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