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

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V]_LIM_.KI ETAL.PAPERSpolarizations. In the kantele, bending the tuning peg at where y(n) is the signal to be analyzed and w(n) is aone end of the strings has been shown to induce nonlin- window function (such as the Hamming window). Anearity due to longitudinal forces [20]. This effect has estimate for the pitch is obtained by searching for thebeen simulated successfully in the following way. The maximum of _bk(m)for each k.delay line signal is squared, then filtered using a leaky Typical pitch contours of a guitar and a kantele toneintegrator, and the result is added to the output of the are shown in Fig. 18. The contours have been smoothedstring model. This procedure boosts the even harmonics using a three-point running median filter [40]. In bothof the signal, cases the pitch decreases with time and approaches aIn [38] a passive nonlinear filter was PrOPosed for constant value after about 0.5 s. In the kantele the deproducingeffects similar to mode coupling in string increase of the fundamental frequency is quite substantial..struments. A first-order digital all-pass filter is attached The problem now is to determine the best estimate foto the end of the delay line of the string model. The · for the fundamental frequency in a perceptual sense. Incoefficient of the all-pass filter depends on the sign of practice a good solution is to use the average pitch valuethe delay line signal. Another alternative is to use a after some 500 ms as the nominal value, since it isvariable delay based on a Lagrange interpolator FIR important to have a reliable pitch estimate toward thefilter. This technique was found to be suitable for pro- end of the note. This improves the quality of syntheducingsynthetic signals that exhibit time-varying behav- sized tones.ior in their decay parts. Once the estimate .f0 for the fundamental frequencyhas been chosen, the effective length L of the delay line3 CALIBRATION OF MODEL PARAMETERS can be computed asThe synthesis system includes three parts that corn- fspletely determine the character of the synthetic sound-- L =for-. (13)the string model, the plucking-point equalizer, and theinput sequence. This implies that to calibrate the model 250[ , jto some particular instrument it is needed to estimatethe values for the length of the delay line L, the coeffi-_249.8I__cients of the loop filter Hi(z), the delay M and the param-_249'6I_249.4 Ieter signalrfx(n). of the In plucking-point this section the parameter equalizer, estimation and the input procedures_ 249'2Ithat will extract these values are described. 249_ o'.l o'.2 o13 0'.4 o15 0.6(a)3.1 Estimation of the Delay Line Length LThe delay length L (in samples) determines the funda- 409 1mental frequency fo of the synthetic signal according toEq. (3) so that_4o8fo = (11) _--406 01.! 0'.2 0'.3 01.4 0.5 0.6Time(s)wherefs is the sample rate. For pitch detection we have (b)used a well-known method based on the autocorrelation Fig. 18. Fluctuation of fundamental frequency. (a) Steel-stringfunction. The short-term or windowed autocorrelation gmtar tone. (b) Kantele tone.functio n is defined as [39]1N-1_k(m) = _ _'=o[y(n + k)w(n)y(n + k + m)w(n + m)] , 0 _
PAPERSPHYSICALMODELINGOFPLUCKEDSTRINGINSTRUMENTSWhen the loop filter has also been designed, we can Spectral peak detection results in a sequence of magnisubtractits phase delay from L and define the nominal tude and frequency pairs for each harmonic. The sefractionaldelay Lr as quence of magnitude values for one harmonic is calledthe envelope curve of that harmonic. A straight line isLr = L - 'tH(C00)- floor[L - %/(COo)] (14) matched to each envelope curve on a decibel scale, sinceideally the magnitude of every harmonic should decaywhere cb0 = 2_rjeo. The delay Lr is used as the desired exponentially, that is, linearly on a logarithmic scale.delay in the design of the fractional delay filter F(z). In Measurements show that this idealized case is rarely metpractice the phase delay of this filter can be expressed in practice, and many different kinds of deviations areas xF(Co) = Mr + Lr, where positive integer Mr is the common. It is possible to decrease the error in fit byintegral part of the phase delay that depends on the order starting it at the maximum of the envelope curve [43]of the fractional delay filter and Lr is the fractional part. and by terminating it before the data get mixed with thenoise floor, such as after the decay of about 40 dB. As3.2 Measuring the Frequency-Dependent a result, a collection of slopes I_k for k = 1, 2.....Damping N h is obtained (see Fig. 19). The corresponding loopNext we discuss how to measure damping of a string gain of the string model at the harmonic frequencies isas a function of frequency. As discussed in Section 2.1, computed asall losses, including the reflection at the two ends of thestring, are incorporated in the loop filter Hi(z). In order Gk = 10_Aa°H , k = 1,2 ..... N h (17)to design a loop filter we need to estimate the dampingfactors at the harmonic frequencies of the sound signal, where [3k are the slopes of the envelopes and H is theThis is achieved using the short-time Fourier transform hop size. The sequence Gk determines the prototype(STFT) [19] and tracking the amplitude of each har- magnitude response of the loop filter Hl(o_) at the harmonic,monic frequencies o_k, k = 1, 2 ..... N h, as illustratedThe STFT of y(n) is a sequence of'discrete Fourier in Fig. 20.transforms (DFT) defined as The desired phase response for Hl(_) can be determinedbased on the frequency estimates of the harmon-N- _ ics. We do not, however, try to match the phase responseYm(k)= n=O _ w(n)y(n + mH)e -j'°_ , m = 0, 1, 2,... of the filter. There are two principal reasons for this.(15) First we believe that it is much more important to matchthe time constants of the partials of the synthetic soundwith than the frequencies of the partials, since rather smalldeviations from harmonic frequencies occur in the case2_rk of plucked string instruments. In case we wanted to,r_°k - N ' k = 0, 1, 2 ..... N - 1 ' (16) resynthesize strongly inharmonic tones, the phase responseof the loop filter should be considered. The secwhereN is the length of the DFT, w(n) is a window ond reason is that as we want to use a low-order loopfunction, and H is the hop size or time advance (in filter, the complex approximation of the frequency re-'samples) per frame. In practice we compute each DFT sponse would not be very successful. Thus we restrictusing the FFT algorithm. To obtain a suitable compro- ourselves to magnitude approximation only in the loopmise between time and frequency resolution, we use a filter design.window length of four times the period length of thesignal. The overlap of the windows is 50%, implyingthat H is 0.5 times the window length. We apply exces- . . ' .....sive zero padding by filling the signal buffer with zeros __ _ ' 2 'to reach N = 4096 in order to increase the resolutioning to harmonics can be found from the magnitude spectrumfirst finding the nearest local minimum at both -_cbetween sides of an these assumed two local maximum. minima The is assumed largest magnitude to be the _-_5 ._ _---_____ __inthefrequencydomain.by harmonic peak. The estimates Thespectralpeakscorrespond- for the frequency and :_ _ _._.. 3magnitude of the peak are fine-tuned by applying para- -2cbolic interpolation around the local maximum and solvingfor the value and location of the maximum of this -2_interpolating function (see [41] or [42] for details). Thenumber of harmonics to be detected is typically Nh < -3c 0.05 01! 0.15 012 0.25 0'.3 0.35 0'.4 0.45 0.520 (for the acoustic guitar). The STFT analysis is applied Time(s)to the portion of the signal that starts before the attack Fig. 19. Straight-line fits (dotted lines) of envelope curvesand ends after some 0.5-1 s after the attack. (solid lines) of six lowest harmonics of electric guitar tone.J.AudioEng.Soc.,Vol.44,No.5, 1996May 341
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