Author: Joel B. Harley

Fast, Furious, and Fundamental: The Untold Depth of the Fast Fourier Transform

We tend to treat algorithms as transient tools—clever bits of logic that solve specific problems and then fade into the background. But the Fast Fourier Transform, or FFT, is not that kind of algorithm.

The FFT isn’t just a fast way to compute a Fourier transform. It’s a cornerstone of digital signal processing, and by extension, of modern computation. It’s the engine beneath your voice assistant’s signal chain, the speed behind your MRI scanner’s image reconstruction, and the unsung hero of everything from wireless communication to deep learning.

Yet for many, the FFT begins and ends with a textbook diagram of a radix-2 butterfly. That’s a shame, because the FFT is both deeper and more diverse than its most famous form.

Let’s explore what the FFT really is, how it goes beyond radix-2, and why it remains one of the most influential ideas in the applied mathematical sciences.

October 15, 2025 0

The DSP Survival Guide: What You Really Need to Know for the Midterm

If you’re in the middle of a graduate-level digital signal processing course, congratulations: you’re officially in deep. By now, you’ve wrestled with delta functions, decoded difference equations, squinted at z-plane plots, and maybe even cursed at a phase response graph that made no intuitive sense.

Your exam is coming. You’re reviewing the notes. It’s overwhelming. So here’s a better idea: let’s walk through the big ideas, the ones that actually matter—and not just because they’re on the test. These are the concepts that form the backbone of digital signal processing, and understanding them doesn’t just help you pass a midterm. It gives you tools to analyze the world.

October 8, 2025 0

Beyond the Grid: Why Non-Uniform Sampling Is the Secret Weapon You Didn’t Know You Needed

If you’ve ever taken a signal processing class, you know the first law of the land: sample uniformly, and sample fast enough. The Nyquist-Shannon Sampling Theorem reigns supreme. Uniform sampling is neat. Predictable. And it makes math work like magic.

But here’s the twist: the real world doesn’t always cooperate.

Sensors drift. Heartbeats don’t occur on a grid. Electrode measurements in the brain arrive irregularly. Seismic pulses bounce back whenever they feel like it. Welcome to the untamed frontier of non-uniform sampling, where the old rules don’t exactly apply—but where understanding those rules becomes more important than ever.

This article is for grad students stepping into advanced digital signal processing. You already know the beauty of ideal sampling. Now, get ready to embrace its imperfections—and see how non-uniform sampling is not just a glitch, but a feature in disguise.

October 1, 2025 0

The Hidden Half of Every Signal: Why Phase Response Deserves Your Attention

In every signals and systems class, you learn to take a signal, pass it through a system, and analyze what comes out. Somewhere along the way, you’ll graph a frequency response, plot the magnitude (how much each frequency is amplified), and probably move on without giving much thought to the other half of the story: the phase.

That’s a mistake.

Because while the magnitude response tells you how much, the phase response tells you when—and when it comes to how systems behave, especially in real time, when can be everything.

Phase might seem like a minor detail at first, an optional subplot to the main storyline of system behavior. But dig a little deeper, and you’ll find that phase isn’t just important—it’s essential. In signal processing, control systems, AI, medical imaging, and even virtual reality, knowing the phase response can make or break your design.

So let’s set the record straight. This article is your guide to understanding why phase matters, where it matters, and how to start seeing it not as an afterthought, but as a powerful tool.

September 24, 2025 0

The Frequency Is the Feature: Why Every Modern Engineer Should Master the Fourier Transform

Imagine a symphony playing in a concert hall. Now imagine being able to isolate the violin from the rest of the orchestra, pull out just the cellos, or turn up the oboe solo like you’re adjusting sliders on a digital soundboard. That’s the magic of the Fourier transform—it breaks down complex signals into simpler pieces, telling you what frequencies are inside and how much of each is present.

But here’s the thing: the Fourier transform isn’t just for audio engineers or musicians. It’s everywhere—from how we compress images and videos, to how we train machine learning models, to how doctors interpret brain signals or engineers model vibrations in bridges. It’s one of the most fundamental tools in the entire engineering toolbox. And if you’re a student learning it now, you’re stepping into a world where thinking in frequency is just as important as thinking in time or space.

