QuanTAlib/lib/cycles/homod/homod.md

HOMOD: Ehlers Homodyne Discriminator

The homodyne discriminator locks onto a cycle's frequency by comparing successive analytic signal rotations.

Property	Value
Category	Cycle
Inputs	Source (close)
Parameters	minPeriod (default 6.0), maxPeriod (default 50.0)
Outputs	Single series (Homod)
Output range	Varies (see docs)
Warmup	maxPeriod * 2 bars (default 100)
PineScript	homod.pine

• HOMOD estimates the dominant cycle period of a market using homodyne mixing, a technique from radio engineering where a signal is multiplied by a d...
• Similar: HT_Phasor, HT_Sine | Complementary: Cycle period for context | Trading note: Homodyne discriminator; Ehlers' phase detection for cycle turning points.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

HOMOD estimates the dominant cycle period of a market using homodyne mixing, a technique from radio engineering where a signal is multiplied by a delayed copy of itself to expose the angular phase change between samples. The output is a continuously varying period measurement (in bars) that tracks the market's instantaneous cycle length, enabling adaptive indicator tuning. Developed by John Ehlers, it offers better noise rejection and stability than the raw Hilbert Transform period estimator.

Historical Context

John Ehlers introduced the Homodyne Discriminator in Rocket Science for Traders (2001) and refined it in Cybernetic Analysis for Stocks and Futures (2004). In RF engineering, homodyne detection multiplies a signal with a local oscillator at the same frequency to extract phase information. Ehlers adapted this by multiplying the complex analytic signal z_t = I_t + jQ_t by its own conjugate delayed by one bar, yielding the phase rotation per sample. The angular velocity directly encodes the instantaneous frequency (and hence period). Compared to the raw Hilbert Transform discriminator which estimates phase absolutely, homodyne detection measures phase differences, making it less sensitive to amplitude variations and more stable during noisy market conditions.

Architecture & Physics

1. Pre-Processing (4-Bar WMA)

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

2. Analytic Signal Generation

The Hilbert Transform FIR generates quadrature components using Ehlers' 4-tap approximation with coefficients A = 0.0962 and B = 0.5769:

Detrender_t = A \cdot Smooth_t + B \cdot Smooth_{t-2} - B \cdot Smooth_{t-4} - A \cdot Smooth_{t-6}

In-Phase (I_1) is the detrender delayed by 3 bars. Quadrature (Q_1) is the Hilbert transform of the detrender. Both are further refined:

I_2 = I_1 - jQ, \qquad Q_2 = Q_1 + jI

Smoothed with EMA (\alpha = 0.2).

3. Homodyne Mixing

Multiplying the complex signal by its one-bar-delayed conjugate:

Re_t = I_{2,t} \cdot I_{2,t-1} + Q_{2,t} \cdot Q_{2,t-1} Im_t = I_{2,t} \cdot Q_{2,t-1} - Q_{2,t} \cdot I_{2,t-1}

Both smoothed with EMA (\alpha = 0.2).

4. Period Extraction

\theta = \operatorname{atan2}(Im_t, Re_t) Period_{raw} = \frac{2\pi}{\theta}

Clamped to [MinPeriod, MaxPeriod] and smoothed with EMA (\alpha = 0.33).

5. Complexity

O(1) per bar with O(1) memory. The pipeline consists entirely of fixed-depth IIR filters and short delay lines.

Mathematical Foundation

Parameters

Parameter	Description	Default	Constraint
minPeriod	Minimum detectable period	6.0	> 0
maxPeriod	Maximum detectable period	50.0	> minPeriod

Output Interpretation

Output	Meaning
period	Dominant cycle length in bars
Stable period	Market exhibiting regular cyclical behavior
Period drifting to maxPeriod	Trending market; cycle measurement unreliable
Rapidly fluctuating period	Noisy or transitioning market regime

Performance Profile

Operation Count (Streaming Mode)

Operation	Count per bar	Notes
4-bar WMA	~5	3 MUL + 1 ADD + 1 DIV (precomputed as ×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, jQ)	~14	Two additional 4-tap Hilbert passes
Phasor EMA (I2, Q2)	~8	2 SUB/ADD + 4 FMA
Homodyne mixing	~8	4 MUL + 2 ADD/SUB per Re/Im
Homodyne EMA smoothing	~4	2 FMA for Re, Im
ATAN2	~20	Math.Atan2 transcendental (~15-20 cycles)
Period clamp + EMA	~4	2 comparisons + 1 FMA
Buffer management	~8	4 circular buffer writes + index updates
Total	~85	O(1) fixed; dominated by ATAN2

Batch Mode (SIMD Analysis)

Aspect	Assessment
SIMD vectorizable	No: cascaded IIR filters and Hilbert FIR with sequential state dependencies
Bottleneck	Math.Atan2 transcendental (~20 cycles); Hilbert FIR circular buffer lookups
Parallelism	None: each bar depends on previous bar's I2, Q2, Re, Im state
Memory	O(1): ~7-element circular buffers × 4 + 6 scalar EMA states (~300 bytes)
Throughput	Moderate; ~3× slower than simple EMA due to multi-stage Hilbert pipeline

Resources

• Ehlers, J.F. Rocket Science for Traders. Wiley, 2001.
• Ehlers, J.F. Cybernetic Analysis for Stocks and Futures. Wiley, 2004.
• Haykin, S. Communication Systems. 4th ed., Wiley, 2001. (Homodyne detection theory)