NON-LINEAR CONVOLUTION: A NEW APPROACH FOR THE AURALIZATION OF DISTORTING SYSTEMS

Transcription

1 NON-LINEAR CONVOLUTION: A NEW APPROAC FOR TE AURALIZATION OF DISTORTING SYSTEMS Angelo Farna, Alberto Belln and Enrco Armellon Industral Engneerng Dept., Unversty of Parma, Va delle Scenze 8/A Parma, 00 ITALY TTP://pcfarna.eng.unpr.t

2 Goals for Auralzaton Transform the results of objectve electroacoustcs measurements to audble sound samples sutable for lstenng tests Tradtonal auralzaton s based on lnear convoluton: ths does not replcates fathfully the nonlnear behavour of most transducers The new method presented here overcomes to ths strong lmtaton, provdng a smplfed treatment of memory-less dstorton

3 Methods We start from a measurement of the system based on exponental sne sweep (Farna, 08th AES, Pars 000) Dagonal Volterra kernels are obtaned by postprocessng the measurement results These kernels are employed as FIR flters n a multple-order convoluton process (orgnal sgnal, ts square, ts cube, and so on are convolved separately and the result s summed)

4 Exponental sweep measurement The exctaton sgnal s a sne sweep wth constant ampltude and exponentally-ncreasng frequency

5 Raw response of the system Many harmonc orders do appear as colour strpes

6 Deconvoluton of system s mpulse response The deconvoluton s obtaned by convolvng the raw response wth a sutable nverse flter

7 Multple mpulse response obtaned th rd st nd The last peak s the lnear mpulse response, the precedng ones are the harmonc dstorton orders

8 Auralzaton by lnear convoluton Convolvng a sutable sound sample wth the lnear IR, the frequency response and temporal transent effects of the system can be smulated properly

9 What s mssng n lnear convoluton? No harmonc dstorton, nor other nonlnear effects are beng reproduced. From a perceptual pont of vew, the sound s judged cold and nnatural A comparatve test between a strongly nonlnear devce and an almost lnear one does not reveal any audble dfference, because the nonlnear behavor s removed for both

10 Theory of nonlnear convoluton The basc approach s to convolve separately, and then add the result, the lnear IR, the second order IR, the thrd order IR, and so on. Each order IR s convolved wth the nput sgnal rased at the correspondng power: y(n) M M M ( ) x( n ) h ( ) x ( n ) h ( ) x ( n )... h The problem s that the requred multple IRs are not the results of the measurements: they are nstead the dagonal terms of Volterra kernels

11 Volterra Volterra kernels kernels and and smplfcaton smplfcaton The general Volterra seres expanson s defned as: ( ) ( ) ( ) ( ) ( ) M 0 M 0 M 0 n x n x, h n x h y(n) ( ) ( ) ( ) ( ) M 0 M 0 M 0... n x n x n x,, h Ths explans also nonlnear effect wth memory, as the system output contans also products of prevous sample values wth dfferent delays

12 Memoryless dstorton followed by a lnear system wth memory Nose n(t) nput x(t) Not-lnear system K[x(t)] dstorted sgnal w(t) lnear system w(t) h(t) output y(t) The frst nonlnear system s assumed to be memory-less, so only the dagonal terms of the Volterra kernels need to be taken nto account. Furthermore, we neglect the nose, whch s effcently rejected by the sne sweep measurement method.

13 Volterra kernels from the measurement results y(t) y(t) The measured multple IRs h can be defned as: [ ω ] h' sn[ ω ] h' sn[ ω ]... h' sn var var var We need to relate them to the smplfed Volterra kernels h: [ ω ] h sn [ ω ] h sn [ ω ]... h sn var var var Trgonometry can be used to expand the powers of the snusodal terms: sn ( ω τ) cos( ω τ) sn ( ω τ) sn( ω τ) sn( ω τ) sn sn 8 ( ω τ) cos( ω τ) cos( ω τ) 8 6 ( ω τ) sn( ω τ) sn( ω τ) sn( ω τ) 8 6

14 Fndng the connecton pont Gong to frequency doman by takng the FFT, the frst equaton becomes: [ ω] X[ ω] ' [ ω] X[ ω/ ] ' [ ω] X[ ω/ ]... Y( ω) ' Dong the same n the second equaton, and substtutng the trgonometrc expressons for power of snes, we get: Y( ω) 6 8 X X 8 [ ω ] jx[ ω/ ] 6 [ ω/ ] jx[ ω/ ] X[ ω/ ]... The terms n square brackets have to be equal to the correspondng measured transfer functons of the frst equaton

15 Soluton Soluton Thus we obtan a lnear equaton system: [ ] 6 ' 8 j ' 6 ' j ' 8 ' We can easly solve t, obtanng the requred Volterra kernels as a functon of the measured multple-order IRs: ' 6 ' j 8 ' 0 ' ' j 8 ' j ' ' '

16 Non-lnear convoluton As we have got the Volterra kernels already n frequency doman, we can effcently use them n a multple convoluton algorthm mplemented by overlap-and-save of the parttoned nput sgnal: nput x(t) x x x Normal sgnal Squared sgnal Cubc sgnal Quartc sgnal h (t) h (t) h (t) h (t) output y(t)

17 Software mplementaton Although today the algorthm s workng off-lne (as a mx of manual CoolEdt operatons and some Matlab processng), a more effcent mplementaton as a CoolEdt plugn s beng worked out: Ths wll allow for real-tme operaton even wth a very large number of flter coeffcents

18 Audble evaluaton of the performance Orgnal sgnal Lnear convoluton These last two were compared n a formalzed blnd lstenng test Lve recordng Non-lnear mult convoluton

19 Subjectve lstenng test A/B comparson Lve recordng & non-lnear auralzaton selected subjects musc samples 9 questons -dots horzontal scale Smple statstcal analyss of the results A was the lve recordng, B was the auralzaton, but the lstener dd not know ths 9% confdence ntervals of the responses

20 Concluson Statstcal parameters more advanced statstcal methods would be advsable for gettng more sgnfcant results Queston Number Average score.67 * Std. Dev. (dentcal-dfferent) (better tmber)..96 (more dstorted).0. 9 (more pleasant).0.6 Fnal remarks - The CoolEdt plugn s planned to be released n two months t wll be downloadable from TTP:// - The sound samples employed for the subjectve test are avalable for download at TTP://pcangelo.eng.unpr.t/publc/AES0 - The new method wll be employed for realstc reproducton n a lstenng room of the behavour of car sound systems