⚡ electrical engineering

Fractional Order Filters with √f Characteristic

This paper derives fractional order filters with $\sqrt{f}$ and $1/\sqrt{f}$ transfer functions by spatially discretizing a partial differential equation from Maxwell's equations to model the skin effect, and presents a corresponding digital implementation via bilinear transformation that is benchmarked against the Oustaloup recursive filter.

Original authors: Hans Georg Brachtendorf

Published 2026-09-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hans Georg Brachtendorf

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the hidden world of electricity flowing through metal, there is a peculiar phenomenon that changes how signals behave depending on their speed. When an electric current travels through a conductor, it does not always fill the entire wire evenly. At very high speeds, the current is pushed toward the outer surface, a behavior known as the skin effect. This effect is not just a curiosity; it fundamentally alters how electrical signals are filtered and processed. Engineers have long sought ways to create electronic components that mimic a specific mathematical relationship where the signal's strength changes in proportion to the square root of its frequency. Such components are vital for cleaning up noise in sensitive measurements, controlling complex mechanical systems, and modeling how biological tissues interact with electromagnetic waves. For decades, the best tools to achieve this have been complex approximations that work well only within limited ranges, often requiring intricate arrangements of poles and zeros to flatten the phase response.

A researcher at the University of Applied Sciences Upper Austria has now proposed a different path, one that starts not with circuit approximations, but with the fundamental laws of physics themselves. By taking the equations that describe how electric and magnetic fields interact inside a metal rod and breaking them down into small, manageable steps, the author has derived a new way to build these special filters. Instead of forcing a standard circuit to act like a square-root filter, this method builds a filter that naturally behaves that way because it is a direct digital representation of the physical skin effect. The work demonstrates that by carefully spacing these calculation steps—placing them closer together near the surface of the rod where the action happens—one can create a highly accurate model that works over a much wider range of frequencies than previous methods.

The journey begins with a simple physical picture: a long, rectangular metal rod. The researcher assumes the material is uniform and that the electric and magnetic fields inside it follow the standard rules of electromagnetism, specifically Maxwell's equations. In the regime where the skin effect is strong, the depth to which the current penetrates the metal is very small compared to the rod's thickness. This physical reality dictates a specific relationship between frequency and signal loss. To turn this continuous physical law into something a computer can use, the researcher divides the rod into a series of tiny slices. This process, known as spatial discretization, transforms the smooth, flowing equations of physics into a chain of ordinary differential equations. These equations can then be mapped directly onto an electrical circuit made of resistors and capacitors, creating a behavioral model that mimics the rod's response to an incoming signal.

A crucial insight in this work is how the slices are arranged. If the slices were all the same size, the model would require an impractical number of them to capture the rapid changes happening right at the surface of the metal. Instead, the researcher uses a non-uniform grid, where the slices become progressively smaller as they approach the surface. This clustering of calculation points mirrors the physical reality that the most significant changes occur in the thinnest layer of the conductor. The study finds that the ratio between the sizes of these adjacent slices is critical. If the slices grow too quickly in size as one moves away from the surface, the model becomes unstable and produces unwanted ringing effects. The simulations suggest that the ratio of adjacent spacings should be kept below a specific mathematical constant, with a value around 2.5 appearing to offer the best balance between accuracy and the range of frequencies covered.

Once the analog model is established, the next step is to translate it into a digital format that can run on modern processors. The researcher tested several standard methods for converting continuous-time systems into discrete-time ones. One popular method, known as the bilinear transformation, emerged as the clear winner. Unlike other techniques that introduce artificial damping or require higher-order calculations that complicate hardware implementation, the bilinear transformation preserves the stability and accuracy of the original physical model. The resulting digital filter behaves exactly like the analog skin-effect model, maintaining the desired square-root relationship across a broad spectrum. In simulations, this new approach proved superior to the widely used Oustaloup recursive filter, a standard technique for approximating fractional-order systems. While the Oustaloup method places poles and zeros on a logarithmic scale to flatten the phase response, the skin-effect model derived here naturally achieves a higher phase accuracy and remains valid over a frequency range that is larger than that of the Oustaloup model.

The results of the simulations are striking. For a model using just eight grid points, the new filter maintains its characteristic behavior over a span of roughly 6,500 times the starting frequency. This is a significant improvement over existing methods, which often struggle to maintain accuracy across such wide bands. The study confirms that the physical intuition behind the skin effect provides a robust foundation for designing these filters, bypassing the need for complex optimization algorithms. The author notes that while the current work focuses on the specific case of the square root of frequency, the underlying method of discretizing partial differential equations could potentially be extended to other fractional orders in the future. For now, the work offers a clear, physically grounded alternative for engineers needing precise control over signal processing, proving that sometimes the best way to solve a digital problem is to look closely at the physics of the real world.

Technical Summary: Fractional Order Filters with $\sqrt{f}$ Characteristic

Problem Statement

The paper addresses the design and analysis of fractional order filters characterized by a transfer function proportional to $s^{\alpha}$ where $\alpha = \pm 1/2$ (corresponding to $\sqrt{f}$ and $1/\sqrt{f}$ frequency behaviors). These filters are essential for applications such as preemphasis/deemphasis, modeling $1/f$ (flicker) noise power spectra, fractional order control, and simulating the dispersive properties of media like biological tissues.

