But what about changing frequencies?
Everything we've seen so far has one thing in common: the frequencies stay constant. Each sine wave has a fixed pitch that doesn't change over time. Each circle spins at a constant speed.
But what happens when the frequency changes?
Think about a bird's chirp. It doesn't stay on one note – the pitch sweeps up or down. Or think about a police siren, sliding between two notes. Or the sound of a car driving past you – the pitch seems to drop as it passes (that's the Doppler effect!).
These are all examples of chirps – signals where the frequency changes over time.
This is a chirp – the frequency sweeps from low to high.
Here's the problem: if we try to analyze this with a regular Fourier transform, we get a mess.
The Fourier transform tells us "this signal contains frequencies from here to here" – which is technically true, but not very helpful. It's like describing a melody by saying "it uses these notes" without mentioning the order or timing.
What we really want to know is: which frequencies happen when?
Enter the Spectrogram
One solution is to look at small windows of time, one after another. For each window, we do a Fourier transform and see what frequencies are present right then.
This is called a spectrogram (or sometimes Short-Time Fourier Transform). Time goes left to right, frequency goes bottom to top, and brightness shows how much of each frequency is present at each moment.
For our chirp, you can see a diagonal line – the frequency rises over time. Much more informative than the blob we got from regular Fourier!
Adjust the window size – smaller windows give better time resolution but blurrier frequencies.
There's a catch though. See that slider? There's always a trade-off:
- Smaller windows → better at locating when something happens, worse at identifying what frequency
- Larger windows → better at identifying frequencies, worse at locating time
This is actually a fundamental limit, related to the Heisenberg Uncertainty Principle in physics! You can't have perfect resolution in both time and frequency at once.
The Chirplet: A Better Basis
Here's where things get interesting. What if instead of matching our signal against constant-frequency sine waves, we matched it against changing-frequency waves?
A chirplet is exactly that: a short burst of sound where the pitch can slide up or down.
Think of it this way:
- A sine wave asks: "How much of this steady pitch is in the signal?"
- A chirplet asks: "How much of this pitch-slide is in the signal, right now?"
Left: a sine wave (constant frequency). Right: a chirplet (sliding frequency).
Chirplet Epicycles
Remember how Fourier analysis uses circles spinning at constant speeds? For chirplets, we use circles that speed up or slow down!
Watch the circle on the right. Instead of spinning at a constant rate, it's accelerating – spinning faster and faster. And it only "exists" for a short time (notice how it fades in and out).
Adjust the "chirp rate" – how fast the frequency changes. Zero gives you a regular Fourier circle.
When the chirp rate is zero, we get back to a regular Fourier component. But when it's positive, the circle speeds up (up-chirp). When it's negative, it slows down (down-chirp).
Matching Chirps
Now here's the real power. When we have a signal with a chirp in it, we can find a chirplet that matches it perfectly.
Try to match the chirplet (orange) to the signal. When they align, the match meter lights up!
When the chirplet's parameters (time, frequency, chirp rate) align with the signal, they "resonate" – the match is strong. When they're different, the match is weak.
Why This Matters
Let's see chirplets in action on a more complex signal. Here we have two chirps crossing each other:
With a regular spectrogram, it's hard to tell what's going on at the crossing point. But chirplets can lock onto each chirp separately because they can match the rate of change of frequency, not just the frequency itself.
Real-World Applications
Chirplets aren't just a mathematical curiosity. They're used in:
Radar and Sonar Bats and dolphins use frequency-modulated chirps for echolocation. Radar systems do the same thing – sending out chirps and listening for echoes. Chirplet analysis helps extract information from these signals.
Gravitational Waves In 2015, LIGO detected gravitational waves for the first time. The signal was a "chirp" – two black holes spiraling into each other, with the frequency rising rapidly until they merged. The famous "chirp" is exactly the kind of signal chirplets were made for!
Music and Speech Many musical instruments produce sounds with changing frequencies. Vibrato, glissando, and pitch bends are all chirps. Speech also contains rapid frequency transitions as we move between vowels and consonants.
The Big Picture
Let's put it all together:
| Transform | What it matches | Best for |
|---|---|---|
| Fourier | Constant-frequency sine waves | Steady tones, periodic signals |
| Wavelet | Localized bursts of frequency | Transient events, multi-scale analysis |
| Chirplet | Sliding-frequency bursts | Signals with changing frequency |
The chirplet transform is a generalization – when the chirp rate is zero, you get back to wavelets and Fourier analysis. But when frequency is changing, chirplets give you a much cleaner picture.
Interactive Playground
Now it's your turn! Draw a signal with chirps and see how Fourier and chirplet analysis compare:
Draw a signal with varying frequencies!
Fourier Analysis
Chirplet Analysis
Conclusion
So let's recap:
- Fourier transforms split signals into constant-frequency components
- But many real signals have changing frequencies – chirps!
- The chirplet transform uses sliding-frequency atoms instead
- Chirplet epicycles are circles that speed up or slow down
- This gives us better analysis of real-world signals like radar, speech, music, and even gravitational waves
The progression from Fourier to wavelets to chirplets is about getting better and better at representing the kinds of signals we actually encounter in nature – where nothing stays perfectly constant.
Further Reading
To learn more about chirplets and time-frequency analysis:
The Chirplet Transform (Wikipedia) The original paper was by Steve Mann at MIT, who coined the term.
Time-Frequency Analysis (Wikipedia) An overview of different ways to analyze signals in both time and frequency.
LIGO Gravitational Waves Learn about how LIGO detected the famous "chirp" from colliding black holes.
Credits
This interactive explanation builds on Jez Swanson's excellent Fourier transform visualization. The chirplet extension was created to help explain how we can analyze signals with changing frequencies.