FDA in the Frequency Domain

Viewing Eq.(7.2) in the frequency domain, the ideal differentiator transfer-function is $H(s)=s$ , which can be viewed as the Laplace transform of the operator $d/dt$ (left-hand side of Eq.(7.2)). Moving to the right-hand side, the z transform of the first-order difference operator is $(1-z^{-1})/T$ . Thus, in the frequency domain, the finite-difference approximation may be performed by making the substitution

$$s \eqsp \frac{1-z^{-1}}{T} \protect$$ (8.3)

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

$$z \eqsp \frac{1}{1 - sT} \eqsp 1 + sT+ (sT)^2 + \cdots \, .$$

As discussed in §8.3.1, the FDA is a special case of the matched $z$ transformation applied to the point $s=0$ .

Note that the FDA does not alias, since the conformal mapping $s = {1 - z^{-1}}$ is one to one. However, it does warp the poles and zeros in a way which may not be desirable, as discussed further on p. below.