| 摘要: | 在現代高速電路與通訊系統中，為了確保系統設計的正確性與訊號品質，常需對傳輸通道進行精確的暫態模擬。由於實際電路多以頻率域方式進行量測與模擬，如何有效地將頻域資料轉換為可用於時域模擬的模型，成為一項極具挑戰性的課題。巨觀建模技術便是在此背景下被提出，其核心目標是將原始的頻率響應資料近似為具備數值穩定性與物理可解釋性的有理轉移函數模型，進而加速時域模擬的效率，並維持一定的擬合準確度。
本研究探討並實作一種新穎的巨觀建模方法—正交有理近似法。此方法透過建立一組正交多項式基底，解決傳統使用單項式基底所導致的病態矩陣問題，避免因高次多項式導致的數值不穩定。同時，針對有理模型中分母未知參數所造成的非線性問題，本方法採用一種具備收斂特性的迭代機制，將其轉換為可逐步求解的線性問題，大幅簡化計算流程。為使方法能應用於實際高維資料情境，本文進一步提出三項運算加速策略：包括使用切片取代矩陣堆疊以降低記憶體重分配開銷、利用特徵值分解快速求解最小奇異向量，以及透過多執行緒平行運算來加快大型多埠系統中各子矩陣之處理速度。這些技術整合後構成「快速正交有理近似法」，可顯著提升整體演算法的運行效率。
為驗證本方法的有效性，本文設計多組模擬與量測案例，涵蓋從雙埠到一百埠的大型傳輸網路，並將所實作方法與現今常見的向量擬合法與洛伊納矩陣法進行比較。實驗結果顯示，在控制相同擬合誤差的前提下，本方法可使用較少的極點數達到相近甚至更佳的擬合準確度，並在執行時間上顯著優於其他方法，特別是在高埠數系統中更具效能優勢。此外，透過轉換為狀態空間模型與極點殘數形式，亦可進一步合成出等效電路結構，以實現高效率的遞迴捲積模擬。
綜合而言，正交有理近似法具備高穩定性、高準確度與高運算效率的優勢，不僅適用於多種高頻電路建模情境，也具備良好的工程實用性與擴充潛力，未來可應用於高速介面設計、自動建模流程與商用模擬工具整合等領域。
;In modern high-speed circuits and communication systems, accurate transient simulation of transmission channels is essential to ensure design correctness and signal integrity. Since real-world circuits are often characterized in the frequency domain through measurement or simulation, converting frequency-domain data into models suitable for time-domain simulation has become a significant challenge. Macromodeling techniques were developed to address this issue, aiming to approximate the original frequency response data as rational transfer functions that are both numerically stable and physically interpretable, thereby improving simulation efficiency while maintaining a desired level of fitting accuracy.
This study investigates and implements a novel macromodeling technique—Orthogonal Rational Approximation (ORA). By constructing a set of orthogonal polynomial bases, ORA overcomes the numerical instability caused by ill-conditioned matrices arising from monomial bases, which are common in traditional methods. To address the nonlinearity resulting from unknown denominator coefficients in rational models, the proposed method employs an iterative scheme with convergence properties, transforming the problem into a sequence of linear least-squares problems and significantly simplifying the computation. To enable practical application in high-dimensional data scenarios, this work introduces three acceleration strategies: using slicing operations instead of matrix stacking to reduce memory reallocation overhead, employing eigenvalue decomposition to efficiently obtain the smallest singular vector, and applying multithreaded parallel computation to accelerate the processing of submatrices in large multi-port systems. These optimizations are integrated into a fast implementation of ORA, significantly enhancing the overall execution efficiency.
To verify the effectiveness of the implemented method, several simulation and measurement-based test cases were designed, covering transmission networks ranging from two-port to one-hundred-port configurations. The results are compared against commonly used Vector Fitting and Loewner Matrix approaches. Experimental results show that, under the same fitting error constraint, the proposed method achieves comparable or even better accuracy with fewer poles and significantly shorter execution times, especially in high-port-count scenarios. Furthermore, by converting the model into a state-space form and expressing it in pole-residue format, the resulting transfer function can be synthesized into an equivalent circuit suitable for efficient recursive convolution.
In summary, the Orthogonal Rational Approximation method demonstrates superior stability, accuracy, and computational efficiency. It is well-suited for a wide range of high-frequency circuit modeling applications and offers strong practical value and extensibility. It holds promise for future applications in high-speed interface design, automated modeling workflows, and integration with commercial simulation tools. |
