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# Optical Communications from a Fourier Perspective

## Fourier Theory and Optical Fiber Devices and Systems

- 1st Edition - November 17, 2023
- Authors: Palle Jeppesen, Bjarne Tromborg
- Language: English
- Paperback ISBN:9 7 8 - 0 - 4 4 3 - 2 3 8 0 0 - 0
- eBook ISBN:9 7 8 - 0 - 4 4 3 - 2 3 8 0 1 - 7

Optical Communications from a Fourier Perspective: Fourier Theory and Optical Fiber Devices and Systems covers Fourier theory and signal analysis over photonic component… Read more

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Request a sales quote*Optical Communications from a Fourier Perspective: Fourier Theory and Optical Fiber Devices and Systems* covers Fourier theory and signal analysis over photonic components, including time lenses in optical communication. Sections cover wave propagation in optical waveguides based on Maxwell equations and the nonlinear Schrödinger equation. Optical Fourier transform in the form of time lens is covered, for example in modulation format conversion and spectrum magnification, and couplers and their use for optical discrete Fourier transformation are also discussed. Other important subjects are discussed such as shot noise, thermal noise and also the basics of four wave mixing in relation to time lenses. Detailed derivations and a deeper background for the chapters are provided in appendices where appropriate. Some of the theory is more generally applicable beyond optical communication and is of relevance also for communications engineering. The Fourier theory dimension of the book presents the relationship between Fourier series and Fourier integrals and also the related Laplace transform.

- Introduces Fourier theory and signal analysis tailored to applications in optical communications devices and systems
- Provides a strong theoretical background and a ready resource for researchers and advanced students in optical communication and optical signal processing
- Starts from basic theory and then develops descriptions of useful applications

