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NONNON--LINEAR CONVOLUTION: A NEW LINEAR CONVOLUTION: A NEW APPROACH FOR THE AURALIZATION APPROACH FOR THE AURALIZATION
OF DISTORTING SYSTEMSOF DISTORTING SYSTEMS
Angelo Farina, Alberto Bellini and Enrico Armelloni
Industrial Engineering Dept., University of Parma, Via delle Scienze 181/AParma, 43100 ITALY – HTTP://pcfarina.eng.unipr.it
GoalsGoals forfor AuralizationAuralizationTransform the results of objective electroacousticsmeasurements to audible sound samples suitablefor listening tests
Traditional auralization is based on linearconvolution: this does not replicates faithfully the nonlinear behaviour of most transducers
The new method presented here overcomes to thisstrong limitation, providing a simplified treatment of memory-less distortion
MethodsMethodsWe start from a measurement of the system based on exponential sine sweep (Farina, 108th AES, Paris 2000)Diagonal Volterra kernels are obtained by post-processing the measurement resultsThese kernels are employed as FIR filters in a multiple-order convolution process (originalsignal, its square, its cube, and so on are convolved separately and the result is summed)
ExponentialExponential sweepsweep measurementmeasurement
The excitation signal is a sine sweep with constantamplitude and exponentially-increasing frequency
RawRaw responseresponse ofof the systemthe system
Many harmonic orders do appear as colour stripes
DeconvolutionDeconvolution ofof systemsystem’’s s impulseimpulse responseresponse
The deconvolution is obtained by convolving the raw response with a suitableinverse filter
Multiple Multiple impulseimpulse responseresponse obtainedobtained
The last peak is the linear impulse response, the precedingones are the harmonic distortion orders
1st
2nd
3rd5th
AuralizationAuralization byby linearlinear convolutionconvolution
Convolving a suitable sound sample with the linear IR, the frequencyresponse and temporal transient effects of the system can be simulatedproperly
⊗
=
WhatWhat’’s s missingmissing in in linearlinear convolutionconvolution ??
No harmonic distortion, nor other nonlineareffects are being reproduced.From a perceptual point of view, the sound is
judged “cold” and “innatural”A comparative test between a strongly nonlinear device and an almost linear one does not reveal any audible difference, because the nonlinear behavior is removed for both
TheoryTheory ofof nonlinearnonlinear convolutionconvolutionThe basic approach is to convolve separately, and then add the result, the linear IR, the second orderIR, the third order IR, and so on.Each order IR is convolved with the input signalraised at the corresponding power:
( ) ( ) ( ) ( ) ( ) ( ) .....inxihinxihinxih)n(y1M
0i
33
1M
0i
22
1M
0i1 +−⋅+−⋅+−⋅= ∑∑∑
−
=
−
=
−
=
The problem is that the required multiple IRs are not the results of the measurements: they are instead the diagonalterms of Volterra kernels
Volterra Volterra kernelskernels and and simplificationsimplification
The general Volterra series expansion is defined as:
( ) ( ) ( ) ( ) ( )∑ ∑∑−
=
−
=
−
=+−⋅−⋅+−⋅=
1M
0i
1M
0i21212
1M
0i111
1 21
inxinxi,ihinxih)n(y
( ) ( ) ( ) ( )∑ ∑ ∑−
=
−
=
−
=+−⋅−⋅−⋅+
1M
0i
1M
0i
1M
0i3213213
1 2 3
.....inxinxinxi,i,ih
This explains also nonlinear effect with memory, as the system output contains also products of previous sample values withdifferent delays
MemorylessMemoryless distortiondistortion followedfollowed byby a a linearlinearsystem system withwith memorymemory
The first nonlinear system is assumed to be memory-less, so only the diagonal terms of the Volterra kernels need tobe taken into account.
Not-linearsystemK[x(t)]
Noise n(t)
input x(t)+
output y(t)linear systemw(t)⊗h(t)
distorted signalw(t)
Furthermore, we neglect the noise, which is efficientlyrejected by the sine sweep measurement method.
