Bilinear Transform derivative

A reference for the theory of the bi-linear method of filter design can be found in Gold and Rader (5). Mathematically, the Z-transform for the derivative is given as

$$Y(Z)=\dfrac{2}{\Delta t}\cdot \dfrac{\left( 1-Z \right) }{\left( \left( 1+stab \right) +Z \right) } \cdot X(Z)$$ (49)

where $\Delta t$ is the sample interval, $X(Z)$, is the input signal Z-transform, and $Y(Z)$ is the output (ie. derivative of X), Z-transform. Feedback is used in an auto-regressive-moving-average (ARMA) operator to realize this equation in the time domain:

$$y_j=\left( \dfrac{2}{\Delta t} \cdot \left( x_j - x_{j-1}\right) - y_{j-1} \right) / \left( 1 + stab \right) ~~.$$ (50)

Figure 63: Plot of file bdifwavV.seg, differentiated wavV.seg simulates what a velocity geophone might see. Only near offset signals are shown for easier comparison.