⚡ electrical engineering

Optimum and Adaptive Complex-Valued Bilinear Filters

This paper investigates the extension of real-valued bilinear filters to complex-valued domains by deriving them from real-valued counterparts and introducing novel fully complex-valued adaptive filters (including Wiener, least squares, and recursive variants) for identifying complex-valued nonlinear systems such as MISO and Hammerstein models.

Original authors: Bernhard Plaimer, Matthias Wagner, Oliver Lang, Mario Huemer

Published 2026-07-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Bernhard Plaimer, Matthias Wagner, Oliver Lang, Mario Huemer

Original paper licensed under CC BY 4.0 (http://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

Imagine you are trying to tune a radio to catch a song from a distant station. Usually, the radio is a simple machine: you turn a knob, and the sound gets clearer. In the world of signal processing, this is like using a linear filter. It works great when the world is predictable and straight. But the real world is messy! Sometimes, the signal gets twisted, distorted, or "bent" in complicated ways. This is where nonlinear systems come in.

To fix these twisted signals, engineers use bilinear filters. Think of a bilinear filter not as a single knob, but as a team of two people working together. One person (let's call them Vector H) holds a map of the past, and the other (Vector G) holds a map of the current situation. To get the right answer, they have to multiply their maps together. If they get it right, the song comes through crystal clear.

For a long time, this team only spoke Real. They dealt with simple numbers like 1, 2, or -5. But in the high-tech world of radar and modern communications, signals aren't just simple numbers; they are complex. They have a "real" part and an "imaginary" part (a bit like having both a volume knob and a phase knob). The problem? The old "Real" team didn't know how to handle this double-sided complexity.

The Old Way: Splitting the Team

The paper first looks at how people tried to solve this by splitting the complex signal into two separate teams: one for the "Real" part and one for the "Imaginary" part.

The 2R Team: They use two separate real filters. It's simple, but it's like trying to dance with a partner while looking away from them. If the Real and Imaginary parts of the signal are connected (correlated), this team stumbles. They can't see the whole picture.
The 4R Team: To fix the stumbling, they added two more filters, making a team of four. Now they can see the connection between the Real and Imaginary parts. It works better, but it's clumsy. It's like using four separate calculators to do one math problem. It gets the job done, but it's slow and uses a lot of energy.

The New Way: The Fully Complex Super-Team

The authors of this paper say, "Why split the team? Let's give everyone a full set of complex tools." They introduce Fully Complex-Valued Bilinear Filters.

Imagine a new, super-smart team where every member speaks the full language of "Complex" (Real + Imaginary) fluently. Instead of splitting the signal, they handle it all at once. The paper introduces five new types of this super-team:

C-BWF (The Wiener Filter): The "perfect planner" who knows the statistics of the signal ahead of time.
C-BLS (The Least Squares Filter): The "data cruncher" that learns from a big pile of past examples.
C-BLMS (The Least Mean Squares Filter): The "adaptive learner" that tweaks its settings bit-by-bit as the signal comes in.
C-BNLMS (The Normalized LMS): The "smart learner" that adjusts its learning speed automatically so it doesn't get confused by loud or quiet signals.
C-BRLS (The Recursive Least Squares Filter): The "fast tracker" that updates its memory instantly as new data arrives.

Why is this a big deal?

The paper runs simulations (computer experiments) to see how these new filters perform. Here is what they found:

Speed and Efficiency: The new C-BLMS filter is a champion of efficiency. While the old "4R" team needed a massive amount of math (multiplications) to do the job, the new complex team does it with fewer steps. For example, if the signal has a certain size, the new filter might need around 9LM + 6M + 3L + 4 real math steps, whereas the old 4R method needed 12LM + 4L + 8M + 8. That's a significant saving in energy and time.
Better Performance: The paper suggests that the new filters don't just save energy; they also learn faster. The old linear methods (which ignore the bilinear nature) might take forever to settle down, but the new bilinear ones zoom to the correct answer.
Handling the "Unmodelable": The authors point out a critical flaw in the old "4R" method. Some complex systems are so weird that you literally cannot describe them using four separate real filters. The new fully complex filters are the only ones that can model these tricky systems correctly.

What the Paper Says "No" To

The paper is very clear about what doesn't work or isn't the focus:

No to "Just Linear": You can't just use a standard linear filter on these complex bilinear systems. It's like trying to fix a twisted hose by just turning the water on harder. It might work for a simple system, but for these specific complex ones, it fails to capture the true behavior.
No to "Splitting is Perfect": While the "4R" method (splitting into four real filters) is better than the "2R" method, the paper argues it is still not the best solution. It's too heavy and, in some cases, mathematically impossible to represent the system accurately.

How Sure Are They?

The authors are confident in their math. They have proved the equations for how these filters update themselves. They have also proved that their new filters converge (meaning they will eventually find the right answer and stop changing) for the Wiener and LMS versions.

However, when it comes to the C-BRLS (the fast tracker), the paper is a bit more cautious. Because they had to make some smart approximations to make the math work, they haven't written a strict mathematical proof that it always converges. Instead, they say, "Look at our simulations; they show it works very well in practice." So, while the theory is solid for most, the fastest version is backed by strong computer experiments rather than a 100% ironclad proof.

The Bottom Line

This paper is like handing a complex, double-sided puzzle to a new team of solvers who can see both sides at once. The old team had to split the puzzle into pieces and solve them separately, which was slow and sometimes impossible. The new Fully Complex-Valued Bilinear Filters solve the puzzle together, faster, with less energy, and they can even handle the weird pieces that the old team couldn't touch. It's a step forward for anyone trying to tune into a clear signal in a noisy, complex world.

