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Coefficient-of-Determination Fourier Transform

This paper introduces a novel algorithm that converts temporal data into a high-resolution spectral representation by analyzing the Coefficient of Determination between the data and artificial sinusoidal functions, enabling the generation of spectral data at any user-defined resolution that can be accurately transformed back into the temporal domain.

Original authors: Matthew David Marko

Published 2026-05-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Matthew David Marko

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 have a recording of a complex sound, like a choir singing together. You want to know exactly which notes (frequencies) they are hitting and how loud each note is. Usually, to do this, you use a tool called a Fourier Transform. Think of the traditional Fourier Transform as a very strict librarian who only allows you to check out books in a specific, pre-set order. If your data (the choir recording) is a bit "fuzzy" or low-quality, the librarian can only give you a blurry list of notes, and you can't ask for more detail without starting over.

This paper introduces a new, smarter librarian named CFT (Coefficient-of-Determination Fourier Transform). Here is how it works, using simple analogies:

1. The Problem: The "Aliasing" Trap

The author starts by pointing out a common problem. If you take a picture of a spinning fan with a slow camera, the fan might look like it's spinning backward or standing still. In math, this is called the "Nyquist limit." If your data points are too far apart in time, standard tools (like the Fast Fourier Transform or FFT) get confused. They can't tell the difference between a low note and a high note that happens to look the same at those specific moments. It's like trying to guess a song's melody by only hearing every 10th second of it; you might think it's a slow ballad when it's actually a fast rock song.

2. The Solution: The "Guess-and-Check" Detective

Instead of using a rigid formula, the CFT algorithm acts like a detective playing a game of "Guess the Song."

The Setup: The computer takes your messy, low-quality data (the choir recording).
The Game: It generates thousands of perfect, artificial "test songs" (pure sine waves) at every single frequency you want to check. You can ask it to check frequencies as closely spaced as you like, even if your original data was very coarse.
The Comparison: For every test song, the computer asks: "How much does this test song look like my messy data?"
- It doesn't just look at the shape; it calculates a Coefficient of Determination ( $R^2$ ). Think of this as a "Match Score" from 0 to 1.
- If the test song matches the data perfectly, the score is 1.
- If it's a complete mismatch, the score is 0.
- If it's the exact opposite, the score is -1.
The Result: The computer saves these match scores. High scores mean that specific frequency is present in your data. By doing this for both "sine" and "cosine" versions of the test songs, it figures out not just the volume (magnitude) but also the timing (phase) of the notes.

3. The Superpower: Reversibility

The most unique feature of this method is that it is reversible.

Traditional methods: Once you turn your data into a spectrum using standard tools, you often can't turn it back perfectly, especially if the data was low-resolution. It's like shredding a document and trying to tape it back together; you lose pieces.
CFT method: Because the algorithm builds the spectrum based on how well the pieces fit together, it can take that spectrum and perfectly reconstruct the original sound. The paper shows that if you turn the data into a spectrum and then turn it back, it looks almost identical to the original (a match score of 0.99999).

4. What the Tests Showed

The author tested this "detective" against the old "librarian" (standard FFT) and another advanced method (NDFT).

The Test: They created sounds with specific notes (frequencies) and added some static noise to make it hard to hear.
The Result:
- The old tools (FFT) got confused by the noise and the low resolution, often missing the notes or guessing the wrong pitch.
- The CFT detective found the correct notes with incredible precision, even distinguishing between notes that were very close together.
- When the CFT turned the data back into a sound, it sounded almost exactly like the original, even with the noise.

Summary

In short, this paper proposes a new way to analyze data that doesn't rely on rigid mathematical rules. Instead, it uses a correlation game: "Does this specific frequency fit my data?"

Benefit: You can ask for as much detail (resolution) as your computer can handle, regardless of how "fuzzy" your original data was.
Key Feature: Unlike other methods, you can take the result and turn it back into the original data without losing information.
Limitation: It is a "numerical" method, meaning it's a clever calculation, not the "pure" mathematical definition of a Fourier Transform. However, for practical engineering and science, it works better than the traditional tools when data is limited.

The author concludes that this method is a powerful tool for finding the true "notes" hidden in messy, low-quality data, and it can be trusted to give you the original picture back when you're done.

Technical Summary: Coefficient-of-Determination Fourier Transform (CFT)

Problem Statement
The paper addresses the limitations of traditional Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) methods when applied to temporal data with limited resolution. In standard DFT/FFT implementations, spectral resolution is strictly proportional to the temporal sampling rate. Consequently, if the temporal data is sparse or has a low sampling rate, the resulting spectral domain cannot resolve frequencies higher than the Nyquist rate, nor can it distinguish between frequencies that alias to the same discrete points (e.g., frequencies $f=1$ and $f=9$ appearing identical at a specific $\delta t$ ). While methods like Non-uniform Discrete Fourier Transforms (NDFT) exist to handle specific sampling issues, the author notes that recovering the original temporal function from these spectral representations often requires computationally intensive matrix analysis, if it is possible at all.

