Filters: Math I Still Use—Even When I’m Not Using the Math

By Delia Grenville, Ph.D.

By Grade 9, I already knew I wanted to study engineering. But I didn’t realize until much later—while writing this, in fact—that engineering gave me stories about math for free.

In engineering, you’re not just solving for x. You’re constantly weighing what’s known, what’s assumed, and what you can safely ignore. You get used to uncertainty. You learn to recognize when a familiar pattern is forming—and which tool gives you the clearest view through the mess.

You’ve probably said it (or heard it said): Why do I need to learn calculus? It’s not going to help me do my taxes. I get it. But for me, math wasn’t about taxes. It was about perspective.

And that’s why filters stuck with me.

In school, “filters” showed up in nearly every engineering discipline I explored. Filters help you take something noisy, scattered, or chaotic and make it usable

And that’s not just an engineering concept. That’s everyday life. And life’s messy.

There’s no shame in messiness. Better to accept that norm and have ways to deal with it. But without a system, it’s hard to know where to begin. That’s where filters come in.

I remember all the times my family and I had to deal with a cluttered playroom and I'm now realizing that we were instinctually leveraging my math knowledge to efficiently declutter.

I would have the kids start sorting: blocks in one bucket, wheels in another, and a third for all the random bits that only go together because—even though no one really knows what they’re for—you don’t throw them away, just in case.

And just like that, we’d have turned chaos into something usable. We filtered through the noisy clutter by separating and grouping the toys.

Two ideas stuck with me

Of the many math stories I encountered, two stuck with me most clearly: the Fourier Transform and the Laplace Transform.

One helped me understand what’s inside a system. The other helped me understand how that system behaves over time.

Before we get to the Greek symbols, let's talk about the why—and the story the equations are telling us.

Why Fourier?

You know that sound—that whirring, pulsing, or rattling that tells you something’s off. A laptop fan. A ceiling vent. A car engine that just doesn’t sound right.

In engineering, the differences in those sounds mean something, and it’s by using the Fourier Transform that those differences can be found.

It’s how engineers spot failure before it shows up on a dashboard.

Why Laplace?

If Fourier shows you what’s in a signal, Laplace helps you understand what the system does.

Most real-world systems don’t behave in clean, repetitive patterns. They start, surge, overshoot, and settle. Laplace is how we model that behavior over time.

Remember that cluttered room? It looks different when your student comes home from college versus when you’re just doing a quick tidy-up before relatives arrive. In systems design, we would see one scenario as a reset. The other is a response. Laplace helps engineers model both.

If you’ve ever watched a thermostat overshoot before leveling out—or cruise control surge before locking in—you’ve seen Laplace in action.

A Diagnostic and a Forecast

If Fourier is the tool for detection, Laplace is the tool for prediction.

If Fourier is a stethoscope, Laplace is a heart monitor.

Let me break that down:

A stethoscope lets you hear what’s already happening. Because of Fourier, you’re listening for murmurs, hums, irregularities.
A heart monitor shows how your system is behaving over time. Because of Laplace, you can track patterns, model responses, and predict whether you’re stabilizing or drifting into trouble.

Pulling It All Together

If Fourier helps you isolate and understand what’s in the signal, Laplace helps you predict how that signal will behave next.

They’re both ways of navigating noise. They’re both ways of translating complexity into clarity.

The Next Time You Sort Out Your Pantry…

...and you predict that you’ll be doing the same thing again next year—realize this:

You just did complex math, with integrals, in your head.

The decluttering and organizing? That was your Fourier Transform. The decision to set a reminder, build a system, and plan for the next reset? That was your Laplace Transform.

Math is only doing what we already do—just more consciously. And sometimes, when we take the time to notice, we realize we’ve been using these tools all along.

For those who really love the details and all those yummy Greek symbols, here’s Fourier:

Fourier Transform (continuous form):

F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i 2 \pi \omega t} \, dt

And here’s Laplace:

Laplace Transform (continuous form):

F(s) = \int_{0}^{\infty} f(t) e^{-st} \, dt

🔍What the Formulas Are Actually Doing

You’ve seen the symbols. Now let’s look at what they’re really doing—chunk by chunk, using the same metaphors we’ve been working with.

🩺 Fourier Transform

F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i 2 \pi \omega t} \, dt

Think of this like a stethoscope. You’re listening for a specific sound hidden in the noise.

F(ω) – “How much of a particular frequency (like a hum or buzz) is present?”
→ This is the result—the amount of a specific frequency found in the original mess.
f(t) – “What does the messy signal look like over time?”
→ This is your raw data: the sound, the vibration, the observation.
e^(-iωt) – “This part checks for a signature wave pattern.”
→ It compares your messy signal to a known frequency—like asking, “Do I hear a violin in this orchestra?”
∫ ... dt – “Let’s check for that pattern across the whole timeline.”
→ We’re scanning the entire history to measure how much that frequency shows up.

💓 Laplace Transform

F(s) = \int_{0}^{\infty} f(t) e^{-st} \, dt

This one’s more like a heart monitor. You’re not just detecting what's there—you’re tracking how it responds over time.

F(s) – “How does the system behave?”
→ The output shows how the system grows, settles, or reacts to change.
f(t) – “What are we observing?”
→ Still your original signal—what's unfolding over time.
e^(-st) – “This part applies time-weighting.”
→ It emphasizes what’s happening early vs. later. Is it fading? Building?
∫ ... dt – “Now let’s sum it up.”
→ It pulls together the weighted story to give a complete view of the system’s behavior.

👉 You don’t need to memorize these.
Just know:

Fourier is for what’s in there
Laplace is for how it plays out

Delia Grenville, Ph.D., is a business consultant and certified coach who has worked for Fortune 100 companies including Intel and Oracle. She brings her background in engineering and human factors to help leaders translate complexity into clarity.

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