在時間延遲估測(time delay estimation, TDE)的問題中, 當訊號源因具有相關性而呈
現特殊頻譜分佈的時候, 使用傳統的時域適應性限制性和非限制性最小均方值時間延遲
估測演算法 (LMS TDE algorithm), 其收斂速度將會變得緩慢, 使得時間延遲估測的品
質嚴重變差。事實上, 收斂速度受到訊號源頻譜分佈的影響很大, 而且, 時間延遲估測的
品質會受到背景雜訊的影響。
為了解決以上所提到的問題, 在本論文中, 提出了一種轉換域適應性限制性的方法,
稱為適應性限制性離散餘弦轉換最小均方值時間延遲估測演算法 (adaptive constrained
DCT-LMS TDE algorithm)。我們將證明利用這個新的限制性演算法, 使用直接延遲估
測公式 (direct delay estimation formula) 來做非整數的時間延遲估測後, 其時間延遲估
測結果將會比傳統的時域限制性和非限制性演算法及轉換域非限制性演算法還要好。
最後,為了更進一步降低在適應性非限制性離散餘弦轉換最小均方值演算法的特徵
值分佈 (eigenvalue spread), 我們將觀察利用梯形架構 (escalator structure)來實現哥瑞姆
-史密特正交化(Gram-Schmidt Orthogonalization)的方式。結果發現在沒有使用限制性方
式的時候會有時間延遲估測的偏差產生, 因此加了哥瑞姆-史密特正交化的方式無法去
除背景雜訊所帶來的影響。

n the problem of time delay estimation (TDE), the desired source signals of interest are
correlated and with a specific spectral distribution. In such cases, the convergence speed using
the conventional approaches, viz., time domain adaptive constrained and unconstrained LMS
TDE algorithms, becomes slowly and the performance of TDE will be degraded, dramatically.
In fact, the convergence rate depends highly on the distribution of spectral density of the
desired signal sources. Also, the performance of TDE is affected by the background noises,
accordingly.
To circumvent the problem described above, in this thesis, a transformed domain adaptive
constrained filtering scheme, refers to the constrained adaptive DCT-LMS algorithm, for TDE
is devised. We show that this new proposed constrained algorithm, with the so-called direct
delay estimation formula, for non-integer TDE does perform better than the conventional time
domain adaptive constrained and unconstrained LMS TDE algorithms and the unconstrained
adaptive DCT-LMS TDE algorithm.
Finally, to further reduce the spread of eigenvalue in the unconstrained adaptive
DCT-LMS algorithm, the Gram-Schmidt orthogonalizer approach realizing by the adaptive
Escalator is investigated. It indicates that bias of TDE will occur without using the constraint
of weight vector. That is, it could not be used to alleviate the effect due to background noises.

Contents
Acknowlegement i
Abstract ii
Contents iii
List of Figures and Tables v
Chapter 1 Introduction 1
Chapter 2 Time Domain Adaptive Constrained LMS Filtering Algorithm for
Time Delay Estimation
2.1 Introduction 4
2.2 Conventional Adaptive Constrained LMS Filtering Algorithm for TDE 5
2.2.1 The Conventional Adaptive LMS TDE Filtering Algorithm 8
2.2.2 The Conventional Adaptive Constrained LMS TDE Filtering Algorithm 9
2.3 Computer Simulation Results 11
Chapter 3 Adaptive Constrained DCT-LMS Filtering Algorithm for Time Delay
Estimation
3.1 Introduction 18
3.2 Transform Domain Adaptive Normalized LMS Filtering Algorithm for TDE 19
3.3 Adaptive Constrained DCT-LMS Filtering Algorithm for TDE 24
3.4 Computer Simulation Results 30
3.4.1 Stationary Time Delay Case 30
3.4.2 Time-Varying Delay Case 31
Chapter 4 Adaptive Gram-Schmidt DCT-LMS TDE Filtering Algorithm
4.1 Introduction 39
4.2 Escalator Algorithm for Gram-Schmidt Orthogonalization 40
4.3 Adaptive Gram-Schmidt DCT-LMS TDE Filtering Algorithm 44
4.4 Computer Simulation Results 49
Chapter 5 Conclusions 56
Appendix A 58
References 60

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