AMFM - Ehlers AM Detector / FM Demodulator

// AMFM: Ehlers AM Detector / FM Demodulator
// John F. Ehlers — "A Technical Description of Market Data for Traders"
// TASC, May 2021 (AM + FM) / June 2021 (strategy application)
// Source: https://www.mesasoftware.com/papers/AMFM.pdf
//
// Decomposes price movement into amplitude (AM) and frequency (FM) components
// using digital signal processing techniques from radio engineering.
//
// AM Detector: extracts volatility envelope from whitened spectrum
//   Deriv = Close - Open  (whitening — removes DC, handles gaps)
//   Envel = Highest(|Deriv|, 4)  (envelope detection)
//   AM = SMA(Envel, 8)  (smoothed volatility)
//
// FM Demodulator: extracts timing/phase from whitened spectrum
//   Deriv = Close - Open
//   HL = clamp(10 * Deriv, -1, +1)  (hard limiter — strips amplitude)
//   FM = SuperSmoother(HL, Period)  (integration via 2-pole Butterworth IIR)

//@version=6
indicator("AMFM - Ehlers AM Detector / FM Demodulator", shorttitle="AMFM",
     overlay=false, precision=4)

// ── Inputs ────────────────────────────────────────────────────────────
int period = input.int(30, "FM Super Smoother Period", minval=1)

// ── Common: Whitened derivative ───────────────────────────────────────
float deriv = close - open

// ── AM Detector ──────────────────────────────────────────────────────
float envel = ta.highest(math.abs(deriv), 4)
float am    = ta.sma(envel, 8)

// ── FM Demodulator ───────────────────────────────────────────────────
// Hard limiter: 10x gain then clamp to ±1 strips all amplitude info
float hl = math.max(-1.0, math.min(1.0, 10.0 * deriv))

// Super Smoother (2-pole Butterworth IIR) — integrates the hard-limited signal
float a1 = math.exp(-1.414 * math.pi / period)
float b1 = 2.0 * a1 * math.cos(1.414 * math.pi / period)
float c2 = b1
float c3 = -(a1 * a1)
float c1 = 1.0 - c2 - c3

var float fm = 0.0
fm := bar_index < 2 ? deriv : c1 * (hl + nz(hl[1])) / 2.0 + c2 * nz(fm[1]) + c3 * nz(fm[2])

// ── Plots ────────────────────────────────────────────────────────────
plot(am, "AM", color=color.new(color.orange, 0), linewidth=2)
plot(fm, "FM", color=color.new(color.aqua, 0), linewidth=2)
hline(0, "Zero", color=color.gray, linestyle=hline.style_dotted)