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Terminated String Impedance

Note that the impedance of the terminated string, seen from one of its endpoints, is not the same thing as the wave impedance $ R=\sqrt{K\epsilon }$ of the string itself. If the string is infinitely long, they are the same. However, when there are reflections, they must be included in the impedance calculation, giving it an imaginary part. We may say that the impedance has a ``reactive'' component. The driving-point impedance of a rigidly terminated string is ``purely reactive,'' and may be called a reactance7.1). If $ f(t)$ denotes the force at the driving-point of the string and $ v(t)$ denotes its velocity, then the driving-point impedance is given by (§7.1)

$\displaystyle R(j\omega) \isdefs \left.\frac{F(s)}{V(s)}\right\vert _{s=j\omega},

where $ F(s)$ and $ V(s)$ denote the Laplace transforms of $ f(t)$ and $ v(t)$. In the case of a rigidly terminated string above, as well as in any system in which all energy delivered to the system is ultimately reflected back to the input, the impedance is purely imaginary at every frequency (a ``pure reactance''), as is easy to show:

$\displaystyle R(s) \isdefs \frac{F(s)}{V(s)}
\eqsp \frac{F^{+}+F^{-}}{V^{+}+V^...
\eqsp R\frac{1+e^{-s2L/c}}{1-e^{-s2L/c}}

where $ L$ denotes the string length. Let $ P=2L/c$ denote the period of string vibration. Then on the frequency axis $ s=j\omega$ we have

$\displaystyle R(j\omega)
\eqsp R\frac{1+e^{-j\omega P}}{1-e^{-j\omega P}}
\eqsp R\frac{2\cos(\omega P/2)}{2j\sin(\omega P/2)}
\eqsp -jR\,\cot(\omega P/2).

Thus, the driving-point impedance of a rigidly terminated string is purely reactive (imaginary), with alternating poles and zeros along the $ j\omega $ axis. Impedance will be discussed further in §7.1 below.

Computational Savings

To illustrate how significant the computational savings can be, consider the simulation of a ``damped guitar string'' model in Fig.6.11. For simplicity, the length $ L$ string is rigidly terminated on both ends. Let the string be ``plucked'' by initial conditions so that we need not couple an input mechanism to the string. Also, let the output be simply the signal passing through a particular delay element rather than the more realistic summation of opposite elements in the bidirectional delay line. (A comb filter corresponding to pluck position can be added in series later.)

Figure 6.11: Discrete simulation of the rigidly terminated string with distributed resistive losses. The $ N$ loss factors $ g$ are embedded between the delay-line elements.

In this string simulator, there is a loop of delay containing $ N = 2L/X=
f_s/f_1$ samples where $ f_1$ is the desired pitch of the string. Because there is no input/output coupling, we may lump all of the losses at a single point in the delay loop. Furthermore, the two reflecting terminations (gain factors of $ -1$) may be commuted so as to cancel them. Finally, the right-going delay may be combined with the left-going delay to give a single, length $ N$, delay line. The result of these inaudible simplifications is shown in Fig. 6.12.

Figure 6.12: Discrete simulation of the rigidly terminated string with consolidated losses (frequency-independent). All $ N$ loss factors $ g$ have been ``pushed'' through delay elements and combined at a single point.

If the sampling rate is $ f_s=50$ kHz and the desired pitch is $ f_1=100$ Hz, the loop delay equals $ N=500$ samples. Since delay lines are efficiently implemented as circular buffers, the cost of implementation is normally dominated by the loss factors, each one requiring a multiply every sample, in general. (Losses of the form $ 1-2^{-k}$, $ 1-2^{-k}-2^{-l}$, etc., can be efficiently implemented using shifts and adds.) Thus, the consolidation of loss factors has reduced computational complexity by three orders of magnitude, i.e., by a factor of $ 500$ in this case. However, the physical accuracy of the simulation has not been compromised. In fact, the accuracy is improved because the $ N$ round-off errors per period arising from repeated multiplication by $ g$ have been replaced by a single round-off error per period in the multiplication by $ g^N$.

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Stiff String Synthesis Models
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Animation of Moving String Termination and Digital Waveguide Models