EhlersSuperSmoother

Summary

EhlersSuperSmoother is RTTA's streaming two-pole Super Smoother low-pass filter by John F. Ehlers. It attenuates high-frequency market noise with substantially less lag than a comparable exponential moving average while remaining causal and suitable for bar-by-bar streaming use.

Update API

result = rtta.EhlersSuperSmoother(period=10, fillna=True).update(price)

Parameter	Default	Meaning
`period`	`10`	Critical period \(P\) of the low-pass (minimum 2)
`fillna`	`True`	If `False`, return NaN until `period` samples have been seen

The update(...) call consumes one scalar price observation and returns the current filtered value. advance(...) is not exposed on this class; use update or batch / batch_array for bulk series.

Theory Of Operation

Ehlers designs the Super Smoother as a two-pole recursive low-pass whose poles lie on a circle set by the critical period \(P\). The continuous-time prototype uses \(\sqrt{2}\,\pi / P\) as the angle of the complex conjugate pole pair; the discrete filter coefficients \(c_1,c_2,c_3\) follow from the exponential and cosine of that angle, with \(c_1\) chosen so DC gain is unity.

The input is a two-sample average \((x_t + x_{t-1})/2\) before the recursive section, which reduces the one-bar stair-step of discrete price. The first two samples seed the filter state to the raw price so the recurrence has finite history without a long warm-up buffer.

Compared with an EMA of similar noise rejection, Super Smoother rolls off faster past the critical period and produces a cleaner smooth for cycle indicators that chain after it (for example the roofing filter low-pass stage).

Recurrence

Let \(x_t\) be the input price and \(P=\max(\texttt{period},2)\) the critical period. Precompute once in the constructor:

\[\theta = \frac{\sqrt{2}\,\pi}{P},\qquad a_1 = e^{-\theta},\qquad b_1 = 2 a_1 \cos(\theta)\]

\[c_2 = b_1,\qquad c_3 = -a_1^{2},\qquad c_1 = 1 - c_2 - c_3\]

State after each bar: previous price \(x_{t-1}\) and previous two filter outputs \(y_{t-1}\), \(y_{t-2}\). For the first two samples (\(t < 2\)):

\[y_t = x_t\]

Thereafter:

\[y_t = c_1 \cdot \frac{x_t + x_{t-1}}{2} + c_2\, y_{t-1} + c_3\, y_{t-2}\]

When fillna=False and fewer than \(P\) samples have been processed, the return value is NaN; otherwise \(y_t\) is returned.

Implementation Notes

Implemented in src/rtta/indicator.cpp in class EhlersSuperSmoother.
Coefficients use the constant \(\sqrt{2}\approx 1.4142135623730951\) and \(\pi\) matching the C++ source.
State variables: price1_, filt1_, filt2_, sample count_.
Output is a scalar double, not a result struct.

Summary

Update API

Theory Of Operation

Recurrence

Implementation Notes

Reference