## The DFT: An Owners' Manual for the Discrete Fourier TransformJust as a prism separates white light into its component bands of colored light, so the discrete Fourier transform (DFT) is used to separate a signal into its constituent frequencies. Just as a pair of sunglasses reduces the glare of white light, permitting only the softer green light to pass, so the DFT may be used to modify a signal to achieve a desired effect. In fact, by analyzing the component frequencies of a signal or any system, the DFT can be used in an astonishing variety of problems. Among the applications of the DFT are digital signal processing, oil and gas exploration, medical imaging, aircraft and spacecraft guidance, and the solution of differential equations of physics and engineering. The DFT: An Owner's Manual for the Discrete Fourier Transform explores both the practical and theoretical aspects of the DFT, one of the most widely used tools in science and engineering. |

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### Contents

Introduction | 1 |

The Discrete Fourier Transform | 15 |

Properties of the | 65 |

Multi Multidimensional DFTs | 144 |

Errors | 179 |

A few A Few Applications of the | 236 |

Relate | 310 |

Quadrature and the | 357 |

The Fast Fourier Transform | 379 |

Table of DFTs | 401 |

409 | |

419 | |

### Common terms and phrases

algorithm aliasing approximate the Fourier array assume band-limited boundary conditions Chapter Chebyshev Chebyshev polynomials coefficients ck complex DFT computed convergence convolution cosine decrease defined denote derivatives DFT approximations DFT coefficients DFTs of length difference equation discrete Fourier transform endpoints error example f(xn Figure filter finite Fourier coefficients Fourier series frequency domain frequency grid frequency modes function given grid points grid spacing IDFT input sequence integral interpolation interval A/2 inverse DFT inverse Fourier transform Laplace transform linear matrix method notation orthogonal polynomials output piecewise points xn Poisson Summation Formula problem Radon transform RDFT real DFT real-valued reciprocity relations replication representation result sampled sequence of length Show shown sine solution spatial domain spike train symmetric DFTs symmetry theorem trapezoid rule trigonometric trigonometric polynomial TV-point two-dimensional DFT vector wave wavelength z-transform zero