## The Spectral Analysis of Time SeriesTo tailor time series models to a particular physical problem and to follow the working of various techniques for processing and analyzing data, one must understand the basic theory of spectral (frequency domain) analysis of time series. This classic book provides an introduction to the techniques and theories of spectral analysis of time series. In a discursive style, and with minimal dependence on mathematics, the book presents the geometric structure of spectral analysis. This approach makes possible useful, intuitive interpretations of important time series parameters and provides a unified framework for an otherwise scattered collection of seemingly isolated results. The books strength lies in its applicability to the needs of readers from many disciplines with varying backgrounds in mathematics. It provides a solid foundation in spectral analysis for fields that include statistics, signal process engineering, economics, geophysics, physics, and geology. Appendices provide details and proofs for those who are advanced in math. Theories are followed by examples and applications over a wide range of topics such as meteorology, seismology, and telecommunications. Topics covered include Hilbert spaces; univariate models for spectral analysis; multivariate spectral models; sampling, aliasing, and discrete-time models; real-time filtering; digital filters; linear filters; distribution theory; sampling properties ofspectral estimates; and linear prediction. Key Features * Hilbert spaces * univariate models for spectral analysis * multivariate spectral models * sampling, aliasing, and discrete-time models * real-time filtering * digital filters * linear filters * distribution theory * sampling properties of spectral estimates * linear prediction |

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### Contents

1 | |

29 | |

Chapter 3 Sampling Aliasing and DiscreteTime Models | 66 |

Chapter 4 Linear FiltersGeneral Properties with Applications to ContinuousTime Processes | 79 |

Chapter 5 Multivariate Spectral Models and Their Applications | 119 |

Chapter 6 Digital Filters | 165 |

Chapter 7 Finite Parameter Models Linear Prediction and RealTime Filtering | 210 |

Chapter 8 The Distribution Theory of Spectral Estimates with Applications to Statistical Inference | 257 |

Chapter 9 Sampling Properties of Spectral Estimates Experimental Design and Spectral Computations | 294 |

354 | |

359 | |

Probability and Mathematical Statistics | 367 |

### Other editions - View all

The Spectral Analysis of Time Series: Probability and ..., Volume 22 L. H. Koopmans Limited preview - 2014 |

### Common terms and phrases

applications asymptotic autoregression band-pass filters bandwidth calculate Chapter coefficient of coherence complex complex-valued components computed condition confidence interval Consequently continuous spectrum continuous-time convolution filter defined degrees of freedom denote determined digital filter discrete discussion equation equivalent example expression fast Fourier transform filter with transfer follows Fourier series frequency graph Hannan Hilbert space important inner product input integral lag window linear filter linear subspace low-pass filter matrix method Moreover moving average moving average representation multivariate obtain one-sided moving average operations output peaks periodic function polynomial prediction predictor problem properties random variables real-valued regression sample satisfy sequence series analysis side lobes smoothed periodogram estimator spectral density function spectral distribution spectral estimators spectral parameters spectral representation spectral window spectrum analysis symmetric tapering theorem theory transfer function transfer function B(A uncorrelated values variance vector weakly stationary process weighted covariance estimator white noise white noise process zero