Generalized Bilinear Transformation (GBT)

• GBT is a parametric mapping technique that converts continuous analog transfer functions to discrete digital systems while optimizing magnitude and phase accuracy.
• It employs a hexagonal geometric integration method using a design parameter α to balance backward and forward contributions, thereby unifying classical methods like backward-Euler and Tustin.
• GBT establishes explicit stability criteria and enables systematic frequency-domain error minimization, facilitating precise digital filter design and robust embedded signal processing.

The Generalized Bilinear Transformation (GBT) is a parametric one-parameter family of first-order mappings for discretizing continuous-time (analog) transfer functions into discrete-time (digital) systems. It generalizes the widely used backward-Euler (rectangular) and Tustin (bilinear/trapezoidal) s-to-z mappings via an explicit design parameter, $\alpha$, which allows for principled tradeoffs between magnitude and phase accuracy in the digital approximation of analog systems. The GBT’s new geometric derivation provides a direct physical interpretation for $\alpha$, along with an explicit stability region and a systematic framework for error optimization and experimental design verification (Chen et al., 5 Nov 2025).

1. Mathematical Formulation and Connections

The GBT defines the mapping between continuous-time Laplace variable $s$ and discrete-time $z$ via

$$s = \frac{1}{T} \frac{z-1}{\alpha z + (1-\alpha)}, \qquad \alpha \in [0,1],$$

where $T$ is the sampling period. This mapping unifies several classical rules:

• $\alpha=1$: Backward-Euler (rectangular) rule,

$s = \frac{1}{T}\frac{z-1}{z}.$

• $\alpha=0.5$: Tustin (bilinear/trapezoidal) rule,

$s = \frac{2}{T}\frac{z-1}{z+1}.$

An algebraic reparameterization shows that the Al-Alaoui operator,

$$s = \frac{2}{T} \frac{z-1}{(1+a)z + (1-a)}, \qquad a \in [0,1],$$

is equivalent to GBT with $\alpha = (1 + a)/2$.

This formulation subsumes established discretization schemes as limiting or special cases while providing a continuum for optimizing performance.

2. Geometric Derivation and Physical Meaning of $\alpha$

The GBT’s alternative derivation employs a "hexagonal" integration approximation for discretizing the fundamental integrator

$u(t) = \int e(t) \, dt$

over one sampling period $[ (n-1)T,\ nT ]$. The approximation uses two rectangles:

• Forward rectangle, width $(1-\alpha)T$ at $e(n-1)$,
• Backward rectangle, width $\alpha T$ at $e(n)$.

The resulting discrete update is: $$u(n) = u(n-1) + (1-\alpha) e(n-1) T + \alpha e(n) T.$$

$\alpha$ quantifies the proportion of the integral’s area contributed by the backward rectangle: $$\alpha = \frac{S_{\text{bw}}}{S_{\text{fw}} + S_{\text{bw}}}$$

where $S_{\text{fw}}$ and $S_{\text{bw}}$ are areas of the forward and backward rectangles, respectively. Thus, $\alpha$ is precisely the “percent backward-rectangle” of the composite area—directly linking the parameter’s geometric, algorithmic, and physical significance.

• $\alpha \to 1$: Fully backward (backward-Euler).
• $\alpha \to 0.5$: Equal contribution (trapezoidal/Tustin).
• $\alpha \to 0$: Fully forward (unstable for most applications).

3. Stability Criterion and Domain Mapping

For the digital system to faithfully reproduce analog stability, the left-half $s$-plane $\operatorname{Re}[s] \leq 0$ must map inside the unit circle $|z|<1$. An explicit analytic condition yields a stability domain for $\alpha$: $\alpha \in [0.5, 1.0]$ This ensures that, under the GBT mapping, stable analog poles are mapped within the stability region of the z-plane. The derivation relies on parametric equations relating $z = \gamma + j\zeta$ and $s = \sigma + j\omega$, imposing the locus constraint

$$(\gamma - [1 - \tfrac{1}{2\alpha}])^2 + \zeta^2 \leq (\tfrac{1}{2\alpha})^2.$$

Setting $\alpha \geq 0.5$ guarantees all left-half plane $s$ map inside $|z|=1$.

