QuanTAlib/lib/cycles/ht_phasor/HtPhasor.md

HT_PHASOR: Ehlers Hilbert Transform Phasor Components

Phasor components decompose price into in-phase and quadrature parts, mapping the cycle as a rotating vector.

Property	Value
Category	Cycle
Inputs	Source (close)
Parameters	None
Outputs	Single series (HT_PHASOR)
Output range	Varies (see docs)
Warmup	LOOKBACK bars
PineScript	phasor.pine

• HT_PHASOR decomposes the price signal into two orthogonal components, InPhase (I) and Quadrature (Q), using the Hilbert Transform.
• 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_PHASOR decomposes the price signal into two orthogonal components, InPhase (I) and Quadrature (Q), using the Hilbert Transform. Together these form a complex phasor Z = I + jQ that describes the instantaneous amplitude and phase of the dominant market cycle. Compatible with TA-Lib's HT_PHASOR function, this dual-output indicator provides the fundamental building blocks for cycle analysis, phasor crossover timing, and instantaneous amplitude measurement.

Historical Context

John Ehlers introduced phasor decomposition of market data in Rocket Science for Traders (2001). In electrical engineering, a phasor represents a sinusoidal signal as a rotating complex vector, separating the cycle's "position" (InPhase) from its "velocity" (Quadrature). Ehlers recognized that this decomposition is the mathematical foundation for all his cycle indicators: HT_SINE, HT_DCPERIOD, HT_DCPHASE, and HOMOD all derive from these same I/Q components. TA-Lib exposes HT_PHASOR to give advanced users direct access to the analytic signal for custom cycle analysis. The InPhase output is delayed by 3 bars to align with the Quadrature component's effective lag from the Hilbert Transform FIR.

Architecture & Physics

1. WMA Smoothing

SmoothPrice_t = \frac{4P_t + 3P_{t-1} + 2P_{t-2} + P_{t-3}}{10}

2. Hilbert Transform FIR

Using Ehlers' coefficients (A = 0.0962, B = 0.5769), the 4-tap discrete Hilbert approximation generates the detrender, and from it the fundamental In-Phase and Quadrature components (I_1, Q_1). Further Hilbert transforms of these produce jI and jQ.

3. Phasor Components

I_{2,t} = I_{1,t} - jQ_t, \qquad Q_{2,t} = Q_{1,t} + jI_t

Both smoothed with EMA (\alpha = 0.2):

I_t = 0.2 \cdot I_{2,t} + 0.8 \cdot I_{t-1} Q_t = 0.2 \cdot Q_{2,t} + 0.8 \cdot Q_{t-1}

4. Phase Relationship

Q leads I by 90°. When I peaks, Q crosses zero downward. When I crosses zero upward, Q peaks. The instantaneous amplitude is A = \sqrt{I^2 + Q^2} and the instantaneous phase is \phi = \arctan(Q/I).

5. Complexity

O(1) per bar. Fixed Hilbert cascade with circular buffers. Warmup: 32 bars (TA-Lib lookback).

Mathematical Foundation

Parameters

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

Phasor Crossover Signals

Condition	Signal
Q crosses I from below	Bullish (anticipates cycle trough)
Q crosses I from above	Bearish (anticipates cycle peak)
\sqrt{I^2 + Q^2} increasing	Cycle amplitude growing
\sqrt{I^2 + Q^2} decreasing	Cycle amplitude fading (trend or noise)

Output Interpretation

Output	Range	Meaning
InPhase	unbounded	Cycle component aligned with price
Quadrature	unbounded	Rate of change (velocity) of cycle

Performance Profile

Operation Count (Streaming Mode)

Operation	Count per bar	Notes
4-bar WMA	~5	3 MUL + 1 ADD + 1 MUL(×0.1)
Hilbert FIR (detrender)	~7	4-tap FIR: 4 MUL + 3 ADD
Hilbert FIR (Q1)	~7	Same 4-tap structure on det buffer
Hilbert FIR (jI)	~7	4-tap on I1 history
Hilbert FIR (jQ)	~7	4-tap on Q1 history
Phasor EMA (I2, Q2)	~8	2 SUB/ADD + 4 FMA
Buffer management	~10	4 circular buffer writes + index arithmetic
Total	~51	O(1) fixed; no transcendentals (no period/phase extraction)

Batch Mode (SIMD Analysis)

Aspect	Assessment
SIMD vectorizable	No: cascaded IIR EMA smoothing creates sequential dependencies
Bottleneck	Circular buffer indexed lookups for 4 Hilbert FIR passes
Parallelism	None: each bar's phasor depends on previous bar's EMA state
Memory	O(1): 4 circular buffers (7 elements each) + 2 scalar EMA states (~240 bytes)
Throughput	Fastest of the HT family; no transcendental calls (no ATAN/SIN/COS)

Resources

• Ehlers, J.F. Rocket Science for Traders. Wiley, 2001.
• TA-Lib TA_HT_PHASOR() reference implementation.
• Ehlers, J.F. Cybernetic Analysis for Stocks and Futures. Wiley, 2004.