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Parallel Impedance Biquads

In the current situation, computing the junction-velocity $ v_J(n)$ from the incoming waves $ v^+_i(n)$ using the parallel biquad expansion Eq.(C.104) for $ R_J(z)$ , we split each term of Eq.(C.102) into its instantaneous and delayed components [25]:C.12

$\displaystyle \frac{B_i(z)}{A_i(z)}$ $\displaystyle =$ $\displaystyle \frac{b_{i0} + b_{i1}z^{-1}+ b_{i2}z^{-2}}{1 + a_{i1}z^{-1}+ a_{i2}z^{-2}}$ (C.106)
  $\displaystyle =$ $\displaystyle b_{i0} + \frac{(b_{i1}-b_{i0} a_{i1})z^{-1}+ (b_{i2}-b_{i0}a_{i2})z^{-2}}{1 + a_{i1}z^{-1}+ a_{i2}z^{-2}}$ (C.107)
  $\displaystyle \isdef$ $\displaystyle b_{i0} + \frac{B_i^d(z)}{A_i(z)}
\protect$ (C.108)

so that
$\displaystyle R_J(z)$ $\displaystyle \isdef$ $\displaystyle R_J^0 + R_J^d(z), \;$where (C.109)
$\displaystyle R_J^0$ $\displaystyle \isdef$ $\displaystyle \sum_{i=1}^N b_{i0},\;$and (C.110)
$\displaystyle R_J^d(z)$ $\displaystyle \isdef$ $\displaystyle \sum_{i=1}^N B_i^d(z)/A_i(z).$ (C.111)

$\displaystyle R_J^+$ $\displaystyle \isdef$ $\displaystyle \sum_{i=1}^N R_i,\;$and (C.112)
$\displaystyle F_J^+(z)$ $\displaystyle \isdef$ $\displaystyle \sum_{i=1}^N R_i V^+_i(z).$ (C.113)

Then Eq.(C.97) can be written
$\displaystyle V_J(z)$ $\displaystyle =$ $\displaystyle \frac{2F_J^+(z)}{R_J^+ + R_J(z)}$ (C.114)
  $\displaystyle =$ $\displaystyle \frac{2F_J^+(z)}{\underbrace{R_J^+ + R_J^0}_{\isdeftext R_0} + R_J^d(z)}$ (C.115)
  $\displaystyle =$ $\displaystyle \frac{2F_J^+(z)/R_0}{1 + R_J^d(z)/R_0}.$ (C.116)

This structure can be realized as shown in Fig.C.30 and derived above. This form is convenient for encoding in FAUST [471]. Here, $ R_0$ denotes the sum of all incoming wave impedances $ R_i$ plus the instantaneous impedance of the load $ R_J^0$ .

More directly derived, we can write Eq.(C.97) as

$\displaystyle V_J(z) \left[R_0+R_J^d(z)\right]$ $\displaystyle =$ $\displaystyle 2 F_J^+, \;$or (C.117)
$\displaystyle V_J(z)\, R_0$ $\displaystyle =$ $\displaystyle 2F_J^+(z) - R_J^d(z)\, V_J(z),$ (C.118)

This expression can be taken by inspection to the time domain in terms of the parallel biquads $ R_J^d(z)$ to yield the following difference equation:
$\displaystyle v_J(n)$ $\displaystyle =$ $\displaystyle \frac{1}{R_0} \left[2f_J^+(n) - f_J^d(n)\right],\;$where (C.119)
$\displaystyle f_J^d(n)$ $\displaystyle \isdef$ $\displaystyle {\cal Z}_n^{-1}\left\{\,R_J^d(z)\, V_J(z)\,\right\}$ (C.120)
  $\displaystyle \isdef$ $\displaystyle \sum_{i=1}^N f_{iJ}^d(n),\;$where (C.121)
$\displaystyle f_{iJ}^d(n)$ $\displaystyle \isdef$ $\displaystyle \quad\! b_{i1}^d\, v_J(n-1) + b_{i2}^d\, v_J(n-2)$  
    $\displaystyle - a_{i1} \, f_{iJ}^d(n-1) - a_{i2} \, f_{iJ}^d(n-2),$ (C.122)

and $ b_{i1}^d \isdeftext b_{i1}-b_{i0} a_{i1}$ and $ b_{i2}^d \isdeftext b_{i2}-b_{i0}*a_{i2}$ , as in Eq.(C.108) above.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University