# Blind Phase Search with Angular Quantization Noise Mitigation for Efficient Carrier Phase Recovery
## 2 Principle of the Proposed Angular Quantization Noise Filter in F-BPS  
The operation principle of the proposed F-BPS scheme is illustrated in Figure 1. As in the C-BPS, a block of 2M + 1 symbols is rotated by a number of test phases B. The resulting rotated blocks of symbols for each of the test phases are then fed into a decision circuit module where the closest constellation points in the original constellation are determined. The squared distances of 2M + 1 symbols to its closest constellation points in the original constellation are calculated for each of the rotated block of symbols. The sum of 2M + 1 square distances is considered for averaging the impact of additive white Gaussian noise (AWGN), which is known as the C-BPS block filter [2]. The angular rotation providing the minimum distance metric sum is considered to be the phase noise estimator for the symbol in the middle of the block. An unwrap module is then required to minimize cycle slip occurrence. The phase noise estimator after the unwrap module corresponds to the final estimator in the C-BPS algorithm (\(\hat{\beta}\)). It is worth noting that \(\hat{\beta}\) has been selected from a set of discrete angular values. Therefore, \(\hat{\beta}\) contains quantization noise and, when it is employed for phase noise compensation as in the C-BPS, results in angular quantization noise in the corrected signal. Figure 2a shows the frequency noise spectrum (FN-PSD) of the phase tracked by the C-BPS algorithm (\(\hat{\beta}\)) for different values of test phases in a sliding window CPR approach. Increased frequency noise in the C-BPS phase estimator is observed with decreased number of test phases. This is due to the discrete phase search nature of the C-BPS algorithm, introducing quantization noise in its phase noise estimator. The first observed dip in Figure 2a (see \(\beta\) = 64 curve) corresponds to the C-BPS block filter bandwidth which defines the estimation bandwidth of the C-BPS. It is observed how the out-of-band frequency components increase as the number of employed test phases reduces due to angular quantization noise. Figure 2b shows the temporal quantized phase evolution tracked by the C-BPS algorithm (\(\hat{\beta}\)) for different numbers of test phases and a sliding window CPR approach. The discrete angular jumps due to angular quantization in the C-BPS are bigger for lower values of employed test phases which translates into larger quantization noise as illustrated correspondingly in  
Figure 1: Block diagram of the C-BPS CPR scheme and the proposed F-BPS scheme employing the angular quantization noise filter. Figure 2a. The proposed F-BPS scheme shown in Figure 1 uses an angular quantization noise filter consisting of a low pass filter operation performed on the C-BPS phase noise estimator to mitigate its out-of-band angular quantization noise. The proposed angular quantization noise filter in the F-BPS must be designed such that it removes the out-of-band noise enhancement while preserving the in band information of the original C-BPS phase estimator contained within the C-BPS block filter bandwidth (see Figure 2a). The improved F-BPS phase noise estimator (\(\hat{\beta}_{f}\)) after the proposed angular quantization noise filter is then used to compensate for the phase noise of the original signal as illustrated in Figure 1.  
In practice, the need of the proposed filter is justified as the angular resolution of the C-BPS phase estimator (\(\beta\)) is significantly smaller than the resolution of the digital signal processor used to perform the algorithm. As an example, 32 and 64 possible test phase values or less (equivalent to <5 and <6 resolution bits, respectively) are recommended for CPR in the C-BPS for 16 QAM and 64 QAM respectively which, in a practical implementation of the algorithm, might be much lesser [2; 3; 4; 5; 6; 7; 8; 9; 10; 11]. However, current technologies on ADC/DAC integrated in the ASICs as part of modern transceivers are shown to be able to achieve 8 bits of resolution [13].