Before this:The Fourier transform
The FFT in practice
Key takeaways The FFT (Fast Fourier Transform) is a fast algorithm for the DFT, quick enough to run live on a radio stream. It splits the bandwidth into bins — more bins mean finer frequency resolution, but each FFT then spans more time. A waterfall is just many FFTs stacked over time. The FFT is how software radio sees the spectrum.
The Fourier transform is the idea; the FFT is how you actually compute it fast enough to matter. This lesson covers the practical knobs and what they trade off.
Fast enough to be useful
A direct DFT of N samples costs on the order of N² operations — hopeless for the millions of samples a second an SDR produces. The FFT computes the identical result in about N·log N operations. For a 4,096-point transform that’s the difference between ~16 million operations and ~50 thousand — the leap that makes real-time spectrum analysis possible on ordinary hardware.
Bins and resolution
An FFT of size N divides the sampled bandwidth into N bins of equal width:
bin width = sample rate / FFT size
At 2.4 MS/s with a 4,096-point FFT, each bin is about 586 Hz wide. Want to resolve signals closer together? Use more bins. But there’s a catch built into physics.
The time–frequency tradeoff
To get finer frequency resolution you need a bigger FFT, and a bigger FFT needs more samples, which cover a longer stretch of time. So you can know precisely what frequency or precisely when — but not both at once:
| FFT size (at 2.4 MS/s) | Bin width | Time span |
|---|---|---|
| 1,024 | ~2.3 kHz | ~0.43 ms |
| 4,096 | ~586 Hz | ~1.7 ms |
| 16,384 | ~146 Hz | ~6.8 ms |
Fine frequency detail costs time resolution, and vice versa. Choosing the FFT size is choosing where on that trade you want to sit — a short FFT to catch brief bursts, a long one to separate close carriers.
The waterfall
Compute an FFT, draw its bins as a row of colours (bright = strong), drop down a line, compute the next FFT, and repeat. That scrolling picture is a waterfall — time running down the screen, frequency across it.
The RF path’s FFT & waterfall lesson looks at reading these displays; here the point is that the picture is literally a stack of FFTs, and its sharpness in frequency versus time is the tradeoff above. Windowing — the next lesson — cleans up how each FFT row looks.
Quick check: you double the FFT size at a fixed sample rate. What happens?
Recap
- The FFT computes the DFT in ~N·log N operations — fast enough for live radio.
- It splits bandwidth into bins; bin width = sample rate ÷ FFT size.
- Finer frequency resolution needs a bigger FFT, which spans more time — a fundamental tradeoff.
- A waterfall is many FFTs stacked over time — how SDR software shows the spectrum.
Next up: why real FFTs smear energy, and how window functions fix it.
Frequently asked questions
What is an FFT bin?
An FFT splits the sampled bandwidth into a fixed number of equal-width slots called bins, each reporting the energy in a small range of frequencies. The number of bins equals the FFT size. More bins over the same bandwidth means finer frequency resolution — but each FFT then needs more samples, which means it covers a longer slice of time.
What is a waterfall display?
A waterfall stacks many FFTs computed over successive slices of time, drawing each as a horizontal line where colour or brightness shows energy per frequency bin, and scrolling downward. It lets you watch signals appear, disappear, and move across the spectrum — the standard way SDR software visualizes activity.