Synthesis of echoes to restore the distance (ii) Minimal time analysis between two sounds y(t)= h(t)∗ x(t) () S: it is the source in terms of signal x(t). where h(t) is the impulse response (IR) of the acoustic Ear: it is the receiver in terms of signal y(t) [ ]. channels [, ]. e purpose of the measurement is to find the (IR) 3. Utilized Equipment h(t) which completely characterizes the acoustic channel. In acoustic rooms, these IR have a duration of the order of one e equipment used in the test bench is described as follows: second, and the desired dynamics is of the order of dB. e conventional measurement technique that generates an (i) Two-channel real-time frequency analyzer of type impulse and records the RI does not allow to simultaneously  B & K(B& K= Bruel&Kjaer). We usedthe meet all these requirements [, ]. latter to, on the one hand, generate the excitation  Advances in Acoustics and Vibration Operation time Room Time Shi ft x(t) Acoustic Channel h(t) register x(n) Sampling Signals of clock y(t) x(n) and y(n) Production of x(n) F  : Acoustic channel []. x(t) Results h(n) Acoustic channel y(n) Recording (n) F  : Flowchart of the measurement method []. Direct 4. Principle of Measurement of Reflection Impulse Responses e general flowchart of the measurement method is given in Figure . e main element of this method is the identifica- F  : Example of application [ ]. tion of the impulse response h(t) from the two signals x (t) and y (t) [, ]. signal which is a pseudorandom white noise x (n) of 5. Table of Fast Recursive Leas Squares frequency band equal to  KHz and amplitude  volts Algorithm (FRLS) [9] effective. On the other hand, this analyzer allows the acquisition in real time and in synchronism with the 𝛾 𝛽 𝛼 (i) Proper choice of𝜇 .𝜇 .𝜇 𝑎𝑛𝑑𝜆 excitation x(n) of the response signal of the room y(n) [, ]. (ii) Variables available at the moment t: (ii) A loudspeaker of type B & K that transforms the excitation signal x(n) into a sound pressure that will 𝑎 ,𝑏 ,∁ ,𝛾 ,𝛼 ,𝛽 ,𝐻 () 𝑁,−𝑡 1 𝑁,−𝑡 1 𝑁,−𝑡 1 𝑁,−𝑡 1 𝑁,−𝑡 1 𝑁,−𝑡 1 𝑁,−𝑡 1 be broadcast in the room [, ]. (iii) A rotating arm microphone of type  B & K (iii) New information:𝑥 𝑎𝑛𝑑𝑦 that converts acoustic pressures into electrical signals 𝑡 𝑡 usable on the spectrum analyzer. (iv) Modeling of𝑥 𝑎𝑛𝑑𝑥 : 𝑡 𝑡−𝑁 e microphone-speaker pair defines the acoustic channel of the room [, ]. 𝑒 =𝑥 −𝑎 𝑋 𝑁,𝑡 𝑡 𝑁,−𝑡 1 𝑁,−𝑡 1 (iv) A precision integrating modular sound meter of type .𝛼 =𝜆𝛼 +𝛾 e B & K that measures the level of ambient noise in N,t N,t−1 N,t−1 N,t the room during measurement [, ]. 𝜆𝛼 𝑁,−𝑡 1 𝛾 = 𝛾 𝑁+1,𝑡 𝑁,−𝑡 1 e measurement of the impulse response of the 𝑁,𝑡 acoustic channel is carried out in two steps: 0 1 𝑁,𝑡 𝑐 =[ ]− [ ] 𝑁+1,𝑡 (i) Synchronous acquisition of the signals x(n) and y(n) 𝜆𝛼 𝑐 −𝛼 𝑁,−𝑡 1 𝑁,−𝑡 1 𝑁,−𝑡 1 with the equipment described above. 𝑎 =𝑎 −𝑒 𝛾 𝑐 𝑁,𝑡 𝑁,−𝑡 1 𝑁,𝑡 𝑁,−𝑡 1 𝑁,−𝑡 1 (ii) Identification in deferred time of the impulse re- sponse. 𝑟 =𝑥 −𝑏 𝑋 𝑁,𝑡 𝑡−𝑁 𝑁,𝑡 𝑁,−𝑡 1 Advances in Acoustics and Vibration e excitation signal x(n) of the loudspeaker (of type ) is a stationary pink noise generated by the spectrum Analyzer analyzer (of type  ) whose sampling frequency is fs = kHz. e two signals x(n) and y(n) are separated and stored on disk in separate files for further processing. e nonstation- arity of the acoustic channel results in nonstationarity in the signal y (n) picked up by the microphone. To check for non- Microphone Loudspeaker stationarities caused by moving a