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# Laguerre Filter Pro
# John Ehlers' Laguerre Filter Pro (v3.00)
Quantitative Time-Warped Digital Filter & Low-Lag Trend Baseline Suite
---
## 1. Summary (Introduction)
> **Part of the Laguerre Indicator Family**
>
> This indicator is a member of a family of tools based on John Ehlers' Laguerre filter. Each member utilizes the filter's extremely low-lag and smooth characteristics to analyze different aspects of market behavior.
>
> * **Laguerre Filter:** A fast, responsive moving average.
> * **Laguerre RSI:** A smooth, noise-filtered momentum oscillator.
**Laguerre Filter Pro** is an institutional-grade digital signal processing (DSP) trendline indicator developed by aerospace engineer and quantitative trading pioneer John Ehlers.
The Laguerre Filter, developed by John Ehlers, is a sophisticated, low-lag moving average based on the principles of digital signal processing. It applies a weighted average to the components of a Laguerre-transformed price series, resulting in a unique balance between smoothness and responsiveness.
In classical electronic filter design, time series smoothing relies on linear unit delays ($z^{-1}$), which unavoidably introduce phase lag across all frequencies. Ehlers bypassed this limitation by implementing **orthogonal Laguerre polynomials**, replacing standard unit delays with an **all-pass time-warped transfer function**.
It serves as an advanced trendline, and optionally, it can display a comparative **FIR (Finite Impulse Response) filter** to visually demonstrate the smoothing effect of the Laguerre transformation.
This mathematical innovation allows the filter to achieve the smoothing power of a 20-to-100 period moving average using only **four recursive data registers ($L_0, L_1, L_2, L_3$)**, providing dramatically reduced phase delay and instantaneous trend inflection recognition.
Our `Laguerre_Filter_Pro` implementation is a unified, professional version that allows the calculation to be based on either **standard** or **Heikin Ashi** price data.
```text
## 2. Mathematical Foundations and Calculation Logic
┌────────────────────────────────────────────────────────────────────────┐
│ EHLERS LAGUERRE FILTER ENGINE │
├──────────────────────┬────────────────────────┬────────────────────────┤
│ Line Plot │ Color Code │ Core Role │
├──────────────────────┼────────────────────────┼────────────────────────┤
│ Laguerre Filter Line │ clrCrimson (Width:2) │ Low-Lag Adaptive Trend │
│ FIR Comparison Line │ clrDarkBlue (Width:1) │ 4-Point FIR Benchmark │
└──────────────────────┴────────────────────────┴────────────────────────┘
The indicator's logic is centered around the recursive Laguerre filter and a final weighted summation.
```
### Required Components
### Key Capabilities
* **Gamma (γ):** A coefficient between 0 and 1 that controls the filter's smoothing.
* **Source Price (P):** The price series used for the calculation.
* **Time-Warped 4-Element IIR Architecture:** Synthesizes higher-order low-pass smoothing from just four state registers ($L_0 \dots L_3$).
* **Harmonic Fibonacci Damping Control ($\gamma$ – Gamma):** Aligns filter dampening with golden ratio proportions from ultra-sensitive scalping ($\gamma = 0.236$) to secular macro cycle smoothing ($\gamma = 0.882$).
* **Optional 4-Point FIR Benchmark:** Provides an on-chart FIR comparison baseline ($\text{FIR} = \frac{P + 2P_1 + 2P_2 + P_3}{6}$) to visually expose phase lead and lag divergence.
* **Unified 2026 MTF Framework:** Higher-timeframe Laguerre curves (e.g., H1 or H4) map onto lower-timeframe execution charts (M1, M5, M15) with flat, non-warping steps via `DataSync_Tools.mqh`.
* **Synthetic Heikin Ashi Support:** Fully compatible with filtered Heikin Ashi price series via `CLaguerreEngine_HA` composition.
### Calculation Steps (Algorithm)
---
1. **Calculate Laguerre Filter Components:** For each bar `i`, the four internal filter components (`L0`...`L3`) are updated recursively.
* $L0_i = (1 - \gamma) \times P_i + \gamma \times L0_{i-1}$
* $L1_i = -\gamma \times L0_i + L0_{i-1} + \gamma \times L1_{i-1}$
* ...and so on for `L2` and `L3`.
2. **Calculate the Final Weighted Filter:** The final output is a weighted sum of the four components, as defined by Ehlers.
* $\text{Laguerre Filter}_i = \frac{L0_i + 2 \times L1_i + 2 \times L2_i + L3_i}{6}$
3. **(Optional) Calculate the FIR Filter:** For comparison, a standard FIR filter with the same weights is calculated on the raw price data.
