# BLMA: Blackman Window Moving Average

> *If you want to filter noise, don't just average it - window it.*

| Property         | Value                            |
| ---------------- | -------------------------------- |
| **Category**     | Trend (FIR MA)                        |
| **Inputs**       | Source (close)                          |
| **Parameters**   | `period`                      |
| **Outputs**      | Single series (Blma)                       |
| **Output range** | Tracks input                     |
| **Warmup**       | `period` bars                          |
| **PineScript**   | [blma.pine](blma.pine)                       |
| **Signature**    | [blma_signature](blma_signature.md) |

- BLMA is a FIR filter that applies a triple-cosine Blackman window function from digital signal processing to financial time series.
- Best suited as a long-term trend filter due to its superior noise suppression (-58 dB sidelobes) at the cost of ~N/2 lag.
- **Similar:** [WMA](../wma/wma.md), [TRIMA](../trima/trima.md) | **Complementary:** Trend confirmation | **Trading note:** Blackman-windowed MA; low sidelobe leakage for clean spectral response.
- Validated against reference implementations using the standard Blackman window formula.

The Blackman Window Moving Average (BLMA) applies a triple-cosine window function from digital signal processing to financial time series. Originally developed by **Ralph Beebe Blackman** at Bell Labs in the 1950s for spectral analysis, this filter provides superior noise suppression compared to standard moving averages by minimizing spectral leakage.

## Historical Context

In the early days of signal processing, engineers struggled with **spectral leakage** where energy from one frequency bleeds into others during analysis. Simple rectangular windows (like SMA) caused significant leakage. Blackman proposed a window function with tapered edges that drastically reduced this effect. In trading, "leakage" manifests as market noise distorting the trend signal. BLMA adapts this DSP innovation to create a trend filter that is remarkably smooth yet responsive to significant moves.

## Architecture & Physics

BLMA is a Finite Impulse Response (FIR) filter. Unlike Exponential Moving Averages (IIR) which have infinite memory, BLMA considers only the last $N$ bars.

The "physics" of BLMA relies on its bell-shaped weighting curve. The weights are highest in the center of the window and taper to zero at both ends (newest and oldest data). This symmetry means BLMA has a lag of approximately $N/2$, but it effectively suppresses high-frequency noise (jitter) that often plagues other averages.

### The Zero-Edge Effect

Because the Blackman window tapers to zero at the edges ($w[0] \approx 0$ and $w[N-1] \approx 0$), the most recent price data has very little immediate impact on the indicator value. This creates a "smoothness" that filters out sudden spikes, but it also introduces a specific type of lag where the indicator is slow to react to a sudden trend reversal until the price move enters the "fat" part of the window (the center).

## Mathematical Foundation

The Blackman window weights $w(n)$ for a period $N$ are calculated as:

$$ w(n) = 0.42 - 0.5 \cos\left(\frac{2\pi n}{N-1}\right) + 0.08 \cos\left(\frac{4\pi n}{N-1}\right) $$

Where $0 \le n \le N-1$.

The BLMA value is the weighted average:

$$ BLMA_t = \frac{\sum_{i=0}^{N-1} P_{t-i} \cdot w(i)}{\sum_{i=0}^{N-1} w(i)} $$

## Performance Profile

### Operation Count (Streaming Mode, Scalar)

**Constructor (one-time weight precomputation):**

| Operation | Count | Cost (cycles) | Subtotal |
| :--- | :---: | :---: | :---: |
| COS | 2N | 40 | 80N |
| MUL | 4N | 3 | 12N |
| ADD/SUB | 3N | 1 | 3N |
| **Total (init)** | — | — | **~95N cycles** |

For period=20: ~1,900 cycles (one-time).

**Hot path (per bar):**

| Operation | Count | Cost (cycles) | Subtotal |
| :--- | :---: | :---: | :---: |
| MUL | N | 3 | 3N |
| ADD | N | 1 | N |
| DIV | 1 | 15 | 15 |
| **Total** | **2N + 1** | — | **~4N + 15 cycles** |

For period=20: ~95 cycles per bar.

**Hot path breakdown:**
- Weighted sum: `∑(buffer[i] × weights[i])` → N MUL + N ADD
- Normalization: `sum / wSum` → 1 DIV (wSum precomputed)

### Batch Mode (SIMD)

The convolution is highly vectorizable:

| Operation | Scalar Ops | SIMD Ops (AVX2) | Speedup |
| :--- | :---: | :---: | :---: |
| Weighted products | N | N/8 | 8× |
| Horizontal sum | N | log₂(8) | ~N/3× |

**Batch efficiency (512 bars, period=20):**

| Mode | Cycles/bar | Total | Notes |
| :--- | :---: | :---: | :--- |
| Scalar streaming | ~95 | ~48,640 | O(N) per bar |
| SIMD batch | ~25 | ~12,800 | Vectorized dot product |
| **Improvement** | **~4×** | **~36K saved** | — |

### Quality Metrics

| Metric | Score | Notes |
| :--- | :---: | :--- |
| **Accuracy** | 10/10 | Precise DSP windowing |
| **Timeliness** | 4/10 | Significant lag (N/2) due to symmetric window |
| **Overshoot** | 10/10 | Never overshoots (FIR property) |
| **Smoothness** | 10/10 | Excellent noise suppression (-58dB side-lobes) |

### Zero-Allocation Design

The implementation uses a pre-calculated weights array and a circular buffer (`RingBuffer`) to store price history. The `Update` method performs the weighted sum without allocating any new memory on the heap. For the static `Calculate` method, `stackalloc` is used for weights and temporary buffers for small periods (up to 256), ensuring high performance.

## Validation

BLMA is validated against a reference implementation using the standard Blackman window formula.

| Library | Status | Notes |
| :--- | :--- | :--- |
| **QuanTAlib** | ✅ | Matches theoretical formula. |
| **PineScript** | ✅ | Matches PineScript reference logic. |

### Common Pitfalls

* **Lag**: BLMA has more lag than EMA or WMA because it suppresses the most recent data. It is a smoothing filter, not a leading indicator.
* **Warmup**: During the first $N$ bars, the window expands dynamically. The full noise-suppression characteristics are only achieved after $N$ bars.