- Cover image
- Title page
- Table of Contents
- Copyright
- Preface
- Acknowledgments
- Bibliography
- Chapter 1: The Dirac delta function and Heaviside step function
- Abstract
- 1.1. Definition of the Dirac delta function
- 1.2. Dirichlet integrals
- 1.3. Properties of the delta function
- 1.4. The Heaviside step function
- 1.5. Summary
- Bibliography
- Chapter 2: Fourier series, Parseval's theorem, FFT and Cooley–Tukey algorithm
- Abstract
- 2.1. Fourier series
- 2.2. Train of rectangular pulses
- 2.3. Train of delta pulses and the comb function
- 2.4. Parseval's power theorem for periodic signals
- 2.5. The discrete Fourier transform
- 2.6. FFT based on Cooley–Tukey algorithm
- 2.7. Summary
- Bibliography
- Chapter 3: Fourier integrals and series
- Abstract
- 3.1. The Fourier transformation and the Fourier integral theorem
- 3.2. From Fourier series to integrals
- 3.3. From Fourier integrals to series
- 3.4. Notation
- 3.5. Summary
- Chapter 4: Properties of the Fourier transform and Heaviside's step function
- Abstract
- 4.1. Important properties
- 4.2. Important examples of Fourier transforms
- 4.3. Relation between pulse and spectral widths
- 4.4. Fourier transform of the Heaviside step function
- 4.5. Integration theorem and other examples involving the step function
- 4.6. Sampling theorem for aperiodic functions
- 4.7. Numerical calculation of the Fourier transform
- 4.8. Numerical calculation of the inverse Fourier transform
- 4.9. Summary
- Bibliography
- Chapter 5: Complex signal, complex envelope, and Hilbert transform
- Abstract
- 5.1. Complex signal
- 5.2. Complex envelope of a bandpass signal
- 5.3. IQ modulation and demodulation
- 5.4. The complex signal and Hilbert transform
- 5.5. Summary
- Bibliography
- Chapter 6: Correlation functions, spectral density, Wiener–Khinchine theorem
- Abstract
- 6.1. Correlation functions and Wiener–Khinchine theorem for deterministic energy signals
- 6.2. Correlation functions and Wiener–Khinchine theorem for deterministic power signals
- 6.3. Random processes, ensemble and probability density averages, stationary and ergodic processes
- 6.4. Autocorrelation function and Wiener–Khinchine theorem for wide-sense stationary random process, effective bandwidth, and noise considerations
- 6.5. Shot noise
- 6.6. Thermal noise
- 6.7. Summary
- Bibliography
- Chapter 7: Linear, time-invariant systems
- Abstract
- 7.1. Impulse response and transfer functions
- 7.2. Linear electrical circuits
- 7.3. Convolution and RMS widths
- 7.4. Summary
- Bibliography
- Chapter 8: Transfer matrices and frequency filters
- Abstract
- 8.1. Transfer functions, transfer matrices, and quadrature filter
- 8.2. Frequency filters
- 8.3. Summary
- Chapter 9: Laplace transforms, transfer functions, Nyquist criterion
- Abstract
- 9.1. The Laplace transform and its inverse
- 9.2. Important properties of Laplace transform
- 9.3. Important examples of Laplace transforms
- 9.4. Transfer functions for passive and active LTI systems
- 9.5. Laplace-method applied on series resonant circuit
- 9.6. Transimpedance front end
- 9.7. Nyquist criterion
- 9.8. Summary
- Bibliography
- Chapter 10: Maxwell's equations, optical waveguides, and Poynting vector
- Abstract
- 10.1. Maxwell's equations and Helmholtz equation
- 10.2. Wave propagation in planar optical waveguide
- 10.3. Wave propagation in a step-index optical fiber
- 10.4. Wave propagation in a graded-index optical fiber
- 10.5. Derivation of the wave equation for both rectangular and circular waveguide
- 10.6. Poynting vector and photodetection
- 10.7. Summary
- Bibliography
- Chapter 11: Pulse propagation in optical fibers
- Abstract
- 11.1. Fundamental equations for a general pulse shape
- 11.2. Introduction of a new time reference
- 11.3. Fiber impulse response
- 11.4. Introduction of a Gaussian input pulse
- 11.5. Propagation of a Gaussian pulse
- 11.6. Material and waveguide dispersion
- 11.7. Pulse propagation in a dispersion compensating fiber
- 11.8. Summary
- Bibliography
- Chapter 12: Split-step Fourier method and nonlinear Schrödinger equation
- Abstract
- 12.1. The nonlinear Schrödinger equation
- 12.2. Operators for dispersion and nonlinearity
- 12.3. Split-step Fourier method
- 12.4. Fundamental soliton solution to the NLSE
- 12.5. Summary
- Bibliography
- Chapter 13: Introduction to modulation formats
- Abstract
- 13.1. Intensity modulation and direct detection
- 13.2. ASK, PSK, and FSK binary modulation
- 13.3. Multilevel modulation formats
- 13.4. Differential binary and quadrature PSK
- 13.5. Short introduction to PDM and WDM
- 13.6. Summary
- Bibliography
- Chapter 14: Required bandwidth for heterodyne and homodyne detection
- Abstract
- 14.1. Baseband signal
- 14.2. Modulated transmitter
- 14.3. Heterodyne detection
- 14.4. Homodyne detection
- 14.5. Summary
- Bibliography
- Chapter 15: Bandpass noise
- Abstract
- 15.1. Introduction
- 15.2. Gaussian bandpass noise
- 15.3. Formulas for autocorrelation functions and power spectral densities
- 15.4. Autocorrelation and cross-correlation functions of quadrature components
- 15.5. Power spectral density functions of quadrature components
- 15.6. Proof of statistical independence of quadrature components
- 15.7. Bandlimited white noise
- 15.8. Summary
- Bibliography
- Chapter 16: Bit error rate
- Abstract
- 16.1. Receiver noise in direct ASK detection
- 16.2. Bit error rate for ASK direct detection
- 16.3. ASK direct detection and the quantum limit
- 16.4. Synchronous ASK heterodyne detection
- 16.5. Synchronous ASK homodyne detection
- 16.6. Synchronous BPSK heterodyne detection
- 16.7. Synchronous BPSK homodyne detection
- 16.8. Summary
- Bibliography
- Chapter 17: Pulse shaping using optical Fourier transform
- Abstract
- 17.1. Output pulse shaping using Cd configuration
- 17.2. Output spectrum shaping using a dC configuration
