單元量化功率和自我相關函數估計: Cramér-Rao界和自適應方法

Abstract

在到達方向 (DOA)、頻譜分析和雷達應用中，單元類比數位轉換器的自相關估計越來越受歡迎。因為單元類比數位轉換器的成本效益高、功耗低和硬體設計簡單。傳統上，我們利用埃爾米特定律 (Hermite Law) 進行自相關估計，但其準確性和效率並不理想。以，我們的研究改進和優化埃爾米特定律演算法，以提高自相關估計的準確性以及加快演算法的速度。 為了提高埃爾米特定律演算法在自相關估計中的準確性，我們使用克拉馬-羅限（Cramér-Rao Bound）分析估計功率和相關係數的最佳臨界值 (threshold)。結果顯示，功率和相關係數的最佳臨界值並不一致。因此，選擇一個臨界值讓功率估計和相關係數估計同時達到最佳是困難的。然而，結合兩者克拉馬-羅限的結果，在相同臨界值下，我們發現 0.7085 會有最低的克拉馬-羅限。因此，我們認為0.7085 作為臨界值的選擇是最好的。 此外，我們使用迭代方法和近似埃爾米特定理來加速埃爾米特定理。對於前者，我們使用迭代方法避免計算多個根。對於後者，我們將埃爾米特定理簡化為閉合形式的近似埃爾米特定理。兩種方法都可以減少計算時間。 最後，我們進行了一個模擬實驗來分析迭代方法。其結果表明，迭代方法比原先的埃爾米特定律演算法快 3 倍多。

One-bit autocorrelation estimation has gained attention in the direction of arrival, spectral analysis, and radar applications, attributed to its cost efficiency, lower power consumption, and simpler hardware design. Traditionally, researchers have employed the Hermite Law for autocorrelation estimation, but its accuracy and efficiency are suboptimal. Our study primarily focuses on improving and optimizing the Hermite Law algorithm to enhance the accuracy of autocorrelation estimation, reduce hardware costs, and speed up the algorithm. To improve the accuracy of the Hermite Law algorithm in autocorrelation estimation, we use the Cramér-Rao Bound (CRB) to analyze the optimal threshold for estimating power and correlation coefficient. The results show that the optimal power and correlation coefficient thresholds are different. Therefore, choosing a threshold that simultaneously allows for the best power and correlation coefficient estimation is challenging. However, by combining CRB results, we find that 0.7085 is the optimal point in the CRB results, indicating the minimum values among the worst values of CRB. Thus, 0.7085 is the best choice for the optimal threshold. Additionally, we use iterative methods and the Approximate Hermite Law to speed up the Hermite Law. For the former, we used iterative methods to avoid calculating multiple roots. For the latter, we simplified the Hermite Law into the Approximate Hermite Law as a closed form. Both approaches reduce computational time. Finally, we conducted a simulation experiment to analyze the iterative methods. The results indicated that the iterative methods are over 3 times faster than the Hermite Law algorithm.