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# John Ehlers' Butterworth Acceleration Pro Suite (Standard & MTF)
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## 1. Summary (Introduction)
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The **John Ehlers' Butterworth Acceleration Pro Suite** is an institutional-grade, low-latency trend-deceleration and trend-exhaustion tracking engine. It consists of two highly optimized indicators: `Butterworth_Acceleration_Pro` (Standard) and `Butterworth_Acceleration_Pro` (Multi-Timeframe variant).
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In physical dynamics, an object must decelerate before it can reverse direction. Similarly, in financial markets, the force of buying or selling pressure decelerates before price forms a structural swing pivot. While the first derivative (Velocity or Slope) shows the direction and speed of a trend, the **second derivative (Acceleration)** measures the *rate of change of that velocity*.
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Standard market acceleration indicators (such as the second difference of raw price) are un-tradable because mathematical differentiation exponentially amplifies high-frequency market noise (chatter).
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This suite resolves this noise-amplification bottleneck by calculating the second difference of John Ehlers' higher-order **Butterworth Filter**:
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$$\text{Acceleration}_t = \text{Filter}_t - 2 \times \text{Filter}_{t-1} + \text{Filter}_{t-2}$$
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Because the 2-pole and 3-pole Butterworth Filter topologies are mathematically engineered for a **maximally flat passband response** with an extremely sharp roll-off, they completely eliminate the noise floor. Taking the second difference of this pristine baseline results in an exceptionally smooth, fourier-stable acceleration wave.
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By classifying this clean acceleration wave into a symmetrical 5-zone thermal matrix (using blues for bullish acceleration and reds/corals for bearish acceleration), the suite allows quant traders to identify the exact market inflection points (inflexiós pontok) with near-zero lag.
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---
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## 2. Mathematical & Quant Foundations
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The indicator calculates the second difference of a stateful, recursive 2-pole or 3-pole IIR Butterworth Filter.
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### A. Recursive Butterworth Baseline Formulas
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On each bar $t$, the decimal price $P_t$ (Standard or Heikin Ashi) is processed through the selected poles configuration:
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#### 1. 2-Pole Butterworth Filter
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The dampening coefficients are calculated using the critical period ($T$):
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$$a = e^{-\frac{\sqrt{2}\pi}{T}}, \quad b = 2a \cos \left(\frac{\sqrt{2}\pi}{T} \right), \quad c_1 = \frac{1 - b + a^2}{4}$$
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$$\text{Filter}_t = b \times \text{Filter}_{t-1} - a^2 \times \text{Filter}_{t-2} + c_1 \times (P_t + 2 \times P_{t-1} + P_{t-2})$$
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#### 2. 3-Pole Butterworth Filter
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The 3-pole configuration offers an even sharper roll-off (vágási meredekség) using three recursive states:
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$$a = e^{-\frac{\pi}{T}}, \quad b = 2a \cos \left(\frac{1.738\pi}{T} \right), \quad c = a^2, \quad c_1 = \frac{(1 - b + c)(1 - c)}{8}$$
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$$\text{Filter}_t = (b + c) \text{Filter}_{t-1} - (c + bc) \text{Filter}_{t-2} + c^2 \text{Filter}_{t-3} + c_1 (P_t + 3 P_{t-1} + 3 P_{t-2} + P_{t-3})$$
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### B. The Acceleration Equation
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The Acceleration ($A_t$) represents the change in Slope ($\text{Slope}_t - \text{Slope}_{t-1}$). It is derived mathematically as:
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$$A_t = (\text{Filter}_t - \text{Filter}_{t-1}) - (\text{Filter}_{t-1} - \text{Filter}_{t-2})$$
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$$A_t = \text{Filter}_t - 2 \times \text{Filter}_{t-1} + \text{Filter}_{t-2}$$
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Due to requiring two lookback periods of history, calculations are strictly restricted to indices $t \ge 2$.
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### C. 5-Zone Symmetrical Thermal Acceleration Matrix
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The acceleration $A_t$ is classified into a specific visual state based on an adjustable noise threshold ($\epsilon$):
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| Color Index | Market State | Mathematical Condition | Visual Representation |
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| :---: | :--- | :--- | :--- |
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| **`0.0`** | **Neutral / Noise** | $A_t \le \epsilon$ | **`clrGray`** (No directional acceleration) |
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| **`1.0`** | **Strong Bullish Acceleration** | $A_t > \epsilon \quad \text{AND} \quad A_t > A_{t-1}$ | **`clrDodgerBlue`** (Buying force is accelerating) |
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| **`2.0`** | **Weak Bullish Deceleration** | $A_t > \epsilon \quad \text{AND} \quad A_t \le A_{t-1}$ | **`clrLightSkyBlue`** (Buying force is slowing down) |
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| **`3.0`** | **Strong Bearish Acceleration** | $A_t < -\epsilon \quad \text{AND} \quad A_t < A_{t-1}$ | **`clrCrimson`** (Selling force is accelerating) |
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| **`4.0`** | **Weak Bearish Deceleration** | $A_t < -\epsilon \quad \text{AND} \quad A_t \ge A_{t-1}$ | **`clrCoral`** (Selling force is slowing down) |
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---
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## 3. Recommended Calibration & Volatility Presets
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Because second differences yield extremely small fractional values, configuring the neutral noise threshold ($\epsilon$) correctly is key to isolating consolidations:
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| Asset Class | Timeframe | Poles Selection | Period ($T$) | Threshold ($\epsilon$) | Quant Tactical Objective |
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| :--- | :--- | :--- | :---: | :---: | :--- |
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| **Major FX Pairs** | M15 / H1 | `POLES_TWO` | `20` | `0.000010` | **Execution Reversal.** Catches early intraday cyclical pivots with minimum fourier delay. |
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| **Equity Indices** | H1 / H4 | `POLES_THREE` | `15` | `0.000050` | **Volatility Contraction.** Sharp roll-off is ideal for breakout trading. |
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| **Cryptocurrencies** | H4 / Daily | `POLES_THREE` | `25` | `0.000250` | **Exhaustion Detection.** Bypasses heavy retail noise to expose macro exhaustion points. |
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---
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## 4. Visual & Technical Highlights
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* **Micro-Point Precision Settings:**
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Since second differences represent tiny fractional shifts, the indicator automatically increases the chart decimal display to four decimals past standard digit precision (`_Digits + 4`) to prevent visual rounding anomalies:
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```mql5
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IndicatorSetInteger(INDICATOR_DIGITS, _Digits + 4);
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```
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* **Performance-First O(1) Updates:**
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By avoiding dynamic memory allocation (`new`/`delete`) inside the `OnCalculate()` tick loop, the indicator prevents heap fragmentation. Calculations are performed on the stack using localized, state-safe dynamic structures.
