The Fast Fourier Transform and its application to tidal oscillations

The Fast Fourier Transform and its application to tidal oscillations^*

A. dos Santos Franco; Norman J. Rock^**

Instituto Oceanográfico da Universidade de São Paulo

SYNOPSIS

This paper proposes a new way of tidal spectral analysis based on the Cooley-Tukey algorithm, known as the Fast Fourier Transform. The Fast Fourier Transform analysis is used to compute both the harmonic constants of the tide and the power spectrum.The latter is obtained by means of a weighted sum. A new way is also derived to obtain the formula giving the number of the degrees of freedom,on which is based the confi dence interval corresponding to the noise spectrum.

RESUMO

Este trabalho propõe um novo caminho para a analise espectral da maré baseada no algoritmo de Cooley-Tukey. A análise através da "Transformação Rápida de Fourier" (Fast Fourier Transform - FFT) é empregada tanto para calcular as constantes harmônicas da maré quanto para a obtenção do espectro de energia. Este é calculado por meio de uma soma ponderada. Também é dada uma nova dedução da fórmula que exprime o número de graus de liberdade em que se baseia o intervalo de confiança correspondente ao espectro do ruído. O trabalho foi redigido em inglês a fim de facilitar o intercâmbio de informações.

Full text available only in PDF format.

Texto completo disponível apenas em PDF.

ACKNOWLEDGEMENTS

Grateful appreciation is extended to the "Instituto de Física", USP, for allowing unrestricted use of the IBM 360 computer installation.

Also sincere thanks are due to the "Instituto de Comunicações Elétricas" for the free use of the IBM 1130 for high speed plotting of spectra.

A special thank you to Mr. Sylvio José Correa for his valuable collaboration in the preparation of computer programs in Appendices V, VI.

LIST OF SYMBOLS USED

a,b cosine/sine component A, B cosine/sine matrices A(Θ) auto-correlation function for time lag Θ c vector denoting phase and amplitude of oscillation C_S spectral estimate at frequency s.Δf (Δf = fundamental frequency) F_J nearest Fourier frequency to J^th tidal constituent J subscript denoting tidal constituent K(Θ) cross-correlation function for time lag Θ m maximum number of lags n subscript denoting Fourier number N number of values in Fourier series p index denoting Fourier number q angular frequency (speed number) Q number of tidal constituents r phase lag reckoned from the time origin R amplitude of tidal constituent s index denoting discrete frequency of spectral estimate Ŝ_xy cross-spectral estimate between series x and y t time v (t) gaussian noise as a function of time x (t) time series y (t) tidal heights as a function of time z index for values of discrete weighting function α damping coefficient for weighting function γ raw Fourier spectral estimate estimate ε estimate vector denoting phase and amplitude of random oscillation Θ time lag Φ, ψ weighting functions µ_o r.m.s. amplitude of white noise ν degrees of freedom τ sampling interval ω angular speed difference/sum ξ, η cosine/sine of residual noise

(Received 10/15/1970)

APPENDICES

SET OF PROGRAMS FOR TIDAL ANALYSIS

by the

"Instituto Oceanográfico" Method

N.B. The programs are presented separately to provide greater flexibility to the user. In practice, however, programs in Appendices I, II, III, IV, are inter-linked and executed sequentially by the computer.

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APPENDIX I

FAST FOURIER TRANSFORM

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APPENDIX II

MATRIX GENERATION

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APPENDIX III

CALCULATION OF TIDAL COMPONENTES a_j AND b_j., HARMONIC CONSTANTS H&g AND CORRECTION OF FOURIER COEFFICIENTS FOR TIDAL EFFECTS TO OBTAIN RESIDUALS ξ_n AND η_n

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APPENDIX IV

POWER SPECTRAL ANALYSIS

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*

This is the reprint with minor corrections and improved programs. The former reprint is obsolete.

**

Visiting Professor at the Instituto Oceanográfico da Universidade de São Paulo on a Ford Foundation and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) grants.

Publication Dates

• Publication in this collection
  12 June 2012
• Date of issue
  1971

History

• Received
  15 Oct 1970