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Add new moving average implementations: LTMA, MCNMA, NLMA, NMA, NYQMA, RAIN, and TRAMA
- LTMA (Linear Trend Moving Average): Introduces a predictive moving average using dual cascaded EMAs for trend estimation. - MCNMA (McNicholl EMA): Implements a zero-lag TEMA using a cascaded EMA structure for enhanced responsiveness. - NLMA (Non-Lag Moving Average): Utilizes a damped cosine kernel to achieve reduced lag in moving averages. - NMA (Natural Moving Average): Adapts smoothing based on volatility profiles using a square-root kernel. - NYQMA (Nyquist Moving Average): Applies the Nyquist-Shannon theorem to prevent aliasing in cascaded moving averages. - RAIN (Rainbow Moving Average): Combines multiple SMA layers with weighted averages for multi-scale smoothing. - TRAMA (Trend Regularity Adaptive Moving Average): Adapts smoothing based on the frequency of new highs and lows in price data.
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Miha Kralj
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# SSFDSP: Ehlers SSF Detrended Synthetic Price
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> "The Super-Smoother filter provides Butterworth-quality noise rejection—combine two of them and you isolate cycles with surgical precision."
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The SSF-Based Detrended Synthetic Price (SSFDSP) is an advanced oscillator by John Ehlers. It creates a synthetic, detrended price series by subtracting a half-cycle Super-Smoother from a quarter-cycle Super-Smoother, providing superior noise rejection and reduced lag compared to EMA-based DSP.
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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.
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## Historical Context
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Ehlers introduced the concept of "Synthetic Price" to remove the DC (trend) component from market data, isolating cyclic energy. While earlier versions used EMAs, the SSF variant exploits the 2-pole Butterworth characteristics of the Super-Smoother Filter to achieve cleaner separation between trend and cycle.
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The SSF provides zero phase lag at the cutoff frequency, making it ideal for cycle isolation in noisy market data.
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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.
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## Architecture & Physics
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The indicator computes the difference between two Super-Smoother filters tuned to fractions of the dominant cycle period.
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### 1. Filter Periods
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$$
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P_{fast} = \max(2, \text{round}(P / 4))
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$$
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From the user-specified dominant cycle period $P$:
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$$
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P_{slow} = \max(3, \text{round}(P / 2))
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$$
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$$P_{fast} = \max(2, \lfloor P / 4 + 0.5 \rfloor)$$
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$$P_{slow} = \max(3, \lfloor P / 2 + 0.5 \rfloor)$$
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### 2. Super-Smoother Coefficients
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$$
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\alpha = \frac{\pi\sqrt{2}}{period}
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$$
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For each filter period $p$:
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$$
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c_2 = 2e^{-\alpha}\cos(\alpha)
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$$
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$$\alpha = \frac{\pi\sqrt{2}}{p}$$
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$$
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c_3 = -e^{-2\alpha}
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$$
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$$c_2 = 2 e^{-\alpha} \cos(\alpha)$$
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$$
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c_1 = 1 - c_2 - c_3
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$$
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$$c_3 = -e^{-2\alpha}$$
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$$c_1 = 1 - c_2 - c_3$$
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### 3. SSF Recursion
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$$
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SSF_t = c_1 \cdot \frac{P_t + P_{t-1}}{2} + c_2 \cdot SSF_{t-1} + c_3 \cdot SSF_{t-2}
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$$
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$$SSF_t = c_1 \cdot \frac{P_t + P_{t-1}}{2} + c_2 \cdot SSF_{t-1} + c_3 \cdot SSF_{t-2}$$
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The 2-bar input averaging provides an additional anti-aliasing stage.
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### 4. SSFDSP Output
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$$
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SSFDSP = SSF_{fast} - SSF_{slow}
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$$
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$$SSFDSP_t = SSF_{fast,t} - SSF_{slow,t}$$
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## Performance Profile
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### 5. Complexity
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### Operation Count (Streaming Mode, per Bar)
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$O(1)$ 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.
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| FMA (SSF updates) | 4 | 4 | 16 |
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| MUL (coefficients) | 2 | 3 | 6 |
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| ADD/SUB (input avg, output) | 3 | 1 | 3 |
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| **Total** | **9** | — | **~25 cycles** |
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## Mathematical Foundation
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### Complexity Analysis
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### Parameters
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- **Streaming:** O(1) per bar—fixed 2-pole IIR filters
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- **Memory:** O(1)—only filter state variables
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- **Warmup:** ~2 × slow period for convergence
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- **Note:** Recursive dependencies prevent SIMD vectorization
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| Parameter | Description | Default | Constraint |
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|-----------|-------------|---------|------------|
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| `period` | Expected dominant cycle period | 40 | $\geq 4$ |
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## Validation
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### Super-Smoother Frequency Response
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| Library | Status | Notes |
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| :--- | :---: | :--- |
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| TA-Lib | N/A | Not standard |
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| Skender | N/A | Not standard |
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| PineScript | ✅ | Matches Ehlers' reference logic |
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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.
