Complex Orthogonal Decomposition (C.O.D.) using Python

This paper introduces the Complex Orthogonal Decomposition (C.O.D.) method for extracting spatial and temporal modes from oscillatory signals, providing theoretical foundations and Python-based examples to demonstrate its effectiveness in analyzing spatio-temporal data where phase information is critical.

The Gist

Imagine you are watching a school of fish swim. To the naked eye, they look like a chaotic, wiggly mess. But if you could freeze time and look closely, you'd see that every fish is moving in a very specific, rhythmic pattern. The problem is, their bodies aren't just wiggling back and forth like a simple pendulum; the wave of motion travels down their bodies in complex ways.

Standard tools for analyzing waves (like a standard Fourier transform) are like trying to describe a complex painting using only straight horizontal and vertical lines. They work great for simple, repeating patterns, but they struggle with the messy, organic reality of a swimming fish or a sloshing wave in a tank. They force the data into a box that doesn't fit.

This paper introduces a new, smarter tool called Complex Orthogonal Decomposition (C.O.D.). Think of C.O.D. as a "magic decoder ring" for wiggly, moving things.

Here is how it works, broken down into simple concepts:

1. The "Ghost" Signal (The Hilbert Transform)

Real-world signals (like the height of a water wave) are just numbers going up and down. But waves have two hidden secrets: how big they are (amplitude) and where they are in their cycle (phase).

To see these secrets, C.O.D. creates a "ghost" version of the signal. It takes the real wave and adds a "shadow" version of itself that is shifted by 90 degrees.

• Analogy: Imagine a dancer spinning. The real signal is the dancer's position. The "ghost" signal is the dancer's shadow on the wall. By looking at the dancer and the shadow together, you can see the full 3D rotation, not just the 2D movement back and forth. This "complex" signal allows the math to understand the direction the wave is moving.

2. The "Unmixing" Process (The Decomposition)

Once the signal is "ghosted," C.O.D. tries to break it apart into its simplest building blocks. It asks: "Can I describe this whole messy movement as a sum of a few simple, independent patterns?"

• Spatial Modes (The Shape): These are the "frozen" shapes the wave makes. Is it a single bump? A double-hump? A cubic curve?
• Temporal Coefficients (The Rhythm): These are the "timers" that tell the shapes how to move, grow, or shrink over time.

The Magic Trick: C.O.D. ensures that these shapes are orthogonal.

• Analogy: Think of a piano. If you press the "C" key and the "E" key, they make a chord, but they are distinct notes. If you press two keys that are exactly the same, you just get a louder version of the same note. C.O.D. finds the "keys" (shapes) that are completely different from each other, so it can separate a complex chord (the fish swimming) back into its individual notes (the specific body waves).

3. The "Traveling Index" (Standing vs. Moving)

One of the coolest features of this method is the Traveling Index. This is a number between 0 and 1 that tells you what kind of wave you are looking at.

• 0 (Standing Wave): Imagine a jump rope being shaken up and down. The rope goes up and down in place, but the "hump" doesn't move left or right. The index is 0.
• 1 (Traveling Wave): Imagine a snake slithering. The wave moves from head to tail. The index is 1.
• 0.5 (The Mix): Most real-world waves are somewhere in between. C.O.D. gives you a precise number to say, "This wave is 70% traveling and 30% standing."

What the Paper Demonstrates

The authors tested this "magic decoder" on three different scenarios to prove it works:

1. The Fish Tank (Water Waves): They simulated waves in a tank. Even when two different waves were mixed together, C.O.D. instantly separated them, identified their shapes, and told them exactly how much energy each wave had. It was like separating a smoothie back into the original fruit.
2. The Fading Wave: They looked at a wave that was slowly dying out (damping). Standard tools often get confused when a wave changes speed or size over time. C.O.D. handled it gracefully, correctly identifying that it was a single shape that was just getting quieter.
3. The Chameleon Wave (Frequency Modulation): They created a wave that changed its rhythm constantly (like a siren going up and down in pitch). Even though the "music" was changing, the "shape" of the wave stayed the same. C.O.D. realized, "Hey, this is just one shape doing a complex dance," and didn't get tricked into thinking there were many different shapes involved.

