Ehlers 3-Pole Butterworth Filter (BUTTER3)

// Licensed under the Apache License, Version 2.0
// © mihakralj
//@version=6
// Indicator algorithm (C) 2004-2024 John F. Ehlers
indicator("Ehlers 3-Pole Butterworth Filter (BUTTER3)", "BUTTER3", overlay=true)

//@function Calculates 3-Pole Butterworth Lowpass Filter
//@param source Series to calculate Butterworth from
//@param length Cutoff period (related to -3dB frequency)
//@returns 3-pole Butterworth filter value
//@optimized Uses 3-pole IIR Butterworth filter with O(1) complexity per bar
butter3(series float src, simple int length) =>
    var float SQRT3_PI = math.sqrt(3.0) * math.pi
    var float butter3_internal = 0.0
    var float coef1 = 0.0
    var float coef2 = 0.0
    var float coef3 = 0.0
    var float coef4 = 0.0
    var int prev_length = 0
    if prev_length != length
        int p = math.max(2, length)
        float a1 = math.exp(-math.pi / p)
        float b1 = 2.0 * a1 * math.cos(SQRT3_PI / p)
        float c1 = a1 * a1
        coef2 := b1 + c1
        coef3 := -(c1 + b1 * c1)
        coef4 := c1 * c1
        coef1 := (1.0 - b1 + c1) * (1.0 - c1) / 8.0
        prev_length := p
    float ssrc = nz(src, src[1])
    float src1 = nz(src[1], ssrc)
    float src2 = nz(src[2], src1)
    float src3 = nz(src[3], src2)
    float filt1 = nz(butter3_internal[1], src1)
    float filt2 = nz(butter3_internal[2], src2)
    float filt3 = nz(butter3_internal[3], src3)
    if bar_index < 4
        butter3_internal := ssrc
    else
        butter3_internal := coef1 * (ssrc + 3.0 * src1 + 3.0 * src2 + src3) + coef2 * filt1 + coef3 * filt2 + coef4 * filt3
    butter3_internal

// ---------- Main loop ----------

// Inputs
i_length = input.int(20, "Length", minval=2)
i_source = input.source(close, "Source")

// Calculation
butter3_val = butter3(i_source, i_length)

// Plot
plot(butter3_val, "Butterworth 3-Pole", color=color.yellow, linewidth=2)