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The Ideal Plucked String

The ideal plucked string is defined as an initial string displacement and a zero initial velocity distribution [317]. More generally, the initial displacement along the string $ y(0,x)$ and the initial velocity distribution $ {\dot y}(0,x)$, for all $ x$, fully determine the resulting motion in the absence of further excitation.

An example of the appearance of the traveling-wave components and the resulting string shape shortly after plucking a doubly terminated string at a point one fourth along its length is shown in Fig.6.7. The negative traveling-wave portions can be thought of as inverted reflections of the incident waves, or as doubly flipped ``images'' which are coming from the other side of the terminations.

Figure 6.7: A doubly terminated string, ``plucked'' at 1/4 its length.

An example of an initial ``pluck'' excitation in a digital waveguide string model is shown in Fig.6.8. The circulating triangular components in Fig.6.8 are equivalent to the infinite train of initial images coming in from the left and right in Fig.6.7.

There is one fine-point to note for the discrete-time case: We cannot admit a sharp corner in the string since that would have infinite bandwidth which would alias when sampled. Therefore, for the discrete-time case, we define the ideal pluck to consist of an arbitrary shape as in Fig.6.8 lowpass filtered to less than half the sampling rate. Alternatively, we can simply require the initial displacement shape to be bandlimited to spatial frequencies less than $ f_s/2c$. Since all real strings have some degree of stiffness which prevents the formation of perfectly sharp corners, and since real plectra are never in contact with the string at only one point, and since the frequencies we do allow span the full range of human hearing, the bandlimited restriction is not limiting in any practical sense.

Figure 6.8: Initial conditions for the ideal plucked string. The initial contents of the sampled, traveling-wave delay lines are in effect plotted inside the delay-line boxes. The amplitude of each traveling-wave delay line is half the amplitude of the initial string displacement. The sum of the upper and lower delay lines gives the physical initial string displacement.

Note that acceleration (or curvature) waves are a simple choice for plucked string simulation, since the ideal pluck corresponds to an initial impulse in the delay lines at the pluck point. Of course, since we require a bandlimited excitation, the initial acceleration distribution will be replaced by the impulse response of the anti-aliasing filter chosen. If the anti-aliasing filter chosen is the ideal lowpass filter cutting off at $ f_s/2$, the initial acceleration $ a(0,x)\isdeftext {\ddot y}(0,x)$ for the ideal pluck becomes

$\displaystyle a(0,x) = \frac{A}{X}$sinc$\displaystyle \left(\frac{x-x_p}{X}\right)$ (7.13)

where $ A$ is amplitude, $ x_p$ is the pick position, and, as we know from §4.4.1, sinc$ [(x-x_p)/X]$ is the ideal, bandlimited impulse, centered at $ x_p$ and having a rectangular spatial frequency response extending from $ -\pi/X$ to $ \pi/X$. (Recall that sinc$ (\xi)\isdeftext
\sin(\pi\xi)/(\pi\xi)$). Division by $ X$ normalizes the area under the initial shape curve. If $ x_p$ is chosen to lie exactly on a spatial sample $ x_m = mX$, the initial conditions for the ideal plucked string are as shown in Fig.6.9 for the case of acceleration or curvature waves. All initial samples are zero except one in each delay line.

Aside from its obvious simplicity, there are two important benefits of obtaining an impulse-excited model: (1) an extremely efficient ``commuted synthesis'' algorithm can be readily defined (§8.7), and (2) linear prediction (and its relatives) can be readily used to calibrate the model to recordings of normally played tones on the modeled instrument. Linear Predictive Coding (LPC) has been used extensively in speech modeling [296,297,20]. LPC estimates the model filter coefficients under the assumption that the driving signal is spectrally flat. This assumption is valid when the input signal is (1) an impulse, or (2) white noise. In the basic LPC model for voiced speech, a periodic impulse train excites the model filter (which functions as the vocal tract), and for unvoiced speech, white noise is used as input.

In addition to plucked and struck strings, simplified bowed strings can be calibrated to recorded data as well using LPC [428,439]. In this simplified model, the bowed string is approximated as a periodically plucked string.

Figure 6.9: Initial conditions for the ideal plucked string when the wave variables are chosen to be proportional to acceleration or curvature. If the bandlimited ideal pluck position is centered on a spatial sample, there is only a single nonzero sample in each of the initial delay lines.

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The Ideal Struck String
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Moving Rigid Termination