Abstract: The inner tracking system of the ATLAS experiment will be upgraded to a full silicon detector in 2030 for HL-LHC. The new tracking system is called ITk, the Inner Tracker. The ITk requirements include operational efficiency higher than 99% and noise hit occupancy smaller than 0.1%. During the pre-production phase of the ITk project, many short-strip modules were observed to exhibit so-called “Cold Noise (CN)”, wherein clusters of strips displayed very high noise when the modules were operated at temperatures below-35 degrees C. To investigate the CN impact and ensure the quality of module production, huge amount of effort have been put in by the collaboration. This paper focuses on the impact of CN on the tracking performance by examining two short strip modules that exhibit CN: one is non-irradiated, while the other one has been irradiated to the maximum expected end-of-lifetime fluence. For each module, the global and single strip tracking performance are evaluated. The global performance study shows that the non-irradiated module can be operated within specifications with a threshold around 1fC, but it is not possible to operate the irradiated module as required. In the single strip analysis, it was found that while CN does not affect the charge collection, it reduces the operating window and leaves less margin for detector operation. In the non-irradiated module, less than 3% of strips fail the detector requirements in the CN regions. For the irradiated module, about 20% of strips fail the requirements in the low CN region and around 52% fail in the high CN region. The fraction of strips that cannot operate due to CN throughout their lifetime can be predicted according to the measured noise at the required noise occupancy level and its expected median collected charge. In cases when the noise hit occupancy caused by CN is kept below 1% with a threshold smaller than 0.45 fC, at least 60% of strips could meet the operating requirements by the end of detector's lifetime. Thus, the module is likely to satisfy the operating requirements in terms of global efficiency and global noise occupancy.