Theory and Applications of Fourier, Laplace, and Z Transformations


  • Carlo Ciulla Epoka University, Albania



k-space, L-space, z-space, phase, shift


This study presents the mathematics for the implementation of direct and inverse Fourier, Laplace, and Z transformations. This research is at the intersection between signal processing, applied mathematics, and software engineering, and it provides a study guide to implementers. Mathematical concepts and details necessary to transform the math into code are provided as theoretical background. Validation is conducted for the cases when the transforms do intersect, when the transforms do not intersect, and when, in Fourier and Z-transformations, the frequency domain encodes a phase shift which is reconstructed as an image space shift. Coherence between the software implementation of the three transformations is confirmed when: 1. The real component of the complex variable s = σ + i ω is σ = 0, which is the case when Fourier and Laplace transforms are the same. 2. When the magnitude of the complex variable z = r e is r = 1, which is the case when Fourier and Z transforms are the same. Congruency between software implementation of transformations is confirmed comparing departing image and inverse reconstructed image. The novelty of this research is the presentation style of the theory of direct and inverse Fourier, Laplace, and Z transforms. Details provided in this research make this paper a study guide that is not found elsewhere.


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Cooley, J. W. and Tukey, J. W. (1965). An algorithm for the machine calculation of complex Fourier series, Mathematics of Computation, vol. 19, no. 90, 297-301.

Cooley, J. W., Lewis, P. A. and Welch, P. D. (1967). Historical notes on the fast Fourier transform, Proceedings of the IEEE, vol. 55, no. 10, pp. 1675-1677.

Cooley, J. W., Lewis, P. A. and Welch, P. D. (1969). The fast Fourier transform and its applications, IEEE Transactions on Education, vol. 12, no. 1, pp.27-34.

Robinson, E. A. (1982). A historical perspective of spectrum estimation, Proceedings of the IEEE, vol. 70, no. 9, pp.885-907.

Dubner, H. and Abate, J. (1968). Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform, Journal of the ACM, vol. 15, no. 1, pp.115-123.

Stehfest, H. (1970). Algorithm 368: Numerical inversion of Laplace transforms [D5], Communications of the ACM, vol. 13, no. 1, pp.47-49.

Mansfield, P. (1962). Proton magnetic resonance relaxation in solids by transient methods, Doctoral dissertation, Queen Mary, University of London. London, U.K.

Lauterbur, P. C. (1973). Image formation by induced local interactions: examples employing nuclear magnetic resonance, Nature, vol. 242, no. 5394, pp.190-191.

Weeks, W. T. (1966). Numerical inversion of Laplace transforms using Laguerre functions, Journal of the ACM, vol. 13, no. 3, pp.419-429.

Talbot, A. (1979). The accurate numerical inversion of Laplace transforms, IMA Journal of Applied Mathematics, vol. 23, no. 1, pp.97-120.

Garbow, B. S., Giunta, G., Lyness, J. N. and Murli, A. (1988). Software for an implementation of Weeks' method for the inverse Laplace transform, ACM Transactions on Mathematical Software (TOMS), vol. 14, no. 2, pp.163-170.

Murli, A. and Rizzardi, M. (1990). Algorithm 682: Talbot's method of the Laplace inversion problems, ACM Transactions on Mathematical Software (TOMS), vol. 16, no. 2, pp.158-168.

D'amore, L., Laccetti, G. and Murli, A. (1999). An implementation of a Fourier series method for the numerical inversion of the Laplace transform, ACM Transactions on Mathematical Software (TOMS), vol. 25, no. 3, pp.279-305.

Kuhlman, K. L. (2013). Review of inverse Laplace transform algorithms for Laplace-space numerical approaches, Numerical Algorithms, vol. 63, no. 2, pp.339-355.

Khan, M. and Hussain, M. (2011). Application of Laplace decomposition method on semi-infinite domain, Numerical Algorithms, vol. 56, no. 2, pp.211-218.

Kexue, L. and Jigen, P. (2011). Laplace transform and fractional differential equations, Applied Mathematics Letters, vol. 24, no. 12, pp.2019-2023.

Ragazzini, J. R. and Zadeh, L. A. (1952). The analysis of sampled-data systems, Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry, vol. 71, no. 5, pp.225-234.

Jury, E.I. (1973). Theory application of the z-transform method, R.E. Krieger Pub. Co., Huntington, N.Y., U.S.A.

Fazarinc, Z. (2013). Z-transform and its application to development of scientific simulation algorithms, Computer Applications in Engineering Education, vol. 21, no. 1, pp.75-88.

Caldarola, F., Maiolo, M. and Solferino, V. (2020). A new approach to the Z-transform through infinite computation, Communications in Nonlinear Science and Numerical Simulation, vol. 82, 105019.

