A Note on Stable, Physically Realizable, Linear, Time Invariant Systems

A time invariant system is said to be physically realizable if its unit impulse response
h(t)
is zero for
t < 0
. It is often assumed, in treatments of physical realizability, that the Fourier transform of
h(t)
is square integrable
(L_2(-\infty, \infty))
. Some use is then made of the regularity of the transform, for real frequencies, to derive further properties of the transform. The condition that the system be stable leads to a different assumption, that
|h(t)|
be integrable (is
L(-\infty, \infty)
); this implies only uniform continuity of the transform for real frequencies. Physically realizable, stable systems are discussed in this note, and several necessary, and a few sufficient, conditions on the Fourier transform of
h(t)
are summarized. In particular, the behavior of the transform for complex frequencies, the Bode relations, and the Paley-Wiener criterion, are reviewed.