Recently, a system concept for joint communication and positioning has been proposed by the authors. Channel parameter estimation (CPE) is the core part of this system proposal. Parameters of the physical channel, which can be exploited for positioning, are estimated based on the assumption that a priori knowledge about pulse shaping and receive filtering is available. At the same time, channel estimates of the equivalent discrete-time channel model, which are needed for data detection, are obtained inherently. This article focusses on the positioning part of the system proposal. Performance limits for CPE in terms of Cramer-Rao lower bounds are determined for different channel models. The influence of oversampling and of different channel characteristics is investigated. Oversampling proves especially helpful in dense multipath scenarios, which are most challenging. Based on the presented results, oversampling with a factor of two is recommended in order to improve the positioning accuracy. Excessive oversampling like in conventional global positioning system receivers is not necessary.