Analysis of sampling and noise error on the energy measurement of direct digitally acquired pulsed signals

The precise measurement of the energy content of a pulse signal, limited in time and bandwidth, is of great interest in many applications, and can be achieved by direct digitization of the analog signal, followed by calculation of the mean squared amplitude of the data samples. The beam position measurement in a particle accelerator is among the applications, and this technique is proposed for the next generation of read-out electronics for the beam position monitors of the CERN Large Hadron Collider. The analog-to-digital converter is a key component of the signal processing chain, and defines the performance of the measurement through its characteristics and features, in particular the maximum sampling rate and the effective number of bits. This paper analyzes the error on the measurement of the signal energy for a specific pulse waveform within a given time window, caused by the analog-to-digital converter sampling effects due to the finite, unsynchronised sampling rate. In a following step the result is combined with the effect of a noisy analog-to-digital of finite resolution. This gives a general expression for the signal-to-noise ratio, which can be used to find the optimal trade-off between sampling rate and resolution of an analog-to-digital converter for any given single pulse waveform.