Why Should You Care?

If you are a scientist studying fish swimming, ocean waves, or even how a bridge vibrates in the wind, you are dealing with signals that are messy and changing.

• Old Way: You try to force the data into a simple box, and you lose information. You might miss the fact that a fish is swimming more efficiently because of a specific body curve.
• C.O.D. Way: You let the data tell you its own story. It finds the hidden shapes and rhythms automatically, even if they are changing, fading, or mixing together.

The paper also provides Python code (a set of instructions for computers) so that anyone can use this tool. It's like giving everyone a new pair of glasses that lets them see the hidden structure in the chaos of the physical world.

In short: C.O.D. is a sophisticated way of taking a messy, moving picture and breaking it down into its cleanest, most understandable parts, telling you exactly what is moving, how it's moving, and where it's going.

Technical Summary

1. Problem Statement

The paper addresses the challenge of analyzing spatio-temporal signals ($s(t, x)$) that represent oscillatory physical phenomena (e.g., fluid waves, fish swimming kinematics).

• Limitations of Traditional Methods: Standard Fourier analysis assumes spatial periodicity and often fails when the spatial form of the wave is unknown, non-periodic, or complex (e.g., the body wave of a swimming fish). Similarly, standard Proper Orthogonal Decomposition (POD) is typically applied to real-valued data and may not fully capture phase relationships essential for distinguishing between standing and traveling waves.
• The Goal: To extract distinct spatial modes and temporal coefficients from oscillatory signals where phase information is critical, and to quantify the nature of the wave motion (standing vs. traveling).

2. Methodology: Complex Orthogonal Decomposition (C.O.D.)

The authors propose a mathematical framework based on the decomposition of a complex analytic signal rather than the raw real signal. The methodology consists of the following steps:

A. Signal Construction (Hilbert Transform)

1. Analytic Signal: For a real-valued signal $s(t, x)$, a complex analytic signal $s_c(t, x)$ is constructed using the Hilbert transform in the time domain:
   $$s_c(t, x) = s(t, x) + i \mathcal{H}\{s(t, x)\}$$
   In the frequency domain, this corresponds to suppressing negative frequencies and doubling positive ones.
2. Purpose: This transformation separates amplitude and phase, allowing the signal to be represented as a sum of separable complex modes.

B. Modal Decomposition

The complex signal is decomposed into a sum of separable terms:
$$s_c(t, x) = \sum_{j=1}^{\infty} a_j(t) \phi_j(x)$$
Where:

• $\phi_j(x)$ are complex spatial modes (eigenvectors).
• $a_j(t)$ are complex temporal coefficients (Complex Orthogonal Coordinates).

Algorithm (Discrete Implementation):

1. Construct the complex signal matrix $S_c$ from the data matrix $S$ using the Discrete Hilbert Transform (via FFT).
2. Form the temporal covariance matrix $R = \frac{1}{N_t} Z Z^\dagger$, where $Z = S_c^T$.
3. Solve the eigenvalue problem $R \phi_j = \lambda_j^e \phi_j$.
   • $\lambda_j^e$: Eigenvalues representing modal energy.
   • $\phi_j$: Eigenvectors representing spatial modes.
4. Compute temporal coefficients via projection: $a_j = \phi_j^\dagger Z$.

C. Reconstruction

The original real signal is recovered by taking the real part of the complex decomposition:
$$s(t, x) = \Re \left[ \sum a_j(t) \phi_j(x) \right]$$
This can be expanded into a sum of real products of time and space functions.

