利用差分色散關係進行因果性評估與強化

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姓名

王正陽(Cheng-Yang Wang) 查詢紙本館藏

畢業系所

電機工程學系

論文名稱

利用差分色散關係進行因果性評估與強化
(Causality Assessment and Enforcement through Dispersion Relations with Subtractions)

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至系統瀏覽論文 (2027-7-31以後開放)

摘要(中)

透過模擬、量測和各種應用之演算法所獲得之頻率響應(S參數)，在使用前都必須
確認該響應是否滿足因果性。違反因果性意味著頻率響應不滿足真實世界之物理性質，
這會導致信號完整度之相關應用產生錯誤的結果。然而，鮮少有研究對如何正確的判斷
因果性有深入的研究，再者如果我們成功判斷頻率響應違背因果性，如何將該頻率響應
修正成滿足因果性，怎麼樣的修正結果是最好的? 以上都是值得探索的問題。
此研究首先確立因果性的評估方法，在多種評估方法中選擇色散關係(希爾伯特轉換)
作為主要的方式，透過差分色散關係去處理截斷誤差，並估計延遲時間。透過平移延遲
時間，將響應導正到能夠正確判斷因果性的位置。透過上述之方法，我們能夠精確的判
斷頻率響應的因果性，並且能夠獲得違背之頻段。對於量測的數據，我們可以針對違背
的頻段進行重新量測。然而對於模擬的頻率響應，往往我們都會耗費大量的時間去模擬
生成頻率響應。如果要重新模擬，將會浪費大量時間，所以不少人會選擇直接透過演算
法強化因果性。
在處理違背因果性之頻率響應，常用的方法是使用宏觀建模去重建頻率響應，然而
宏觀建模並不一定能夠造出擬合良好的模型。這項研究通過因果性的評估結果，識別出
違背的頻段，並對這些違背的頻段進行修正，以獲得滿足因果性之評估方法的響應。這
個方法由於只修正違背之頻段，理論上可以得到最小修改量的頻率響應，但是實際測試
之結果，並不能保證得到最小修改量的頻率響應，並且強化因果性的效率取決於因果性
之評估嚴格程度。如何確立評估之嚴格程度以及強化因果性的效果，將會是未來值得更
進一步研究的方向。

摘要(英)

Frequency responses (S-parameters) obtained through simulations, measurements, and
various algorithmic applications must be verified for causality before use. Violating causality
implies that the frequency response does not adhere to the physical properties of the real world,
leading to incorrect outcomes in applications related to signal integrity. However, there is
limited research on how to accurately determine causality. Moreover, if we successfully identify
a causality breach in a frequency response, how should it be corrected to satisfy causality, and
what constitutes the best correction? These are questions worth exploring.
This study first establishes a method for assessing causality, choosing the dispersion
relation (Hilbert transform) as the primary method among various evaluation techniques. We
address truncation errors through dispersion relation with subtractions and estimate delay times.
By shifting delay times, we adjust the response to a position where causality can be correctly
assessed. Through these methods, we can precisely determine the causality of frequency
responses and identify the violated bands. For measured data, we can remeasure the violated
bands. However, simulating frequency responses often consumes considerable time. Re
simulation would be time-consuming, leading many to choose to enhance causality directly
through algorithms.
In dealing with frequency responses that violate causality, a common approach is to use
macromodeling to reconstruct the frequency response. However, macromodeling does not
always succeed in accurately creating the model. This study, through the results of causality
assessment and focusing on the violated bands, aims to correct these to achieve a response that meets causality assessment criteria. This method theoretically achieves the minimal
modification of the frequency response by only correcting the violated bands. However,
practical tests do not guarantee the minimal modification of the frequency response, and the
efficiency of enhancing causality depends on the stringency of the causality assessment.
Establishing the stringency of the assessment and the effectiveness of causality enhancement
are directions for further research in the future.

關鍵字(中)

★ 因果性
★ 色散關係

關鍵字(英)

★ Causality
★ Dispersion Relation

論文目次

中文摘要 VI
English Abstract VII
致謝 IX
Outline X
List of Figures XII
List of Tables XVI
List of Publications XVII
Chapter 1 Introduction: Causality and Dispersion Relations 1
1.1General Introduction 1
1.2 Causality 4
1.3 Analyticity 5
1.4 Titchmarsh’s Theorem 6
1.5 Subtractions 13
Chapter 2 Causality Assessment: S - Plane 19
2.1 Basic Idea and Theorem 19
(a) Maximum Modulus Principle 19
(b) Implement Method 21
(c) Nyquist–Shannon sampling theorem 21
2.2 Implementation on Simulation Data 25
(a) Causal pole and Oversampling Rational Function 25
(b) Causal pole and Undersampling Rational Function 26
(c) Non-causal pole and Oversampling Rational Function 28
(d) Non-causal pole and Undersampling Rational Function 29
(f) Djordjevic model and the Constant model 32
Chapter 3 Causality Assessment: Dispersion Relations with Subtractions 34
3.1 Dispersion Relations with Subtractions in Practice 34
3.2 Error Bound 38
(a) Truncation Error 38
(b) Discretization error 45
(c) Tolerable Error Near the Subtractions 48
3.3 The Issue of Delay Systems 49
3.4 Implementation on Simulation Data 54
(a) Dielectric model 54
(b) Roughness model 59
3.5 Implementation on Measurement Data 62
Chapter 4 Causality Enforcement 66
4.1 Basic Idea of Causality Enforcement 66
4.2 Violated Bands Replacement 71
4.3 Result 73
(a) Djordjevic Model with Step Perturbation 73
(b) Constant Model 77
(c) 2x-Thru De-embedding Simulation Data 79
(d) 2x-Thru De-embedding Simulation Data (Positive Phase) 83
(e) 2x-Thru De-embedding Measurement Data 89
Chapter 5 Conclusion 94
Appendix 97
A.1 Dielectric Model 97
(a) Djordjevic model 97
(b) Constant model 99
A.2 Roughness Model 100
(a) Huray model 101
(b) Huray-Bracken model 102
A.3 Filtered Fourier Transform 104
Reference 107

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指導教授

周求致

審核日期

2024-7-22

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