Parallel Impedance Biquads

Parallel Impedance Biquads

In the current situation, computing the junction-velocity from the incoming waves using the parallel biquad expansion Eq.(C.104) for , 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)

Define

$\displaystyle R_J^+$	$\displaystyle \isdef$	$\displaystyle \sum_{i=1}^N R_i,\;$ 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,

denotes the sum of all incoming wave impedances

plus the instantaneous impedance of the load

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

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)