Discovering the Wavelet Transform: As an Infant

Yucong Duan

International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation(DIKWP-SC)

World Artificial Consciousness CIC(WAC)

World Conference on Artificial Consciousness(WCAC)

(Email: duanyucong@hotmail.com)

Introduction

From the earliest days of my life, I was surrounded by a world rich in patterns and signals. The rhythms of sounds, the textures of sights, and the sequences of events all beckoned me to explore and understand them. My innate curiosity drove me to seek the underlying structures within these sensory experiences. Through observation, experimentation, and logical reasoning, I began to uncover the tools needed to analyze and interpret complex signals.

In this narrative, I will detail how, starting from basic experiences as an infant, I independently observed, experimented, and logically deduced the concept of the Wavelet Transform. This journey illustrates how a sophisticated mathematical tool can emerge from simple observations and reasoning, without relying on subjective definitions or prior formal education.

Chapter 1: Observing Time-Varying Signals1.1 Early Auditory ExperiencesEncountering Complex Sounds

• Sounds Around Me: I was enveloped by a variety of sounds—voices, music, environmental noises.

• Observation: Sounds changed over time; they were not constant.

Recognizing Temporal Variations

• Noticing Changes: A song had different notes at different times; a conversation involved changing tones.

• Reflection: Signals vary over time and have different characteristics at different moments.

1.2 Visual Patterns Over TimeWatching Moving Objects

• Observation: Leaves rustling in the wind, people walking by.

• Reflection: Visual signals also change over time, presenting different patterns.

Chapter 2: Understanding Frequency and Time Domains2.1 Exploring Frequency ComponentsExperiencing Musical Notes

• Experiment: Playing notes on a piano.

• Observation: Each key produced a sound of a specific pitch, corresponding to a frequency.

• Semantics: Frequency relates to how high or low a sound is.

Combining Frequencies

• Experiment: Playing chords, combining multiple notes.

• Observation: The resulting sound is more complex, containing multiple frequencies.

2.2 Time and Frequency Trade-offsShort vs. Long Notes

• Observation: Some notes were played quickly, others were sustained.

• Reflection: The duration of a note affects how it is perceived.

Recognizing Limitations of Frequency Analysis

• Question: How can I analyze signals that change over time using frequency components?

Chapter 3: Introducing the Fourier Transform3.1 Understanding the Fourier TransformConceptualizing Signal Decomposition

• Idea: Any complex signal can be decomposed into a sum of sine and cosine functions of different frequencies.

• Semantics: The Fourier Transform converts a time-domain signal into its frequency-domain representation.

Mathematical Expression:

• Fourier Transform:

  F(ω)=∫−∞∞f(t)e−iωtdtF(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dtF(ω)=∫−∞∞​f(t)e−iωtdt

• Inverse Fourier Transform:

  f(t)=12π∫−∞∞F(ω)eiωtdωf(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omegaf(t)=2π1​∫−∞∞​F(ω)eiωtdω

3.2 Limitations of the Fourier TransformLoss of Time Localization

• Observation: The Fourier Transform provides frequency information but loses time localization.

• Reflection: It tells me what frequencies are present but not when they occur.

Need for Time-Frequency Analysis

• Problem: For non-stationary signals (signals whose frequency content changes over time), the Fourier Transform is insufficient.

• Goal: Develop a method to analyze signals in both time and frequency domains simultaneously.

Chapter 4: Short-Time Fourier Transform (STFT)4.1 Introducing WindowingConcept of Windowing

• Idea: Apply the Fourier Transform to small sections (windows) of the signal.

• Method: Multiply the signal by a window function that is non-zero only over a short time interval.

4.2 Limitations of STFTTrade-off Between Time and Frequency Resolution

• Observation: Narrow windows provide good time resolution but poor frequency resolution; wide windows do the opposite.

• Heisenberg's Uncertainty Principle in Signals:

• Statement: It is impossible to obtain arbitrarily precise time and frequency information simultaneously.

Reflection:

• Challenge: Need a method that adapts to different frequency components, providing good time resolution for high frequencies and good frequency resolution for low frequencies.

Chapter 5: Exploring Multiresolution Analysis5.1 Understanding Signal StructuresAnalyzing Different Signal Components

• Observation: Signals often contain both slow-changing (low-frequency) and fast-changing (high-frequency) components.

• Example: A music piece with slow melodies and rapid drum beats.

5.2 Concept of ScaleDefining Scale in Signals

• Scale: Refers to the level of detail or resolution at which a signal is analyzed.

• Semantics: Low scales correspond to high frequencies (fine details); high scales correspond to low frequencies (coarse features).

Chapter 6: Introducing Wavelets6.1 Conceptualizing WaveletsDefining Wavelets

• Wavelet: A waveform of effectively limited duration that has an average value of zero.

• Semantics: Wavelets are functions that can be used to analyze signals at different scales.

Properties of Wavelets

• Localization in Time and Frequency:

• Wavelets are localized in both time and frequency domains.

• Orthogonality:

• Wavelets can form an orthogonal basis for signal decomposition.

6.2 Mother Wavelet and ScalingMother Wavelet

• Definition: A prototype function from which other wavelets are derived through scaling and translation.

• Mathematical Representation:

  ψa,b(t)=1aψ(t−ba)\psi_{a,b}(t) = \frac{1}{\sqrt{a}} \psi\left( \frac{t - b}{a} \right)ψa,b​(t)=a​1​ψ(at−b​)

• $\psi (t)$: Mother wavelet.

• $a$: Scale parameter (dilates or compresses the wavelet).

• $b$: Translation parameter (shifts the wavelet in time).