person into the room, we have identified the signal y(n) from the signal x(n) using the fast recursive least squares algorithm (RLS) with exponential oblivion factor𝜆 . An oblivion factor other than  allows the algorithm to continue tracking the nonstationarities of the acoustic channel. e identification error is given by 𝜀 (𝑛 )=𝑦 (𝑛 )−𝐻 𝑥 (𝑛 ) () F  : Room configuration during measure experiments [ ]. where the adaptation of H is carried out by RLS algorithm with an exponential oblivion factor𝜆 = . and a size 𝑁+1 𝜉 =𝑟 +𝜆𝛽 𝑐 𝑁,𝑡 𝑁,𝑡 𝑁,−𝑡 1 𝑁+1,𝑡 of the impulse response equal to  points allows the algorithm to track the nonstationarities of the acoustic 𝑟 =𝑟 +𝜇 𝜉 𝑁,𝑡 𝑁,𝑡 𝑁,𝑡 channel. e performance criterion used is the time evolution 𝑟 =𝑟 +𝜇 𝜉 𝑁,𝑡 𝑁,𝑡 𝑁,𝑡 energy of the normalized error(𝑦(𝑛)− 𝑦(𝑛))̂ by the energy of the signal. 𝑏 𝑏 𝑟 =𝑟 +𝜇 𝜉 𝑁,𝑡 𝑁,𝑡 𝑁,𝑡 is criterion is given by 𝑐 −𝑏 𝑁,−𝑡 1 𝑁,−𝑡 1 2 2 𝑁+1 𝐽 (𝑛 )=10.log ⟨⟨𝜀 ⟩/⟨𝑦 (𝑛 )⟩⟩ ( [ ]=𝑐 −𝑐 [ ] 10 𝑁 𝑁+1,𝑡 𝑁+1,𝑡 0 1 where <> is a short-term time average over a number of −𝑏 𝑏 =𝑏 −𝑟 𝛾 𝑐 𝑁,𝑡 𝑁,−𝑡 1 𝑁,𝑡 𝑁,𝑡 𝑁,𝑡 consecutive samples (, ...) [ ]. isperformance indexiswidely used inadaptive cancel- −𝛽 lation of acoustic echo []. 𝛽 =𝜆𝛽 +𝛾 ( ) 𝑁,𝑡 𝑁,−𝑡 1 𝑁,𝑡 𝑁,𝑡 7. Simulation Results and Comments ( ) e results obtained in the case of a nonstationary acoustic (v) Filtering of𝑦 : channelare showninFigures , , and . e size ofthe 𝜀 =𝑦 −𝐻 𝑋 impulse response of the acoustic channel is N = . e dis- 𝑁,𝑡 𝑡 𝑁,𝑡 𝑁,−𝑡 1 () tance “d” between the microphone and the loudspeaker, the 𝐻 =𝐻 −𝜀 𝛾 𝑐 𝑁,𝑡 𝑁,−𝑡 1 𝑁,𝑡 𝑁,𝑡 𝑁,𝑡 nature of the excitation signal, and the type of nonstationarity are also indicated. In these figures, the nonstationarities of the acoustic 6. Description of the Different channel appear as an increase in the value of the criterion J(n). Measurements Taken In the stationary case, the J(n) criterion regularly Our measurements were taken in a room  meters long,  decreases towards a minimum value. Here the signal x(n) is meters wide, and . meters high. e sound system during stationary; therefore the increase of the criterion J(n) is due the measurements consists of a rotating arm on which are to an increase of the energy of the error𝜖 (𝑛). is increase mounted the microphone and a speaker that radiates in the in error is due to the inability of the adaptive algorithm direction of the microphone (Figure ). to perfectly track nonstationarities in the acoustic channel. e acoustic changes in the room or nonstationary acous- erefore, the visible lobes on the temporal evolutions of tic channel coupling are caused randomly by the movement of J(n) are the nonstationarities caused by the moving person a person between the microphone and the speaker. Depend- in the room. e analysis of these results shows that the ing on the speed of motion of the person, the acoustic changes notion of the temporal rapidity or temporal slowness of the areconsidered over timeas follows: Slow, Medium,and Fast. nonstationarity of the experimenter is not the same at the e movements of the person between the microphone and scale of the adaptive identification algorithm. is rapid or the speaker introduce nonstationarities of acoustic channel slow mobility of the experimenter results in an increase or that can be considered strong compared to those encountered decrease in the number of energy lobes in the evolution of in real-life situations. the criterion J(n). For the adaptive algorithm, all these results Amp Amp Amp Amp Amp