Campbell derived and defined a quantity called the coefficient rate of a random process that involves the process spectral entropy. In this correspondence, his interpretation is substantiated with two new derivations. One derivation tightens the connection to source bandwidth, while the second derivation implies a specific approach to adaptive coefficient selection in realization-adaptive approaches to compression. After a discussion on the role the coefficient rate plays in adaptive source coding, a quantity called Campbell bandwidth is defined based on its connection to source bandwidth and is contrasted with Fourier bandwidth and Shannon bandwidth. The connection between coefficient rate and reverse water-filling from rate distortion theory is also demonstrated.

In 1960, Campbell derived a quantity that he defined as the coefficient rate of a random process that involves the process spectral entropy. However, no potential applications of the coefficient rate were identified. Two new derivations of Campbell's rate coefficient rate are presented. One derivation solidifies the interpretation of this quantity as a coefficient rate and allows definition of an effective bandwidth for the process. The second derivation implies a new approach for realization adaptive source compression. The coefficient rate can be used for realization adaptive coefficient selection in a sequence of source representations. Furthermore, the effective bandwidth is designated as Campbell bandwidth and contrasted with Fourier bandwidth and Shannon bandwidth. Several specific examples are presented that illustrate the differences among the three quantities.