## Handbook of Fourier Analysis & Its ApplicationsFourier analysis has many scientific applications - in physics, number theory, combinatorics, signal processing, probability theory, statistics, option pricing, cryptography, acoustics, oceanography, optics and diffraction, geometry, and other areas. In signal processing and related fields, Fourier analysis is typically thought of as decomposing a signal into its component frequencies and their amplitudes. This practical, applications-based professional handbook comprehensively covers the theory and applications of Fourier Analysis, spanning topics from engineering mathematics, signal processing and related multidimensional transform theory, and quantum physics to elementary deterministic finance and even the foundations of western music theory. As a definitive text on Fourier Analysis, Handbook of Fourier Analysis and Its Applications is meant to replace several less comprehensive volumes on the subject, such as Processing of Multifimensional Signals by Alexandre Smirnov, Modern Sampling Theory by John J. Benedetto and Paulo J.S.G. Ferreira, Vector Space Projections by Henry Stark and Yongyi Yang and Fourier Analysis and Imaging by Ronald N. Bracewell. In addition to being primarily used as a professional handbook, it includes sample problems and their solutions at the end of each section and thus serves as a textbook for advanced undergraduate students and beginning graduate students in courses such as: Multidimensional Signals and Systems, Signal Analysis, Introduction to Shannon Sampling and Interpolation Theory, Random Variables and Stochastic Processes, and Signals and Linear Systems. |

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### Contents

3 | |

10 | |

3 Fourier Analysis in Systems Theory | 104 |

4 Fourier Transforms in Probability Random Variables and Stochastic Processes | 151 |

5 The Sampling Theorem | 217 |

6 Generalizations of the Sampling Theorem | 242 |

7 Noise and Error Effects | 288 |

8 Multidimensional Signal Analysis | 326 |

9 TimeFrequency Representations | 411 |

### Common terms and phrases

algorithm aliasing Alternating Projections applied autocorrelation bandlimited bandlimited function bandlimited signal bandwidth Bessel function cardinal series causal characteristic function computed constraint convergence convex sets convolution corresponding deﬁned denotes derivative diffraction dimensional Dirac delta equation Evaluate example Exercise ﬁlter finite ﬁrst Fourier analysis Fourier series fractional Fourier transform fX(x ge(t gives GTFR harmonic Hilbert transform IEEE Transactions illustrated in Figure impulse response input integral interpolation interpolation function interpolation noise interval inverse iteration kernel Laplace lost samples matrix multidimensional NINV noise level Note Nyquist rate Optical output parameter periodic function periodicity matrix pixels plane plot POCS Poisson polynomials probability density function R.J. Marks replication restoration result sampling theorem scale Section shift short time Fourier shown in Figure Signal Processing sinc(2Bt sinc(t solution spectrogram spectrum stochastic process time-frequency variance vector wave window zero