Amp Advances in Acoustics and Vibration dB dB 0.96 −0.09 −1.16 −1.09 2 2 −2.52 −2.09 10 10 0.01 0.96 1.92 2.87 3.83 0.01 0.96 1.92 2.87 3.83 samp samp (a) (b) dB 0.28 −0.77 −1.81 0.01 0.96 1.92 2.87 3.83 samp (c) F  : (a) Test of the tracking capacity algorithm with slow variations. (b) Test of the tracking capacity algorithm with medium variations. (c) Test of the tracking capacity algorithm with rapid variations []. dB dB 0.15 0.07 −0.99 −1.10 −2.04 −2.35 0.01 0.96 1.92 2.87 3.83 0.01 0.96 1.92 2.87 3.83 samp samp (a) (b) dB 0.06 −1.24 −2.54 10 0.01 0.96 1.92 2.87 3.83 samp (c) : Nonstationary tests. (a) Near in time with slow variations. (b) With stationary final state with medium variations. (c) Near to an initial state with rapid variations []. canbe considered asfast asthe algorithm cannot perfectly (i) e tracking capacity close to an initial state: the follow the nonstationarities in the acoustic channel slowly or signals do not leaveenough timefor thealgorithm to rapidly variable at the scale of the experimenter. erefore, converge from an initial state. one can say that an adaptive algorithm is more successful (ii) e ability of tracking from a permanent state: the in nonstationary situations when it gives the lowest maxima signals do not leaveenough timefor thealgorithm to possible in J(n) (lobe maxima). reach an acceptable steady state. For the use of the measurements carried out in the tests of the tracking capacity of adaptive algorithms we propose (iii) Nonstationarities close in time: the signals explain the following signals according to the type of the desired that the lobes of nonstationarities caused by the tests: person are close to each other. Amp Amp Amp Advances in Acoustics and Vibration dB dB 10 10 0.68 0.35 −0.34 −0.64 −1.62 10 −1.36 10 0.01 0.96 1.92 2.87 3.83 0.01 0.96 1.92 2.87 3.83 samp samp (a) (b) dB 0.46 −0.77 −2.00 10 0.01 0.96 1.92 2.87 3.83 samp (c) F  : Nonstationary tests. (a) Near to an initial state with slow variations. (b) With stationary permanent state with medium variations. (c) Near in time with rapid variations []. (iv) Nonstationarities with a stationary final state: this in time. We have obtained in this case a better tracking type of signals makes it possible to give an estimate (Figure ). of the dynamics of the nonstationarity with respect to (iii) e third experiment with the following measure- the stationary state. ment parameters: In all the manipulations done in the laboratory, we used the same two-channel frequency analyzer (of type  B&K), as (a) A loudspeaker of type MICDIS well as thesameacoustic medium, butwetested several pairs (b) A distance of m of microphones and loudspeakers. We realized a bench of (b) x(n) that is a pink noise sequence measurement of nonstationary acoustic impulse responses. We quote some practical impulse responses obtained with In this nonstationary acoustic impulse response measure- their own parameters: ment test, we have not given the algorithm enough time to converge. We have created two nonstationary lobes (tests (a) (i) e first experiment with the following measurement and (c)). e signal of test (b) gives the algorithm sufficient parameters: time to reach an acceptable steady state (Figure ). (a) A loudspeaker of type  B & K (iv) e fourth experiment with the following measure- (b) A distance of .m ment parameters: (c) x(n) that is a pink noise sequence (a) A loudspeaker of type MICDIS Convergence curves in this realistic context clearly show (b) A distance of .m that we have given the algorithm time to converge and we (c) x(n) that is a pink noise sequence have caused nonstationarity during the asymptotic phase. For this casewehavegood convergence and better tracking In this measurement test we have several lobes of system (Figure ). nonstationarities, which give the researcher the choice of nonstationarities to test their identification algorithms. On (ii) e second experiment with the following measure- the other hand, these system nonstationarities may allow him ment parameters: to see the utility of a professional speaker and consumer speakers that poorly represent the low frequencies (Figure ). (a) A loudspeaker of type  B & K (b) A distance of m (c)𝑥 (𝑛 ) that is a pink noise sequence 8. Conclusion In this experiment, we have two nonstationary lobes e analysis of these results shows that the notion of time which aim to test the efficiency and robustness of the speed or time slowness of the experimenter nonstationarity identification algorithms to track nonstationary systems close is not the same at the scale of the adaptive identification Amp Amp Amp Advances in Acoustics and Vibration dB dB 0.45 0.30 −0.75 −0.69 2 2 −1.69 10 −1.95 10 0.01 0.96 1.92 2.87 3.83 0.01 0.96 1.92 2.87 3.83 samp samp (a) (b) dB 0.47 −0.70 −1.86 10 0.01 0.96 1.92 2.87 3.83 samp (c) F  : Nonstationary tests. (a) Far in time with slow variations. (b) Near to an initial state with medium variations. (c) Near in time with rapid variations []. algorithm. is rapid or slow mobility of the experimenter [ ] “Condenser microphone and microphone preamplifier for acoustic measurement,” in Data Handbook,Bruel ¨ & Kajaer, leads to an increase or a decrease in the number of energy lobes in the evolution of the J(n) criterion. As far as the [] Sound Level Meter Type 2131 for Acoustic Measurement,Bruel ¨ & adaptive algorithm is concerned, all these results can be Kajaer, . considered as rapid since the algorithm cannot perfectly [] M. Mekarzia and M. Guerti, “Measurement and identification of follow the nonstationarity in the acoustic channel slowly or an acoustic impulse responses,” Journal Building Acoustics,vol. rapidly variable at the experimenter scale. ,no. , pp. –,. erefore, one can say that an adaptive algorithm tracks [] M. Arezki and A. Benallal, “Fast adaptive filtering algorithm better nonstationarities when it gives the lowest maximums for acoustic noise cancellation,” in Proceedings of the World possible in J(n) (lobe maxima). Congress on Engineering 2012,vol.,London,UK,. [] M. Mekarzia, “LATSI speech team of the institute of electronics Data Availability of the university of blida,” - e data used to support the findings of this study are available from the corresponding author upon request. Conflicts of Interest e author declares that he has no conflicts of interest. References [] M. Mekarzia, “Measure and identification of acoustic impulse responses by NLMS-DC,” . [] M. Mekarzia, Measure And Identification of the Acoustic Impulse Responses, Institute of Aeronautics and Space Studies University of Blida, . [ ] M. Mekarzia, Measure and Identification of the acoustic Impulse responses, esis of doctorate February, polythechnic school El Harrach Algerie, . [] Analyzer of the frequencies real time  / instruction, Analyzer of the frequencies real time 2123/2133 instruction,vol. , Bruel ¨ & Kajaer, . [] Loudspeaker 4224 Data,Bruel ¨ & Kajaer, . 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