On the Capacity of an Ensemble of Channels with Differing Parameters

Cables are usually analyzed as if all of the components had a particular set of parameters (for example, nominal, worst case, and so on). Because of manufacturing tolerances, installation differences and various environmental effects, however, transmission parameters actually vary from pair to pair. To account for these variations, this paper takes an 415 416 T H E BELL SYSTEM T E C H N I C A L J O U R N A L , MARCH 1970 approach based on information theory. We consider the cable network to be a statistical population of channels which have parameters that vary from channel to channel. We propose a quality measure for the network based on this model. After defining channel capacity, we present the suggested capacity definition for a group of channels (as outlined in the Abstract). Using this definition, we provide an example in Section IV in which the capacity of a group of channels with stationary Gaussian inputs, additive noise and crosstalk is derived. The capacity for a single such channel is found in the Appendix. In Section 4.1 the crosstalk is assumed to differ from channel to channel; in Section 4.2 the channel attenuation is assumed variable; and in Section 4.3 both the crosstalk and attenuation vary from channel to channel. These results indicate the trade-off between design rate for a transmission system and the expected fraction of channels which will be capable of error-free transmission at the design rate. This technique can also be used to evaluate different parameter distributions as may result from tighter production controls.