Audio Math Explainer

The rule: DSP math is small and composable — each concept builds directly on the last. Start with what the user needs, then connect it to code they're writing.

Step 1 — Identify the concept

Topic	Core idea	Common "why"
Fourier transform / DFT / FFT	Decompose signal into frequency components	"Why does my FFT output look wrong?"
Convolution	Weighted moving average; LTI system response	"How does an IR reverb work?"
Z-transform	Frequency-domain analysis of discrete systems	"How do filter poles/zeros work?"
Biquad / IIR filters	Recursive difference equation with feedback	"Why does my filter ring / go unstable?"
FIR filters	Non-recursive weighted sum	"How do I design a linear-phase filter?"
Windowing	Reduce spectral leakage in DFT	"Why are FFT edges smeared?"
Sample rate / Nyquist	Highest representable frequency = sr/2	"What causes aliasing?"
Decibels	Log scale for amplitude/power ratios	"How do I convert gain to dB?"

If the user's question maps to multiple rows, start with the most fundamental one and build up.

Step 2 — Explain with three anchors

For each concept, give exactly these three things in order:

1. Intuition — one sentence a musician could understand
2. Math — the key formula (inline, no walls of derivation)
3. Code — one-line pseudocode or a real function call from common audio libraries

Keep each anchor to 1–3 lines. Cut anything that doesn't directly answer the question.

Step 3 — Connect to the user's code

After the three anchors:

• Locate where the math appears in their implementation (coefficient calculation, loop structure, buffer size choice, etc.)
• Name the exact variable or line where theory becomes code
• If a value looks wrong, explain which part of the math it violates

Quick connection table

Math concept	Where it shows up in code
DFT basis frequencies	bin_index * sr / N — each FFT bin
Convolution sum	FIR tap loop: y[n] = sum(h[k] * x[n-k])
Z-plane pole radius	IIR feedback coefficients a1, a2
Nyquist limit	if (freq > sampleRate / 2) clamp(...)
Window function	Multiply x[n] *= window[n] before FFT
dB conversion	gain_db = 20 * log10(amplitude)

Output format

### [Concept name]

**Intuition:** [one sentence]

**Math:** `[formula]`

**Code:** `[one-liner]`

**In your code:** [where this appears and what it means for their specific question]

See references/dsp-math-reference.md for full formula tables, gotchas, and property lists.