The inherent discrete phase search nature of the conventional blind phase search (C-BPS) algorithm is found to introduce angular quantization noise in its phase noise estimator. The angular quantization noise found in the C-BPS is shown to limit its achievable performance and its potential low complexity implementation. A novel filtered BPS algorithm (F-BPS) is proposed and demonstrated to mitigate this quantization noise by performing a low pass filter operation on the C-BPS phase noise estimator. The improved performance of the proposed F-BPS algorithm makes it possible to significantly reduce the number of necessary test phases to achieve the C-BPS performance, thereby allowing for a drastic reduction of its practical implementation complexity. The proposed F-BPS scheme performance is evaluated on a 28-Gbaud 16QAM and 64QAM both in simulations and experimentally. Results confirm a substantial improvement of the performance along with a significant reduction of its potential implementation complexity compared to that of the C-BPS.