D. The Travelling Index ($\alpha_j$)

A key innovation is the Travelling Index, a scalar metric $\alpha_j \in [0, 1]$ that quantifies the wave nature of a mode:

• Calculation: Based on the Gram matrix of the real and imaginary parts of the spatial mode $\phi_j$. It uses the ratio of the smallest to the largest singular value (or eigenvalues of the Gram matrix).
• Interpretation:
  • $\alpha_j = 1$: Perfectly traveling wave (circular trace in the complex plane).
  • $\alpha_j = 0$: Purely standing wave (real or imaginary only).
  • $0 < \alpha_j < 1$: Mixed behavior.

E. Non-Uniform Grids

The paper extends the method to non-uniform spatial grids (common in experiments) by introducing a quadrature weight matrix $W$. The inner product and covariance matrix are modified to include these weights ($R_w = \frac{1}{N_t} Z W Z^\dagger$), ensuring energy conservation and orthogonality are preserved despite irregular sampling.

3. Key Contributions

1. Theoretical Framework: A rigorous derivation of C.O.D. for continuous and discrete signals, linking it to the Hilbert transform and eigenvalue problems.
2. Python Implementation: The authors provide a complete Python package (pack_COD) and a suite of test scripts (test_COD_ex_*.py) to make the method accessible for research and pedagogy.
3. Travelling Index: The formalization of a robust metric to distinguish traveling vs. standing waves, which is crucial for analyzing efficiency in biological locomotion (e.g., fish swimming) or structural dynamics.
4. Handling Non-Uniformity: A generalized formulation for non-uniform spatial grids, addressing a common limitation in experimental data processing.

4. Results and Validation

The paper validates the method through three distinct examples:

• Example 1: Water Waves in a Tank (Superposition of Modes)

  • Setup: A signal composed of two standing waves with different frequencies and wavelengths.
  • Result: C.O.D. successfully separated the two modes, recovering their exact spatial shapes and amplitudes. The calculated Travelling Index correctly identified the modes as standing waves ($\alpha \approx 0$).
  • Significance: Demonstrates the method's ability to decouple orthogonal modes even when they overlap in space and time.

• Example 2: Temporally Decreasing Standing Wave (Damping)

  • Setup: A standing wave with exponential amplitude decay ($e^{-\gamma t}$).
  • Result: C.O.D. recovered the single spatial mode with high precision. The Travelling Index remained 0.
  • Nuance: The paper notes that for non-purely oscillatory signals (damped), the Hilbert transform requires an approximation ($\gamma \ll \omega$). While the spatial mode was recovered perfectly, the temporal spectrum showed slight broadening due to this approximation.

• Example 3: Frequency Modulated (FM) Wave

  • Setup: A standing wave where the frequency is modulated over time (chirp signal).
  • Result: Despite the temporal signal containing a broad spectrum of frequencies (Jacobi-Anger expansion), C.O.D. identified a single spatial mode.
  • Significance: This proves that C.O.D. can handle multi-frequency temporal oscillations associated with a single spatial structure, preventing the "mode splitting" that might occur in other decomposition methods. The Travelling Index correctly remained 0.

• Appendix: Non-Uniform Grids

  • Result: Applied to a Chebyshev-distributed grid. The weighted C.O.D. successfully recovered the same spatial modes and energies as the uniform grid case, validating the quadrature weighting approach.

5. Significance

• Physical Interpretability: Unlike standard POD which yields real modes that may be mixtures of traveling and standing waves, C.O.D. yields complex modes that allow for a direct physical interpretation of wave propagation direction and speed via the Travelling Index.
• Efficiency in Biological/Fluid Dynamics: The method is particularly suited for analyzing complex, non-periodic motions (like fish swimming) where the spatial form is unknown and phase is critical.
• Accessibility: By providing open-source Python tools, the authors lower the barrier to entry for researchers in hydrodynamics and structural mechanics to apply advanced signal processing techniques.

In summary, this paper presents C.O.D. as a superior alternative to standard Fourier or POD analysis for oscillatory spatio-temporal data, offering a mathematically rigorous way to separate modes, quantify wave behavior, and handle experimental data irregularities.