dB reminders for reading the bars: +6 dB = double the amplitude, +10 dB = roughly twice as loud, -3 dB = half the power. Speed of sound ~343 m/s at 20 C.
How it works
Audio is chopped into a block of samples whose length is a power of 2 (e.g. 4096).
A window (Hann) is multiplied over the block to taper the edges to zero.
FFT splits the block in half again and again, reusing shared sums (the divide-and-conquer trick) instead of testing every frequency separately.
Out comes a magnitude (and phase) value for each frequency bin.
Magnitudes are converted to dB and drawn as the bars/curve on your analyser.
The next overlapping block is grabbed and the whole thing repeats many times per second, giving smooth live movement.
Real examples
The moving spectrum bars in Smaart, REW, or your phone's RTA app are FFTs running ~20-60 times a second.
A digital mixer's graphic EQ display and feedback-finder use an FFT to show where the energy sits.
Auto-tune and pitch correction use FFT to find the singer's fundamental frequency.
Shazam fingerprints songs by FFT-ing tiny slices and storing the peak frequencies.
Noise-cancelling headphones and room-correction DSP analyse the incoming sound with FFTs before reacting.
How it helps in live sound
Ringing out a room: use a small-medium FFT (1024-2048) so the display reacts fast and you can spot feedback frequencies as they build.
Tuning a system's tonal balance: use a larger FFT (4096-16384) for fine low-frequency detail, even though it updates slower.
Bin width = sample rate / FFT size: at 48 kHz, 16384 points gives 2.9 Hz resolution, needed to see problem notes down low.
Use a Hann window with 50% overlap (analyser default) for honest readings; avoid the 'rectangular' window which smears peaks.
Watch Nyquist: at 48 kHz you can only trust the display up to 24 kHz, plenty for full audio.
Mind the latency floor: a 16k FFT at 48 kHz needs 341 ms of audio per frame, so it lags reality; do not chase tiny transient blips with it.
Everyday analogy
Sorting a huge jar of mixed coins: instead of checking every coin one by one, you tip them through a sorting tray that groups them in seconds, same count, far quicker.
Watch out
Myth: the FFT is a different or less accurate transform. Truth: it gives the identical result to the slow DFT, it is only an algorithm shortcut, the trade-offs come from your chosen FFT size and window, not from the FFT itself.
Fun fact
Without the 1965 Cooley-Tukey FFT a real-time analyser would need roughly 100-4000x more computing power, your phone literally could not draw a live spectrum, the maths was simply too slow before the shortcut.
Key takeaways
FFT = a fast algorithm, not a new transform: same answer as the slow DFT.
Cost drops from N squared to N log2 N, roughly 100x faster at N=1024.
Bin width (Hz) = sample rate / FFT size; bigger FFT = finer detail but more lag.
FFT size must be a power of 2; you only get FFT size / 2 useful bins up to Nyquist.
Always window the block first (Hann) to kill spectral leakage.
It is what makes every live spectrum analyser, tuner and feedback-finder possible.