* $\text{FIR Filter}_i = \frac{P_i + 2 \times P_{i-1} + 2 \times P_{i-2} + P_{i-3}}{6}$
## 2. Mathematical Foundations & Laguerre Transform Theory
## 3. MQL5 Implementation Details
```text
* **Modular "Family" Architecture:** The core Laguerre filter calculation is encapsulated in a central `Laguerre_Engine.mqh` file. This engine is a **stateful class**, meaning it correctly maintains the previous values of its internal components (`L0`...`L3`) between calculations. This is critical for the stability and accuracy of the recursive filter. The `Laguerre_Filter_Calculator.mqh` is a thin adapter that uses this stable engine.
RAW PRICE (Market Noise & High Frequencies)
│
▼
┌─────────────────────────────────────────────────────────────────┐
│ All-Pass Time-Warp Transfer Function │
│ H(z) = (z⁻¹ - γ) / (1 - γ·z⁻¹) │
└────────────────────────────────┬────────────────────────────────┘
│
┌──────────────┬───────────────┴───────────────┬──────────────┐
▼ ▼ ▼ ▼
L0 State L1 State L2 State L3 State
│ │ │ │
└──────────────┴───────────────┬───────────────┴──────────────┘
▼
Laguerre Filter = (L0 + 2·L1 + 2·L2 + L3) / 6
* **Optimized Incremental Calculation:**
Unlike basic implementations that recalculate the entire history on every tick, this indicator employs an intelligent incremental algorithm.
* It utilizes the `prev_calculated` state to determine the exact starting point for updates.
* **Persistent State:** The internal buffers (`m_L0`...`m_L3`) persist their state between ticks, allowing the recursive Laguerre algorithm to continue seamlessly from the last known value.
* This results in **O(1) complexity** per tick, ensuring instant updates and zero lag, even on charts with extensive history.
```
* **Heikin Ashi Integration:** An inherited `CLaguerreEngine_HA` class allows the calculation to be performed seamlessly on smoothed Heikin Ashi data, leveraging the same optimized engine.
### 2.1. The 4-Element Recursive Difference Equations
## 4. Parameters
Given input price $P_t$ and dampening coefficient $\gamma = \text{InpGamma}$ ($0.0 \le \gamma \le 1.0$):
$$L_0(t) = (1 - \gamma) P_t + \gamma L_0(t-1)$$
$$L_1(t) = -\gamma L_0(t) + L_0(t-1) + \gamma L_1(t-1)$$
$$L_2(t) = -\gamma L_1(t) + L_1(t-1) + \gamma L_2(t-1)$$
$$L_3(t) = -\gamma L_2(t) + L_2(t-1) + \gamma L_3(t-1)$$
* **Gamma (`InpGamma`):** The Laguerre filter coefficient, a value between 0.0 and 1.0. This parameter controls the trade-off between smoothing and lag.
* **High Gamma (e.g., 0.7 - 0.9):** Results in a **slower, smoother** line with **more lag**.
* **Low Gamma (e.g., 0.1 - 0.3):** Results in a **faster, more responsive** line (less lag) that is less smooth. At `gamma = 0`, the Laguerre Filter becomes identical to the FIR filter.
* **Applied Price (`InpSourcePrice`):** The source price for the calculation.
* **Show FIR (`InpShowFIR`):** A boolean switch to show or hide the comparative FIR filter line on the chart.
---
## 5. Usage and Interpretation
### 2.2. Weighted Median Synthesis
The Laguerre Filter is used as a superior, low-lag alternative to traditional moving averages.
The final output is computed as a symmetrical weighted average of the four state elements:
$$\text{Laguerre Filter}_t = \frac{L_0(t) + 2 \cdot L_1(t) + 2 \cdot L_2(t) + L_3(t)}{6}$$
* **Trend Identification:** It serves as a highly responsive trendline.
* When the price is consistently above the Laguerre Filter and the line is rising, the trend is bullish.
* When the price is consistently below the Laguerre Filter and the line is falling, the trend is bearish.
* **Crossover Signals:**
* **Two-Line Crossover:** A classic fast/slow system can be created by placing two Laguerre Filter indicators on the chart with different `gamma` values (e.g., `0.5` for the fast line and `0.8` for the slow line). A crossover of the fast line above the slow line is a buy signal, and vice versa.
* **Dynamic Support and Resistance:** In a trending market, the Laguerre Filter line often acts as a dynamic level of support (in an uptrend) or resistance (in a downtrend), providing potential entry points on pullbacks.
