Delay Operator Notation

It is convenient to think of the FDA in terms of time-domain difference operators using a delay operator notation. The delay operator $d$ is defined by

$$d^k x(n) \eqsp x(n-k).$$

Thus, the first-order difference (derivative approximation) is represented in the time domain by $(1-d)/T$ . We can think of $d$ as $z^{-1}$ since, by the shift theorem for $z$ transforms, $z^{-k} X(z)$ is the $z$ transform of $x$ delayed (right shifted) by $k$ samples.

The obvious definition for the second derivative is

$${\hat{\ddot x}}(n) \eqsp \frac{(1-d)^2}{T^2} x(n).$$ (8.4)

However, a better definition is the centered finite difference

$${\hat{\ddot x}}(n) \isdefs \frac{(d^{-1}-1)(1-d)}{T^2} x(n) \eqsp \frac{d^{-1}-2+d}{T^2}x(n) \protect$$ (8.5)

where $d^{-1}$ denotes a unit-sample advance. This definition is preferable as long as one sample of look-ahead is available, since it avoids an operator delay of one sample. Equation (7.5) is a zero phase filter, meaning it has no delay at any frequency, while (7.4) is a linear phase filter having a delay of $1$ sample at all frequencies.