Volterra Volterra kernelskernels fromfrom the the measurementmeasurement resultsresultsThe measured multiple IRs h’ can be defined as:
[ ] [ ] [ ] ...3sin'h2sin'hsin'h)t(y var3var2var1 +ω⋅⊗+ω⋅⊗+ω⊗=
We need to relate them to the simplified Volterra kernels h:
[ ] [ ] [ ] ...sinhsinhsinh)t(y var3
3var2
2var1 +ω⊗+ω⊗+ω⊗=
Trigonometry can be used to expand the powers of the sinusoidal terms:
( ) ( )τ⋅ω⋅⋅−=τ⋅ω 2cos21
21sin2 ( ) ( ) ( )τ⋅ω⋅⋅−τ⋅ω⋅=τ⋅ω 3sin
41sin
43sin3
( ) ( ) ( )τ⋅ω⋅⋅+τ⋅ω⋅⋅−=τ⋅ω 4cos812cos
21
83sin4
( ) ( ) ( ) ( )τ⋅ω⋅⋅+τ⋅ω⋅⋅−τ⋅ω⋅=τ⋅ω 5sin1613sin
165sin
85sin5
FindingFinding the connection the connection pointpointGoing to frequency domain by taking the FFT, the first equation becomes:
[ ] [ ] [ ] [ ] [ ] [ ] ...3/X'H2/X'HX'H)(Y 321 +ω⋅ω+ω⋅ω+ω⋅ω=ω
Doing the same in the second equation, and substituting the trigonometric expressions for power of sines, we get:
[ ] [ ]
[ ] [ ] [ ] ...5/XH1614/XjH
813/XH
165H
41
2/XjH21H
21XH
85H
43H)(Y
5453
42531
+ω⋅⋅+ω⋅⋅⋅+ω⋅⎥⎦⎤
⎢⎣⎡ ⋅−⋅−+
+ω⋅⋅⎥⎦⎤
⎢⎣⎡ ⋅−⋅−+ω⋅⎥⎦
⎤⎢⎣⎡ ⋅+⋅+=ω
The terms in square brackets have to be equal to the corresponding measured transfer functions H’ of the first equation
SolutionSolutionThus we obtain a linear equation system:
[ ]
⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪
⎨
⎧
⋅=
⋅⋅=
⋅−⋅−=
+⋅⋅−=
⋅+⋅+=
55
44
533
422
5311
H161'H
H81j'H
H165H
41'H
HH21j'H
H85H
43H'H
We can easily solve it, obtaining the required Volterra kernels as a function of the measured multiple-order IRs:
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⋅=
⋅⋅−=
⋅−⋅−=
⋅⋅+⋅⋅=
⋅+⋅+=
55
44
533
422
5311
'H16H
'Hj8H
'H20'H4H
'Hj8'Hj2H
'H5'H3'HH
NonNon--linearlinear convolutionconvolutionAs we have got the Volterra kernels already in frequency domain, we can efficiently use them in a multiple convolution algorithm implemented byoverlap-and-save of the partitioned input signal:
Normal signal
input x(t)
x2
output y(t)h1(t)
Squared signal
x3
Cubic signal
x4
Quartic signal
⊗
+
h2(t) ⊗
+
h3(t) ⊗
+
h4(t) ⊗
Software Software implementationimplementationAlthough today the algorithm is working off-line (as a mix ofmanual CoolEdit operations and some Matlab processing), a more efficient implementation as a CoolEdit plugin is being worked out:
This will allow for real-time operation even witha very large number of filter coefficients
AudibleAudible evaluationevaluation ofof the performancethe performanceOriginal signal Linear convolution
Live recording Non-linear multi convolution
These last two were compared in a formalized blind listening test
SubjectiveSubjective listeninglistening testtest
A/B comparisonLive recording & non-linearauralization12 selected subjects4 music samples9 questions5-dots horizontal scaleSimple statistical analysis ofthe resultsA was the live recording, B was the auralization, but the listener did not know this
95% confidence intervalsof the responses
ConclusionConclusionStatistical parameters – more advanced statistical methods
would be advisable for getting more significant results
Final remarks- The CoolEdit plugin is planned to be released in two months – it
will be downloadable from HTTP://www.ramsete.com/aurora
- The sound samples employed for the subjective test are availablefor download at HTTP://pcangelo.eng.unipr.it/public/AES110
- The new method will be employed for realistic reproduction in a listening room of the behaviour of car sound systems
Question Number Average score 2.67 * Std. Dev.1 (identical-different) 1.25 0.763 (better timber) 3.45 1.965 (more distorted) 2.05 1.349 (more pleasant) 3.30 2.16