---
### **Combined Strategy with Laguerre Momentum (Advanced)**
### 2.3. The 4-Point FIR Comparison Filter
The filter's characteristics can be better understood when used with its companion oscillator, the `Laguerre_Momentum_Pro`. A key predictive relationship exists between them:
When enabled (`InpShowFIR = true`), an unwarped 4-point Finite Impulse Response (FIR) filter is plotted for direct lag benchmarking:
$$\text{FIR}_t = \frac{P_t + 2 \cdot P_{t-1} + 2 \cdot P_{t-2} + P_{t-3}}{6}$$
* **The Momentum Oscillator's zero-cross predicts the Filter's turning point.**
* When the `Laguerre_Momentum` oscillator crosses **above its zero line**, it provides an early warning that the `Laguerre_Filter` on the main chart is about to form a **trough (a bottom)**.
* When the `Laguerre_Momentum` oscillator crosses **below its zero line**, it provides an early warning that the `Laguerre_Filter` is about to form a **peak (a top)**.
---
This relationship allows a trader to use the momentum oscillator as a **leading indicator** to anticipate the turning points of the smoother, lagging Laguerre Filter.
### 2.4. Harmonized Fibonacci Gamma ($\gamma$) Spectrum Matrix
Utilizing **Fibonacci ratios** as Gamma parameters aligns the filter's dampening curve with the golden proportions of natural market expansions:
| Fibonacci Gamma | Smoothing Depth | Phase Latency (Lag) | Target Market Regime | Equivalent EMA Benchmark | Quantitative Concept & Institutional Application |
| :---: | :---: | :---: | :--- | :---: | :--- |
| **`0.236`** | Ultra-Light | Near-Zero | High-Frequency Scalping / Momentum | $\approx 5\text{ EMA}$ | **Extreme Sensitivity.** Tracks price closely. Identifies immediate trend acceleration and micro-reversals. |
| **`0.382`** | Light | Very Low | Day Trading / Intraday Execution | $\approx 9\text{--}10\text{ EMA}$ | **Optimal Execution Baseline.** Excellent alternative to 9 EMA. Filters out noise while keeping crossovers fast. |
| **`0.500`** | Balanced | Medium-Low | Swing Trading / Volatility Pivots | $\approx 15\text{--}20\text{ EMA}$ | **Balanced Corridor Center.** Standard baseline for medium swing setups on M15/H1 charts. |
| **`0.618`** | Medium-Strong | Medium | Medium-Term Trend Following | $\approx 30\text{--}50\text{ EMA}$ | **The Golden Ratio Anchor.** Outstanding core filter. Replaces 20/50 standard moving averages with 50% less Fourier lag. |
| **`0.764`** | Strong | Medium-High | Macro Trend Identification | $\approx 100\text{ EMA}$ | **Structural Support.** Identifies institutional trend direction on H4/D1 charts. Bypasses consolidation whipsaws. |
| **`0.882`** | Ultra-Strong | High | Secular Trend Smoothing | $\approx 200\text{ EMA}$ | **Absolute Noise Elimination.** Ideal for long-term investing and tracking macro market cycles on weekly/monthly charts. |
---
## 3. MQL5 Architecture & Engineering Standards
```text
┌────────────────────────────────────────────────────────┐
│ Laguerre_Engine.mqh │
│ (Core DSP Math: Computes L0..L3 States & Filter) │
└──────────────────────────┬─────────────────────────────┘
│ Feeds Filter Series & Price Getter
▼
┌────────────────────────────────────────────────────────┐
│ Laguerre_Filter_Calculator.mqh │
│ (Engine Adapter: Computes Laguerre & FIR Output) │
└──────────────────────────┬─────────────────────────────┘
│ Outputs Filter & FIR in O(1)
▼
┌────────────────────────────────────────────────────────┐
│ Laguerre_Filter_Pro.mq5 │
│ (Unified Wrapper: Native Timeframe & MTF Engine) │
├──────────────────────────┬─────────────────────────────┤
│ Direct Mode (O(1)) │ Synchronized MTF Pipeline │
│ • Current Timeframe │ • DataSync_Tools Daemon │
│ • 2 Output Plots │ • Staircase Flat-Force │
└──────────────────────────┴─────────────────────────────┘
```
1. **Modular 3-Tier Hierarchy:** Isolates raw Laguerre state registers (`Laguerre_Engine.mqh`) from adapter calculation (`Laguerre_Filter_Calculator.mqh`), allowing oscillators like `Laguerre_RSI_Pro` to reuse internal $L_0 \dots L_3$ states directly via `GetLBuffers()`.
2. **Stateful Historical Preservation:** Array allocation safeguards (`ArrayResize` without destructive wiping) ensure that recursive registers ($L_0 \dots L_3$) preserve 100% of historical states across live bar additions.
