> *Take the SMA of an SMA and you get a triangular filter. It is the simplest possible smoothing kernel that has zero phase distortion and no frequency-domain discontinuities. Sometimes simple is exactly what you need.*
SWMA applies triangular (symmetric) weights that peak at the center of the window and taper linearly to the edges. For period $N$, the weight at position $i$ is $w(i) = (N/2 + 1) - |i - N/2|$, producing a tent-shaped kernel. This is mathematically equivalent to convolving two rectangular windows (SMA of SMA), giving SWMA a frequency response that is the square of the SMA's sinc-like response. The result is smoother than SMA with better sidelobe suppression, at the cost of slightly more lag.
## Historical Context
The symmetric (triangular) weighted average is one of the oldest smoothing methods in statistics, predating modern signal processing by centuries. Its equivalence to the double-application of the simple moving average was recognized by Macaulay (1931) in his NBER monograph on time-series smoothing. The TRIMA (Triangular Moving Average) implemented elsewhere in QuanTAlib is the same mathematical operation computed via double SMA composition.
In PineScript, `ta.swma` refers specifically to the 4-point variant with weights $[1, 2, 2, 1]/6$, which is a special case of the general symmetric weighted average. QuanTAlib's SWMA generalizes this to arbitrary periods.
The triangular kernel has a natural Bayesian interpretation: if you believe the "true" signal is equally likely to be any value in a window of width $N/2$, and your observation window is also $N/2$, the posterior belief about the signal value is triangular. This makes SWMA the optimal Bayesian filter under uniform prior and uniform observation noise assumptions.
## Architecture & Physics
### 1. Weight Computation
For a window of length $N$ with half-width $h = (N-1)/2$:
$$
w(i) = h + 1 - |i - h|, \quad i = 0, 1, \ldots, N-1
$$
Weights form a triangle peaking at the center. For even $N$, the peak is a plateau of two equal values.
The weight sum equals $(h+1)^2$ for odd $N$ and $h(h+2)+1$ for even $N$.
### 3. Equivalence to Double SMA
SWMA(N) produces the same output as SMA(M) applied to SMA(M) where $M = \lceil N/2 \rceil$. This means the streaming implementation can compose two SMA instances for O(1) updates, rather than O(N) convolution.
## Mathematical Foundation
The triangular window for length $N$, with $h = (N-1)/2$:
$$
w[i] = h + 1 - |i - h|, \quad i = 0, \ldots, N-1
$$
**Frequency response:**
$$
H_{\text{SWMA}}(f) = H_{\text{SMA}}^2(f) = \left[\frac{\sin(\pi f M)}{\pi f M}\right]^2
$$
where $M = \lceil N/2 \rceil$. The squared sinc provides:
SWMA(N) is an O(N) FIR convolution using symmetric triangular weights (ascending then descending). Weights are precomputed at construction and normalized to sum = 1. The triangular shape gives the center bar the highest weight.
Symmetric folding reduces the effective FMA count to ⌈N/2⌉. For N = 14: 7 FMAs per bar. AVX2 batch throughput: ~N/8 cycles per bar. Among the windowed FIR filters, SWMA has the fewest effective operations due to its simple triangular shape.