4. Distortion Analysis: Magnitude and Phase Errors

Given an analog transfer function

$$G_{\rm an}(s) = K \prod_{i=1}^m \frac{(s+Z_i)}{(s+P_k)},$$

the GBT (with Zero-Order Hold [ZOH] correction) yields discrete-frequency responses via

$$G_{\rm disc}(e^{j\omega T}, \alpha) = G_{\rm an}\left( \frac{1}{T} \frac{e^{j\omega T} - 1}{\alpha e^{j\omega T} + (1-\alpha)} \right) \cdot \frac{\sin(\omega T / 2)}{\omega T / 2} e^{-j\omega T / 2}.$$

Two principal discrete errors are:

• Magnitude error (dB):

$$L_{\rm err}(f, \alpha) = 20 \log_{10} \left| \frac{G_{\rm disc}(e^{j2\pi f T}, \alpha)}{G_{\rm an}(j2\pi f)} \right|$$

• Phase error (radians, after ZOH delay compensation):

$$\phi_{\rm err}(f, \alpha) = \operatorname{arg}\left[G_{\rm disc}(e^{j2\pi f T}, \alpha) e^{+j\pi f T} \right] - \operatorname{arg}[G_{\rm an}(j2\pi f)]$$

Empirical trends:

• Lower $\alpha$ (approaching Tustin) improves phase accuracy but underestimates magnitude.
• Higher $\alpha$ (approaching Euler) gives better magnitude fidelity but causes greater phase lag.

5. Optimal Design of $\alpha$ through Frequency-Domain Error Minimization

The GBT framework enables systematic optimization of $\alpha$ for target application requirements by formulating frequency-error objectives. Three typical scenarios:

• (A) Single Frequency Optimization:

$$\min_{\alpha \in [0.5, 1]} \left| \frac{L_{\rm err}(f_{\rm exp}, \alpha)}{L_{\rm err, max}} \right|$$

(or for phase error).

• (B) Finite Frequency Set with Weights:

$$\min_{\alpha \in [0.5, 1]} Q_B(\alpha) = \frac{ \sqrt{ \sum_{i=1}^N w_i [L_{\rm err}(f_i, \alpha)]^2 } }{ \max_i | L_{\rm err}(f_i, \alpha) | }$$

(and similarly for phase).

• (C) Frequency Interval:

$$\min_{\alpha \in [0.5, 1]} Q_C(\alpha) = \frac{ \int_{f_{\rm start}}^{f_{\rm end}} |L_{\rm err}(f,\alpha)|\, df }{ \int_{f_{\rm start}}^{f_{\rm end}} L_{\rm err, max}(f)\, df }$$

(and similarly for phase).

$\alpha$ is then efficiently identified by direct 1-D search to minimize the application-relevant objective, enabling explicit control over the fundamental tradeoff between magnitude and phase distortions.

6. Application Example: Low-Pass Filter Digitalization and Experimental Results

A practical example considers the discretization of a first-order analog low-pass filter

$$G_{\rm an}(s) = \frac{\omega_c}{s + \omega_c}, \qquad \omega_c = 2\pi \cdot 4.823\;\text{kHz},\quad f_{\rm samp} = 12~\textrm{kHz}~(T = 1/12~\textrm{kHz}).$$

Table summarizing optimal $\alpha$ under three design scenarios:

Scenario	$\alpha_{\text{mag-first}}$	$\alpha_{\text{trade-off}}$	$\alpha_{\text{phase-first}}$
A (75% $f_c$)	0.50	0.575	1.00
B (CEC-weights)	0.50	0.549	1.00
C ([10%,100%] $f_c$)	0.50	0.593	1.00

The resulting discrete difference equation for the digital filter is: $$V_{\rm out}(n) = \frac{\alpha \omega_c T}{1+\alpha \omega_c T} [V_{\rm in}(n)-V_{\rm in}(n-1)] + \frac{ \omega_c T }{ 1+\alpha \omega_c T } [ V_{\rm in}(n-1) - V_{\rm out}(n-1) ] + V_{\rm out}(n-1)$$ Implementation on a TMS320F28P65 (12 kHz sampling) showed magnitude and phase errors matching theory to within 5% across $10\%-100\%$ of $f_c$. Adjusting for hardware-induced delay, phase errors were further reduced by 45%. Increasing $f_{\rm samp}$ to $48$ kHz improved accuracy, as predicted by analysis.

7. Significance and Practical Implications

The Generalized Bilinear Transformation provides a systematic, physically motivated method for discretizing analog transfer functions with tunable fidelity trade-offs between magnitude and phase response. The new hexagonal geometric derivation gives $\alpha$ a direct and interpretable meaning, resolving ambiguities present in earlier literature. The explicit $\alpha$ stability interval ensures robust digital realization, and the optimization strategy generalizes to arbitrary frequency sets and operational intervals. Experimental results and theoretical analysis demonstrate the approach’s utility in practical digital filter design and embedded signal processing implementation (Chen et al., 5 Nov 2025).