Methodology
The author proposes a numerical algorithm termed the Coefficient-of-Determination Fourier Transform (CFT). Unlike the analytical Fourier Transform, which integrates a function against complex exponentials, CFT operates by calculating the statistical correlation between the input temporal data and a series of user-defined artificial sinusoidal functions.

The algorithm proceeds as follows:

Amplitude Definition: The algorithm determines the peak-to-peak range of the input temporal data to establish a base amplitude ( $A$ ) for the spectral series.
Sinusoidal Generation: For every frequency ( $\omega_k$ ) defined by the user within a desired spectral domain, the algorithm generates two reference functions: a cosine function ( $\Phi_k$ ) and a sine function ( $\hat{\Phi}_k$ ).
Coefficient of Determination (CoD) Calculation: The algorithm calculates the Coefficient of Determination ( $R^2$ $R^{2}$ ) between the input data and both the sine and cosine reference functions.
- The CoD is derived from the correlation coefficient ( $R$ ), where $R = \frac{S_{St}}{\sqrt{SS_1 \cdot SS_2}}$ .
- The real spectral component corresponds to the $R^2$ of the cosine match, and the imaginary component corresponds to the $R^2$ of the sine match.
Normalization: The resulting spectral values are normalized against the sum of their absolute values and scaled by the base amplitude $A$ .
Inverse Transform: A defining feature of the CFT is its invertibility. The temporal function is reconstructed by summing the series of sinusoidal waves weighted by the calculated real and imaginary spectral components:
$f(t) = \sum_{k=1}^{N} \{ \text{real}(F(\omega_k)) \cdot \cos(\omega_k \cdot t) \} + \{ \text{imag}(F(\omega_k)) \cdot \sin(\omega_k \cdot t) \}$

Key Contributions

User-Defined Resolution: The primary contribution is the decoupling of spectral resolution from temporal sampling rate. The user can define any frequency range and resolution (e.g., 1 mHz steps) regardless of the input data's temporal density.
Invertibility with Low-Resolution Data: The paper demonstrates that the CFT can transform limited-resolution temporal data into a high-resolution spectral domain and, crucially, transform it back to the temporal domain with high fidelity ( $R^2 > 0.99$ ), a capability the author claims is difficult to achieve with standard NDFT without excessive computational cost.
Phase and Magnitude Recovery: The method successfully captures both spectral magnitude and phase information, even in the presence of random noise and phase shifts.

Results
The paper presents three distinct validation stages:

Aliasing Demonstration: Using functions with frequencies $f=1$ and $f=9$ sampled at $\delta t = 0.1$ , the CFT successfully distinguished the two frequencies and their phase shifts, whereas the NDFT failed to provide a coherent match for the inverse transform. The CFT inverse transform yielded $R^2$ values exceeding 0.996 for all test cases.
Peak Frequency Accuracy: In a test involving four sinusoidal components (20.80 Hz, 38.38 Hz, 61.38 Hz, 77.55 Hz) sampled at 50 Hz (below the Nyquist rate for the higher frequencies), the CFT identified peak frequencies with significantly higher accuracy than both traditional FFT and NDFT. For instance, for the 77.55 Hz component, CFT identified 77.58 Hz, while FFT was limited to 50 Hz and NDFT identified 78.37 Hz. The inverse transform of the CFT data matched the original function with $R^2 = 0.99999$ .
Parametric Study: A study involving 15 random frequencies (2–17 cycles) with random phases, magnitudes, and noise showed that the CFT could predict peak frequencies with accuracy down to tens of mHz. The correlation between original and reconstructed functions consistently yielded $R^2 \geq 0.92$ .

Significance and Claims
The author claims that the CFT offers a practical, working, and invertible method for obtaining high-resolution spectral data from limited-resolution temporal inputs. The paper explicitly states that the CFT is not a "true" analytical Fourier Transform (as defined by the integral in Equation 1); rather, it is a representation of spectral magnitudes based on the correlation coefficient ( $R^2$ ) between the data and sinusoidal bases.

The significance lies in the algorithm's ability to:

Overcome the Nyquist limit for frequency identification in sparse data.
Provide arbitrary spectral resolution limited only by computational resources.
Allow for variable resolution within the spectral domain (focusing on peaks without increasing cost globally).
Enable a robust inverse transform that recovers the original temporal shape, magnitude, and phase, even when the input data is noisy or low-resolution.

The paper concludes that while the CFT is computationally more expensive than traditional DFT or NDFT, its ability to characterize peak frequencies and phases with high accuracy, and its unique invertibility, makes it a superior tool for specific engineering applications involving limited temporal data.