同步自發性耳聲傳射是瞬時誘發耳聲傳射衰減後產生的持續震盪，瞬時誘發耳聲傳射則是耳朵的脈衝響應，可以由短暫音刺激產生。同步自發性耳聲傳射具有監控暫時聽力損失的潛力，因此準確估計並取得高訊噪比的訊號是相當重要的。可用於估計瞬時誘發耳聲發射的一種技術稱為奇異質最佳收縮法，此方法是基於奇異值分解，且透過收縮奇異值大小以達成降噪目的，奇異值的最佳收縮比例可透過最小化損失函數求得，例如 Frobenius 範數。然而，此方法在壓抑同步自發性耳聲傳射訊號的隨機雜訊能力表現不佳，因此在本篇研究中，我們提出了改良版本，運用了頻率域漢克爾矩陣的奇異值分解，並透過最佳收縮法來提升估計同步自發性耳聲傳射的準確度。由於漢克爾矩陣反斜對角線上具有相同元素，故此方法首先會在每一個頻率進行漢克爾矩陣的奇異值分解，並且使用最佳收縮器調整奇異值大小，然後再透過於矩陣反斜對角線取中位數的方式，來取得估測訊號，最後再將訊號轉換回到時域。本篇論文發現，改良版本的演算法於雜訊較高的環境下在提升估測訊號訊噪比表現優異，在同步自發性耳聲傳射方面，改良版本的演算法對比資料取平均的基本方法，在僅分析前100次量測下，可以提升2.26 dB訊噪比。

Synchronized spontaneous (SS) otoacoustic emission (OAE) is the response of the cochlea after transient evoked (TE) OAE decays. TEOAE is the impulse response of the cochlea that can be evoked by a short stimulus. SSOAE has shown potential for monitoring hearing loss. As a result, it is crucial to achieve a high signal-to-noise ratio (SNR) in the estimate of the SSOAE signal. One existing technique used to estimate TEOAE is called optimal shrinkage (OS) of singular values. This method is based on singular value decomposition (SVD), and it denoises the data by shrinking the singular values. This optimal shrinker can be derived by minimizing different types of loss functions, such as the Frobenius norm. Nevertheless, the conventional approach does not perform well in suppressing random noise in the SSOAE spectra. In this thesis, we propose a modified algorithm which performs SVD of Hankel matrices in the frequency domain and applies OS techniques to improve the estimation of SSOAE. Hankel matrix is a special type of matrix that has constant values along each descending anti-diagonal. Therefore, the proposed algorithm first performs SVD on each Hankel matrix that is constructed at each frequency bin; subsequently the singular values are optimally shrunk. Following that, we take the median over each descending anti-diagonal to have an estimate of the SSOAE spectrum. Finally, we compute the inverse Fourier transform to acquire an estimate in the time domain. Our results suggest that this modification has the potential to be an effective noise reduction algorithm for use in noisy environments. When only the first 100 trials were used in the denoising, the proposed algorithm could enhance the SNR by 2.26 dB from an artifact rejection method that is based on calculating the mean.

Contents
1 Introduction 1
1.1 Transient Evoked Otoacoustic Emission and Noise-Induced Hear-
ing Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Clinical Potentials of Synchronized Spontaneous Otoacoustic Emis-
sion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Related Works 6
3 Methods 9
3.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Measurement Protocols: Linear vs. Nonlinear . . . . . . . . . . . . 9
3.3 Signal Modeling and Mathematical Notations . . . . . . . . . . . . 10
3.4 Artifact Rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.5 Optimal Shrinkage of Singular Values . . . . . . . . . . . . . . . . 11
3.6 Frequency Domain Processing . . . . . . . . . . . . . . . . . . . . 13
3.7 Hankel Matrix in the Local Frequency Domain . . . . . . . . . . . 14
3.8 Subjects and Materials . . . . . . . . . . . . . . . . . . . . . . . . 18
3.9 Evaluation of Denoising Algorithms . . . . . . . . . . . . . . . . . 18
4 Experiments and Results 20
4.1 SSOAE Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.1.1 Noise Auto-correlation Estimation . . . . . . . . . . . . . . 20
4.1.2 Noise Spectrum Estimation . . . . . . . . . . . . . . . . . . 21
4.1.3 Find Peaks via Confidence Interval (CI) . . . . . . . . . . . 22
4.2 Denoised Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2.1 Varying Number of Frames . . . . . . . . . . . . . . . . . . 26
4.2.2 Varying Portion of the Recording . . . . . . . . . . . . . . 26
4.2.3 Varying Range of the Signal . . . . . . . . . . . . . . . . . 27
5 Discussion 34
5.1 Auto-Correlation of Real and Simulation Data . . . . . . . . . . . . 34
5.2 Singular Values Distribution . . . . . . . . . . . . . . . . . . . . . 35
5.3 SVD-Based Noise Standard Deviation Estimation . . . . . . . . . . 37
6 Conclusions 40
References 41
Appendix 43
A.1 Suggestions from the oral defense committees . . . . . . . . . . . . 43
A.1.1 吳炤民教授 . . . . . . . . . . . . . . . . . . . . . . . . . . 43
A.1.2 賴穎暉教授 . . . . . . . . . . . . . . . . . . . . . . . . . . 43
A.1.3 白明憲教授 . . . . . . . . . . . . . . . . . . . . . . . . . . 43
A.1.4 劉奕汶教授 . . . . . . . . . . . . . . . . . . . . . .

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