A Relationship Between Total Radiated Power of Difference Patterns and Sum Patterns Using Variance Beamwidth

For a given current distribution along a line source, the relationship between the line source's total radiated power and the radiated power of a line source whose current distribution is the derivative of the original current distribution is established herein using the notion of variance beamwidth. Current distributions of specific interest are symmetrical and antisymmetrical distributions that produce "sum" and "difference" patterns, respectively. An example is provided by using the triangular distribution as the symmetrical current distribution and the bipolar pulse as the antisymmetrical distribution. Error estimates are also provided that highlight the limitations of using Parseval's identity to approximate the total radiated power of a line source. This limitation is exemplified by using the bipolar pulse and uniform distributions as a case study. Numerical data are plotted and assessed to substantiate all theoretical claims.