Z-Transform AI | Gerard Calvo Bartra

Z-transform filters are helpful when managing data between the time and frequency domain, such as EEG signals or music. In this project, I proposed to train Z-transform filters to adapt to the data.

PROJECT

Z-Transform AI

IMPACT AREAS

Challenge: Analyzing EEG signals for motion imagery detection

When developing an AI that could detect motion imagery from EEG data—with the ultimate goal of controlling a virtual character through thought alone—I encountered significant challenges with conventional neural network approaches. The difficulty stemmed from the inherent nature of EEG signals, which contain crucial information in both time and frequency domains.

Traditional approaches were insufficient for several reasons:

Time-domain analysis alone missed critical frequency patterns in the neural signals
Pure frequency-domain methods (like Fourier transforms) lost temporal information essential for detecting motion imagery
Existing public EEG datasets lacked diversity in movement types needed for training robust models
Individual variation in EEG patterns required flexible, adaptive filtering approaches rather than fixed filter banks

I needed an approach that could adaptively analyze both the spectral and temporal characteristics of EEG signals, learning directly from the data which frequencies and timing patterns were most informative for distinguishing different imagined movements.

Approach: Learnable Z-transform filters for adaptive signal processing

After considering several options including wavelet transforms, I developed a novel neural network architecture centered around learnable Z-transform filters. The Z-transform is a powerful tool in digital signal processing that extends the Fourier transform to discrete-time signals, capturing both frequency characteristics and time-domain behavior.

While wavelets excel at pre-structured, multi-resolution analysis, my approach with learnable Z-transforms allowed the network to tailor its filter shapes and timing directly from the data without extensive fine-tuning.

H(z) = \frac{B(z)}{A(z)} = \frac{b_0 + b_1 z^{-1} + \cdots + b_M z^{-M}}{a_0 + a_1 z^{-1} + \cdots + a_N z^{-N}}

The core innovation was treating this digital filter transfer function as a learnable module within a neural network. Rather than manually designing filters, the model would learn the optimal filter coefficients through gradient descent, adapting to the specific characteristics of the data.

Implementation: Differentiable Z-transform filtering in deep learning

The implementation focused on creating a fully differentiable Z-transform layer that could be integrated into modern neural networks and trained end-to-end. The core components include:

Learnable numerator coefficients {bᵢ} of length M+1
Learnable denominator coefficients {aᵢ} of length N+1, with a₀ fixed to 1 to avoid scale ambiguity
FFT-based implementation for computational efficiency
Fully differentiable operations to enable backpropagation

The forward pass follows these steps:

Pad the learned coefficient vectors to match the signal length
Apply FFT to the input signal and coefficient sequences
Form the frequency response H(e^jω) = B_f / (A_f + ε)
Multiply in the frequency domain and apply IFFT to return to the time domain

I extended this basic design with several advanced features:

Channel Attention: Learns channel-wise gating weights via pooled descriptors (average and max), boosting the most informative EEG rhythms
Multi-Filter Design: Learns multiple parallel filters per channel, concatenating their outputs to enrich the representation
Transformer Integration: Processes filtered signals through a Transformer encoder to capture long-range temporal dependencies
Stability-Aware Reparameterization: Constrains denominator coefficients to ensure filter stability
Combined Stable Multi-Filter Layer: Merges multi-filter capacity with stability constraints

Filter Evolution: Visualizing how Z-transform filters adapt during training

One of the most fascinating aspects of the Z-Transform AI architecture is being able to observe how the filters evolve during the training process. Unlike traditional deep learning models where the learned features remain opaque, the Z-transform filters can be directly visualized and interpreted in terms of their frequency responses.

The following visualizations show the filter coefficients and frequency responses as they adapt during training:

Animation showing Z-transform filter adaptation during training — Evolution of filter coefficients as the model learns from EEG data. Notice how the filters gradually specialize to capture specific frequency bands relevant to motor imagery tasks.

These visualizations demonstrate a key advantage of the Z-Transform AI approach: interpretability. We can literally see what signal characteristics the model is learning to focus on. This not only provides insight into the model's decision-making process but also can offer scientific value by revealing which frequency components are most relevant for the task at hand.

Technical Details: Bridging signal processing theory with deep learning

The Z-transform architecture combines classical digital signal processing theory with modern deep learning techniques. Key technical aspects include:

Differentiable implementation of Z-transform filtering using PyTorch's autograd
Stability guarantees through constrained parameterization of filter coefficients
Multi-scale temporal analysis through parallel filters with different orders
Integration with attention mechanisms and transformers for contextual processing
End-to-end optimization that adapts filters directly to the classification task

The stability-aware reparameterization was particularly important for IIR filters (those with feedback terms in the denominator). By enforcing constraints that keep all poles inside the unit circle, we ensure the filters remain stable throughout training—a critical property for reliable model behavior across different inputs.

Applications: Beyond EEG: Versatile time-series analysis

While initially developed for EEG signal analysis, the Z-Transform AI architecture has potential applications across numerous domains that involve time-series data:

EEG and Biomedical Signals: Adaptive feature extraction for motor imagery, seizure detection, and brain-computer interfaces, where spectral bands and timing vary by subject
Audio Processing: Learnable equalization, denoising, or source separation filters that adapt to different recordings or environments
Financial Time Series: Data-driven bandpass or trend-extraction filters to highlight cyclical components or suppress noise for forecasting
Predictive Maintenance and IoT: Vibration or sensor signal conditioning filters that automatically adapt to new machines or changing operating conditions
General Time-Series Forecasting and Anomaly Detection: Learned filter banks that emphasize relevant periodicities or transient events in any sequential data

The architecture's flexibility allows it to discover and exploit domain-specific patterns without requiring expert knowledge to design custom filters for each application.

Impact: Unifying signal processing and deep learning

The Z-Transform AI architecture represents a significant step in bridging traditional signal processing with modern deep learning approaches. This integration offers several advantages:

Interpretability: Unlike black-box neural networks, the learned filter coefficients can be analyzed using standard DSP tools
Efficiency: By incorporating signal processing knowledge into the architecture, the model can learn from smaller datasets
Adaptability: The architecture automatically adapts to the specific characteristics of the data without manual filter design
Theoretical foundation: Builds on well-established signal processing principles while extending them through learning
End-to-end optimization: Filter characteristics are directly optimized for the task rather than designed separately

By marrying classical DSP (Z-transform theory) with end-to-end deep learning, this work provides a flexible, interpretable filtering layer that—unlike fixed filters—can adapt entirely through gradient descent to optimize performance on a given task.

Future Directions: Expanding the capabilities of Z-Transform AI

This project opens up several promising avenues for future research and development:

Real-time BCI applications: Optimizing the architecture for low-latency processing to enable responsive brain-computer interfaces
Adaptive filter banks: Extending the approach to dynamically adjust filter characteristics based on changing signal properties
Multi-modal integration: Combining Z-transform layers with other sensor modalities for richer contextual understanding
Explainable AI: Further developing the interpretability aspects to provide insights into why certain frequency bands are selected for specific tasks
Transfer learning: Investigating how learned filters can transfer between related time-series tasks

The Z-Transform AI architecture represents not just a solution for a specific problem, but a new paradigm for approaching time-series analysis in deep learning. By embedding signal processing principles directly into neural network architectures, we can create models that are more efficient, interpretable, and adaptable across a wide range of applications.

Z-Transform AI | An AI model architecture based on learnable Z-transform filters.