- 17.3. OTDM-to-WDM conversion using a dC configuration
- 17.4. OTDM-to-WDM conversion in WDM-PON
- 17.5. WDM-to-OTDM conversion using a dC conversion
- 17.6. WDM-to-OTDM conversion using a complete dCd configuration
- 17.7. WDM-to-OTDM conversion using a Cd configuration
- 17.8. WDM-to-OTDM conversion using a complete CdC configuration
- 17.9. Optical Nyquist transmission using time-domain optical Fourier transformation
- 17.10. Summary
- Bibliography
- Chapter 18: Spectrum magnification
- Abstract
- 18.1. Gaussian input pulse
- 18.2. Rectangular input pulse
- 18.3. Spectrum magnification of WDM signals
- 18.4. Summary
- Bibliography
- Chapter 19: Optical Fourier transformation, dispersion compensation, jitter suppression
- Abstract
- 19.1. Dispersion compensation for general input pulse shape
- 19.2. Time lens used for jitter suppression in direct detection receiver
- 19.3. Trends and opportunities for time lenses
- 19.4. Summary
- Bibliography
- Chapter 20: Regeneration of WDM phase-modulated signals
- Abstract
- 20.1. FWM basics
- 20.2. Nonlinear equations for nonstatic fields
- 20.3. Degenerate FWM
- 20.4. Degenerate signal and idler. Phase sensitive amplifier
- 20.5. Time lenses and WDM phase regeneration for BPSK modulation
- 20.6. Summary
- Bibliography
- Chapter 21: Time–space duality, dispersion and diffraction, time lens
- Abstract
- 21.1. Gaussian pulse propagating in a dispersive fiber
- 21.2. Introduction to the Gaussian beam
- 21.3. Relations between dispersion of a narrow pulse and diffraction of a narrow beam
- 21.4. Comparison with thin optical cell with quadratic index variation
- 21.5. Comparison with thin optical lens
- 21.6. Summary
- Bibliography
- Chapter 22: Couplers and their use for optical DFT
- Abstract
- 22.1. Scattering matrices
- 22.2. Optical DFT using an N×N coupler
- 22.3. Summary
- Bibliography
- Chapter 23: Multicarrier modulation, OFDM, DFT, Nyquist modulation
- Abstract
- 23.1. Multicarrier modulation and OFDM
- 23.2. OFDM using IDFT and DFT
- 23.3. Nyquist modulation format
- 23.4. Conversion of DWDM signals to Nyquist channel based on complete CdC optical Fourier transformation
- 23.5. OFDM-to-Nyquist-WDM conversion based on complete CdC optical Fourier transformation
- 23.6. Summary
- Bibliography
- Chapter 24: Optical orthogonal frequency division modulation
- Abstract
- 24.1. “On-the-fly” inverse Fourier transformation
- 24.2. Transfer function of combined serial-to-parallel converter and OIFT circuit
- 24.3. OIFT based on a circuit synthesized of delay interferometers
- 24.4. Transmission system based on OIFT circuits
- 24.5. Summary
- Bibliography
- Appendix A: Acronyms
- Appendix B: Analytic complex functions, Cauchy's integral theorem, and integral formula
- B.1. Simple application of the Cauchy integral theorem
- B.2. Application of Cauchy integral theorem and integral formula
- B.3. The principal value integral of an analytic function
- B.4. Analyticity and causality
- B.5. Example of inverse Fourier of H(ω)
- B.6. Cauchy's argument principle
- Bibliography
- Appendix C: Fourier series expansion of the comb function
- Appendix D: Calculation of the integral of a Gaussian function
- Appendix E: Time–bandwidth product
- Appendix F: Examples of MATLAB® programs
- F.1. The Fourier transforms in Fig. 4.8
- F.2. Fourier transforms using window functions
- F.3. Solution of the nonlinear Schrödinger equation (NLSE) by the split step Fourier method (SSFM)
- Appendix G: Proof of autocorrelation going to zero for large values of τ
- Appendix H: Autocorrelation of digital signal
- Appendix I: Ensemble averaging and pdf averaging
- Appendix J: Probability density functions and jointly Gaussian random variables
- J.1. Probability density functions
- J.2. Jointly Gaussian random variables
- Bibliography
- Appendix K: Poisson statistics
- K.1. Poisson statistics derived from differential equations
- K.2. Poisson statistics derived from binomial distribution
- Appendix L: RMS widths of convoluted signals
- Appendix M: Example of Fourier–Mellin integral determined by Cauchy integral theorem
- Appendix N: Proof of Thévenin's and Norton's theorems
- Appendix O: Propagation of a Gaussian pulse in a single-mode fiber
- O.1. Introduction of new time reference
- O.2. Ae(z,t) for Gaussian input
- O.3. Pulse width of output power
- O.4. Minimum pulse width
- Appendix P: Carrier envelope offset
- Bibliography
- Appendix Q: Bandwidth of random ASK and PSK signals
- Q.1. Binary ASK
- Q.2. Binary PSK
- Q.3. Summary
- Bibliography
- Appendix R: Solution of coupled wave equations
- Appendix S: Matched filter
- Bibliography
- Bibliography
- Bibliography
- Index

- No. of pages: 550
- Language: English
- Edition: 1
- Published: November 17, 2023
- Imprint: Elsevier
- Paperback ISBN: 9780443238000
- eBook ISBN: 9780443238017

PJ

### Palle Jeppesen

Palle Jeppesen is professor emeritus at the Technical University of Denmark (DTU) and a researcher with many years’ experiences in optical fiber communications, lasers, fibers, systems, and ultra-high-speed optical communications. He has been a member of the Danish Technical Research Council, the Scientific Council for the Danish National Encyclopedia, and on the boards of a number of large corporate entities. His current research interests are optical signal processing, optical multi-level modulation formats, and terabit optical communication.

BT

### Bjarne Tromborg

Bjarne Tromborg was formerly a research and teaching professor at the Technical University of Denmark (DTU) best known for his work in particle physics and photonics. He has been a member of the Danish Natural Science Research Council and many technical program committees. He has published widely on the topics of optoelectronics, semiconductor lasers, and optical communications

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