One-sided functions

  All physical systems share the property that they do not respond before they are excited. Thus the impulse response of any physical system is a one-sided time function (it vanishes before t = 0). In system theory such a filter function is called realizable. In wave propagation this property is associated with causality in that no wave may begin to arrive before it is transmitted. The lag-time point t = 0 plays a peculiar and an important role. For this reason, many subtle matters will be much more clearly understood with sampled time than with continuous time. When a filter responds at and after lag time t = 0, we will say the filter is realizable or causal. The word causal is appropriate in physics where stress may cause (practically) instantaneous strain and vise versa, but one should revert to the more precise words realizable or one-sided when using filter theory to describe economic or social systems where simultaneity is quite different from cause and effect.