Parallel Admittance Biquads

Similarly, given any filter realization of the load admittance $\Gamma_J(z) = 1/R_J(z)$ , we can split $\Gamma_J$ into its instantaneous and delayed components as $\Gamma_J(z) = \Gamma_J^0 + \Gamma_J^d(z)$ , and analogously obtain

$$V_J(z) = \frac{2F_J^+(z)}{R_J^+ + R_J(z)} \,\mathrel{\mathop=}\,\frac{2F_J^+(z)}{R_J^+ + \frac{1}{\Gamma_J(z)}} \,\mathrel{\mathop=}\,\frac{2F_J^+(z)}{R_J^+ + \frac{1}{\Gamma_J^0 + \Gamma_J^d(z)}}$$ (C.123)

$$= \frac{2F_J^+(z)}{R_J^+ + \frac{1/\Gamma_J^0}{1 + \Gamma_J^d(z)/\Gamma_J^0}} \,\mathrel{\mathop=}\,\frac{2F_J^+(z)}{R_J^+ + \frac{1}{\Gamma_J^0} + \frac{1}{\Gamma_J^0}\left[\frac{1}{1 + \Gamma_J^d(z)/\Gamma_J^0} - 1\right]}$$ (C.124)

$$= \frac{2F_J^+(z)}{R_0 + \frac{1}{\Gamma_J^0}\left[\frac{-\Gamma_J^d(z)/\Gamma_J^0}{1+\Gamma_J^d(z)/\Gamma_J^0}\right]} \,\mathrel{\mathop=}\,\frac{2F_J^+(z)/R_0}{1 + \frac{1}{R_0\Gamma_J^0}\left[\frac{-\Gamma_J^d(z)/\Gamma_J^0}{1+\Gamma_J^d(z)/\Gamma_J^0}\right]}$$ (C.125)

where $R_0\isdeftext R_J^+ + R_J^0$ , and clearly $\Gamma_J^0 = R_J^0$ . We can define $\Gamma_0\isdeftext 1/R_0$ but be careful not to interpret it as the sum of $\Gamma_J^0$ and the branch admittances, since we are doing a series junction. Now we can draw the block diagram in Fig.C.31 by inspection, and we obtain an interesting nested feedback structure placing $\Gamma_J^d(z)$ in the inner feedback loop.

Figure C.31: Feedback structure implementing a series junction in terms of a prescribed admittance realization.

Figure C.31 can readily be encoded in the FAUST language by extracting a unit-sample delay from the admittance filter and ``pushing'' it to the right through its input summer, which splits it into the $v_J$ output tap and the inner feedback loop. This makes both feedback loops valid in FAUST using the tilde (`~') operator. In terms of obvious definitions:

vJ = fJp : *(2*G0) : ( + ~
       ( *(G0/GJ0) : ( + : GJd/GJ0 : *(-1)) ~ _ : *(-1)));

where the delay in GJd has been pulled out:

GJd = _ <: par(i, M, GiJd(i+1)) :> _ ;
GiJd(i) = fi.tf2(b1d(i), b2d(i), 0, a1(i), a2(i) ); // SHIFTED