QuanTAlib/lib/cycles/ht_sine/HtSine.md

HT_SINE: Ehlers Hilbert Transform SineWave (also known as SINE)

Property	Value
Category	Cycle
Inputs	Source (close)
Parameters	None
Outputs	Single series (HT_SINE)
Output range	Varies (see docs)
Warmup	LOOKBACK bars

TL;DR

• HT_SINE extracts the dominant market cycle phase and outputs both Sine and LeadSine (45° phase advance) for cycle timing.
• No configurable parameters; computation is stateless per bar.
• Output range: Varies (see docs).
• Requires LOOKBACK bars of warmup before first valid output (IsHot = true).
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

HT_SINE extracts the dominant market cycle phase and outputs both Sine and LeadSine (45° phase advance) for cycle timing. The crossover of these two waves identifies turning points in ranging markets up to one-eighth of a cycle early. Compatible with TA-Lib's HT_SINE function, the indicator builds on the full Hilbert Transform cascade (phasor extraction, homodyne period estimation, DFT phase accumulation) to produce dual bounded [-1, +1] oscillators that track cycle position rather than price amplitude.

Historical Context

John Ehlers introduced the Hilbert Transform SineWave in Rocket Science for Traders (2001) as part of his signal processing framework for financial markets. Traditional oscillators (RSI, Stochastic) respond to price amplitude, inherently lagging reversals. HT_SINE measures cycle phase directly, theoretically providing zero-lag detection of cycle turning points. The LeadSine output advances the phase by 45°, creating a built-in early warning system: when LeadSine diverges from Sine, a reversal is approaching. The dual-line design provides both confirmation (crossover) and anticipation (LeadSine leading). The indicator is most effective in ranging markets with well-defined cycles; in strong trends, the two lines travel in parallel ("snake pattern"), correctly indicating that no cyclical reversal is imminent.

Architecture & Physics

1. Hilbert Transform Cascade

The full TA-Lib Hilbert pipeline: 4-bar WMA smoothing, Hilbert FIR with coefficients A = 0.0962, B = 0.5769, phasor extraction (I_2, Q_2), EMA smoothing (\alpha = 0.2).

2. Homodyne Period Estimation

Re_t = 0.2(I_{2,t} \cdot I_{2,t-1} + Q_{2,t} \cdot Q_{2,t-1}) + 0.8 \cdot Re_{t-1} Im_t = 0.2(I_{2,t} \cdot Q_{2,t-1} - Q_{2,t} \cdot I_{2,t-1}) + 0.8 \cdot Im_{t-1} Period = \frac{2\pi}{\arctan(Im / Re)}

Clamped to [6, 50], then smoothed (\alpha = 0.33).

3. DC Phase via DFT Accumulation

Over the smoothed period P:

RealPart = \sum_{i=0}^{P-1} \sin\!\left(\frac{2\pi i}{P}\right) \cdot SmoothPrice_{t-i} ImagPart = \sum_{i=0}^{P-1} \cos\!\left(\frac{2\pi i}{P}\right) \cdot SmoothPrice_{t-i} \phi_t = \arctan\!\left(\frac{RealPart}{ImagPart}\right)

With quadrant correction and phase unwrapping.

4. Output Generation

Sine_t = \sin(\phi_t) LeadSine_t = \sin(\phi_t + 45°)

5. Complexity

O(P) per bar where P is the smoothed period (typically 6-50), due to the DFT accumulation loop. Fixed-size circular buffers (50 + 44 + 64 elements) give O(1) space. Warmup: 63 bars (31 + 32 for TA-Lib compatibility).

Mathematical Foundation

Parameters

Parameter	Description	Default	Constraint
(none)	No user-configurable parameters

All constants are fixed by the TA-Lib specification.

Pseudo-code

function HT_SINE(source):
    // Full Hilbert cascade (same as HT_DCPHASE)
    // Produces: smoothPeriod, smoothPriceBuf, dcPhase

    for each bar (after warmup):
        // Phase from DFT accumulation (see HT_DCPHASE)
        φ ← computeDCPhase(smoothPeriod, smoothPriceBuf)

        // Convert phase to radians
        φ_rad ← φ · (π / 180)

        // Dual sine output
        sine     ← sin(φ_rad)
        leadSine ← sin(φ_rad + π/4)    // 45° lead

        emit sine, leadSine

Crossover Signals

Pattern	Signal
Sine crosses above LeadSine	Bullish: cycle turning up from trough
Sine crosses below LeadSine	Bearish: cycle turning down from peak
Lines parallel, both rising	Uptrend in progress (not cycling)
Lines parallel, both falling	Downtrend in progress (not cycling)
LeadSine diverges first	Early warning of approaching reversal

Output Interpretation

Output	Range	Meaning
Sine	[-1, +1]	Current cycle phase position
LeadSine	[-1, +1]	45° advanced cycle phase (early warning)

Performance Profile

Operation Count (Streaming Mode)

Operation	Count per bar	Notes
Hilbert cascade (WMA + 4×FIR + phasor + homodyne)	~84	Same pipeline as HT_DCPERIOD
DFT sin/cos accumulation	~4P	P sin + P cos evaluations + 2P FMA
Phase ATAN extraction	~15	Math.Atan transcendental
Phase adjustment + unwrapping	~5	Quadrant correction + wrapping
Final SIN (sine)	~15	Math.Sin transcendental
Final SIN (leadSine)	~15	Math.Sin(φ + π/4) transcendental
Total (P=20 typical)	~214	O(P) dominated by DFT + 3 transcendentals
Total (P=50 worst case)	~454	Heaviest of the HT family

Batch Mode (SIMD Analysis)

Aspect	Assessment
SIMD vectorizable	Partially: DFT inner loop vectorizable; final sin calls are scalar
Bottleneck	DFT loop (P sin/cos calls) + 3 final transcendentals per bar
Parallelism	DFT accumulation independent; dual sin output trivially parallel
Memory	O(P): ~50-element smooth price buffer + ~44-element det buffer + Hilbert state (~1.3 KB)
Throughput	Slowest HT variant; ~2.5× HT_DCPHASE due to extra sin evaluations

Resources

• Ehlers, J.F. Rocket Science for Traders. Wiley, 2001.
• TA-Lib TA_HT_SINE() reference implementation.
• Ehlers, J.F. Cybernetic Analysis for Stocks and Futures. Wiley, 2004.
• Hilbert, D. Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Teubner, 1912.