QuanTAlib/lib/cycles/dsp/dsp.md

DSP: Ehlers Detrended Synthetic Price

Property	Value
Category	Cycle
Inputs	Source (close)
Parameters	period (default 40)
Outputs	Single series (Dsp)
Output range	Varies (see docs)
Warmup	slowPeriod * 3 bars

TL;DR

• DSP creates a zero-centered oscillator by subtracting a half-cycle EMA from a quarter-cycle EMA, isolating the dominant cyclical component of price...
• Parameterized by period (default 40).
• Output range: Varies (see docs).
• Requires slowPeriod * 3 bars of warmup before first valid output (IsHot = true).
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

DSP creates a zero-centered oscillator by subtracting a half-cycle EMA from a quarter-cycle EMA, isolating the dominant cyclical component of price while cancelling longer-term trends. Developed by John Ehlers, the indicator is grounded in cycle theory rather than arbitrary period selection, making it a principled alternative to MACD for cycle-aware trading. Bias-corrected EMAs ensure accurate amplitude during warmup.

Historical Context

John Ehlers introduced the Detrended Synthetic Price as part of his cycle analytics framework. While MACD uses fixed periods (12/26), DSP calibrates its two EMAs to specific fractions of the dominant cycle period: quarter-cycle for the fast component and half-cycle for the slow. Subtracting aligned filters at these frequencies effectively bandpass-isolates the cycle of interest while suppressing both high-frequency noise and low-frequency trend. The "synthetic" label reflects that the output is a constructed signal that exposes cyclical energy invisible in raw price.

Architecture & Physics

1. Component Periods

From the user-specified dominant cycle period P:

P_{fast} = \max(2, \lfloor P / 4 + 0.5 \rfloor) P_{slow} = \max(3, \lfloor P / 2 + 0.5 \rfloor)

2. Alpha Coefficients

Standard EMA smoothing factors:

\alpha_{fast} = \frac{2}{P_{fast} + 1}, \qquad \alpha_{slow} = \frac{2}{P_{slow} + 1}

3. EMA Updates with Bias Correction

Raw EMA recursion:

EMA_{raw,t} = \alpha \cdot P_t + (1 - \alpha) \cdot EMA_{raw,t-1}

Warmup bias correction (prevents initial distortion):

EMA_t = \frac{EMA_{raw,t}}{1 - (1 - \alpha)^n}

where n is the number of bars processed.

4. DSP Output

DSP_t = EMA_{fast,t} - EMA_{slow,t}

5. Complexity

O(1) per bar with O(1) memory. Two EMA state variables plus two bias correction accumulators.

Mathematical Foundation

Parameters

Parameter	Description	Default	Constraint
period	Dominant cycle period	40	\geq 4

Pseudo-code

function DSP(source, period):
    pFast ← max(2, round(period / 4))
    pSlow ← max(3, round(period / 2))
    αFast ← 2 / (pFast + 1)
    αSlow ← 2 / (pSlow + 1)

    emaFastRaw ← 0
    emaSlowRaw ← 0
    decayFast ← 1.0    // (1 - αFast)^n
    decaySlow ← 1.0    // (1 - αSlow)^n

    for each price in source:
        emaFastRaw ← FMA(αFast, price, (1 - αFast) * emaFastRaw)
        emaSlowRaw ← FMA(αSlow, price, (1 - αSlow) * emaSlowRaw)

        decayFast *= (1 - αFast)
        decaySlow *= (1 - αSlow)

        emaFast ← emaFastRaw / (1 - decayFast)
        emaSlow ← emaSlowRaw / (1 - decaySlow)

        dsp ← emaFast - emaSlow
        emit dsp

Output Interpretation

Condition	Meaning
DSP > 0	Fast EMA above slow: bullish cycle phase
DSP < 0	Fast EMA below slow: bearish cycle phase
Zero crossing	Cycle phase transition point
Divergence from price	Cycle energy waning; potential trend exhaustion

Performance Profile

Operation Count (Streaming Mode)

Operation	Count	Cost (cycles)	Subtotal
ADD/SUB	3	1	3
MUL	4	3	12
FMA	2	4	8
DIV	2	15	30
Total	11	—	~53 cycles

O(1) per bar. Two EMA updates (fast + slow) using FMA, plus warmup bias-correction divisions. After warmup completes, the DIV cost drops to zero, reducing steady-state to ~23 cycles.

Quality Metrics

Metric	Score	Notes
Accuracy	9/10	Bias-corrected EMAs eliminate warmup distortion
Timeliness	8/10	Quarter-cycle EMA responds quickly; half-cycle provides reference
Smoothness	8/10	Dual EMA differencing inherently smooths noise
Memory	10/10	O(1) state: 6 scalar values in record struct

Resources

• Ehlers, J.F. Cybernetic Analysis for Stocks and Futures. Wiley, 2004.
• Ehlers, J.F. Rocket Science for Traders. Wiley, 2001.