在壓縮感測 (Compressive Sensing, CS) 技術中，其核心目標是以低於香農 奈奎斯特取樣速率 (Shannon-NyquistSamplingRate) 的方式，實現對原始信號的 高精度重建。為了進一步節約資源並提高數據處理效率，我們深入探討其中的單 一位元壓縮感測 (1-BitCS) 技術。該技術的特點是每次感測僅用單一位元表示。
在本文中，我們構建了一個單一位元壓縮感測系統模型，並考慮感測與傳輸 過程中可能出現的位元翻轉錯誤。此種情況在現實場景中常見，且對現有的重建 方法產生了顯著影響。基於此，我們深入研究在接收端資訊有限的條件下，是否 能獲取額外有助於重建的資訊。我們改進了現有的重建方法，提出一種以估計原 信號非零集合為核心的構想。在進行信號重建之前，找出原信號的非零位置，並 針對這些位置進行信號還原，從而有效減??重建信號與原信號非零位置之間的差 異。
當估計原信號非零集合的效果良好且能夠提升重建表現時，我們進一步探討 了原信號非零值數量是否為必要的先驗資訊。在未知原信號非零值數量的情況下， 我們基於統計分析，利用原信號零與非零位置的估計值差異來判斷其是否屬於非 零集合。模擬結果顯示，零與非零位置之間的估計值差異越大，對非零集合的判 斷越準確。我們將所得的非零集合資訊融入現有方法中，實現了更精確的信號重 建。該方法相較於現有方法，不僅在重建效果上更為精準，對抗位元翻轉的能力 也顯著提升，且更符合現實應用場景的需求。
;In compressive sensing (CS) technology, the goal is to reconstruct the original signal using a sampling rate lower than the Shannon-Nyquist rate. To conserve resources and enhance data processing efficiency, we discuss 1-bit Compressive Sensing (1-bit CS) further. 1-bit CS is characterized by representing each measurement with only a single bit.
In this thesis, we construct a 1-bit CS system model and consider the occurrence of bit-flipping error during the sensing and transmission processes. This condition is commonly encountered in real-world scenarios and has a significant impact on existing reconstruction methods. Bit-flipping significantly impacts existing reconstruction methods. Based on this, we investigate whether it is possible to obtain additional information that aids reconstruction under the condition of limited information at the receiver. We improve existing reconstruction methods by proposing a concept on estimating the support of the original signal. Before reconstructing the signal, we first locate the support of the original signal and then reconstruct the signal based on these positions, effectively reducing the difference between the support of the estimated signal and the original signal.
When the support of the estimated signal is accurate and improves reconstruction performance, we further explore whether prior knowledge of the number of non-zero values in the original signal is necessary. In cases where the number of non-zero values in the original signal is unknown, we use statistical analysis to differentiate estimated values between zero and non-zero positions, determining whether they belong to the support. The results indicate that the greater the difference between the estimated values of zero and non-zero positions, the more accurate the judgment of the support. We integrate the obtained support information into existing methods to achieve more precise signal reconstruction. Compared to existing methods, the proposed approach not only provides more accurate reconstruction results but also significantly enhances resistance to bit-flipping error. Additionally, it meets the demands of real-world applications better.
