QuanTAlib/lib/cycles/ht_phasor/HtPhasor.md

HT_PHASOR: Ehlers Hilbert Transform Phasor Components

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

Pseudo-code

function HT_PHASOR(source):
    A ← 0.0962; B ← 0.5769
    smoothBuf ← CircularBuffer(7)
    detBuf, q1Buf, i1Buf ← CircularBuffers

    I2 ← 0; Q2 ← 0

    for each price in source:
        // WMA smooth
        smooth ← (4·price + 3·p[1] + 2·p[2] + p[3]) / 10
        smoothBuf.Add(smooth)

        // Hilbert FIR (adaptive)
        det ← A·smooth[0] + B·smooth[2] - B·smooth[4] - A·smooth[6]
        Q1 ← A·det[0] + B·det[2] - B·det[4] - A·det[6]
        I1 ← det[3]

        // Hilbert of I1 and Q1
        jI ← A·I1[0] + B·I1[2] - B·I1[4] - A·I1[6]
        jQ ← A·Q1[0] + B·Q1[2] - B·Q1[4] - A·Q1[6]

        // Phasor components (EMA smoothed)
        I2 ← 0.2·(I1 - jQ) + 0.8·I2
        Q2 ← 0.2·(Q1 + jI) + 0.8·Q2

        emit InPhase = I2, Quadrature = Q2

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

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.