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### Frequency Domain Interpretation

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: • the mapping maps all real frequencies to real frequencies uniquely. In partcular dc dc, and infinite frequency maps to .

One way of examining the frequency mapping more closely is by looking at on the unit circle, i.e., where . This yields: • notice in particular the behaviour of the mapping near dc ( = 0): 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).

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