QuanTAlib/lib/cycles/ssfdsp/Ssfdsp.md

SSFDSP: Ehlers SSF Detrended Synthetic Price

Property	Value
Category	Cycle
Inputs	Source (close)
Parameters	period (default 40)
Outputs	Single series (SsfDsp)
Output range	Varies (see docs)
Warmup	slowPeriod * 2 bars

TL;DR

• SSFDSP isolates the dominant cycle by subtracting a half-cycle Super-Smoother from a quarter-cycle Super-Smoother, producing a zero-centered oscill...
• Parameterized by period (default 40).
• Output range: Varies (see docs).
• Requires slowPeriod * 2 bars of warmup before first valid output (IsHot = true).
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

SSFDSP isolates the dominant cycle by subtracting a half-cycle Super-Smoother from a quarter-cycle Super-Smoother, producing a zero-centered oscillator with superior noise rejection compared to the EMA-based DSP. The 2-pole Butterworth characteristic of the Super-Smoother filter provides zero phase lag at the cutoff frequency and sharper rolloff than exponential smoothing, making SSFDSP the preferred variant for cycle-aware trading when the approximate dominant period is known.

Historical Context

John Ehlers introduced the concept of Detrended Synthetic Price in Cybernetic Analysis for Stocks and Futures (2004) as a principled method for removing the DC (trend) component while preserving cyclical energy. The original DSP used EMAs, which have a gradual frequency rolloff and non-zero phase lag. The SSF variant substitutes Super-Smoother filters, which are 2-pole Butterworth low-pass designs with matched coefficients that eliminate the Gibbs phenomenon (ringing) common in sharper filters. The result is a cleaner cycle extraction: the SSF's steeper rolloff better separates the quarter-cycle and half-cycle frequency bands, producing tighter zero crossings and more reliable turning point identification than EMA-DSP.

Architecture & Physics

1. Filter 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. Super-Smoother Coefficients

For each filter period p:

\alpha = \frac{\pi\sqrt{2}}{p} c_2 = 2 e^{-\alpha} \cos(\alpha) c_3 = -e^{-2\alpha} c_1 = 1 - c_2 - c_3

3. SSF Recursion

SSF_t = c_1 \cdot \frac{P_t + P_{t-1}}{2} + c_2 \cdot SSF_{t-1} + c_3 \cdot SSF_{t-2}

The 2-bar input averaging provides an additional anti-aliasing stage.

4. SSFDSP Output

SSFDSP_t = SSF_{fast,t} - SSF_{slow,t}

5. Complexity

O(1) per bar. Two independent 2-pole IIR filters with O(1) memory. Warmup: approximately 2 \times P_{slow} for convergence. Recursive dependencies prevent SIMD vectorization.

Mathematical Foundation

Parameters

Parameter	Description	Default	Constraint
period	Expected dominant cycle period	40	\geq 4

Super-Smoother Frequency Response

The SSF has -3 dB attenuation at the cutoff period, -12 dB/octave rolloff (2-pole), and zero phase lag at the cutoff. This is equivalent to a critically-damped Butterworth filter.

Pseudo-code

function SSFDSP(source, period):
    pFast ← max(2, round(period / 4))
    pSlow ← max(3, round(period / 2))

    // Fast SSF coefficients
    αf ← √2·π / pFast
    c2f ← 2·exp(-αf)·cos(αf)
    c3f ← -exp(-2·αf)
    c1f ← 1 - c2f - c3f

    // Slow SSF coefficients
    αs ← √2·π / pSlow
    c2s ← 2·exp(-αs)·cos(αs)
    c3s ← -exp(-2·αs)
    c1s ← 1 - c2s - c3s

    ssfFast_1 ← 0; ssfFast_2 ← 0
    ssfSlow_1 ← 0; ssfSlow_2 ← 0
    p_prev ← 0

    for each price in source:
        // Input averaging
        avg ← (price + p_prev) / 2

        // Fast SSF update
        ssfFast ← c1f·avg + c2f·ssfFast_1 + c3f·ssfFast_2

        // Slow SSF update
        ssfSlow ← c1s·avg + c2s·ssfSlow_1 + c3s·ssfSlow_2

        // SSFDSP
        ssfdsp ← ssfFast - ssfSlow

        // Shift state
        ssfFast_2 ← ssfFast_1; ssfFast_1 ← ssfFast
        ssfSlow_2 ← ssfSlow_1; ssfSlow_1 ← ssfSlow
        p_prev ← price

        emit ssfdsp

DSP vs SSFDSP

Aspect	DSP (EMA-based)	SSFDSP (Super-Smoother)
Filter type	1-pole IIR (exponential)	2-pole Butterworth
Rolloff	-6 dB/octave	-12 dB/octave
Phase lag at cutoff	Non-zero	Zero
Noise rejection	Moderate	Superior
Turning points	Rounded	Sharper

Output Interpretation

Condition	Meaning
SSFDSP > 0	Bullish cycle phase
SSFDSP < 0	Bearish cycle phase
Zero crossing	Cycle phase transition
Divergence with price	Cycle energy waning; trend exhaustion
Amplitude shrinking	Cycle losing dominance; transition to trend

Performance Profile

Operation Count (Streaming Mode)

Operation	Count per bar	Notes
Input averaging	~2	1 ADD + 1 MUL(×0.5)
Fast SSF (2-pole IIR)	~5	1 MUL(c1f) + 2 FMA(c2f, c3f)
Slow SSF (2-pole IIR)	~5	1 MUL(c1s) + 2 FMA(c2s, c3s)
Subtraction (output)	~1	1 SUB
State shift	~5	5 register moves
Total	~18	O(1) fixed; pure FMA arithmetic, zero transcendentals

Batch Mode (SIMD Analysis)

Aspect	Assessment
SIMD vectorizable	No: both SSF filters are recursive 2-pole IIR with sequential state dependencies
Bottleneck	None significant; pure multiply-accumulate with precomputed coefficients
Parallelism	None: each bar depends on two previous bars' filter state
Memory	O(1): 4 scalar filter states + 1 previous price (~40 bytes)
Throughput	Among fastest cycle indicators; comparable to dual-EMA DSP; no transcendentals at runtime

Resources

• Ehlers, J.F. Cybernetic Analysis for Stocks and Futures. Wiley, 2004.
• Ehlers, J.F. Cycle Analytics for Traders. Wiley, 2013.
• Butterworth, S. "On the Theory of Filter Amplifiers." Experimental Wireless, 7, 1930.