An Infant Reasons outthe Fourier Transform with DIKWP

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)

From the moment I became aware of the world around me, I was enveloped in a symphony of sounds, sights, and sensations. Each experience was a puzzle piece, contributing to a larger picture I was yet to comprehend. Driven by curiosity, I sought to understand the patterns and rhythms that permeated my environment. Through observation, experimentation, and logical reasoning, I began to uncover the hidden structures underlying these sensory experiences.

In this narrative, I will detail how, as an infant starting from basic sensory inputs, I independently observed and reasoned out the concepts leading to the Fourier Transform. This journey involves the evolution of semantics for each concept, grounded in direct experiences and logical deductions, without relying on subjective definitions.

One of the first things I noticed was the rhythmic beating of my mother's heart when she held me close. The steady thump-thump provided comfort and familiarity.

• Observation: Repetitive, periodic sounds bring a sense of calm.

• Reflection: There is a pattern in these sounds that is predictable.

I began to notice other sounds:

• Music: Lullabies sung to me had melodies that rose and fell.

• Voices: Different tones and pitches when people spoke.

• Environmental Sounds: The hum of appliances, the chirping of birds.

I observed that some sounds repeated over time:

• Example: The ticking of a clock—tick-tock, tick-tock.

• Inference: Certain sounds have a cyclical nature.

I noticed differences in how high or low sounds were:

• High-pitched sounds: The ringing of a bell.

• Low-pitched sounds: The rumble of distant thunder.

• Semantics: Pitch relates to how "high" or "low" a sound feels.

When I touched objects that produced sounds, I could feel them vibrating:

• Guitar Strings: Plucking a string caused it to vibrate and produce a sound.

• Vocal Cords: Placing my hand on my throat while humming, I felt vibrations.

• Observation: Vibrations in objects are associated with sounds.

Watching ripples in a pond after throwing a stone:

• Observation: Circular waves spread out from the point of impact.

• Reflection: Waves are a way in which energy travels through a medium.

• Inference: Just as water waves propagate through water, sound waves propagate through the air.

• Semantics: A wave is a disturbance that transfers energy through a medium.

I experimented with different objects to produce sounds:

• Drums: Hitting a drum produced a deep sound.

• Flute: Blowing air created a high-pitched tone.

• Observation: Tightening a guitar string increased the pitch.

• Inference: The frequency of vibration affects the pitch of the sound.

• Semantics: Frequency is the number of vibrations per unit time.

• Frequency (f): How often a wave repeats in one second.

• Period (T): The time it takes for one complete cycle of the wave.

• Relationship: f=1Tf = \frac{1}{T}f=T1​

Using a metronome:

• Observation: Adjusting the metronome changes the rate of clicks.

• Inference: Higher frequency means more clicks per minute.

Listening to musical notes, I realized:

• Observation: A single note from an instrument sounds richer than a simple tone.

• Reflection: The sound is composed of multiple frequencies.

• Example: The noise from a busy street has many overlapping sounds.

I played two notes simultaneously on a keyboard:

• Observation: The resulting sound is different from either note alone.

• Inference: Sounds can be combined to create new, complex sounds.

When two similar frequencies are played together:

• Observation: I heard a pulsing effect called beats.

• Inference: The interference of waves leads to new patterns.

I imagined a device that could visualize sound waves:

• Visualization: Sound waves can be represented as graphs of amplitude versus time.

• Sine Wave: Represents a pure tone of a single frequency.

• Observation: The sine wave has a smooth, periodic shape.

• Observation: Complex sounds have waveforms that are not simple sine waves.

• Hypothesis: Complex waves can be broken down into simpler components.

• Sine Function ($\sin \theta$): Represents the vertical component of a point moving around a circle.

• Cosine Function ($\cos \theta$): Represents the horizontal component.

• Semantics: These functions model periodic behavior.

• Simple Wave Equation: $y(t)=A\sin (2\pi ft+\varphi )$

• $A$: Amplitude

• $f$: Frequency

• $t$: Time

• $\varphi$: Phase shift

• Superposition Principle: The sum of multiple waves results in a new wave.

