Physical Modeling of Plucked String Instruments with Application to ...

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VALIMAKI ETAL.PAPERSrecursive filters. This method was tested in the context the filter. For/-/l(Z) tO be a stable low-pass transfer funcofstring modeling [27]. An extensive review of digital tion, we require that - 1 < al < 0. The numerator 1 +'filter approximations for fractional delay has been pre- a I scales the frequency response (divided by g) at 0 tosented by Laakso et al. [25]. unity, thus allowing control of the gain at co = 0 usingReal strings are more or less stiff so that the wave the coefficient g. We require that Igl_< 1.propagation is dispersive, that is, the total loop delay is Fig. 10 shows the magnitude response and the groupfrequency dependent. In principle, a single filter may delay of the loop filter HI(z) [Eq. (6)] with three differentimplement both the magnitude and the phase in the string values of coefficient a1. These values have been chosenmodel loop, but in practice it is easier to use three filters so that they represent cases found in practice. In this examtocontrol the three parts separately-- a linear-phase all- ple g = 0.99. (A change in g merely shifts the magnitudepass approximation F(z)to control the fundamental fre- response curve vertically.) Note that the group delay ofquency, the loop filter Hi(z) to control the magnitude the loop filter is very small in all three cases.response, and an all-pass filter D (z) to adjust the frequen- Fig. 11 shows the impulse response of the string modelcy-dependent overall phase delay. See [29], [30] for a S(z) with the three loop filters of Fig. 10. Here the lengthdiscussion on the implementation of dispersive waveguidemodels using all-pass filters. For low strings, especiallyor crucial on the to piano the timbre [31], of [32], sound dispersivity synthesis. isFor important higher xp(n) '_'ri_ )_Ht(z_ Il L y(n)strings the dispersivity is relatively small. In mostplucked string instruments the effect due to dispersion Fig. 9. Block diagram of the string model S(z). F(z)--FIR orcan be neglected and the filter D(z) may be left out of all-pass-type fractional delay filter; Hl(z)--loo p filter.the model.To conclude our discussion on the frequency depen- i' 'al:-o00, _dence of the string length, let us express L((o) by means 41=---¢_iofL(co) the three = L I filters + 'tH(CO) discussed, + 'TF(00 ) "{- 'rD(la) ) (5) _0.95i _'0'9 tai_085'0 0.05 0:l 0.15 0'.2 0.25 0'.3 0.35 0.4 0.45 0.5where L[ is the integer number of unit delays in the·delay N.... lized Frequencyline, and Xn(CO),xr(co)m, and 'rD(m)are the phase delays(a)of the loop filter Hi(z), the fractional delay filter F(z), _______ . , ..,_and the dispersion filter D(z), respectively. Fig. 9 illus- 0.052.3trateSLoopthe In the original implementationFilter KS model of the thestring loop model filter Hl(z S(z). ) is a _0_-0o5........{_l_ -- -- _'"'"_ _ !ltwo-tap averager which is easy to implement without Normalized Frequency----__i_alal =-0.01= -0.001' -0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5multiplications [3]. This simple filter, however, cannot (b)match the frequency-dependent damping of a physical Fig. 10. Magnitude response and group delay of one-pole loopstring. We feel that an FIR filter is not suited to this filter for three different values of a I. g = 0.99 in all cases.purpose since its order should be rather high to matchthe desired characteristics. Instead we suggest the useal = -0.001, g = 0.99, L = 19o a,m ,owpass.,terforsimu,ati. 'ILIILL Li III1I thcharacteristics of a physical string. The same decision ?was made in [12]. From [17], [33] it can be concluded _0.5that a second- or third-order all-pole filter could be suita- < 0_10 100 260 300 400 560 600 700 800 900 1000ble for a loop filter. It is important to use the simplest Sample Indexsatisfying loop filte r since one such filter is needed for 41=-0.ol.g=o.99. L=19static filter, but its coefficients should be changed as a _0_tll 1the function modelof of theach string string.Also, length and other loop playing filteris param- not a < o/tit i ].,j '_ i 'eters. This is easiest to achieve using a low-order IIR 0 100 200 300 400 500 600 700 800 900 1000· Sample Indexfilter. We found that a reasonable approximation mayal = -O.O5,g =0.99, L = 19be obtainedusinga first-orderall-polefilter, i .........1 + a I (6) _0.5
PAPERS PHYSICAL MODELING OF PLUCKED STRING INSTRUMENTSof the delay line is L = 19. It is seen that the impulse tic guitar, respectively. The impulse response can beresponse decays more rapidly as the magnitude of al seen to be long enough so that the temporal envelope canincreases,beassumedto be perceptuallyimportant.Thesituationiscomparable to the perception of reverberation in a small2.4 Modeling the Body room. This implies that a digital filter approximationFrom the viewpoint of signal processing, the body of should not be designed exclusively based on magnitudetheacoustic guitar and the transfer function from the response criteria. The spectral envelope seen in Fig. 13 isbridge to a specific direction may be considered as a relatively flat, but there are a large number of resonanceshigh-order linear filter. A set of transfer functions would starting from the lowest resonance around 100 Hz.be needed to simulate the directivity pattem of a string We have studied several digital filter approximationsinstrument [34]. In most physical models only one transfer for modeling the body of the acoustic guitar [18]. Anfunction has been included. This simplified approach simu- FIR filter model of th6 body response must be at leastlates the radiation of the sound to one point in front of the 50 to 100 ms long (more than 1000 taps) to yield goodsound hole of the string instrument. This kind of model synthetic sound. Linear prediction (LPC) analysis sugwillthus at best produce a sound reminiscent of a recording gests an all-pole filter model of order 500 or more. Bothof a musical instrument in a relatively anechoic room. In of these are computationally too expensive for real-timeSection 2.5 we discuss different methods to incorporate implementation using a single DSP processor. We alsodirectivity characteristics into physical models, designed reduced-order IIR filters that approximate theThe transfer function from the bridge to the listener frequency resolution of the human auditory system, butcan be measured approximately by exciting the bridge even these did not reduce the computational load enoughwith an impulse hammer and registering the radiated without audible degradation of the sound quality. A moresound. Figs. 12 and 13 show the impulse response and detailed discussion of the body modeling problem isthe magnitude spectrum of the body of a classical acous- given elsewhere [23], [35].To overcome the inherently heavy computational loadof the filter-based body models a novel method wasinvented [7], [9]-[ 11]. Let us consider the string instru-0_ ment model of Fig. 14(a). The output signal y(n) of thissystem can be expressed as0.60.4 y(n) = e(n) * h(n) * b(n) (7).g$. 02 where the asterisk denotes the operation of linear discrete
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Physical Modeling of Plucked String Instruments with Application to ...

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