Chapter 7: Developing the Continuous Wavelet Transform (CWT)7.1 Defining the CWTMathematical Expression:

W(a,b)=∫−∞∞f(t)1aψ∗(t−ba)dtW(a, b) = \int_{-\infty}^{\infty} f(t) \frac{1}{\sqrt{a}} \psi^*\left( \frac{t - b}{a} \right) dtW(a,b)=∫−∞∞​f(t)a​1​ψ∗(at−b​)dt

• $f(t)$: Signal being analyzed.

• ψ∗(t)\psi^*(t)ψ∗(t): Complex conjugate of the mother wavelet.

• $W(a,b)$: Wavelet coefficients representing the signal at scale $a$ and position $b$.

7.2 Interpreting the CWTTime-Scale Representation

• Observation: The CWT provides a representation of the signal in terms of time and scale (frequency).

• Reflection: It allows for the analysis of signals at different resolutions.

Advantages Over STFT

• Adaptability: Wavelets adjust their time and frequency resolution according to the scale.

• Better Time Resolution at High Frequencies: Wavelets provide finer time resolution where needed.

Chapter 8: Discrete Wavelet Transform (DWT)8.1 Sampling Scales and PositionsDiscretizing the Parameters

• Idea: Choose discrete values for scale $a$ and position $b$ to create a computationally efficient transform.

• Dyadic Sampling: Use scales and positions based on powers of two.

• a=2−ja = 2^{-j}a=2−j, b=k⋅2−jb = k \cdot 2^{-j}b=k⋅2−j

• $j,k$: Integers representing scale and position indices.

8.2 Implementing the DWTMultilevel Decomposition

• Process: Decompose the signal into approximation and detail coefficients at each level.

• Approximation Coefficients (Low Frequencies): Captures the coarse features.

• Detail Coefficients (High Frequencies): Captures the fine details.

Filter Banks

• Concept: Use a pair of filters—a low-pass filter (scaling function) and a high-pass filter (wavelet function)—to separate frequency components.

Chapter 9: Applications and Insights9.1 Signal CompressionData Reduction

• Observation: Many signals can be represented efficiently using a few significant wavelet coefficients.

• Application: Compressing signals by discarding negligible coefficients.

9.2 Noise ReductionDenoising Signals

• Method: Thresholding detail coefficients to remove noise while preserving important features.

• Result: Cleaner signals with minimal loss of essential information.

9.3 Feature ExtractionAnalyzing Non-Stationary Signals

• Application: Using wavelets to detect transients, edges, and singularities in signals.

• Examples: Heartbeat analysis in ECG signals, fault detection in machinery.

Chapter 10: Reflecting on the Discovery10.1 Advantages of Wavelet TransformTime-Frequency Localization

• Understanding: Wavelets provide a more precise analysis of signals whose frequency content changes over time.

• Benefit: Ability to zoom in on short-lived high-frequency events.

10.2 Appreciation of Mathematical ToolsUnified Framework

• Insight: The Wavelet Transform unifies concepts from time-domain and frequency-domain analyses.

• Reflection: Mathematics offers powerful tools to interpret and understand complex phenomena.

Conclusion

Through observation and logical reasoning, I embarked on a journey that led me to discover the Wavelet Transform. Starting with the recognition of time-varying signals and the limitations of existing analysis methods, I sought a solution that could provide both time and frequency information simultaneously. By conceptualizing wavelets and developing both the Continuous and Discrete Wavelet Transforms, I found a versatile tool for signal analysis.

This journey demonstrates that complex mathematical concepts can emerge naturally from simple experiences. By evolving the semantics of each concept explicitly and grounding them in reality, advanced ideas become accessible without the need for subjective definitions or prior formal knowledge.

The Wavelet Transform not only deepened my understanding of signal processing but also highlighted the elegance and utility of mathematical thinking in solving real-world problems.

Epilogue: Implications for Learning and AI

This narrative illustrates how foundational mathematical principles can be understood through direct interaction with the environment and logical reasoning. In the context of artificial intelligence and cognitive development, it emphasizes the importance of experiential learning and the evolution of semantics from core experiences.

By enabling AI systems to recognize patterns, adapt to varying contexts, and apply multiresolution analysis, we can enhance their ability to process complex, non-stationary data. This approach avoids reliance on predefined definitions and promotes the natural discovery of mathematical relationships, leading to more robust and flexible AI models.

Note: This detailed narrative presents the conceptualization of the Wavelet Transform as if I, an infant, independently observed and reasoned it out. Each concept is derived from basic experiences, emphasizing the natural progression from recognizing time-varying signals to developing a sophisticated mathematical tool for their analysis. This approach demonstrates that with curiosity and logical thinking, foundational knowledge about advanced mathematics can be accessed and understood without relying on subjective definitions.

References

1. International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation (DIKWP-SC),World Association of Artificial Consciousness(WAC),World Conference on Artificial Consciousness(WCAC). Standardization of DIKWP Semantic Mathematics of International Test and Evaluation Standards for Artificial Intelligence based on Networked Data-Information-Knowledge-Wisdom-Purpose (DIKWP ) Model. October 2024 DOI: 10.13140/RG.2.2.26233.89445 .  https://www.researchgate.net/publication/384637381_Standardization_of_DIKWP_Semantic_Mathematics_of_International_Test_and_Evaluation_Standards_for_Artificial_Intelligence_based_on_Networked_Data-Information-Knowledge-Wisdom-Purpose_DIKWP_Model

2. Duan, Y. (2023). The Paradox of Mathematics in AI Semantics. Proposed by Prof. Yucong Duan:" As Prof. Yucong Duan proposed the Paradox of Mathematics as that current mathematics will not reach the goal of supporting real AI development since it goes with the routine of based on abstraction of real semantics but want to reach the reality of semantics. ".