Difference and Averaging Operators

In time domain simulation applications, as in digital filtering, the fundamental operations which may be applied to a time series $u^{n}$ are shifts. The forward and backward shifts are defined as

$$e_{t+}u^{n} = u^{n+1}\qquad e_{t-}u^{n} = u^{n-1}$$ (2.2)

and are to be regarded as applying to the time series $u^{n}$ at all values of the index $n$.

A family of useful difference and averaging operations may be derived from these elementary shifts. For example, various approximation to the first derivative operator may be given as

$$\delta_{t+} = \frac{1}{k}\left(e_{t+}-1\right)$$ (2.3)

$$\delta_{t-} = \frac{1}{k}\left(1-e_{t-}\right)$$ (2.4)

$$\delta_{t+} = \frac{1}{2k}\left(e_{t+}-e_{t-}\right)$$ (2.5)

These are often called the forward, backward, and centered difference approximations, respectively.