Ozaktas, H. M. and Kutay, M. A. (2001). The fractional Fourier transform, In: 2001 IEEE European Control Conference (ECC), pp.1477-1483.

Hassanzadeh, H. and Pooladi-Darvish, M. (2007). Comparison of different numerical Laplace inversion methods for engineering applications, Applied Mathematics and Computation, vol. 189, no. 2, pp.1966-1981.

Rabiner, L., Schafer, R. W. and Rader, C. (1969). The chirp z-transform algorithm, IEEE Transactions on Audio and Electroacoustics, vol. 17, no. 2, pp.86-92.

Oppenheim, A.V., Wilsky, A.S. and Nawab, S.H. (1997) Signals & Systems, 2nd Edition, Prentice Hall, Upper Saddle River, N.J., U.S.A.

Kumi, K., Berisha, R., Biçaku, X., Uruçi, R. and Ciulla, C. (2022). Verification of congruency and coherence of Fourier, Laplace, and Z-transformations, Int. J. Student Project Reporting – forthcoming.

Zayed, A. I. (1996). On the relationship between the Fourier and fractional Fourier transforms, IEEE Signal Processing Letters, vol. 3 no. 12, pp.310-311.

Zhang, Y., Wang, S., Yang, J. F., Zhang, Z., Phillips, P., Sun, P. and Yan, J. (2017). A comprehensive survey on fractional Fourier transform, Fundamenta Informaticae, vol. 151 no. (1-4), pp.1-48.

Frigo, M. and Johnson, S. G. (1998). FFTW: An adaptive software architecture for the FFT, In: Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP'98, vol. 3, pp.1381-1384.

Gilbert, A. C., Indyk, P., Iwen, M. and Schmidt, L. (2014). Recent developments in the sparse Fourier transform: A compressed Fourier transform for big data, IEEE Signal Processing Magazine, vol. 31, no. 5, pp.91-100.

Nair, P., Popli, A. and Chaudhury, K. N. (2017). A fast approximation of the bilateral filter using the discrete Fourier transform, Image Processing On Line, vol. 7, pp.115-130.

Anger, J. and Meinhardt-Llopis, E. (2017). Implementation of local Fourier burst accumulation for video deblurring, Image Processing On Line, vol 7, pp.56-64.

Delbracio, M. and Sapiro, G. (2015). Removing camera shake via weighted Fourier burst accumulation, IEEE Transactions on Image Processing, vol. 24, no. 11, pp.3293-3307.

Wahls, S. and Poor, H. V. (2015). Fast numerical nonlinear Fourier transforms, IEEE Transactions on Information Theory, vol. 61, no. 12, pp.6957-6974.

Turitsyn, S. K., Prilepsky, J. E., Le, S. T., Wahls, S., Frumin, L. L., Kamalian, M. and Derevyanko, S. A. (2017). Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives, Optica, vol. 4, no. 3, pp.307-322.

Kazimierczuk, K. and Orekhov, V. (2015). Non uniform sampling: post Fourier era of NMR data collection and processing, Magnetic Resonance in Chemistry, vol. 53 no. 11, pp.921-926.

Fenn, M., Kunis, S. and Potts, D. (2007). On the computation of the polar FFT, Applied and Computational Harmonic Analysis, vol. 22, no. 2, pp.257-263.

Abate, J. and Whitt, W. (1992). The Fourier-series method for inverting transforms of probability distributions, Queueing Systems, vol. 10, no. 1, pp.5-87.

Abate, J. and Whitt, W. (1992). Numerical inversion of probability generating functions, Operations Research Letters, vol. 12, no. 4, pp.245-251

Abate, J. and Whitt, W. (1995). Numerical inversion of Laplace transforms of probability distributions, ORSA Journal on Computing, vol. 7, no. 1, pp.36-43.

Fusai, G., Germano, G. and Marazzina, D. (2016). Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options, European Journal of Operational Research, vol. 251, no. 1, pp.124-134.

Phelan, C. E., Marazzina, D., Fusai, G. and Germano, G. (2018). Fluctuation identities with continuous monitoring and their application to the pricing of barrier options, European Journal of Operational Research, vol. 271, no. 1, pp.210-223.

Phelan, C. E., Marazzina, D., Fusai, G. and Germano, G. (2019). Hilbert transform, spectral filters and option pricing, Annals of Operations Research, vol. 282, no. 1, pp.273-298.

Ciulla, C. (2021). Inverse Fourier transformation of combined first order derivative and intensity-curvature functional of magnetic resonance angiography of the human brain, Computer Methods and Programs in Biomedicine, vol. 211, 106384.




How to Cite

Ciulla, C. (2022). Theory and Applications of Fourier, Laplace, and Z Transformations. The Journal of CIEES, 2(1), 32–43.