The equation for the inductor, assuming zero initial conditions, transforms to
where is the complex frequency variable. Taking transforms of the sequences and in the backward-difference scheme yields:
Thus we can think of our discretized scheme as one obtained under the mapping . So here we are mapping from the plane to the plane. The following figure illustrates where real continuous time frequencies (the axis) are mapped:
dc ( ) mapped to dc ( )
infinite frequency mapped to ( )
We can write the scheme for the trapezoid rule as follows:
One way of examining the frequency mapping more closely is by looking at on the unit circle, i.e., where . This yields:
where, since is odd, there are no even-order terms in its series expansion. Thus, the trapezoid rule is a second-order accurate approximation to a derivative, in the limit of small (i.e., near dc).