QuanTAlib/lib/cycles/homod/homod.md

HOMOD: Homodyne Discriminator Dominant Cycle

Overview and Purpose

The Homodyne Discriminator (HOMOD) is a cycle measurement technique introduced by John F. Ehlers in Rocket Science for Traders (2001) and expanded in the November 2000 Traders’ Tips column. It applies a Hilbert Transform framework to detect the instantaneous dominant cycle present in price data while minimizing lag.

Unlike fixed-length filters, HOMOD continuously adapts to current market rhythm by converting the in-phase and quadrature components into a complex phasor pair, multiplying them homodynally, and extracting period information from the resulting phase angle. This makes it ideal for adaptive indicators and systems requiring dynamic lookback lengths.

Core Concepts

• Homodyne Multiplication: Complex multiply of current and prior phasors to isolate instantaneous frequency
• Hilbert FIR Kernel: Ehlers 0.0962/0.5769 coefficients producing 90° phase shift with minimal distortion
• Quadrature Rotation: Phase-advanced components (jI, jQ) enabling orthogonal phasor construction
• Cycle Clamping: Limiting detected periods to realistic bounds (default 6–50 bars)
• Warmup Compensation: Exponential correction ensuring stable output from bar one

Common Settings and Parameters

Parameter	Default	Function	When to Adjust
Source	hlc3	Input series analyzed for cycle period	Switch to close for end-of-day signals or to custom synthetic blends
Min Period	6	Lower bound for detected cycle length	Increase to ignore ultrashort noise-dominated cycles
Max Period	50	Upper bound for detected cycle length	Raise for weekly/monthly studies; lower for intraday scalping

Pro Tip: Align downstream indicators (e.g., RSI, moving averages) to the live HOMOD period by rounding to the nearest integer—this maintains resonance with the market’s dominant rhythm.

Calculation and Mathematical Foundation

Explanation: HOMOD smooths price, applies a Hilbert Transform to obtain in-phase (I) and quadrature (Q) components, rotates them by 90°, forms phasors, multiplies each phasor by its predecessor, and derives period length from the resulting phase angle. Subsequent smoothing and clamping stabilize measurements.

Technical formula:

1. Weighted smoothing and detrending

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

   
   Detrender_t = \left(0.0962\,SP_t + 0.5769\,SP_{t-2} - 0.5769\,SP_{t-4} - 0.0962\,SP_{t-6}\right)\cdot B_t
   

   where B_t = 0.075\cdot Period_{t-1} + 0.54.

2. Quadrature pair and phase advance

   
   Q1_t = (0.0962\,Det_t + 0.5769\,Det_{t-2} - 0.5769\,Det_{t-4} - 0.0962\,Det_{t-6})\cdot B_t
   

   
   I1_t = Det_{t-3}
   

   
   jI_t = (0.0962\,I1_t + 0.5769\,I1_{t-2} - 0.5769\,I1_{t-4} - 0.0962\,I1_{t-6})\cdot B_t
   

   
   jQ_t = (0.0962\,Q1_t + 0.5769\,Q1_{t-2} - 0.5769\,Q1_{t-4} - 0.0962\,Q1_{t-6})\cdot B_t
   

3. Phasor construction

   
   I2_t = 0.2\,(I1_t - jQ_t) + 0.8\,I2_{t-1},\quad Q2_t = 0.2\,(Q1_t + jI_t) + 0.8\,Q2_{t-1}
   

4. Homodyne product and smoothing

   
   Re_t = 0.2\,(I2_t I2_{t-1} + Q2_t Q2_{t-1}) + 0.8\,Re_{t-1}
   

   
   Im_t = 0.2\,(I2_t Q2_{t-1} - Q2_t I2_{t-1}) + 0.8\,Im_{t-1}
   

5. Period extraction, clamp, warmup

   
   \theta_t = \operatorname{atan2}(Im_t, Re_t)
   

   
   Period^\*_{t} = \frac{2\pi}{\theta_t}
   

   
   Period_t = \operatorname{clip}(|Period^\*_t|,\ Min,\ Max)
   

   
   SmoothPeriod_t = SmoothPeriod_{t-1} + 0.33\,(Period_t - SmoothPeriod_{t-1})
   

Interpretation Details

• Cycle Tracking

  • 6–12 bars: fast oscillatory regimes suited to scalping and short-term countertrend trades
  • 12–30 bars: medium cycles aligning with swing-trading horizons
  • 30–60 bars: slow cycles highlighting macro rhythm or trend exhaustion zones

• Adaptive Parameterization

  • Use rounded SmoothPeriod as the lookback for RSI, stochastic, ATR channels, etc.
  • Match moving-average lengths to maintain coherence between filters and underlying price rhythm.