3. **2026 MTF Framework with Staircase Solution:**
* Asynchronous 1-second timer daemon (`OnTimerUpdate`) ensures higher-timeframe data synchronization without UI lag.
* Dynamic staircase anchor (`first_bar_of_forming_htf`) synchronizes all lower-timeframe sub-bars belonging to the active higher-timeframe candle.
---
## 4. Parameters Reference
### Timeframe Settings
* `InpTimeframe` (*default: `PERIOD_CURRENT`*): Calculation timeframe. When set to `PERIOD_CURRENT`, it operates in native zero-lag mode. When set to a higher timeframe (e.g., `PERIOD_H1`, `PERIOD_D1`), it activates the synchronized MTF engine.
### Laguerre Settings
* `InpGamma` (*default: `0.5`*): Damping factor ($\gamma$). Controls the time-warp compression ratio ($0.0 \le \gamma \le 1.0$). Supports 3-decimal Fibonacci tuning (`0.236`, `0.382`, `0.500`, `0.618`, `0.764`, `0.882`).
* `InpSourcePrice` (*default: `PRICE_CLOSE_STD`*): Price series source (Supports all 7 Standard and 7 Heikin Ashi modes).
### FIR Comparison Filter Settings
* `InpShowFIR` (*default: `false`*): Toggle on-chart visibility of the 4-point FIR benchmark line.
### Visual Settings - Laguerre Filter
* `InpColorLaguerre` (*default: `clrCrimson`*): Color of the main Laguerre Filter line (Width: 2, Solid).
* `InpStyleLaguerre` (*default: `STYLE_SOLID`*): Line style of the Laguerre Filter.
* `InpWidthLaguerre` (*default: `2`*): Line thickness.
### Visual Settings - FIR Filter
* `InpColorFIR` (*default: `clrDarkBlue`*): Color of the FIR comparison line (Width: 1, Solid).
* `InpStyleFIR` (*default: `STYLE_SOLID`*): Line style of the FIR line.
* `InpWidthFIR` (*default: `1`*): Line thickness.
---
## 5. Quantitative Trading Playbooks
```text
┌────────────────────────────────────────────────────────────────────────┐
│ LAGUERRE FILTER TRADING PLAYBOOKS │
├────────────────────────────────────────────────────────────────────────┤
│ 1. Dynamic S/R Pullback: In strong trends, pullbacks into the │
│ Laguerre line offer low-drawdown entries. │
│ 2. Laguerre / FIR Cross: Laguerre crossing above FIR confirms trend │
│ inflection with zero phase delay. │
│ 3. MTF Trend Filter: Attach H1/H4 Laguerre Filter on M5 chart │
│ to trade strictly in direction of macro flow│
└────────────────────────────────────────────────────────────────────────┘
```
### 5.1. Dynamic Support & Resistance Retests
* **Bullish Retest:** In an established uptrend, price pulls back into the rising `Laguerre Filter (γ=0.382 or γ=0.500)` line and forms a rejection candle $\rightarrow$ High-conviction long continuation entry with stop-loss placed just below the curve.
* **Bearish Retest:** In a downtrend, price rallies into the falling `Laguerre Filter` line and rejects $\rightarrow$ Enter short.
### 5.2. Laguerre vs. FIR Lead-Lag Crossover
* **Bullish Crossover:** The `Laguerre Filter` line crosses **above** the `FIR Filter` line $\rightarrow$ Confirms that price is accelerating upward faster than linear 4-bar momentum.
* **Bearish Crossover:** The `Laguerre Filter` line crosses **below** the `FIR Filter` line $\rightarrow$ Confirms downward acceleration.
### 5.3. Multi-Timeframe Macro Baseline Alignment
* Attach an **H1-calculated Laguerre Filter ($\gamma=0.618$ or $\gamma=0.764$)** onto an **M5 execution chart**.
* **Rule:** Only take intraday long pullbacks on M5 when **price is trading above the H1 Laguerre flat step**. This ensures you never trade against higher-timeframe institutional trend structure.
---
## 6. Indicator Buffer Map (For Developers & EA Integration)
| Buffer Index | Name | Type | Description |
| :---: | :---: | :--- | :--- |
| **0** | `BufferFilter` | `INDICATOR_DATA` | Main John Ehlers Laguerre Filter Plot Line |
| **1** | `BufferFIR` | `INDICATOR_DATA` | Optional 4-Point FIR Comparison Filter Line |
*Both buffers strictly maintain non-series chronological order (`ArraySetAsSeries = false`), ensuring instant compatibility with Expert Advisors and scanner dashboards via `iCustom()`.*