While existing approximation techniques, such as the Oustaloup recursive filter and the Charef method, are widely used, they rely on placing poles and zeros on a logarithmic scale to flatten phase or magnitude responses. The paper seeks an alternative derivation rooted in physical laws rather than purely mathematical approximation, aiming to derive a behavioral model that naturally exhibits the desired fractional order characteristics through the discretization of Maxwell's equations.

Methodology

1. Physical Derivation from Maxwell's Equations

The core methodology begins with Maxwell's equations for a linear, isotropic, and homogeneous conducting rod. Under the assumption that displacement current is negligible ( $J \gg \partial D/\partial t$ ) and utilizing a quasistationary approach for a rectangular rod where length $l \gg$ width $b$ and thickness $d$ , the equations reduce to a system of partial differential equations (PDEs) describing the electric ( $E_z$ ) and magnetic ( $H_x$ ) fields.

The paper leverages the known physical phenomenon that the skin depth ( $\Delta x$ ) of a conducting material follows a $1/\sqrt{f}$ law in the strong skin effect regime (where skin depth is small compared to geometry). The goal is to discretize these PDEs to create a filter that mimics this behavior.

2. Spatial Discretization and Circuit Modeling

The PDEs are discretized using a finite difference (FD) scheme.

Uniform Grid: Initially, an equidistant grid is considered, leading to a system of ordinary differential-algebraic equations (DAEs). This system is mapped to passive behavioral circuit models (ladder networks) consisting of inductors and resistors (LR) or capacitors and conductors (GC).
Non-Uniform Grid: Recognizing that a uniform grid requires an impractical number of points to capture the skin effect at high frequencies, the paper proposes a non-uniform grid. The grid spacing is smallest at the surface ( $y = d/2$ ) where the skin effect is most pronounced. The ratio of adjacent spacings is defined by a parameter $c$ .
Circuit Realization: The discretized equations are realized as equivalent passive circuits. The paper notes that while the LR model is natural, the GC model (using capacitors and conductors) is more suitable for SPICE-like simulators as it reduces the number of unknowns in Modified Nodal Analysis (MNA).

3. Digital Transformation

To obtain a digital equivalent, the analog filter is transformed using the bilinear transformation (trapezoidal rule) and compared against Backward Differentiation Formulas (BDF).

Bilinear Transformation: Preferred because it maps the imaginary axis to the unit circle without introducing numerical damping, a side effect observed in BDF methods. It also results in a lower filter order compared to higher-order BDF methods ( $p \ge 2$ ).
Implementation: The paper discusses efficient implementation strategies for the resulting difference equations, including the introduction of auxiliary vectors to avoid multipliers in specific recursive steps.

Key Contributions

Physical Derivation of Fractional Filters: The paper derives $\sqrt{f}$ and $1/\sqrt{f}$ filters directly from the discretization of Maxwell's equations for a conducting rod, providing a physical basis for fractional order behavior rather than relying solely on pole-zero placement heuristics.
Non-Uniform Grid Optimization: It demonstrates that using a non-uniform grid with a specific spacing ratio ( $c$ ) significantly reduces the system order while preserving accuracy. The paper identifies that the ratio of adjacent grid spacings should be less than Euler's constant to avoid ringing effects, with $c=2$ to $c=2.5$ found to be optimal.
Behavioral Circuit Models: The derivation yields passive ladder circuit models (both LR and GC) that are inherently stable, as the Ohmic losses ensure poles remain in the left open plane.
Digital Implementation: The paper provides a clear path to digital realization via the bilinear transformation, offering a stable and efficient digital filter structure.

Results

Comparison with Oustaloup: Simulation results for a filter order of $K=11$ show that the skin depth model derived in this paper offers a larger frequency range of approximate validity and higher phase accuracy compared to the Oustaloup recursive filter.
Frequency Range: For a non-uniform grid with $K=8$ points and $c=2$ , the model exhibits the expected $\sqrt{f}$ behavior (linear slope of 0.5 in a double logarithmic plot of impedance) over approximately four decades of frequency. While theoretical estimates suggested a wider range, simulations indicate a practical validity range of roughly $f_1/10$ .
Parameter Sensitivity: Tests confirm that parameter values for the grid spacing ratio $c$ between 2 and Euler's constant ( $\approx 2.718$ ) yield the best trade-off between system order, range of validity, and exactness. Values of $c$ exceeding Euler's constant result in undesirable ringing effects.
Digital Performance: The digital counterparts, implemented via the bilinear transformation, maintain the square-root behavior for about four decades of normalized angular frequency, confirming the effectiveness of the transformation.

Significance and Claims

The paper claims that the proposed method offers a superior alternative to existing recursive approximations like Oustaloup's, particularly regarding the range of validity and phase accuracy. By grounding the filter design in the physical skin effect of conductors, the resulting models are not only mathematically sound but also physically interpretable as passive networks.

The author emphasizes that the non-uniform discretization strategy allows for accurate modeling of the skin effect with significantly fewer grid points (and thus lower computational cost) than uniform discretization would require. The preference for the bilinear transformation is justified by its ability to preserve stability and avoid numerical damping, making it suitable for hardware implementation.

Finally, the paper positions this work as a specific case of fractional order filtering ( $\alpha = \pm 1/2$ ) and suggests that the underlying methodology of discretizing partial differential equations could be generalized in future work to fractional filters with arbitrary orders $\alpha$ beyond $\pm 1/2$ .

Technical Summary: Fractional Order Filters with f\sqrt{f}f​ Characteristic