Under the bilinear transform, the maps to (half the sampling rate). It is therefore no surprise that given ( ), inspection of Fig.F.37 reveals that any alternating sequence (sinusoid sampled at half the sampling rate) will circulate unchanged in the loop on the right, which is now isolated. Let denote this alternating sequence. The loop on the left receives and adds to it, i.e., . If we start out with and , we obtain , or
However, the physical spring force is well behaved, since
As a check, the mass force is found to be
which agrees with the spring, as it must.