• Regime Analysis

  • Stable plateau in period → consistent cycle regime
  • Rising period → trend elongation or consolidation broadening
  • Falling period → volatility expansion, choppy markets, or nascent rotational phases

Limitations and Considerations

• Warmup Demand: Requires ~60 bars for fully stable phasor history; early readings should be treated cautiously
• Trend Dominance: Persistent directional moves degrade cycle definition, causing erratic period swings
• Noise Sensitivity: Despite smoothing, extremely noisy instruments may oscillate near Min Period consistently
• Clamp Bias: Hard limits prevent detection of cycles outside bounds; adjust for instruments with known longer rhythms
• Computational Intensity: Multiple FIR taps and state variables raise per-bar workload versus simpler averages

Performance Profile

Operation Count (Streaming Mode, per Bar)

Operation	Count	Cost (cycles)	Subtotal
ADD/SUB	~25	1	25
MUL	~30	3	90
DIV	2	15	30
ATAN2	1	80	80
Total	~58	—	~225 cycles

Breakdown:

• Weighted smooth (4-point): 3 MUL + 3 ADD + 1 DIV = 17 cycles
• Detrender FIR (4 taps): 5 MUL + 3 ADD = 18 cycles
• Q1 FIR (4 taps): 5 MUL + 3 ADD = 18 cycles
• jI/jQ FIRs (8 taps total): 10 MUL + 6 ADD = 36 cycles
• I2/Q2 IIR phasor smoothing: 4 MUL + 4 ADD = 16 cycles
• Homodyne Re/Im: 6 MUL + 4 ADD = 22 cycles
• Period extraction (atan2 + div): 1 ATAN2 + 1 DIV = 95 cycles

Complexity Analysis

Mode	Complexity	Notes
Streaming	O(1)	Fixed FIR taps (6-deep) + IIR states
Batch	O(n)	Linear scan, constant work per bar

Memory: ~128 bytes (6-bar FIR history × 4 series + IIR states)

SIMD Analysis

Optimization	Applicable	Notes
AVX2 vectorization	Limited	FIR taps vectorizable, IIRs sequential
FMA	✅	Hilbert kernel: 0.0962×x + 0.5769×x[2] - ...
Batch parallelism	❌	IIR feedback prevents cross-bar parallelism

Optimization Notes: The atan2 call dominates (~35% of cost). Consider:

• Fast atan2 approximation if <1° accuracy acceptable
• Precompute 2π constant, use reciprocal for division

Quality Metrics

Metric	Score	Notes
Accuracy	9/10	Hilbert Transform is mathematically rigorous
Timeliness	7/10	FIR kernel introduces ~3 bar delay
Overshoot	8/10	Smoothed period output is stable
Smoothness	8/10	IIR smoothing reduces jitter

References

• Ehlers, J. F. (2001). Rocket Science for Traders: Digital Signal Processing Applications. Wiley.
• Ehlers, J. F. (2000). Traders’ Tips – Homodyne Discriminator. Technical Analysis of Stocks & Commodities.
• blackcat1402. (2023). Ehlers Homodyne Discriminator Period Measurer (TradingView script).
• MrTools. (2025). Homodyne Discriminator.mq4. Forex-Station Forums.
• Mladen. (2019). Adaptive Lookback Indicators – Homodyne Update. MQL5 Forums.
• 3Jane. (2024). tindicators hd.cc Implementation. GitHub.

Validation Sources

Validation Sources:
Patterns: §2, §6, §7, §16, §17, §18, §19
Wolfram: "atan2(y,x)"
External: "TradingView Homodyne Discriminator","Forex-Station Homodyne Discriminator","MQL5 Adaptive Lookback Homodyne","tindicators hd.cc"
Planning: phases=function,main_loop,docs,index