基於埃爾米特定理的單位元可適性自相關函數估計與分析

Abstract

單位元自相關函數重建 (One-bit ACF reconstruction) 旨在利用單位元量化數據來估計原始訊號的自相關函數 (autocorrelation function)。目前已有一些方法重建位量化訊號之自相關函數，如: 修正反正弦律 (modified arcsine law) 和埃爾米特定理 (Hermite law)。修正反正弦律選用時變閥值的量化器，且涉及積分項；埃爾米特定理選用恆定閥值量化器，卻涉及無窮多項加總。為了緩解無窮項加總，有學者提出埃爾米定律的諧波近似 (harmonic approximation)。然而，修正反正弦律以及埃爾米特定理在資料量不足的情況下，估計的均方誤差 (MSE) 浮動都很大。 在本文中，我們引用諧波近似來逼近埃爾米特定理，以降低無窮級數的計算複雜度，同時保留恆定量化閾值的優勢。選用恆定量化閾值，我們就不需要產生隨機程序作為位量化器的閥值。我們基於諧波近似構造最佳化問題，該問題的最佳解即為提出的自相關函數估計子。其目標函是對應於單位量化資料之平均值與自相關函數估計子的均方誤差線性組和。 此外，我們應用變數變換，以消除估計自相關函數時對標準差與相關係數的自然限制。透過這種目標函數的設計，可以一次更新原始的自相關函數，而不需分為兩步驟進行。在本論文中，我們採用數值最佳化求解問題。接下來，我們利用指數加權移動平均（EWMA）來設計單位量化資料的區塊可適性 (block adaptive) 平均值與相關矩陣估計子。為了降低計算複雜度並可適性地更新自相關函數估計結果，我們亦將隨機梯度下降法（SGD）應用於該最佳化問題。 我們證明了在本論文中基於 EWMA 的估計子是不偏的 (unbiased)。此外，我們進一步推導了平均值估計子的變異數，其變異數受區塊長度控制。最後透過模擬實驗證明，與修正反正弦律相比，我們的估計子無需蒐集大量樣本即可達到良好的效果。提出的估計子每次更新所需的運行時間不到修正反正弦律的一半，除此之外，在適應信號分布的快速變化方面也更具優勢。

One-bit autocorrelation (ACF) estimation reconstructs the original signal’s ACF with received one-bit quantized data. Some present theorems specify the relations among one-bit correlation, such as the modified arcsine law and Hermite law. The modified arcsine law adopts time-varying quantization thresholds and involves integral-based terms. On the other hand, Hermite law employs a constant quantization threshold but involves an infinite sum. The harmonic approximation of the Hermite law has been proposed to relieve the endless sum. However, the modified arcsine and Hermite laws exhibit significant mean-square error (MSE) fluctuations in estimation for insufficient data. In this thesis, we employ a harmonic approximation of the Hermite law to reduce the computational complexity of infinite series while retaining the advantage of a positive threshold. With a constant threshold, we do not have to generate a random process for the quantizer. We formulate the optimization problem based on the harmonic approximation. The optimal solution to the problem is the proposed ACF estimator. The objective function is a linear combination of MSEs corresponding to mean and ACF estimators of one-bit data. Besides, we apply variable transformations to eliminate the natural constraints on standard deviation and correlation coefficients estimated to recover ACF. With the objective function design, the original ACF can be estimated in one step rather than two stages. In this thesis, the problem is addressed using numerical optimization. Next, we utilize the exponentially weighted moving average (EWMA) to design block adaptive mean and correlation matrix estimators of one-bit data. To reduce computational complexity and adaptively update the ACF estimation, we also apply stochastic gradient descent (SGD) to the optimization problem. In the thesis, we prove that the EWMA-based estimators are unbiased. We further derive the variance of the mean estimator, which is controlled by block length. Finally, through simulation experiments, we demonstrate that our estimator does not require collecting many samples compared to the modified arcsine law. Moreover, the run time of each update is less than half that of the modified arcsine law. The proposed estimator is also more capable of adapting to rapid changes in the signal distribution.