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lib/trends_FIR/nyqma/Nyqma.md

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 # NYQMA: Nyquist Moving Average
 
+| Property         | Value                            |
+| ---------------- | -------------------------------- |
+| **Category**     | Trend (FIR MA)                        |
+| **Inputs**       | Source (close)                          |
+| **Parameters**   | `period` (default 89), `nyquistPeriod` (default 21)                      |
+| **Outputs**      | Single series (Nyqma)                       |
+| **Output range** | Tracks input                     |
+| **Warmup**       | 1 bar                          |
+
+### TL;DR
+
+- NYQMA combines a primary LWMA (Linear Weighted Moving Average) with a secondary LWMA applied to the first, using lag-compensating extrapolation: $\...
+- Parameterized by `period` (default 89), `nyquistperiod` (default 21).
+- Output range: Tracks input.
+- Requires 1 bar of warmup before first valid output (IsHot = true).
+- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
+
 > "Manfred Dürschner applied the Nyquist-Shannon sampling theorem to cascaded moving averages: the second smoothing period must not exceed half the first, or you get aliasing artifacts. Respect the theorem and the ghost signals disappear."
 
 NYQMA combines a primary LWMA (Linear Weighted Moving Average) with a secondary LWMA applied to the first, using lag-compensating extrapolation: $\text{NYQMA} = (1+\alpha) \cdot \text{MA}_1 - \alpha \cdot \text{MA}_2$, where $\alpha = N_2 / (N_1 - N_2)$. The Nyquist constraint $N_2 \leq \lfloor N_1/2 \rfloor$ ensures the second smoothing does not introduce aliasing artifacts into the output. This produces a lag-reduced moving average grounded in sampling theory rather than ad-hoc coefficient tuning. Streaming update is O(1) per bar via composed Wma instances; batch mode uses stackalloc/ArrayPool with FMA in the extrapolation loop.