2SPEAKERS ARRAYS FOR A PHYSICAL MODELLING PIANO GOAL: the 3D sound of a real “grancoda” piano3D Techniques:Wave Field SynthesisBeam FormingCrossTalk Cancellation
3Spatialization Effects (1/4) Stereo:Is the “reference” technique.2. Recursive Ambiophonic Crosstalk Elimination (RACE):Cancellation of the signal coming from the R speaker and arriving to the Left ear using, in the L speaker, a copy of that signal with a proper delay and a proper attenuationThe cancellation signal arrives to the Right ear: it needs to be cancelled with another copy coming from the R speaker……… and so on
4Spatialization Effects (2/4) 30°3. Beam Forming:2 plane waves from +30° and -30°.Speaker arrays generate sound fields simulating sound sources placed to a finite or infinite distance. This effect is created by means of different gains and delays.
5Spatialization Effects (3/4) 4. Convolution with real IRs:Points of percussion: F#1, G#3, C6, F#7.5. Gains and delays:4 point sources placed at a fixed distance from the listener.Calculation of the delay from source to speaker.
6Piano recording in anechoic room Impulse Responses of the sound board
7Neumann stereo micsEigenmike®31 Bruel&Kjaer microphonesNeumann Dummy Head
8MICROPHONE ARRAYS: TYPES AND PROCESSING Linear Array Planar Array Spherical ArrayProcessing Algorithm
9General ApproachprocessorV outputsM inputsWhatever theory or method is chosen, we always start with M microphones, providing M signals xm, and we derive from them V signals yvAnd, in any case, each of these V outputs can be expressed by:
10Traditional approaches The processing filters hm,v are usually computed following one of several, complex mathematical theories, based on the solution of the wave equation (often under certaing simplifications), and assuming that the microphones are ideal and identicalIn some implementations, the signal of each microphone is processed through a digital filter for compensating its deviation, at the expense of heavier computational load
11Traditional Spherical Harmonics approach Spherical Harmonics (H.O.Ambisonics)Virtual microphonesA fixed number of “intermediate” virtual microphones is computed, then the dynamically-positioned virtual microphones are obtained by linear combination of these intermediate signals.
12Novel approachNo theory is assumed: the set of hm,v filters are derived directly from a set of impulse response measurements, designed according to a least-squares principle.In practice, a matrix of impulse responses is measured, and the matrix has to be numerically inverted (usually employing some regularization technique).This way, the outputs of the microphone array are maximally close to the ideal responses prescribedThis method also inherently corrects for transducer deviations and acoustical artifacts (shielding, diffractions, reflections, etc.)
13Novel approachThe microphone array impulse responses cm,d , are measured for a number of D incoming directions.d=1…D sourcesm=1…M mikesckiWe get a matrix C of measured impulse responses for a large number P of directions
14Target DirectivityThe virtual microphone which we want to synthesize must be specified in the same D directions where the impulse responses had been measured. Let’s choose a high-order cardioid of order n as our target virtual microphone.This is just a direction-dependent gain. The theoretical impulse response coming from each of the D directions is:
15Novel approach c1,d(t) pd(t) c2,d(t) cM,d(t) Applying the filter matrix H to the measured impulse responses C, the system should behave as a virtual microphone with wanted directivitym = 1…Mmicrophonesd = 1…Ddirectionsδ(t)c1,d(t)pd(t)h1(t)c2,d(t)A2,vh2(t)A1,vδ(t)AM,vcM,d(t)hM(t)Target functionδ(t)But in practice the result of the filtering will never be exactly equal to the prescribed functions pd…..
16Novel approachWe go now to frequency domain, where convolution becomes simple multiplication at every frequency k, by taking an N-point FFT of all those impulse responses:We now try to invert this linear equation system at every frequency k, and for every virtual microphone v:This over-determined system doesn't admit an exact solution, but it is possible to find an approximated solution with the Least Squares method
17Least-squares solution We compare the results of the numerical inversion with the theoretical response of our target microphones for all the D directions, properly delayed, and sum the squared deviations for defining a total error:PThe inversion of this matrix system is now performed adding a regularization parameter b, in such a way to minimize the total error (Nelson/Kirkeby approach):It revealed to be advantageous to employ a frequency-dependent regularization parameter bk.
18Spectral shape of the regularization parameter bk At very low and very high frequencies it is advisable to increase the value of b.eHeL
19Real-time synthesis of the filters h It is possible to compute just once the following term:Then, whenever a new set filters is required, this is generated simply applying to R the gains Q of the target microphone:FIR filters realtime synthesis algorithm:[Rk]MxD[Qk]DxVN-point real-IFFTTime-domain windowing[hn]MxV[Hk]MxVThanks to Hermitian symmetry properties, a real-FFT algorithm can be employed
20Critical aspectsLOW frequencies: wavelength longer than array width - no phase difference between mikes - local approach provide low spatial resolution (single, large lobe) - global approach simply fails (the linear system becomes singular)MID frequencies: wavelength comparable with array width -with local approach secondary lobes arise in spherical or plane wave detection (negligible if the total bandwidth is sufficiently wide) - the global approach works fine, suppressing the side lobes, and providing a narrow spot.HIGH frequencies: wavelength is shorter than twice the average mike spacing (Nyquist limit) - spatial undersampling - spatial aliasing effects – random disposition of microphones can help the local approach to still provide some meaningful result - the global approach fails again
21Linear array16 omnidirectional mikes mounted on a 1.2m aluminium beam, with exponential spacing16 channels acquisition system:2 Behringer A/D converters + RMEHammerfall digital sound cardSound recording with Adobe AuditionFilter calculation, off-line processing and visualization with Aurora plugins
22Linear array - calibration The array was mounted on a rotating table, outdoorA Mackie HR24 loudspeaker was usedA set of 72 impulse responses was measured employing Aurora plugins under Adobe Audition (log sweep method) - the sound card controls the rotating table.The inverse filters were designed with the local approach (separate inversion of the 16 on-axis responses, employing Aurora’s “Kirkeby4” plugin)
24Linear array - practical usage The array was mounted on an X-Y scanning apparatusa Polytec laser vibrometer is mounted along the arrayThe system is used for mapping the velocity and sound pressure along a thin board of “resonance” wood (Abete della val di Fiemme, the wood employed for building high-quality musical instruments)A National Instruments board controls the step motors through a Labview interfaceThe system is currently in usage at IVALSA (CNR laboratory on wood, San Michele all’Adige, Trento, Italy)
25Linear array - practical usage The wood panel is excited by a small piezoelectric transducerWhen scanning a wood panel, two types of results are obtained:A spatially-averaged spectrum of either radiated pressure, vibration velocity, or of their product (which provides an estimate of the radiated sound power)A colour map of the radiated pressure or of the vibration velocity at each resonance frequency of the board