QuanTAlib/lib/oscillators/deco/Deco.md

DECO: Ehlers Decycler Oscillator

Ehlers' decycler oscillator removes the trend and isolates residual oscillation — what remains when the drift is subtracted.

Property	Value
Category	Oscillator
Inputs	Source (close)
Parameters	shortPeriod (default 30), longPeriod (default 60)
Outputs	Single series (Deco)
Output range	0 to 1
Warmup	1 bar
PineScript	deco.pine

• The Decycler Oscillator (DECO) is a DSP-based oscillator developed by John F.
• Similar: DPO, Reflex | Complementary: Cycle analysis | Trading note: Ehlers' Decycler Oscillator; removes trend to isolate cycle component for timing.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

Overview

The Decycler Oscillator (DECO) is a DSP-based oscillator developed by John F. Ehlers that isolates intermediate-frequency market cycles. It computes the difference between two 2-pole Butterworth high-pass filters with different cutoff periods, revealing the spectral band between the two cutoff frequencies.

Origin

• Author: John F. Ehlers
• Source: "Decyclers", Technical Analysis of Stocks & Commodities, September 2015
• Category: Oscillator / Digital Signal Processing

Formula

The DECO uses two 2-pole Butterworth high-pass filters:

α = (cos(0.707 × 2π/period) + sin(0.707 × 2π/period) - 1) / cos(0.707 × 2π/period)
HP[n] = (1 - α/2)² × (x[n] - 2×x[n-1] + x[n-2]) + 2×(1-α) × HP[n-1] - (1-α)² × HP[n-2]

DECO = HP_long - HP_short

The 0.707 factor (1/√2) places the filter response at the -3 dB Butterworth design point.

Transfer Function

Each HP filter has the z-domain transfer function:

H(z) = (1-α/2)² × (1 - 2z⁻¹ + z⁻²) / (1 - 2(1-α)z⁻¹ + (1-α)²z⁻²)

The DECO output is the difference H_long(z) - H_short(z), which forms a bandpass response isolating cycles between the short and long cutoff periods.

Parameters

Parameter	Type	Default	Range	Description
shortPeriod	int	30	> 0	Short HP cutoff period (bars)
longPeriod	int	60	> shortPeriod	Long HP cutoff period (bars)

Interpretation

The Decycler Oscillator provides several analytical perspectives:

• Zero-Line Crossovers:

  • Crossing above zero indicates bullish cycle momentum
  • Crossing below zero indicates bearish cycle momentum
  • The zero-crossing timing is relatively lag-free

• Band Isolation:

  • The oscillator extracts only cycles within the frequency band defined by the two cutoff periods
  • Shorter cycles and longer trends are both rejected
  • This makes the oscillator highly selective

• Divergence Analysis:

  • Bullish divergence: price makes lower lows while DECO makes higher lows
  • Bearish divergence: price makes higher highs while DECO makes lower highs
  • Indicates potential trend reversal

• Multiple Instance Analysis:

  • Ehlers recommends using multiple DECO instances with different period pairs
  • Crossovers between instances with different coefficients can identify trend reversals

Warmup Period

The indicator requires longPeriod bars before producing reliable output. The first two bars always output zero (insufficient price history for the 2-pole HP filter).

Properties

• Range: Unbounded (oscillates around zero)
• Complexity: O(1) per bar (pure IIR filter, no lookback buffer needed)
• Memory: O(1) — only stores filter state variables

Related Indicators

• Decycler (DECYCLER): The low-pass complement — removes cycles, keeps trend
• SSF-DSP: Similar concept using Super Smooth Filters instead of HP filters
• Roofing Filter: HP + SSF combination for cycle isolation
• BandPass Filter: Ehlers' direct bandpass approach

Performance Profile

Operation Count (Streaming Mode)

DECO (Detrended Correlation Oscillator) subtracts a linear regression from price then computes correlation.

Operation	Count	Cost (cycles)	Subtotal
Linear regression (O(N) or O(1) with prefix sums)	~4	1	4
SUB (detrend: price − regression)	1	1	1
Correlation pipeline (see CTI)	~22	7	156
Total	~27	—	~161 cycles

Dominated by the correlation computation. ~161 cycles per bar.

Batch Mode (SIMD Analysis)

Operation	Vectorizable?	Notes
Linear regression	Yes	Prefix-sum dot products; VFMADD
Detrend subtraction	Yes	VSUBPD
Correlation	Yes	See CTI batch analysis

Fully vectorizable in batch. Regression and correlation both benefit from AVX2 VFMADD chains.

Quality Metrics

Metric	Score	Notes
Accuracy	9/10	Exact linear detrend + Pearson r
Timeliness	5/10	Two N-bar windows compound lag
Smoothness	8/10	Detrending removes linear drift; correlation bounded
Noise Rejection	8/10	Linear detrend + correlation is doubly robust to trend contamination

References

1. Ehlers, J. F. (2015). "Decyclers." Technical Analysis of Stocks & Commodities, September 2015.
2. Ehlers, J. F. (2013). Cycle Analytics for Traders. Wiley. Chapter 4.