The Three-Multiply Normalized Lattice Section.

A well known two-port unnormalized scattering junction [57], implemented using one multiply and three adds, can be derived by organizing (64) as

$$\begin{array}{l} p_1^-=p_2^+ + p_\Delta\\ p_2^-=p_1^+ + p_\Delta \end{array}$$ (74)

where

$$p_\Delta \stackrel{\triangle}{=}\rho (p_1^+-p_2^+),$$ (75)

and

$$\rho \stackrel{\triangle}{=}\alpha_1-1 = 1-\alpha_2 = \fra... ... \Gamma_2}{\Gamma_1 + \Gamma_2} = \frac{R_2 - R_1}{R_2 + R_1}$$ (76)

is the reflection coefficient seen at an impedance step from $R_1$ to $R_2$, i.e., at port 1. In the ladder/lattice filter literature, only a 4-multiply, 2-add normalized junction appears to be known. A normalized, three-multiply junction can be obtained simply by coupling any one-multiply junction such as (75) with an ideal transformer [92] (see Appendix A), as illustrated in Fig. 7, where

$$g\stackrel{\triangle}{=}\sqrt{\Gamma_2/\Gamma_1}$$ (77)

However, any lossless two-port junction which preserves a physical interpretation will do. For example, a transformer can also be attached to the alpha-parametrized one-multiply junction of Fig. 6 to obtain the three-multiply normalized junction of Fig. 8, having essentially the same properties as that of Fig. 7.

Figure 7: A three-multiply, three-add, normalized scattering junction parametrized in terms of a reflection coefficient.

Figure 8: A three-multiply, three-add, normalized junction parametrized in terms of an alpha parameter.

Note that a high quality sinusoidal oscillator requiring only one multiply can be formed by adding a unit delay on either side of a three-multiply junction and reflectively terminating (in which case the transformer multiplies cancel) [103].

One-multiply two-port scattering junctions enforce structural losslessness in finite precision arithmetic since the allpass complementarity condition (67) is always true. A normalized junction obtained by transformer-normalizing a one-multiply junction is also structurally lossless, provided the transformer multipliers are implemented in extended precision. In practice, normalized junctions of this type are made passive by using any non-amplifying rounding rule at the output. Denoting the ideal transformer coefficients by $g^-{\tiny\stackrel{\triangle}{=}}\sqrt{\Gamma_2/\Gamma_1}$ and $g^+=1/g^-$, the passivity condition on the transformer with quantized coefficients $\hat{g}^- = g^- - \epsilon ^-$ and $\hat{g}^+ = g^+ -\epsilon ^+$ is $\hat{g}^+\hat{g}^-\leq 1$, even in the time-varying case.

Since one of the transformer coefficients is always greater than $1$, fixed-point implementations require support for an integer part, unlike the fractional fixed-point format used in most DSP chips. Since the transformer commutes with the junction in the two-port case, the larger coefficient can, without loss of generality, be chosen to apply to the signal entering the junction. As an example of the increased internal dynamic range required, if $\rho \in [-1+\epsilon ,1-\epsilon ]$, the transformer coefficients may become as large as $\sqrt{2/\epsilon - 1}$. If $\epsilon$ is the ``machine epsilon,'' i.e., $\epsilon = 2^{-(n-1)}$ for typical $n$-bit two's complement arithmetic normalized to lie in $[-1,1)$, then the dynamic range of the transformer coefficients is bounded by $\sqrt{2^n-1}\approx 2^{n/2}$. In summary, while transformer-normalized junctions trade a multiply for an add in the two-port case, they require up to $50$% more bits of dynamic range within the junction adders.