Explicit formula for elements in ideal pulse compression transmission line networks

The explicit formula
Z_m =m(m +1)/2
is proved for the characteristic impedances of a cascade of
n
transmission lines whose open-circuited voltage transfer function is optimally designed to compress a time-limited signal by the factor
n+1
. In the proof, use is made of the relationship between the synthesis procedure and the generation of sequences of polynomials orthogonal on the unit circle through a recurrence formula. Furthermore, it is shown that the impedance variation in the resulting realization is very close to the global minima for these ideal pulse compression networks with the additional desirable feature of a monotonic line realization.