Functionals of Brownian bridges arising in the current mismatch in D/A-converters

Digital-to-analog converters (DAC) transform signals from the abstract digital domain to the real analog world. In many applications, DAC’s play a crucial role. Due to variability in the production, various errors arise that influence the performance of the DAC. We focus on the current errors, which describe the fluctuations in the currents of the various unit current elements in the DAC. A key performance measure of the DAC is the Integrated Non-linearity (INL), which we study in this paper. There are several DAC architectures. The most widely used architectures are the thermometer, the binary and the segmented architectures. We study the two extreme architectures, namely, the thermometer and the binary architectures. We assume that the current errors are i.i.d. normally distributed, and reformulate the INL as a functional of a Brownian bridge. We then proceed by investigating these functionals. For the thermometer case, the functional is the maximal absolute value of the Brownian bridge, which has been investigated in the literature. For the binary case, we investigate properties of the functional, such as its mean, variance and density. 1 Current mismatch in digital-to-analog converters Digital-to-Analog converters (DAC) transform signals from the abstract digital domain to the real analog world. For many applications, this conversion enables the usage of the computational power of robust digital electronics. For example, digital audio and video, digital control, and telecommunications are fields that require digital-to-analog conversion. The advantageous intelligence of the applications in these fields is implemented with digital logic, e.g. microprocessors, and used via DA conversion in the real analog world. However, the DAC errors, at the end of the application chain, may decrease the performance of the whole system. Therefore, predicting and controlling these errors is crucial. This requirement is further emphasized in the highly integrated mixed-signal Systems-on-a-Chip (SoC), as we will explain now. Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands. [email protected], [email protected] Department of Electrical Engineering, Eindhoven University of Technology, EH 5.15, 5600 MB Eindhoven, The Netherlands. [email protected]