The Most “Divinely Inspired” Equation in Mathematics
— And You Use It Every Single Day
OTV Precision Sensing | Technical Blog
Take a look at this equation. Don’t worry — by the end of this article, you will understand exactly what it does and why it’s beautiful.
FOURIER TRANSFORM
F(ω) = ∫-∞∞ f(t) · e-iωt dt
Forward: Time Domain → Frequency Domain
f(t) = 1/2π ∫-∞∞ F(ω) · eiωt dω
Inverse: Frequency Domain → Time Domain (lossless reconstruction)
eiθ = cos(θ) + i · sin(θ)
Euler’s Formula: the bridge between complex exponentials and trigonometry
In 1822, when French mathematician Joseph Fourier proposed this equation, the reviewers Lagrange and Laplace deemed it “not sufficiently rigorous” and rejected his paper.
Two centuries later, this rejected equation has become one of the most-used mathematical tools in modern civilization. Every song on your phone, every WiFi connection, every JPEG photo, every MRI scan — they all run on this formula.
What it says is remarkably simple: any signal — no matter how complex — can be broken down into a sum of pure sine waves. And vice versa: from those sine waves, you can perfectly reconstruct the original signal, losing zero information.
1. Two Languages: Time Domain vs. Frequency Domain
The Fourier Transform is essentially a translator between two ways of describing a signal.
Time Domain — The World We’re Used To
Horizontal axis = time. Vertical axis = amplitude. You can see when something is loud, but you can’t tell what tones are present.
Frequency Domain — A Musician’s View
Horizontal axis becomes frequency (low to high pitch). Vertical axis = strength of each frequency. Instantly, you can see which tones are present and how strong they are.
The Fourier Transform: Translator Between Worlds
2. How Does the Equation “Sift Out” Individual Frequencies?
Once you understand time and frequency are two different worlds, the next question is: what does the equation actually do, mechanically, to pick out a specific frequency from a messy curve?
The answer lies in three words: multiply, integrate, orthogonality.
2.1 The Core Operation: Probe With a Test Wave, Then Measure
The Fourier Transform uses a pure sine wave of a known frequency as a “probe” to test the original signal:
For every candidate frequency ω, compute:
F(ω) = ∫ f(t) · e-iωt dt
Original signal × “Probe wave” at frequency ω → Energy at that frequency
Using Euler’s formula to expand e-iωt reveals exactly what’s happening:
F(ω) = ∫ f(t) · cos(ωt) dt − i ∫ f(t) · sin(ωt) dt
Real part = correlation with cosine (symmetric component)
Imaginary part = correlation with sine (anti-symmetric component)
It uses two rulers simultaneously — a cosine ruler for symmetric components, a sine ruler for anti-symmetric ones. Together, they capture both amplitude (how strong) and phase (where it starts).
2.2 The Magic: Orthogonality — Different Frequencies Don’t Interfere
You might wonder: if a signal contains dozens of frequencies mixed together, why doesn’t measuring one get contaminated by the others?
Sine waves have a remarkable mathematical property — orthogonality. Multiply two sine waves of different frequencies together and integrate over a full period: the result is exactly zero.
∫02π sin(n·t) · sin(m·t) dt = 0 (when n ≠ m)
Different-frequency sine waves — multiplied then integrated — cancel out completely.
What does this mean in practice? Imagine a noisy room with a 50Hz bass, a 440Hz tone, and a 3500Hz whistle playing simultaneously. When the Fourier Transform “probes” for 440Hz:
1. Multiply the mixed signal by a pure 440Hz sine wave
2. The 440Hz component “resonates” with itself — the integral is large
3. The 50Hz component × 440Hz probe — orthogonality makes the integral zero
4. Same for 3500Hz — zero interference
The final integral only retains information about the 440Hz component — all other frequencies “auto-mute”. The Fourier Transform repeats this for every frequency, one by one. Orthogonality acts as a perfect filter.
