Finite Difference Approximation
A finite difference approximation (FDA) approximates derivatives with finite differences, i.e.,
for sufficiently small .8.5
Equation (7.2) is also known as the backward difference approximation of differentiation.
Viewing Eq.(7.2) in the frequency domain, the ideal differentiator transfer-function is , which can be viewed as the Laplace transform of the operator (left-hand side of Eq.(7.2)). Moving to the right-hand side, the z transform of the first-order difference operator is . Thus, in the frequency domain, the finite-difference approximation may be performed by making the substitution
in any continuous-time transfer function (Laplace transform of an integro-differential operator) to obtain a discrete-time transfer function (z transform of a finite-difference operator).
The inverse of substitution Eq.(7.3) is
It is convenient to think of the FDA in terms of time-domain difference operators using a delay operator notation. The delay operator is defined by
The obvious definition for the second derivative is
However, a better definition is the centered finite difference
where 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 sample at all frequencies.