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

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VALIMAKI ET AL. PAPERS3.3 Loop Filter DesignLet us denote the numerator of Eq. (6) byIt is known that our hearing is sensitive to the change A = g(1 + al) (20)of decay rate of a sinusoid, and in practice we measurethe time constant of some 20 lowest harmonics with the and HI(to) with the numerator removed byintent of matching the frequency response of the loopfilterfilter,Hl(Z)it istoclearthosethatdata.thereSincearewemoreuse arestrictionsfirst-order loopthan//l(to) - HI(t°)A (21)unknowns in this problem, and only an approximatesolution is possible. Then Eq. (18) can be rewritten asIn principle the loop filter should be designed basedon an auditory criterion, but since that would be compli- tq- 1cated, we decided to use weighted least-squares design. E = _ W(GD[IAPi(%)I - Gk]2 . (22)k=0The error function to be minimized in the magnitudeonlyapproximation can be defined as The gain g of the loop filter at low frequencies canNh-l be chosen based on the loop gain values of the lowestE = ___ W(GO[lHl(toOI - Gk]2 (18) harmonics. In many cases it is good enough to set g =k=0 Go, whereas sometimes the average of two or three lowestloop gain values gives a better result.where N h is the number of frequency points where the The value for the coefficient a I that minimizes E canloop gain Gk is approximated, % = kO0 are the central be found by differentiating Eq. (22) with respect to a 1.frequencies of the N h lowest harmonics, and W(Gk) is This yieldsa nonnegative error weighting function. It is reasonableltO choose a weighting function W_ Ut _ that gives a largerOENh - 1- 2A0 _'. W(GO01[-/1(%)1'' [IAo[/l(to0l- Gk].weight to the errorsdecaying harmonicsin the time constants of the slowlysince our hearing tends to focus onOa1 k=o Oaltheirdecay.A candidateforsucha weightingfunctionis (23)Nh-lBy substituting Eqs. (6) and (19), we can writeOE _ 2A° _ Aocos tok(1 + alcos%) -3 - Gkcos ton(1 + alcos%) -l (24)Oal k=o 1 -- GkThe aim is now to find the zero of this function. In1 practice we find a near-optimal solution in the followingW(GO - 1 - Gk' (19) way. The value of the derivative is evaluated, and dependingon the sign of the result, a I is changed by asmall increment, the derivative is evaluated again, andWe require that 0 _< Gk < 1 for all k, which is a physisoon. After the derivative has reached a very smallcally reasonable assumption since the system to be modvalue,the iteration is terminated and the final value foreled is passive and stable.a1 is used in the synthesis model. We have verifiedthe convergence of this design procedure in practice by....... analyzing signals generated using the synthesis model.0.995 data yields the same filter parameters--within numericalaccuracy--as were used in the synthesis. Also analysis '9 Theloopfilterthatwasdesignedbasedonthe the°'99I / I [ T"_ _-- _ match with natural tones has been found to be satisfac-·S['_°'985[ I '"'-( I_tory in many cases. Fig. 20 illustrates the loop filter094r I I I I I design fora typical kantele tone.0.975 It iswellunderstoodthatwhena stringisputtovibra-90.97 tion by plucking it, the sound signal will lack those harmoniesthat have a node at the plucking point (see, for.0'9651 example, [17]). However, in general the string is not0 96_/ ' 015 '_'.s ' _' _'._ , _ , _ plucked exactly at the node of any of the lowest harmon-' . _.5 2 2.5 3 3.5 ics, and since the amplitude of the higher harmonics isFrequency(kHz)Fig. 20. Loop filter design for kantele. Circles--loop gains considerably small anyway, it is not possible to accu-G2 at harmonic frequencies; solid line--the magnitude re- rately detect the plucking point by simply searching forsponse [H_(z)l2. the lacking harmonics in the magnitude spectrum. An-342 d. Audio Eng. Soc., Vol. 44, No. 5, 1996 May
