This paper studies a computationally efficient algebraic graph theory engine for simulating time-domain one-dimensional waves in a multi-segment transmission line, such as for reflectometry applications. Efficient simulation of time-domain signals in multi-segment transmission lines is challenging because the number of propagation paths (and therefore the number of operations) increases exponentially with each new interface. We address this challenge through the use of a frequency-domain, algebraic graphical model of wave propagation, which is then converted to the time domain via the Fourier transform. We use this model to achieve an exact, stable, and computationally efficient (O(NQ), where N is the number of segments and Q is the bandwidth) approach for studying one-dimensional wave propagation. Our approach requires the reflection and transmission coefficients for each interface and each segment's complex propagation constant. We compare our simulation results with known analytical solutions.
2. Cawley, P., "Practical guided wave inspection and applications to structural health monitoring," Proc. of the Australasian Congress on Applied Mechanics, 10, Brisbane, Dec. 2007.
3. Moilanen, P., "Ultrasonic guided waves in bone," IEEE Trans. Ultrason. Ferroelectr. Freq. Control, Vol. 55, No. 6, 1277-1286, 2008.
4. Smith, P., C. Furse, and J. Gunther, "Analysis of spread spectrum time domain reflectometry for wire fault location," IEEE Sens. J., Vol. 5, No. 6, 1469-1478, Dec. 2005.
5. Lowe, M. J. S., "Matrix techniques for modeling ultrasonic waves in multilayered media," IEEE Trans. Ultrason. Ferroelectr. Freq. Control, Vol. 42, No. 4, 525-542, Jul. 1995.
6. Furse, C., P. Smith, C. Lo, Y. C. Chung, P. Pendayala, and K. Nagoti, "Spread spectrum sensors for critical fault location on live wire networks," Struct. Control Health Monit., Vol. 12, No. 3–4, 257-267, Jul. 2005.
7. Santos, E. J. P. and L. B. M. Silva, "Calculation of scattering parameters in multiple-interface transmission-line transducers," Measurement, Vol. 47, 248-254, Jan. 2014.
8. Sumithra, P. and D. Thiripurasundari, "Review on computational electromagnetics," Advanced Electromagnetics, Vol. 6, No. 1, 42-55, Mar. 2017.
9. Chakraborty, A. and S. Gopalakrishnan, "A spectrally formulated finite element for wave propagation analysis in layered composite media," Int. J. Solids Struct., Vol. 41, No. 18, 5155-5183, Sep. 2004.
10. Alterman, Z. and F. C. Karal, "Propagation of elastic waves in layered media by finite difference methods," Bulletin of the Seismological Society of America, Vol. 58, No. 1, 367-398, Feb. 1968.
11. Hoefer, W. J. R., "The transmission-line matrix method --- Theory and applications," IEEE Trans. Microw. Theory Tech., Vol. 33, No. 10, 882-893, Oct. 1985.
12. Bohlen, T. and E. Saenger, "Accuracy of heterogeneous staggered-grid finite-difference modeling of Rayleigh waves," Geophysics, Vol. 71, No. 4, T109-T115, Jul. 2006.
13. Liu, T., C. Zhao, and Y. Duan, "Generalized transfer matrix method for propagation of surface waves in layered azimuthally anisotropic half-space," Geophysical Journal International, Vol. 190, No. 2, 1204-1212, Aug. 2012.
14. Katsidis, C. C. and D. I. Siapkas, "General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference," Appl. Opt., Vol. 41, No. 19, 3978-3987, Jul. 2002.
15. Troparevsky, M. C., A. S. Sabau, A. R. Lupini, and Z. Zhang, "Transfer-matrix formalism for the calculation of optical response in multilayer systems: From coherent to incoherent interference," Opt. Express, Vol. 18, No. 24, 24 715-24 721, Nov. 2010.
16. Cormen, T. H., C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, MIT Press, 2009.
17. Farmaga, I., P. Shmigelskyi, P. Spiewak, and L. Ciupinski, "Evaluation of computational complexity of finite element analysis," Proc. of the International Conference The Experience of Designing and Application of CAD Systems in Microelectronics, 213-214, Feb. 2011.
18. Davidson, D. B. and R. W. Ziolkowski, "Body-of-revolution finite-difference time-domain modeling of space-time focusing by a three-dimensional lens," J. Opt. Soc. Am. A, JOSAA, Vol. 11, No. 4, 1471-1490, Apr. 1994.
19. Pozar, D. M., Microwave Engineering, 4th Ed., Wiley, Nov. 2011.
20. Mason, S. J., "Feedback theory --- Some properties of signal flow graphs," Proceedings of the IRE, Vol. 41, No. 9, 1144-1156, Sep. 1953.
21. Schutt-Aine, J. E., "Transient analysis of nonuniform transmission lines," IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 39, No. 5, 378-385, May 1992.
22. Acar, C., "Nth-order voltage transfer function synthesis using a commercially available active component: Signal-flow graph approach," Electron. Lett., Vol. 32, No. 21, 1933-1934, Oct. 1996.
23. Biggs, N., N. L. Biggs, and E. N. Biggs, Algebraic Graph Theory, Vol. 67, Cambridge University Press, 1993.
24. Li, Z., Z. Duan, G. Chen, and L. Huang, "Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint," IEEE Transactions on Circuits and Systems I: Regular Papers, Vol. 57, No. 1, 213-224, Jan. 2010.
25. Sezer, M. E. and D. D. Iljak, "Nested-decompositions and clustering of complex systems," Automatica, Vol. 22, No. 3, 321-331, 1986.
26. George, A., J. R. Gilbert, and J. W. Liu, Graph Theory and Sparse Matrix Computation, Vol. 56, Springer Science & Business Media, 2012.
27. Dhillon, I. S. and D. S. Modha, "Concept decompositions for large sparse text data using clustering," Machine Learning, Vol. 42, No. 1–2, 143-175, 2001.
28. Shuman, D. I., S. K. Narang, P. Frossard, A. Ortega, and P. Vandergheynst, "The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains," IEEE Signal Processing Magazine, Vol. 30, No. 3, 83-98, May 2013.
29. Sandryhaila, A. and J. M. Moura, "Big data analysis with signal processing on graphs: Representation and processing of massive data sets with irregular structure," IEEE Signal Processing Magazine, Vol. 31, No. 5, 80-90, 2014.
30. Puschel, M. and J. M. F. Moura, "Algebraic signal processing theory: Foundation and 1-D time," IEEE Trans. Signal Process., Vol. 56, No. 8, 3572-3585, Aug. 2008.
31. Davis, T. A., S. Rajamanickam, and W. M. Sid-Lakhdar, "A survey of direct methods for sparse linear systems," Acta Numerica, Vol. 25, 383-566, 2016.
32. Bunch, J. R. and D. J. Rose, Sparse Matrix Computations, Academic Press, 2014.
33. Luebbers, R., F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, "A frequency-dependent finite-difference time-domain formulation for dispersive materials," IEEE Trans. Electromagn. Compat., Vol. 32, No. 3, 222-227, Aug. 1990.
34. Davis, T. A., "Algorithm 832: UMFPACK v4.3 --- An unsymmetric-pattern multifrontal method," ACM Trans. Math. Softw., Vol. 30, No. 2, 196-199, Jun. 2004.