2.3 Back to the Equation — Now You Can Read It
Look at the opening equation again. Its meaning is now clear:
F(ω) = ∫-∞∞ f(t) · e-iωt dt
For every possible frequency ω:
Probe the signal with a test wave at ω → multiply → integrate →
Same frequency “resonates” → non-zero output → F(ω) = amplitude + phase at ω
Different frequencies cancel via orthogonality → zero → no interference
Sweep through all ω → the complete spectrum
3. Watch: How Sine Waves Build a Square Wave
The Fourier Transform claims any periodic waveform can be built from sine waves. Here’s a square wave being assembled from 1st (fundamental) + 3rd + 5th + 7th harmonics — the white waveform gets closer to a square with each harmonic added.
4. It’s Hiding in Every Corner of Your Life
MP3 compression works via FFT: transform audio to the frequency domain, discard frequencies the human ear can’t detect, and shrink the file to 1/10th its size with almost no perceptible quality loss. Noise-cancelling headphones do the same — FFT analyzes ambient noise, generates an inverted anti-noise wave, and cancels it out.
WiFi (OFDM modulation) distributes data across dozens of orthogonal subcarriers transmitted in parallel. Those subcarriers are generated using IFFT — your router performs thousands of Fourier transforms every second.
MRI scanners collect raw data in the spatial frequency domain. A 2D Fourier Transform reconstructs those signals into the anatomical images doctors read. Without FFT, MRI data would be an uninterpretable mess of electromagnetic readings.
JPEG compression uses the Discrete Cosine Transform (DCT), a close relative of the Fourier Transform. It converts 8×8 pixel blocks to the frequency domain — human eyes are insensitive to high-frequency detail, so those components get discarded, shrinking the file by ~90%.
A star’s brightness wobbles minutely due to orbiting planets. FFT extracts these periodic signals from years of observational data — the frequency and phase directly reveal the planet’s orbital period and position.
An equation rejected by the French Academy of Sciences now underpins a large fraction of modern digital infrastructure. That is the power of a truly “divinely inspired” mathematical insight.
5. The Man Behind the Equation: Joseph Fourier
“Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.”
— Joseph Fourier, 1822
6. How the Fourier Transform Makes Encoders More Precise
In OTV’s precision inductive encoders, the Fourier Transform isn’t academic theory — it’s embedded in the signal processing pipeline, running in real time.
6.1 Raw Signals: A Pair of Quadrature Sine Waves
The encoder’s sensing coils produce two signals:
sin channel: Ssin(θ) = A·sin(N·θ) cos channel: Scos(θ) = A·cos(N·θ)
N = pole pairs θ = mechanical angle A = amplitude
6.2 Harmonic Distortion: What Steals Your Precision
Ideally these are pure sine waves. In reality, manufacturing tolerances, coil misalignment, and EMI introduce harmonic distortion — unwanted frequency components that directly degrade angular accuracy.
| Harmonic | Source | Effect on Angle Error |
|---|---|---|
| 2nd | Amplitude imbalance between sin/cos | 2 cycles per electrical period |
| 3rd | Coil non-linearity | 4 cycles per electrical period |
| 4th | DC offset in signal conditioning | 2 cycles per electrical period |
6.3 FFT-Based Calibration: Turning “Dirty” Signals Into Precision
1. Acquire: Mount encoder on a reference stage. Record sin/cos signals at high sample rate over one full revolution.
2. FFT Analyze: Transform the error signal from time domain to frequency domain. Identify the amplitude and phase of each harmonic.
3. Store Calibration: Write harmonic coefficients into the encoder’s onboard non-volatile memory.
4. Compensate in Real Time: During normal operation, subtract the stored harmonic pattern from the raw signal.
Result: Raw uncompensated accuracy of ~±0.2° improves to ±0.006° after FFT calibration — a 30× improvement.
Precision, at the Intersection of Mathematics and Engineering
The Fourier Transform began with a heat equation and crossed two centuries to end up embedded in a 3.6mm-thin encoder chip.
OTV applies FFT-based harmonic compensation in the INE Series inductive encoders, achieving ±0.006° accuracy. Whether standard products or custom solutions — we welcome the conversation about your precision sensing needs.
OTV Precision Sensing | www.otvsensing.com