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# Even Better Sinewave Indicator

This study calculates and displays an Even Better Sinewave Indicator (by John Ehlers) for the data given by the **Input Data** Input.

Let \(X\) be a random variable denoting the **Input Data**, and let \(X_t\) be the value of \(X\) at Index \(t\). Let the **Length** Input be denoted as \(n)\.

We begin by computing a constant that depends on the Length, which we denote as \(\alpha^{(1)}(n)\).

\(\displaystyle{\alpha^{(1)}(n) = \frac{1 - \sin\left(\frac{360^\circ}{n}\right)}{\cos\left(\frac{360^\circ}{n}\right)}}\)If \(\cos\left(\frac{360^\circ}{n}\right) = 0\), then \(\alpha^{(1)}(n) = 0\).

We denote the High Pass Filter at Index \(t\) as \(HP_t(X,n)\), and we compute it as follows.

\(\displaystyle{HP_t(X,n) = \frac{1}{2}\left(1 + \alpha^{(1)}(n)\right)\left(X_t - X_{t - 1}\right) + \alpha^{(1)}(n) \cdot HP_{t - 1}(X,n)}\)**Note**: The lowest possible setting for \(n\) is \(5\). This is because \(\alpha^{(1)} = 0\) for \(n = 2\), and \(\alpha^{(1)}\) is undefined for \(n = 4\). The lower limit of \(5\) is intended to prevent users from using either of these problematic settings.

Next, we compute a 2-bar Simple Moving Average of \(HP_t(X,n)\), as follows.

\(\overline{HP}_t(X,n) = SMA_t(HP(X,n),3)\)

We then denote the the Wave Amplitude and Wave Power as \(A_t(X,n)\) and \(P_t(X,n)\), respectively, and we compute them using a 2-Pole Super Smoother Filter on \(\overline{HP}_t(X,n)\), as follows.

\(A_t(X,n) = SSF^{(2)}_t\left(\overline{HP}(X,n),10\right)\)\(P_t(X,n) = \left[A_t(X,n)\right]^2\)

We then compute the three-bar Simple Moving Averages of the Amplitude and Power, denoting these averages respectively as \(\overline{A}_t(X,n)\) and \(\overline{P}_t(X,n)\). We compute them as follows.

\(\overline{A}_t(X,n) = SMA_t(A(X,n),3)\)\(\overline{P}_t(X,n) = SMA_t(P(X,n),3)\)

Finally, we denote the **Even Better Sinewave Indicator** as \(EBSWI_t(X,n)\), and we compute it as follows.

If \(\overline{P}_t(X,n) = 0\), then \(EBSWI_t(X,n) = 0\).

#### Inputs

*Last modified Monday, 26th September, 2022.