QuanTAlib/lib/cycles/sine/Sine.md

SINE: Ehlers Sine Wave

SINE extracts the dominant cycle from price data using cascaded signal processing: a high-pass filter removes the trend, a Super-Smoother filter removes noise, and a Hilbert Transform FIR decomposes the filtered signal into In-Phase and Quadrature components for power-normalized sine wave output. The result oscillates between -1 and +1, representing the normalized position within the current cycle. Unlike HT_SINE which derives phase from the full TA-Lib Hilbert cascade, this Ehlers implementation uses explicit detrending and bandpass stages for cleaner cycle isolation.

Historical Context

John Ehlers introduced the Sine Wave indicator in Cybernetic Analysis for Stocks and Futures (2004) as a refined approach to cycle extraction. The design philosophy separates three signal processing concerns into distinct filter stages: (1) trend removal via high-pass filtering sets the long-wavelength cutoff, (2) aliasing prevention via Super-Smoother sets the short-wavelength cutoff, and (3) cycle extraction via Hilbert Transform generates the quadrature decomposition. This staged approach produces cleaner output than attempting all three simultaneously (as in the HT_SINE). The Sine Wave output at extremes (\pm 1) indicates the cyclical component is stretched and likely to revert, while zero crossings indicate phase transitions. The indicator is particularly valuable for mean-reversion strategies in ranging markets.

Architecture & Physics

1. High-Pass Filter (Detrending)

A single-pole high-pass filter removes low-frequency trends below the cutoff:

\alpha_{HP} = \frac{1 - \sin(2\pi / P_{HP})}{\cos(2\pi / P_{HP})} HP_t = \frac{1 + \alpha_{HP}}{2}(P_t - P_{t-1}) + \alpha_{HP} \cdot HP_{t-1}

2. Super-Smoother Filter (Noise Removal)

A 2-pole Butterworth low-pass removes high-frequency noise:

a = e^{-\sqrt{2}\pi / P_{SSF}} b = 2a \cos(\sqrt{2}\pi / P_{SSF}) c_1 = 1 - b + a^2, \quad c_2 = b, \quad c_3 = -a^2 Filt_t = \frac{c_1}{2}(HP_t + HP_{t-1}) + c_2 \cdot Filt_{t-1} + c_3 \cdot Filt_{t-2}

3. Hilbert Transform FIR

Discrete Hilbert approximation extracts quadrature component:

Q_t = 0.0962 \cdot Filt_{t-3} + 0.5769 \cdot Filt_{t-1} - 0.5769 \cdot Filt_{t-5} - 0.0962 \cdot Filt_{t-7} I_t = Filt_t

4. Power Normalization

Power_t = I_t^2 + Q_t^2 Sine_t = \frac{I_t}{\sqrt{Power_t}}

When Power \approx 0, output is zero.

5. Complexity

O(1) per bar. Fixed filter stages with ring buffers of 2 (source) + 2 (HP) + 8 (filtered) = 12 elements. Warmup: \max(P_{HP}, P_{SSF}) + 8 bars.

Mathematical Foundation

Parameters

Parameter	Description	Default	Constraint
hpPeriod	High-pass filter cutoff period	40	\geq 1
ssfPeriod	Super-smoother filter period	10	\geq 1

Tuning Relationship

Typically P_{SSF} \approx P_{HP} / 4 to P_{HP} / 2. The high-pass defines the trend/cycle boundary; the super-smoother defines the noise/cycle boundary. Together they create a bandpass that isolates the frequency range of interest.

Pseudo-code

function SINE(source, hpPeriod, ssfPeriod):
    // Precompute HP coefficient
    α_hp ← (1 - sin(2π/hpPeriod)) / cos(2π/hpPeriod)

    // Precompute SSF coefficients
    a ← exp(-√2·π / ssfPeriod)
    b ← 2·a·cos(√2·π / ssfPeriod)
    c₁ ← (1 - b + a²) / 2

    hp_prev ← 0; p_prev ← 0
    filt_1 ← 0; filt_2 ← 0
    filtBuf ← CircularBuffer(8)

    for each price in source:
        // High-pass
        hp ← 0.5·(1 + α_hp)·(price - p_prev) + α_hp·hp_prev

        // Super-smoother
        filt ← c₁·(hp + hp_prev) + b·filt_1 - a²·filt_2

        // Hilbert FIR quadrature
        filtBuf.Add(filt)
        Q ← 0.0962·filtBuf[3] + 0.5769·filtBuf[1]
           - 0.5769·filtBuf[5] - 0.0962·filtBuf[7]
        I ← filt

        // Power normalization
        power ← I² + Q²
        sine ← (power > 0) ? I / √power : 0

        // Shift state
        hp_prev ← hp; p_prev ← price
        filt_2 ← filt_1; filt_1 ← filt

        emit sine

SINE vs HT_SINE

Aspect	SINE	HT_SINE
Detrending	Explicit high-pass filter	Implicit in Hilbert cascade
Noise removal	Explicit Super-Smoother	4-bar WMA only
Period tuning	User-configurable (hpPeriod, ssfPeriod)	Fixed (TA-Lib spec)
Output	Single (Sine only)	Dual (Sine + LeadSine)
Phase source	I/Q power normalization	DFT phase accumulation

Output Interpretation

Condition	Meaning
Sine \approx +1	Cycle peak (potential short / mean-reversion)
Sine \approx -1	Cycle trough (potential long / mean-reversion)
Zero crossing up	Bullish phase transition
Zero crossing down	Bearish phase transition
Erratic output	Strong trend overwhelming cycle extraction

Performance Profile

Operation Count (Streaming Mode)

Operation	Count per bar	Notes
High-pass filter	~4	1 SUB + 1 MUL + 1 FMA
Super-Smoother (2-pole IIR)	~5	1 ADD + 2 FMA + 1 MUL
Hilbert FIR (quadrature)	~7	4-tap FIR: 4 MUL + 3 ADD
I² + Q² (power)	~3	2 MUL + 1 ADD
SQRT + normalization	~4	1 SQRT + 1 DIV + 1 branch
Buffer management	~3	1 circular buffer write + index update
State shift	~4	4 register moves
Total	~30	O(1) fixed; single SQRT is only transcendental

Batch Mode (SIMD Analysis)

Aspect	Assessment
SIMD vectorizable	No: HP and SSF are recursive IIR with sequential state dependencies
Bottleneck	Math.Sqrt in power normalization (~15 cycles); rest is pure arithmetic
Parallelism	None: each bar's HP/SSF output depends on previous bar
Memory	O(1): 8-element ring buffer + 4 scalar state variables (~96 bytes)
Throughput	Very fast; slightly faster than EBSW (no 3-bar averaging, no clamp)

Resources

• Ehlers, J.F. Cybernetic Analysis for Stocks and Futures. Wiley, 2004.
• Ehlers, J.F. Cycle Analytics for Traders. Wiley, 2013.