1	Realtime auralization employing a not-linear, not-time-invariant convolver Angelo Farina ¹, Adriano Farina ² 1) Industrial Engineering Dept., University of Parma, Via delle Scienze 181/A Parma, 43100 ITALY – HTTP://www.angelofarina.it 2) Liceo Classico G.D. Romagnosi, via Marialuigia 1, Parma, 43100 ITALY HTTP://www.adrianofarina.it
2	Goals for Auralization Transform the results of objective electroacoustics measurements to audible sound samples suitable for listening tests
3	What’s linear convolution ? Also called “FIR filtering” Convolution is the mathematical operation performed when filtering a waveform x employing as filter coefficients the samples of a second waveform, usually denoted as h and called “impulse response” The samples of the input waveform are multiplied by the samples of the impulse response h, and the results accumulated (summed):
4	Auralization by linear convolution Convolving linearly a suitable sound sample with the Imp.Resp., the frequency response and temporal transient effects of the system can be simulated properly
5	Auralization by linear convolution The beginnings: hardware DSP-based convolution units
6	Auralization by linear convolution The AMBIOPHONICS Institute: the home of convolution
7	Software Convolution: BruteFIR and AlmusVCU Open-source software for Linux by Anders Torger – AES 24° Conference
8	Auralization Software Nowadays many sytems or software plugins can perform satisfactorily the Linear Convolution operation, and are widely employed for audio processing
9	Linear Impulse Responses Huge collections of impulse responses have been measured in famous theatres and concert halls all around the world, as well for renowned audio processing gear. Recent advancements in the measuring technique, making use of the Exponential Sine Sweep signal, and the usage of multiple sound sources and microphones, make it possible to capture with minimal noise a detailed “acoustical photo” of existing rooms. See, for example, the results of the Waves project on: www.acoustics.net Furthermore, all room acoustic modelling software is nowadays equipped with a “rendering” tool, which exports impulse responses suitable to be employed for auralization by linear convolution. These synthetic IRs are even cleaner and with greater dynamic range than any measured IR
10	What’s missing in linear convolution ? No harmonic distortion, nor other nonlinear effects are being reproduced. From a perceptual point of view, the sound is judged “cold” and “innatural” A comparative test between the simulation of a strongly nonlinear device and an almost linear one does not reveal any audible difference, because the nonlinear behavior is removed for both
11	How can we perform not-linear convolution? There are actually two competing approaches available in the audio industry: IR-switching technique Diagonal Volterra Kernel
12	Method 1 (IR switching) A very simple idea: a different IR is employed depending on the amplitude of each sample of the signal to be filtered The method is quite old: the first published papers are those of Bellini and Farina (1998) and Michael Kemp (1999) Several impulse responses are measured, employing test signals of different amplitudes, and stored for later usage. It is mandatory to implement the convolution as direct form in time domain, as each sample has to be processed with a different IR.
13	Measurement of multiple IRs Michael Kemp employed a step function, with several steps of decreasing amplitude
14	Measurement of multiple IRs Farina e Bellini did employ a sequence of MLS repetitions, each with decreasing amplitude, providing better S/N ratio in real-world measurements
15	Implementation (Michael J. Kemp) Focusrite did release recently Liquid Channel, the first “dynamic convolver” implementing the IR-switching technique
16	Implementation (Farina/Bellini) A FIR filtering algorithm, with the set of coefficients chosen depending on the sample amplitude, was implemented on a Sharc EZ-KIT 20161 board, and employed for car-audio applications
17	Example The “not linear device” is emulated by the DISTORTION plugin of Adobe Audition, followed by sound playback and simultaneous recording over the loudspeaker and microphone of a laptop PC
18	Example This is the multiple MLS signals after being processed through the not-linear device
19	Example Here the 16 impulse responses measured with MLS of different amplitude (decreasing 3dB each from left to right) are shown
20	Audible evaluation of the performance Original signal
21	Method 2 – Diagonal Volterra Kernels We 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 results These kernels are employed as FIR filters in a multiple-order convolution process (original signal, its square, its cube, and so on are convolved separately and the result is summed)
22	Exponential sweep measurement The test signal is a sine sweep with constant amplitude and exponentially-increasing frequency
23	Raw response of the system Many harmonic orders do appear as colour stripes
24	Deconvolution of system’s impulse response The deconvolution is obtained by convolving the raw response with a suitable inverse filter
25	Multiple impulse response obtained The last peak is the linear impulse response, the preceding ones are the harmonic distortion orders
26	Memoryless distortion followed by a linear system with memory The first nonlinear system is assumed to be memory-less, so only the diagonal terms of the Volterra kernels need to be taken into account.
27	Theory of nonlinear convolution The basic approach is to convolve separately, and then add the result, the linear IR, the second order IR, the third order IR, and so on. Each order IR is convolved with the input signal raised at the corresponding power:
28	Efficient non-linear convolution As we have got the Volterra kernels already in frequency domain, we can efficiently use them in a multiple convolution algorithm implemented by overlap-and-save of the partitioned input signal:
29	Software implementation A small Italian startup company, Acustica Audio, developed a VST plugin based on the Diagonal Volterra Kernel method, named Nebula
30	Software implementation Nebula is also equipped with a companion application, Nebula Sampler, designed for automatizing the measurement of a not linear system with the Exponential Sine Sweep method:
31	Time-variant systems Nebula can sample also time-variant systems, such as flangers or compressors, by repeating the sine sweep measurement several times, along a repetition cycle or changing the signal amplitude
32	Reconstruction accuracy Nebula is actually limited to Volterra kernels up to 5^th order, and consequently does not emulate high-frequency harmonics:
33	Audible evaluation of the performance Original signal
34	Subjective listening test A/B comparison Live recording & non-linear auralization 12 selected subjects 4 music samples 9 questions 5-dots horizontal scale Simple statistical analysis of the results A was the live recording, B was the auralization, but the listener did not know this
35	Results Statistical parameters – more advanced statistical methods would be advisable for getting more significant results
36	Another example Original signal
37	Conclusions Traditional Linear Convolution is a powerful technique, but its reconstruction of the real world suffers for the limitations due to Linearity and Time Invariance Not-linear convolution is possible with two competing techniques: IR switching and Diagonal Volterra Kernels. The latter provides much more efficiency in computational load, and allows for easy emulation also of time-variant systems The Nebula software makes this technology available to everyone
38	Remarks