PAPERS PHYSICAL MODELING OF PLUCKED STRING INSTRUMENTSother practical problem may occur because of nonlinear using a time-varying inverse filter, which cancels thebehavior of the string. Namely, the amplitude of vibra- impulse response of the string. This filter can be' detionof a weak harmonic can gain energy from other signed based on the STFT analysis discussed above.modes so that its amplitude begins to rise, reaching a However, satisfactory results are obtained when usingmaximum about 100 ms after the attack, and then begins carefully chosen tones that behave well, that is, thereto decay [44]. This can often be seen in the analysis of are no strong nonlinear effects nor beating present. Fig.harmonic envelopes of the guitar. For these reasons we 21 shows the spectrum of a mandolin tone and the magnibelievethat a more comprehensive understanding of the tude response of the inverse filter. The magnitude speceffectof the plucking point can be achieved by studying trum of the residual (the result of inverse filtering) isthe time-domain behavior of the string in terms of the illustrated in Fig. 22. Note that in this case the harmonicsshort-time autocorrelation function. A frequency- have been canceled quite accurately.domain technique for estimating the plucking point hasbeen reported by Bradley et al. [45]. 4 SYNTHESIS OF PLUCKED STRINGSEstimation of the plucking position is an inherentlydifficult problem since a recorded tone can include con- The residual signals resulting from the analysis andtributions of several delays of approximately the same inverse filtering of an original string instrument tone canmagnitude, such as early reflections from objects near be directly applied as the excitation to the string model,the player, the floor, the ceiling, or a wall. To minimize as described in Sections 2 and 3. The synthesis procedurethe effect of these additional factors, we suggest per- is carried out in the following way.forming the analysis of tones recorded in an anechoic 1) Obtain the residual signal by inverse filtering,chamber. 2) Window the first 100 ms of the residual signalIt is, however, not absolutely mandatory to estimate using, for example, the right half of a Hammingand model the effect of the plucking position. Its contri- window.bution can be left in the excitation signal obtained usinginverse filtering, which is described in the followingsection.03.5 Inverse Filteringfiltering, that is, filtering the signal y (n) which is now _assumed to be the output of the model with the in- _-20 ·hein utsignalxn _,o...!1transfer function of the string model shown in Fig. 9canbeexpressedas 301S(z) = 1 - z -L_F(z)HI(z)' (25)vetted transfer function S-'(z) of the string model. The _ii [If S(z) had zeros outside the unit circle, the inverse filter - ' ' 9 1S-l(z) would be unstable. In that case, inverse filtering Fr*q.... y(mz)would not yield an acceptable result. Fortunately this is Fig. 21. Magnitude spectrum of mandolin tone (solid line) andmagnitude response of inverse filter (dotted line).not a problem in practice, as can be seen by substitutingEq. (6) into Eq. (25). Inverting this equation yieldsS-i(2)=l-[-alz-1--lg(l+alz-+ alz -1 1)z-L' F(z) (26) -it-10This technique is simple to apply, but since the order -_5of the loop filter is low, the harmonics are not canceled 4_20very accurately using the inverse filter S-_(z). The re- vsulting x(n) may suffer from high-frequency noise. The _-25low-order loop filter was chosen primarily since it is then -30efficient to implement the synthesis model. However,inverse filtering is an off-line procedure, where accuracy -35is much more important than efficiency. We could thus -40design a higher order filter to be used in the inverse -45filtering (see for example, [12] or [15]).Another deficiency of the inverse filtering technique I 2 3 4 5 6 7 8 9 l0is that it does not take into account the nonlinear effects Frequency (kHz)of the physical strings. This can be accounted for by Fig. 22. Magnitude spectrum of inverse filtered mandolin tone.d. Audio Eng. Soc., Vol. 44, No. 5, 1996 May 343
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Physical Modeling of Plucked String Instruments with Application to ...

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