# DECO: Ehlers Decycler Oscillator

## 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

## 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.