Technical Summary: Optimum and Adaptive Complex-Valued Bilinear Filters

Problem Statement
The identification of nonlinear systems is a fundamental task in digital signal processing. While linear models (e.g., Wiener, LMS, RLS filters) are well-established, many real-world applications in communications and radar exhibit distinct nonlinear behaviors that require nonlinear models. Specifically, numerous systems can be approximated by bilinear models. However, existing research on bilinear filters has primarily focused on real-valued systems. In many practical scenarios, such as those involving in-phase and quadrature-phase (IQ) imbalance or nonlinear power amplifiers, systems and signals are inherently complex-valued.

A critical gap exists because the definition of bilinearity used in this work concerns the multiplication of the system's parameters (coefficients), rather than the multiplication of input and output signals. Consequently, existing complex-valued nonlinear adaptive filters (e.g., complex Volterra filters or complex neural networks), which typically address nonlinearity with respect to the input signal, cannot be directly applied to general complex-valued bilinear systems. Furthermore, while splitting complex signals into real and imaginary parts (split-complex approach) is a common strategy, it may fail to capture correlations between the real and imaginary components or may not be able to model certain fully complex-valued bilinear systems.

Methodology
The paper investigates the extension of real-valued bilinear filters to the complex domain through four distinct structural approaches:

Split-Complex Approaches (2R and 4R):
- 2R Structure: The input signal is split into real and imaginary parts, processed by two separate real-valued bilinear filters, and recombined. This is a direct extension of real-valued bilinear LMS (BLMS) filters but excludes cross-terms between real and imaginary components.
- 4R Structure: This approach utilizes four real-valued bilinear filters to process the real and imaginary parts of the input and output, including cross-terms. This allows the structure to handle correlations between real and imaginary signal parts, which the 2R structure cannot.
Fully Complex-Valued Structures:
The authors derive novel filters using actual complex-valued coefficient vectors and Wirtinger's calculus to handle the differentiation of real-valued cost functions with respect to complex variables. The proposed filters include:
- Complex-Valued Bilinear Wiener Filter (C-BWF): An optimum filter derived using mean squared error (MSE) minimization for wide-sense stationary (WSS) signals. It employs an alternating iterative approach to solve for the two parameter vectors ( $\hat{h}$ and $\hat{g}$ ).
- Complex-Valued Bilinear Least Squares (C-BLS): A batch-processing filter that minimizes the sum of squared errors without requiring prior statistical knowledge, also solved via alternating updates.
- Complex-Valued Bilinear LMS (C-BLMS): An adaptive filter derived using gradient descent on the instantaneous squared error.
- Complex-Valued Bilinear Normalized LMS (C-BNLMS): An adaptive variant that normalizes the step-size to improve convergence speed and stability.
- Complex-Valued Bilinear Recursive Least Squares (C-BRLS): An adaptive filter utilizing a forgetting factor and recursive updates for faster convergence in time-varying environments.
Mixed Structures:
The paper also derives update equations for mixed complex-valued-real-valued bilinear filters, where one coefficient vector is constrained to be real-valued.

Key Contributions

Extension to Complex Domain: The work systematically extends established real-valued bilinear filters (WF, LS, LMS, NLMS, RLS) to the complex domain.
Novel Filter Derivations: It introduces the first fully complex-valued bilinear filters (C-BWF, C-BLS, C-BLMS, C-BNLMS, C-BRLS) and mixed-complex filters.
Theoretical Analysis: The paper provides convergence analyses for the C-BWF and C-BLMS-based filters, establishing stability boundaries for step-sizes.
Complexity Comparison: A detailed computational complexity analysis is presented, comparing the number of real-valued scalar multiplications required by 2R, 4R, and fully complex-valued structures.
Application Modeling: The filters are applied to identify complex-valued Multiple-Input-Single-Output (MISO) systems and complex-valued Hammerstein models, demonstrating their utility in specific system identification tasks.

Results and Performance
The paper presents simulation results (detailed in Section IV of the full text) supporting the applicability of the proposed filters. Key findings include:

Modeling Capability: The fully complex-valued structures can model general bilinear systems that cannot be represented by 2R or 4R structures due to the inability of the latter to capture all cross-terms inherent in complex multiplication.
Computational Efficiency: While the Big-O complexity order ($O(LM)$) is similar across LMS-based filters, the fully complex-valued C-BLMS requires fewer real-valued scalar multiplications than the 4R structure.
Convergence: The C-BLMS filter typically achieves the same steady-state error in fewer iterations compared to a complex-valued linear LMS (C-LLMS) filter applied to the linearized version of the bilinear model. However, for LS-based and WF-based algorithms, the linear approach scales significantly worse ( $O(L^3M^3)$ ) compared to the bilinear approach ( $O(L^3 + M^3)$ ) due to matrix inversion requirements.
Correlation Handling: The 4R and fully complex-valued structures are shown to be superior to the 2R structure when the real and imaginary parts of the signals are correlated.

Significance
The paper claims that its primary significance lies in providing a comprehensive framework for identifying complex-valued bilinear systems, a task for which no direct solutions previously existed in the literature. By deriving fully complex-valued filters, the authors offer a mathematically compact and computationally efficient alternative to splitting signals into real and imaginary parts. The work bridges the gap between real-valued bilinear system identification and the complex-valued requirements of modern communication and radar systems, offering both theoretical convergence guarantees and practical adaptive algorithms. The results suggest that fully complex-valued bilinear filters are not only theoretically sound but also offer practical advantages in terms of modeling accuracy and computational efficiency for specific system classes.