Ehlers Universal Oscillator (UO)

// Licensed under the Apache License, Version 2.0
// © mihakralj
//@version=6
indicator("Ehlers Universal Oscillator (UO)", "UO", overlay = false)

//@function Ehlers Universal Oscillator — white noise extraction via 2-bar differencing,
//          smoothed by a 2-pole Super Smoother filter, then normalized by AGC peak tracking.
//          Pipeline: WhiteNoise = (Close - Close[2]) / 2 → SuperSmoother(BandEdge) → AGC.
//          Output is bounded [-1, +1].
//@param source Series to analyze
//@param bandEdge Super Smoother cutoff period (>= 2)
//@returns UO oscillator value (bounded ±1)
//@reference Ehlers, J.F. (2015). "Relax, It's Just Noise — Whiter Is Brighter."
//           Technical Analysis of Stocks & Commodities, Jan 2015.
//@optimized O(1) per bar — no buffers needed, pure scalar state
uo(series float source, simple int bandEdge) =>
    if bandEdge < 2
        runtime.error("BandEdge must be at least 2")

    float price = nz(source)

    // --- White noise extraction ---
    float wn = (price - nz(source[2])) / 2.0

    // --- Super Smoother coefficients (2-pole Butterworth at BandEdge) ---
    float a1 = math.exp(-1.414 * math.pi / bandEdge)
    float b1 = 2.0 * a1 * math.cos(1.414 * 180.0 / bandEdge)
    float c2 = b1
    float c3 = -(a1 * a1)
    float c1 = 1.0 - c2 - c3

    // --- 2-pole Super Smoother filter ---
    var float filt  = 0.0
    var float filt1 = 0.0
    var float filt2 = 0.0
    float wn1 = nz(wn[1])
    filt2 := filt1
    filt1 := filt
    filt := c1 * 0.5 * (wn + wn1) + c2 * filt1 + c3 * filt2

    // --- AGC peak tracking ---
    var float peak = 0.0
    peak := 0.991 * peak
    if math.abs(filt) > peak
        peak := math.abs(filt)

    // --- Normalize ---
    float universal = peak > 1e-10 ? filt / peak : 0.0
    universal

// ── Inputs ──
int   p_bandEdge = input.int(20, "BandEdge", minval = 2)
float p_src      = input.source(close, "Source")

// ── Calculation ──
float out = uo(p_src, p_bandEdge)

// ── Plot ──
plot(out, "UO", color.yellow, 2)
hline(0, "Zero", color.gray)
hline(0.5, "+0.5", color.new(color.red, 60))
hline(-0.5, "-0.5", color.new(color.green, 60))