# Towards implementing multi-channels, ring-oscillator-based, Vernier time-to-digital converter in FPGAs: key design points and construction method

## TL;DR

This paper presents a simplified method for implementing multi-channel ring-oscillator-based Vernier TDCs in FPGAs, achieving high resolution with reduced complexity, validated by a 32-channel prototype with excellent performance metrics.

## Contribution

It introduces an efficient fine time interpolator construction method and key design points that reduce implementation complexity for multi-channel RO-based Vernier TDCs in FPGAs.

## Key findings

- Resolution range of 23-37 ps across channels
- DNL within -0.4 to 0.4 LSB, INL within -0.7 to 0.7 LSB
- Successful implementation of 32-channel TDC prototype

## Abstract

For TOF positron emission tomography (TOF PET) detectors, time-to-digital converters (TDCs) are essential to resolve the coincidence time of the photon pairs. Recently, an efficient TDC structure called ring-oscillator-based (RO-based) Vernier TDC using carry chains was reported by our team. The method is very promising due to its low linearity error and low resource cost. However, the implementation complexity is rather high especially when moving to multi-channels TDC designs, since this method calls for a manual intervention to the initial fitting results of the compilation software. In this paper, we elaborate the key points toward implementing high performance multi-channels TDCs of this kind while keeping the least implementation complexity. Furthermore, we propose an efficient fine time interpolator construction method called the period difference recording which only needs at most 31 adjustment trials to obtain a targeted TDC resolution. To validate the techniques proposed in this paper, we built a 32-channels TDC on a Stratix III FPGA chip and fully evaluated its performance. Code density tests show that the obtained resolution results lie in the range of (23 ps ~ 37 ps), the differential nonlinearity (DNL) results lie in the range of (-0.4 LSB ~ 0.4 LSB) and the integral nonlinearity (INL) results lie in the range of (-0.7 LSB ~ 0.7 LSB) for each of the 32 TDC channels. This paper greatly eases the designing difficulty of the carry chain RO-based TDCs and can significantly propel their development in practical use.