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Physical Audio Signal Processing
    Woodwinds
       Single-Reed Instruments
          Single-Reed Theory
             Computational Methods

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Computational Methods

Since finding the intersection of $ G(x)$ and $ p_{\Delta}^{+}-x$ requires an expensive iterative algorithm with variable convergence times, it is not well suited for real-time operation. In this section, fast algorithms based on precomputed nonlinearities are described.

Let $ h$ denote half-pressure $ p/2$, i.e., $ h_m\isdeftext p_m/2$ and $ h_{\Delta}^{+}\isdeftext p_{\Delta}^{+}/2$. Then (6.9) becomes

$\displaystyle p_b^{-}= h_m- \rho(p_{\Delta})\cdot h_{\Delta}^{+}$ (7.11)

Subtracting this equation from $ p_b^{+}$ gives
$\displaystyle p_b^{+}+ \rho h_{\Delta}^{+}- h_m$ $\displaystyle =$ $\displaystyle p_b^{+}- p_b^{-}= p_{\Delta}- p_{\Delta}^{+}$  
$\displaystyle \,\,\Rightarrow\,\, \rho h_{\Delta}^{+}$ $\displaystyle =$ $\displaystyle \underbrace{h_m- p_b^{+}}_{h_{\Delta}^{+}} + p_{\Delta}- 2h_{\Delta}^{+}= p_{\Delta}-h_{\Delta}^{+}$  
$\displaystyle \,\,\Rightarrow\,\, \rho$ $\displaystyle =$ $\displaystyle \frac{p_{\Delta}}{h_{\Delta}^{+}}-1$ (7.12)

The last expression above can be used to precompute $ \rho$ as a function of $ h_{\Delta}^{+}\isdef h_m- p_b^{+}= p_m/2 -p_b^{+}$. Denoting this newly defined function as
$\displaystyle \hat\rho (h_{\Delta}^{+}) = \rho(p_{\Delta}(h_{\Delta}^{+}))$     (7.13)

(6.11) becomes

$\displaystyle p_b^{-}= h_m- \hat\rho (h_{\Delta}^{+})\cdot h_{\Delta}^{+}$ (7.14)

This is the form chosen for implementation in Fig. 6.2 [440]. The control variable is mouth half-pressure $ h_m$, and $ h_{\Delta}^{+}=h_m-p_b^{+}$ is computed from the incoming bore pressure using only a single subtraction. The table is indexed by $ h_{\Delta}^{+}$, and the result of the lookup is then multiplied by $ h_{\Delta}^{+}$. Finally, the result of the multiplication is subtracted from $ h_m$. The cost of the reed simulation is only two subtractions, one multiplication, and one table lookup per sample.

Because the table contains a coefficient rather than a signal value, it can be more heavily quantized both in address space and word length than a direct lookup of a signal value such as $ p_{\Delta}(p_{\Delta}^{+})$ or the like. A direct signal lookup, though requiring much higher resolution, would eliminate the multiplication associated with the scattering coefficient. For example, if