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## Usage & Pitfalls
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### Pseudo-code
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- **Oscillates around zero**—positive values indicate bullish cycle phase
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- **Zero crossings** signal cycle phase changes—entry points in direction of cross
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- **Period mismatch** degrades amplitude and phase accuracy
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- **Smoother than EMA-DSP** with sharper turning points
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- **Divergence** (price highs vs DSP highs) indicates trend exhaustion
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- **Pre-smooth input** for extremely noisy data
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```
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function SSFDSP(source, period):
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pFast ← max(2, round(period / 4))
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pSlow ← max(3, round(period / 2))
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## API
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// Fast SSF coefficients
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αf ← √2·π / pFast
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c2f ← 2·exp(-αf)·cos(αf)
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c3f ← -exp(-2·αf)
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c1f ← 1 - c2f - c3f
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```mermaid
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classDiagram
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class Ssfdsp {
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+int Period
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+double Value
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+bool IsHot
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+Ssfdsp(int period)
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+Ssfdsp(ITValuePublisher source, int period)
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+TValue Update(TValue input, bool isNew)
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+void Reset()
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}
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// Slow SSF coefficients
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αs ← √2·π / pSlow
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c2s ← 2·exp(-αs)·cos(αs)
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c3s ← -exp(-2·αs)
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c1s ← 1 - c2s - c3s
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ssfFast_1 ← 0; ssfFast_2 ← 0
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ssfSlow_1 ← 0; ssfSlow_2 ← 0
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p_prev ← 0
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for each price in source:
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// Input averaging
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avg ← (price + p_prev) / 2
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// Fast SSF update
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ssfFast ← c1f·avg + c2f·ssfFast_1 + c3f·ssfFast_2
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// Slow SSF update
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ssfSlow ← c1s·avg + c2s·ssfSlow_1 + c3s·ssfSlow_2
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// SSFDSP
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ssfdsp ← ssfFast - ssfSlow
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// Shift state
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ssfFast_2 ← ssfFast_1; ssfFast_1 ← ssfFast
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ssfSlow_2 ← ssfSlow_1; ssfSlow_1 ← ssfSlow
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p_prev ← price
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emit ssfdsp
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```
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### Class: `Ssfdsp`
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### DSP vs SSFDSP
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| Parameter | Type | Default | Range | Description |
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| :--- | :--- | :--- | :--- | :--- |
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| `period` | `int` | `40` | `≥4` | Expected dominant cycle period |
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| Aspect | DSP (EMA-based) | SSFDSP (Super-Smoother) |
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|--------|-----------------|------------------------|
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| Filter type | 1-pole IIR (exponential) | 2-pole Butterworth |
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| Rolloff | $-6$ dB/octave | $-12$ dB/octave |
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| Phase lag at cutoff | Non-zero | Zero |
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| Noise rejection | Moderate | Superior |
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| Turning points | Rounded | Sharper |
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### Properties
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### Output Interpretation
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- `Value` (`double`): The current SSFDSP value (oscillates around 0)
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- `IsHot` (`bool`): Returns `true` when warmup is complete
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| Condition | Meaning |
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|-----------|---------|
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| $SSFDSP > 0$ | Bullish cycle phase |
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| $SSFDSP < 0$ | Bearish cycle phase |
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| Zero crossing | Cycle phase transition |
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| Divergence with price | Cycle energy waning; trend exhaustion |
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| Amplitude shrinking | Cycle losing dominance; transition to trend |
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### Methods
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## Resources
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- `Update(TValue input, bool isNew)`: Updates the indicator with a new data point
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## C# Example
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```csharp
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using QuanTAlib;
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// Initialize with a 40-bar dominant cycle assumption
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var ssfdsp = new Ssfdsp(period: 40);
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// Update with streaming data
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foreach (var bar in quotes)
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{
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var result = ssfdsp.Update(new TValue(bar.Date, bar.Close));
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if (ssfdsp.IsHot)
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{
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Console.WriteLine($"{bar.Date}: SSF-DSP = {result.Value:F4}");
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// Zero crossing detection
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if (result.Value > 0 && ssfdsp.Previous.Value <= 0)
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Console.WriteLine(" → Bullish cycle phase");
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else if (result.Value < 0 && ssfdsp.Previous.Value >= 0)
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Console.WriteLine(" → Bearish cycle phase");
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}
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}
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// Batch calculation
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var output = Ssfdsp.Calculate(sourceSeries, period: 40);
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```
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- **Ehlers, J.F.** *Cybernetic Analysis for Stocks and Futures*. Wiley, 2004.
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- **Ehlers, J.F.** *Cycle Analytics for Traders*. Wiley, 2013.
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- **Butterworth, S.** "On the Theory of Filter Amplifiers." *Experimental Wireless*, 7, 1930.
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