Spectral analysis is a broad topic. Most scientists and engineers who deal with signals are quite comfortable with the concepts of time-frequency relationships. They are familiar with the Fourier transform and the common theorems such as Parseval’s, convolution, delay, etc. The fast Fourier transform, or FFT, is widely used and understood. The FFT gives a value of the Fourier transform at all integer multiples of the reciprocal record length T. This spacing in frequency is also the resolution. The Nyquist frequency or folding frequency is half the sampling frequency. FFT coefficients between the folding frequency fs/2 and the sampling frequency fs are identical to those between –fs/2 and 0, due to the periodicity of the coefficients in the frequency domain.
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Covariance Analysis for Seismic Signal Processing
This reference is intended to give the geophysical signal analyst sufficient material to understand the usefulness of data covariance matrix analysis in the processing of geophysical signals. A background of basic linear algebra, statistics, and fundamental random signal analysis is assumed. This reference is unique in that the data vector covariance matrix is used throughout. Rather than dealing with only one seismic data processing problem and presenting several methods, we will concentrate on only one fundamental methodology—analysis of the sample covariance matrix—and we present many seismic data problems to which the methodology applies.