QuanTAlib/lib/cycles/ebsw/ebsw.md

EBSW: Ehlers Even Better Sinewave

Even Better Sinewave refines cycle detection by combining bandpass filtering with adaptive gain normalization.

Property	Value
Category	Cycle
Inputs	Source (close)
Parameters	hpLength (default 40), ssfLength (default 10)
Outputs	Single series (Ebsw)
Output range	Varies (see docs)
Warmup	Math.Max(hpLength, ssfLength) + 3 bars (default 43)
PineScript	ebsw.pine

• EBSW is a refined cycle oscillator that combines a high-pass filter (trend removal), a Super-Smoother filter (noise removal), and Automatic Gain Co...
• Parameterized by hplength (default 40), ssflength (default 10).
• Output range: Varies (see docs).
• Requires Math.Max(hpLength, ssfLength) + 3 bars (default 43) of warmup before first valid output (IsHot = true).
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

EBSW is a refined cycle oscillator that combines a high-pass filter (trend removal), a Super-Smoother filter (noise removal), and Automatic Gain Control to produce a normalized [-1, +1] output representing the current position within the dominant market cycle. Developed by John Ehlers as an improvement over the original Hilbert Transform SineWave, it provides cleaner turning point detection without requiring complex phase extraction mathematics.

Historical Context

Ehlers' original SineWave indicator relied on the Hilbert Transform to extract phase, but direct Hilbert Transforms proved unstable on real market data due to amplitude sensitivity and convergence issues during strong trends. The "Even Better" SineWave, published in Cycle Analytics for Traders (2013), simplifies the approach: instead of complex phase math, it uses a tuned bandpass filter (high-pass cascaded with a 2-pole low-pass) to isolate the dominant cycle, then normalizes the result via RMS-based AGC. The 3-bar averaging in both the wave and power calculations acts as a simple anti-aliasing stage. The result is a more robust tool for identifying turning points in both trending and ranging markets.

Architecture & Physics

1. High-Pass Filter (Trend Removal)

A single-pole high-pass filter removes frequencies below the cutoff:

\alpha_1 = \frac{1 - \sin(2\pi / P_{HP})}{\cos(2\pi / P_{HP})} HP_t = \frac{1 + \alpha_1}{2}(P_t - P_{t-1}) + \alpha_1 \cdot HP_{t-1}

2. Super-Smoother Filter (Noise Removal)

A 2-pole Butterworth low-pass attenuates high-frequency aliasing noise:

a = e^{-\sqrt{2}\pi / P_{SSF}} b = 2a \cos(\sqrt{2}\pi / P_{SSF}) Filt_t = \frac{(1 - b + a^2)}{2}(HP_t + HP_{t-1}) + b \cdot Filt_{t-1} - a^2 \cdot Filt_{t-2}

3. Wave and Power Calculation

Three-bar averaging for both signal and energy:

Wave_t = \frac{Filt_t + Filt_{t-1} + Filt_{t-2}}{3} Power_t = \frac{Filt_t^2 + Filt_{t-1}^2 + Filt_{t-2}^2}{3}

4. Normalization (AGC)

EBSW_t = \frac{Wave_t}{\sqrt{Power_t}}

Result is clamped to [-1, +1]. When Power \approx 0, output is zero.

5. Complexity

O(1) per bar. Fixed cascaded IIR filters with O(1) memory (only filter state variables and 3-bar history for wave/power).

Mathematical Foundation

Parameters

Parameter	Description	Default	Constraint
hpLength	High-pass filter period (detrending cutoff)	40	\geq 1, \neq 4
ssfLength	Super-smoother filter period (noise cutoff)	10	\geq 1

Precomputed Coefficients

α₁ = (1 - sin(2π/hpLength)) / cos(2π/hpLength)
a  = exp(-√2·π / ssfLength)
b  = 2·a·cos(√2·π / ssfLength)
c₁ = (1 - b + a²) / 2

Output Interpretation

Condition	Meaning
EBSW \approx +1	Cycle peak (potential short entry)
EBSW \approx -1	Cycle trough (potential long entry)
Zero crossing up	Bullish phase transition
Zero crossing down	Bearish phase transition
Railing at \pm 1	Strong directional move overwhelming cycle

Performance Profile

Operation Count (Streaming Mode)

Operation	Count per bar	Notes
High-pass filter	~4	1 SUB + 1 MUL + 1 FMA
Super-Smoother (2-pole IIR)	~5	1 ADD + 2 FMA + 1 MUL
Wave (3-bar average)	~3	2 ADD + 1 MUL
Power (3-bar RMS²)	~5	3 MUL + 2 ADD
SQRT normalization	~4	1 SQRT + 1 DIV + 1 branch
Clamp	~2	2 comparisons
State shift	~4	4 register moves
Total	~27	O(1) fixed; no loops or allocations

Batch Mode (SIMD Analysis)

Aspect	Assessment
SIMD vectorizable	No: HP and SSF are recursive IIR filters with sequential dependencies
Bottleneck	Math.Sqrt in AGC normalization (~15 cycles per call)
Parallelism	None: each bar depends on previous bar's filter state
Memory	O(1): 6 scalar state variables + 2 previous filter values
Throughput	Very fast; comparable to single EMA despite 3-stage pipeline

Resources

• Ehlers, J.F. Cycle Analytics for Traders. Wiley, 2013.
• Ehlers, J.F. Cybernetic Analysis for Stocks and Futures. Wiley, 2004.