Let’s unpack what makes the Fourier transform so powerful—and why it’s still driving innovation in fields you might not expect.

September 17, 2025 0

How to Keep a System from Spiraling Out: Why Every ECE Student Should Care About Control Systems

Take a moment to think about anything that moves or adjusts itself—like a drone flying in the wind, a robot arm reaching for a part, or even your bedroom thermostat switching off the heat. What do they all have in common?

They’re all part of a control system, and they all need to behave in a way that’s stable—meaning they don’t freak out when something changes. And that’s where your signals and systems knowledge starts turning into real-world impact.

Control systems are what allow us to design machines that respond to changes, make decisions, and settle down safely. They’re the tools engineers use to keep things running smoothly in a world that’s constantly pushing and pulling in different directions.

September 10, 2025 0

The Hidden Framework Behind Your Favorite Tech: Why Linear Time-Invariant Systems Still Rule the World

There’s a secret lurking beneath your noise-canceling headphones, your AI-powered photo filter, and even some of the tools used to predict stock markets and brain activity. It’s not AI. It’s not quantum computing. It’s something far older—and, in its way, far more elegant.

We’re talking about linear, time-invariant systems—or LTIs, if you’ve seen them scrawled across lecture boards in engineering classes. If that phrase makes your eyes glaze over, you’re not alone. But here’s the twist: understanding LTIs doesn’t just help electrical engineers or control theorists. It gives you a mental model for how inputs become outputs—in machines, in math, even in thought.

This article is a case for why students—and really, anyone interested in how the modern world works—should give LTIs the credit (and curiosity) they deserve.

September 3, 2025 0

The Inner Product: The Most Important Math You’re Already Using Without Realizing It

If you’ve ever used Spotify, trained a neural network, filtered an image, or Googled anything—you’ve relied on an idea so fundamental it barely gets named in everyday conversation: the inner product.

It’s not flashy. It’s not mysterious. It’s certainly not trending on TikTok. But the inner product—also known as the dot product—is quietly powering the smartest algorithms and signal tools in existence. It’s the mathematical handshake that lets one signal say, “Hey, you look a lot like me.”

If you’re a student wondering what separates hand-wavy intuition from real-world results in signal processing, data science, or machine learning, this article is your invitation to take the inner product seriously—and see how it shows up almost everywhere.

August 27, 2025 0

Why Everyone Should Learn a Bit of Signal Processing

If you’ve ever adjusted an Instagram filter, used noise cancellation on a plane, or asked Siri to play Taylor Swift, then you’ve used signal processing—the mathematical art of massaging, analyzing, and extracting meaning from data that changes over time. It’s behind your music, your fitness tracker, your MRI scan, and maybe even your job application. And while it might sound like an electrical engineer’s pet topic, signal processing is actually a foundational—and surprisingly flexible—tool across tech, science, and modern careers.

So if you’re a student wondering what to do with that Fourier Transform assignment, or why your professor keeps talking about “filtering out noise,” read on. Signal processing isn’t just useful—it’s everywhere.

August 22, 2025 0

Hot Tech for Cold Cancers: How Microwave Imaging Is Reinventing Breast Screening

Picture your chest as a crowded subway car: packed, complicated, and full of different signals—some harmless, others alarming. Traditional mammography snapshots it like a grainy train schedule. Enter microwave imaging, which floods this crowded space with gentle electromagnetic pulses (1–10 GHz), listens to how they bounce back, and pieces together a map of tissue properties. It’s like a radar that detects suspicious riders without shaming them for squeezing on too tight.

Microwave imaging for breast cancer combines electromagnetics, inverse problems, and signal processing—a playground for math nerds who want to turn reflections into medical breakthroughs. And behind much of this progress are longtime efforts at McGill University and newer advances from the University of Utah. Let’s unpack how it all works—and why it’s still struggling to go mainstream today.

August 20, 2025 0

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