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* **Double-Smoothed VWMA Signal Line:**
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When configured to VWMA, the indicator converts the platform volume arrays to a `double` cache array, applying volume weighting to the Signal MA. This ensures crossovers are backed by institutional transaction volume.
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---
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## 5. Advanced MQL5 MTF Implementation Details
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Operating high-order recursive filters like the Butterworth 3-pole across multiple timeframes requires robust engineering:
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### A. Non-Warping Staircase Solution
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To prevent the active, forming HTF candle from drawing a warped diagonal slope on lower timeframe charts, the indicator runs a backward-scanning block-force loop. It identifies the beginning of the active forming HTF block and rewrites the entire block flat on every tick:
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```mql5
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int first_bar_of_forming_htf = rates_total - 1;
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while(first_bar_of_forming_htf > 0 &&
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iBarShift(_Symbol, g_calc_timeframe, time[first_bar_of_forming_htf], false) == 0)
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{
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first_bar_of_forming_htf--;
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}
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first_bar_of_forming_htf++; // Anchor start of current HTF period block
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if(start > first_bar_of_forming_htf)
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start = first_bar_of_forming_htf;
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```
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### B. High-Order IIR State Mocking
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Since the Butterworth Filter relies on deep historical states ($\text{Filter}_{t-1}, \text{Filter}_{t-2}, \text{Filter}_{t-3}$), calling calculations continuously on the live forming bar on every tick can cause feedback decay. To solve this, the MTF engine uses **State Mocking** during live ticks by passing `prev_calculated = g_htf_count`, which updates only the active live register while keeping historical closed states completely locked.
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---
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## 6. Quantitative Trading Strategies
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### A. The Calculus Inflection Strategy (0-Line Reversal)
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According to calculus, when the second derivative (Acceleration) crosses the zero line, the first derivative (Velocity) has reached its absolute peak, and the underlying curve is at an inflection point.
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1. **Indicator Setup:**
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* **Butterworth Acceleration Pro:** Period = `20`, Poles = `POLES_THREE`, Threshold = `0.000010`.
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* **Signal Line:** Disabled.
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2. **Execution Rules:**
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* **BUY Trigger:** Enter Long when the histogram crosses **above the zero line** (turning from Crimson/Coral to DodgerBlue). This indicates that downward pressure has exhausted and buying acceleration has taken control.
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* **SELL Trigger:** Enter Short when the histogram crosses **below the zero line** (turning from DodgerBlue/LightSkyBlue to Crimson).
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3. **Strategic Edge:** By entering on the zero-crossing of the *second derivative* (Acceleration), you enter the market at the exact inflection point of Ehlers' filter, capturing the trend far earlier than standard MACD or EMA crossover systems.
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```text
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[ Bearish Accel (Crimson) ] ==> [ ZERO CROSS ] ==> [ Bullish Accel (Blue) ]
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(Downward Force Exhausted) (BUY ENTRY TRIGGERED)
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```
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### B. The Volume-Weighted Deceleration Exit Strategy
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This strategy uses the transition from strong acceleration to weak deceleration, backed by volume, to exit trend-following trades at the absolute peak before the price reverses.
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1. **Indicator Setup:**
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* **Butterworth Acceleration Pro:** Period = `20`, Poles = `POLES_TWO`, Threshold = `0.000010`.
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* **Signal MA:** Enabled, Period = `5`, Type = `VWMA`.
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2. **Execution Rules:**
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* **Bullish Exit (Exit Longs):** When holding a Long position and the histogram is in **Strong Bullish Acceleration** (`clrDodgerBlue`), exit the trade immediately when the histogram bar transitions to **Weak Bullish Deceleration** (`clrLightSkyBlue`) AND **crosses below the VWMA Signal Line**.
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* **Bearish Exit (Exit Shorts):** When holding a Short position and the histogram is in **Strong Bearish Acceleration** (`clrCrimson`), exit the trade immediately when the histogram bar transitions to **Weak Bearish Deceleration** (`clrCoral`) AND **crosses above the VWMA Signal Line**.
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3. **Strategic Advantage:** Traditional trailing stops require price to drop significantly before triggering an exit, giving back a large portion of accrued profits. This strategy detects when the *volume-backed acceleration* of the trend begins to slow down, allowing you to exit at the optimal crest of the wave.
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