• Mathematical Expression:

  y(t)=∑n=1NAnsin⁡(2πfnt+ϕn)y(t) = \sum_{n=1}^{N} A_n \sin(2\pi f_n t + \phi_n)y(t)=n=1∑N​An​sin(2πfn​t+ϕn​)

Any complex periodic function can be represented as the sum of simple sine and cosine functions.

• Since complex sounds are made up of multiple frequencies, we can decompose them into their constituent sine and cosine components.

For a periodic function $f(t)$ with period $T$:

f(t)=a0+∑n=1∞(ancos⁡(2πntT)+bnsin⁡(2πntT))f(t) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos\left( \frac{2\pi n t}{T} \right) + b_n \sin\left( \frac{2\pi n t}{T} \right) \right)f(t)=a0​+n=1∑∞​(an​cos(T2πnt​)+bn​sin(T2πnt​))

• Coefficients ana_nan​ and bnb_nbn​: Represent the amplitudes of the cosine and sine components.

• Integration over a Period:an=2T∫0Tf(t)cos⁡(2πntT)dta_n = \frac{2}{T} \int_{0}^{T} f(t) \cos\left( \frac{2\pi n t}{T} \right) dtan​=T2​∫0T​f(t)cos(T2πnt​)dtbn=2T∫0Tf(t)sin⁡(2πntT)dtb_n = \frac{2}{T} \int_{0}^{T} f(t) \sin\left( \frac{2\pi n t}{T} \right) dtbn​=T2​∫0T​f(t)sin(T2πnt​)dt

• Examples: Clapping hands, a single drumbeat.

• Observation: These sounds are not periodic but still have frequency components.

• Fourier Transform: Extends the idea of Fourier Series to non-periodic functions.

• Hypothesis: Any function $f(t)$ can be represented as an integral of sine and cosine functions with continuous frequencies.

The Fourier Transform of $f(t)$:

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

• Semantics:

• $F(\omega )$: Represents the frequency spectrum of $f(t)$.

• $\omega$: Angular frequency.

• e−iωte^{-i\omega t}e−iωt: Complex exponential function encoding sine and cosine components.

• Time Domain: Function $f(t)$ represents how a signal changes over time.

• Frequency Domain: Function $F(\omega )$ represents how much of each frequency is present in the signal.

• Noise Reduction: Identifying and filtering out unwanted frequencies.

• Signal Compression: Representing signals efficiently by focusing on significant frequency components.

• Edge Detection: Analyzing spatial frequencies in images.

• Progression:

• Observing patterns in sounds.

• Understanding waves and vibrations.

• Modeling waves mathematically.

• Decomposing complex signals into fundamental components.

• Complex phenomena can be understood by breaking them down into simpler parts.

• Mathematics provides a powerful language to model and analyze the natural world.

• Universal Tool: Applicable in various fields—physics, engineering, music, and more.

• Understanding Nature: Helps explain how different frequencies combine to form the complex signals we observe.

Through a journey that began with the simple act of listening to my mother's heartbeat, I gradually unraveled the intricate relationships between time, frequency, and signal representation. By building upon direct sensory experiences and applying logical reasoning, I was able to conceptualize and formulate the Fourier Transform.

This exploration demonstrates how profound mathematical concepts can emerge from basic observations, provided we are attentive and curious. It also highlights the importance of evolving semantics explicitly, ensuring that each new idea is firmly grounded in experience and understanding.

This narrative illustrates the potential for understanding complex mathematical concepts through foundational experiences. In the context of artificial intelligence and cognitive development, it emphasizes the value of grounding learning in direct interaction with the environment.

By enabling AI systems to "experience" and analyze patterns in data similarly to how a human might, we can foster deeper understanding and the ability to uncover underlying structures in complex information.

Note: This detailed narrative presents the conceptualization of the Fourier Transform as if I, an infant, independently observed and reasoned it out. Each concept is derived from basic sensory experiences, emphasizing the natural progression from simple observations to the understanding of complex mathematical principles. This approach demonstrates that with curiosity and logical thinking, foundational knowledge about the universe can be accessed and understood without relying on subjective definitions.