QuanTAlib/lib/cycles/amfm/Amfm.md

AMFM: Ehlers AM Detector / FM Demodulator

Treat price like a radio wave — demodulate amplitude for volatility, demodulate frequency for timing.

Property	Value
Category	Cycle
Inputs	TBar (Open, Close)
Parameters	period (default 30)
Outputs	Dual: AM (≥ 0) + FM (≈ [-1,+1])
Output range	AM: ≥ 0; FM: bounded ≈ [-1,+1]
Warmup	max(12, period) bars
PineScript	amfm.pine

• AMFM decomposes price movement into amplitude (AM) and frequency (FM) components using DSP techniques from radio engineering — AM measures volatility, FM tracks timing of price variations.
• Similar: EEO, DSO | Complementary: Moving averages for trend confirmation | Trading note: AM provides volatility context; FM zero crossings signal direction changes. FM is more robust for strategy optimization (smoother parameter surface).
• No external validation libraries implement AMFM. Validated through self-consistency and behavioral testing.

Ehlers applies radio engineering signal processing to financial data, treating the whitened price derivative (Close − Open) as a modulated carrier. The AM detector extracts the amplitude envelope (volatility) using peak detection and smoothing. The FM demodulator strips amplitude information via a hard limiter (10× gain clamped to ±1), then integrates the result through a Super Smoother filter to recover the frequency/timing component. The FM demodulator produces more robust trading strategies because removing amplitude variation creates a smoother optimization parameter surface.

Historical Context

John F. Ehlers published "A Technical Description of Market Data for Traders" in the May 2021 issue of Technical Analysis of Stocks & Commodities. The article applies classical radio engineering concepts — amplitude modulation (AM) and frequency modulation (FM) — to financial time series analysis. In the June 2021 follow-up, "Creating More Robust Trading Strategies With The FM Demodulator," Ehlers demonstrated that incorporating the FM demodulator into a simple momentum strategy produced significantly smoother parameter optimization surfaces, leading to more robust strategy configurations.

Architecture & Physics

Stage 1: Whitening (Common to Both)

\text{Deriv} = \text{Close} - \text{Open}

Using Close − Open instead of Close − Close[1] removes intraday gap effects, producing a zero-mean whitened derivative.

Stage 2a: AM Detector (Amplitude Envelope)

\text{Envel} = \max(|\text{Deriv}|, 4\text{ bars}) \text{AM} = \text{SMA}(\text{Envel}, 8)

The 4-bar rolling maximum captures the amplitude envelope, and the 8-bar SMA smooths it into a volatility measure.

Stage 2b: FM Demodulator (Frequency/Timing)

\text{HL} = \text{clamp}(10 \cdot \text{Deriv}, -1, +1)

The hard limiter applies 10× gain then clips to ±1, stripping all amplitude information and preserving only the sign/timing.

a_1 = e^{-1.414\pi / \text{Period}}, \quad b_1 = 2a_1\cos(1.414\pi / \text{Period}) c_2 = b_1, \quad c_3 = -a_1^2, \quad c_1 = 1 - c_2 - c_3 \text{FM} = \frac{c_1}{2}(\text{HL} + \text{HL}[1]) + c_2 \cdot \text{FM}[1] + c_3 \cdot \text{FM}[2]

The Super Smoother integrates the hard-limited signal, recovering the frequency modulation component.

Performance Profile

Operation Count (Streaming Mode, Scalar)

Operation	Count	Notes
Subtraction (Deriv)	1	Close − Open
Abs + compare (envelope)	5
SMA update	2	Running sum add/remove
Division (SMA)	1	sum / 8
Multiply + clamp (HL)	3	10×Deriv + 2 comparisons
FMA × 2 (SSF)	2	2-pole recursive filter
Total per bar	~14	Constant O(1)

Batch Mode (SIMD Analysis)

The IIR Super Smoother stage prevents full vectorization. Batch mode uses stackalloc circular buffers to avoid heap allocation.

Validation

Validated through self-consistency tests (streaming ≡ batch) and behavioral tests.

Behavioral Test Summary

Test	Expected Result
Constant OHLC	AM → 0, FM → 0
Strong uptrend (C > O)	AM > 0, FM > 0
Strong downtrend (C < O)	AM > 0, FM < 0
NaN/Inf input	Finite output (fallback)
Bar correction (isNew)	State restored correctly

Common Pitfalls

1. AM vs FM semantics: AM measures how much (volatility), FM measures when (timing). They are complementary, not redundant.

2. Hard limiter gain: The 10× multiplier before clamping is hardcoded per Ehlers. Most price derivatives are small enough that 10× pushes them to the ±1 rails, effectively creating a sign function. Do not tune this.

3. FM period: The period parameter only affects the FM Super Smoother cutoff. The AM detector uses fixed 4-bar envelope + 8-bar SMA (per Ehlers' specification).

4. Input requirement: Needs Open and Close prices (TBar input). Close-only data will produce Deriv = 0 if Open defaults to Close.

References

• Ehlers, J. F. (2021). "A Technical Description of Market Data for Traders." TASC, May 2021.
• Ehlers, J. F. (2021). "Creating More Robust Trading Strategies With The FM Demodulator." TASC, June 2021.
• Ehlers, J. F. (2013). Cycle Analytics for Traders. John Wiley & Sons. (Super Smoother definition)
• MESA Software Paper