QuanTAlib/lib/cycles/ssfdsp/Ssfdsp.md

SSFDSP: Ehlers SSF Detrended Synthetic Price

SSF-based detrended synthetic price applies a super smoother before extracting cycles, achieving cleaner periodicity isolation.

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

• 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.

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.