The ability to efficiently assess and track the utility of each sensor signal is crucial to reduce the energy consumption in a wireless sensor network (WSN), e.g., by putting the sensors with low utility to sleep. Methods to track the sensor signal utility have been described for several multichannel signal estimation methods. For linear minimum mean squared error (LMMSE) estimation, the utility of a sensor signal is defined as the predicted increase in the minimum mean squared error when the sensor would be shut down. However, rather than making such a binary decision, more flexible energy-saving methods could be considered where a sensor changes internal parameters such as, e.g., the number of bits per sample, which results in noise injection in the transmitted sensor signal. We propose a generalization of the original definition of sensor signal utility to include this effect, and we show that it can be efficiently computed and tracked at hardly any computational cost compared to the already available LMMSE estimator. In addition, we illustrate how it can be used to assign a number of bits to each sensor with a greedy approach. Simulation results show that a greedy assignment based on the proposed generalized utility leads to improved results compared to the original utility measure.