Triangle Smoothing Derivative

We introduce a new operator, the triangle smoothing derivative, which is obtained by taking the derivative of equation 7 with respect to the radius $R$:

$$\frac{\partial T}{\partial R}(R,\omega) = i\omega \left[\frac{-i ... ...t}{2})}{2R^2\sin^2(\frac{\omega \Delta t}{2})}\right] - \frac{2}{R}T(R,\omega).$$ (8)

To obtain the time domain implementation of the triangle smoothing derivative, we break down equation 8 into three parts to obtain the following three step implementation:

1. A digital filter analogous to triangle smoothing corresponding to $\left[\frac{-i \Delta t \sin(\frac{R\omega \Delta t}{2})}{2R^2\sin^2(\frac{\omega \Delta t}{2})}\right]$ in equation 8

   $$F(Z)=\frac{Z^N-Z^{-N}}{N^2(1-Z)(1-Z^{-1})},$$ (9)

   implemented in time exactly like triangle smoothing with the slight modification of replacing step (1), the recursion following equation 6, with the following recursion:

   $$y_{t} = x_{t-N}-x_{t+N}.$$ (10)

2. Approximating the derivative of the result of step 1 by taking the second-order central difference. This step corresponds to multiplication by $i\omega$ in equation 8.
3. Subtracting from the result of step 2 the result of smoothing normalized by $\frac{2}{R}$. This step corresponds to the term $-\frac{2}{R}T(R,\omega)$ in equation 8.

To approximate the triangle smoothing derivative function for a non-integer smoothing radius, we use the following interpolation:

$$\frac{\partial T}{\partial R}_{non-integer}(R,w) = [(N+1)-R]\frac{\partial T}{\partial R}(N,w) + [(R-N)]\frac{\partial T}{\partial R}(N+1,w).$$ (11)

The weighting coefficients are justified by matching the second-order Taylor expansion of equations 8 and 11 around the zero frequency:

$$\frac{\partial T}{\partial R}(R,w=0)=\frac{\partial T}{\partial R}_{non-integer}(R,w=0) \approx \frac{-1}{6}Rw^2.$$ (12)

Both triangle smoothing and the triangle smoothing derivative have a straightforward non-stationary implementation in the time domain that is a direct extension of the stationary implementation because all equations depend directly on the radius.