Fourier Transform: The Ex I Thought I'd Graduated From
I learned signal processing in graduate school.
I passed the exams. I closed the textbook. I moved on with my life.
With genuine confidence, I believed Fourier transform was over.
This belief did not survive contact with reality.
The Lie We All Tell Ourselves
Fourier transform is usually introduced early and aggressively. You meet it before you’re ready, under time pressure, surrounded by equations that feel abstract and faintly hostile.
You learn to survive it. Not to love it.
So graduation feels like closure. You assume Fourier belongs to the past—a hazing ritual, not something you’ll actually use. Real research, you think, will be about higher-level ideas.
This assumption is spectacularly wrong.
First Return: Wavelets, or Fourier in a Nicer Outfit
Years later, I published a paper titled “Wavelet subband-specific learning for low-dose CT denoising.”
Wavelets sounded different. Refined. Modern. Almost gentle.
But here’s the uncomfortable truth: to understand wavelets properly, you still need Fourier intuition. Energy distribution. Frequency localization. Basis decomposition.
Fourier hadn’t disappeared.
It had just updated its LinkedIn profile.
Second Return: Deep Learning Will Save Us (It Didn’t)
Then came deep learning.
End-to-end optimization. Learned representations. Neural networks discovering features automatically. This felt like liberation—finally moving beyond handcrafted transforms.
And then Fourier showed up again.
Fourier features. Positional encoding. Sinusoidal bases baked into transformers.
At this point, Fourier was no longer a method.
It had become infrastructure.
Even when we stop mentioning it explicitly, we quietly rebuild it inside our models—because, apparently, the universe just really likes sines and cosines.
When Fourier Became Physical
Until then, Fourier had been a mathematical companion. Annoying, but abstract.
In optical coherence tomography (OCT), it became something else entirely.
In OCT, Fourier transform is not optional. It’s not aesthetic. It’s not a preprocessing choice you make over coffee.
It is how depth exists.
The imaging system is designed so that structural information only appears after transforming an interference fringe. Without Fourier transform, there’s no image—just wiggly nonsense.
This was the turning point.
Fourier was no longer following me as theory.
I was encountering it as physics.
“So This Is What I’m Doing With Fourier Now”
Every so often, I pause in the lab and stare at an OCT A-line on the screen.
It looks simple. A fringe. An oscillation. Nothing dramatic.
And then the internal monologue starts:
That oscillation frequency? Depth.
That asymmetry? System instability.
That broadened peak? Something went wrong upstream.
At some point I catch myself thinking:
Huh. So this is what I’m doing with Fourier now.
I’m no longer learning it.
I’m arguing with it.
Years ago, Fourier transform was an exam topic. Now it’s how I decide whether an optical system is behaving—or lying to me.
The Only Explanation I Ever Give
When someone asks what’s actually happening in an OCT A-line, I give the shortest explanation I can.
What we record is a fringe—an oscillation as a function of wavenumber $k$. Very roughly:
\[I(k) \sim \cos(2kz)\]The important part isn’t the cosine. It’s that depth $(z)$ shows up as how fast the signal oscillates with respect to $k$.
- Shallow structures → slow oscillation
- Deeper structures → faster oscillation
So when we apply an inverse Fourier transform, we’re not doing anything mystical. We’re just sorting oscillations by frequency—and the answer shows up as depth.
That’s it. Fringe in, depth out. Fourier doing its one job, again.
No derivation. No drama. Just math being unreasonably effective, as usual.
What the Pictures Are For
At this point in the post, I’d normally show:
- A raw A-line fringe (the wiggly input)
- A conceptual “Fourier happens here” transition
- A reconstructed depth profile (the useful output)
These aren’t educational diagrams.
They’re snapshots of what my daily relationship with Fourier looks like now.
They replace explanation, not supplement it.
The Real Lesson
Looking back, signal processing was the most valuable class I ever took.
Not because it taught me how to compute a Fourier transform.
But because it trained my intuition for how information hides—and how it reappears when you represent it differently.
That intuition survives every paradigm shift.
Deep learning didn’t kill it. It made it more useful.
Acceptance
I no longer try to escape Fourier transform.
I don’t resent it. I don’t glorify it.
I simply recognize it as a recurring character in my professional life—like that one colleague who shows up in every project, whether you invited them or not.
Fourier transform isn’t a phase you outgrow.
It’s something you keep meeting again, in a different form, every time you start doing real work.
Summary
This isn’t a tutorial. It’s not an explanation. There are already countless excellent resources for those.
This is an acknowledgment.
Fourier transform is not something you finish learning.
It’s not following you. You’re just finally learning to see what was always there. The universe speaks in frequencies. Fourier is just the translator. And you can’t fire your translator when they’re right.