3. Signal Processing (Continuous to Discrete) · Concept 1 of 11
It is a way of taking any sound and listing out all the separate pitches (frequencies) that are hidden inside it.
A tangled chord (time) goes in, the Fourier Transform lists out each hidden pitch and its level (frequency) -- those are your EQ's dancing bars.
What it is
A maths tool that splits any sound into the list of pure frequencies hidden inside it, and how loud each one is.
Key facts
- Every sound = a sum of pure sine waves (Fourier's theorem, Joseph Fourier, 1822)
- Output two numbers per frequency: magnitude (how loud) + phase (timing, 0 to 360 degrees)
- Human hearing range: 20 Hz to 20,000 Hz (20 kHz)
- Nyquist rule: sample rate must be at least 2x the top frequency, so 44,100 Hz audio captures up to 22,050 Hz
- FFT (Fast Fourier Transform, Cooley-Tukey 1965) cuts the maths from N-squared to N x log2(N) operations
- FFT size is a power of 2: 512, 1024, 2048 or 4096 samples; an N-sample FFT gives N/2 usable bins
- Frequency resolution = sample rate / FFT size; e.g. 48,000 / 4096 = 11.7 Hz per bin
- Trade-off: bigger FFT = finer frequency detail but slower and blurrier in time
- +6 dB = double amplitude (voltage); +10 dB = roughly double perceived loudness; -3 dB = half power
- A guitar 'A' = 110 Hz fundamental plus harmonics at 220, 330, 440 Hz; the FT reveals that whole stack
How it works
- Feed in a slice of the waveform (amplitude changing over time)
- The transform tests it against many pure sine waves, one frequency at a time
- It measures how strongly each test frequency matches the input
- It outputs a magnitude (level) and phase for every frequency bin
- Plot magnitude vs frequency and you get the spectrum, those dancing bars
- FFT does this fast enough to update the display live, many times a second
Real examples
- Graphic/parametric EQ: the spectrum on screen is an FFT running constantly
- Smaart, REW and RTA apps use FFT to show your PA's frequency response
- Feedback hunting: the spike on the analyser is the ringing frequency to notch out
- Auto-tune and pitch correction detect the note via frequency analysis
- MP3/AAC compression uses FFT-style transforms to bin and discard inaudible frequencies
How it helps in live sound
- Run an RTA (Smaart or a phone app) with pink noise to flatten a room's response
- Watch the spectrum for a sharp narrow spike = feedback; notch it with a tight EQ band
- Use a larger FFT (4096+) to resolve low-end problems below 100 Hz accurately
- Use a smaller FFT for snappy, responsive metering on transient sources like drums
- Log frequency axis: bass hard left, treble hard right, vocals 1-4 kHz in the middle
- Read the analyser in dB, not 'bars'; a 6 dB bump is a real audible problem
Everyday analogy
Like a barista tasting a smoothie and instantly writing down exactly which fruits went in and how much of each.
Watch out
Myth: the spectrum shows the actual notes playing. Truth: it shows every frequency including harmonics and noise, so one note lights up many bars.
Fun fact
The Fast Fourier Transform was made famous in 1965, but Carl Friedrich Gauss had already worked out the same trick around 1805.
Key takeaways
- Any sound is just lots of pure sine tones added together
- Fourier Transform pulls that mix apart into a frequency list
- FFT is the fast computer version that runs your spectrum displays
- Magnitude = how loud each frequency; phase = its timing
- Bigger FFT = sharper frequency, fuzzier timing (and vice versa)
- Every EQ curve, RTA and feedback finder is FFT at work