Fourier Transform

GLOSSARY OF TERMS USED IN TIME SERIES ANALYSIS
OF CARDIOVASCULAR DATA

FOURIER TRANSFORM

Frequency-domain representation of time series as complex-valued function, characterized by a modulus function and a phase function.

The Fourier transform X(f) of the signal x(t) is given by

X(f) contains all the information of the original signal, and x(t) can be obtained from X(f) by the inverse Fourier transformation:

Dealing with discrete time-series x(n), however, the Discrete-time Fourier Transform is used:

whose inverse transformation is:

Computationally efficient algorithms for the calculation of the discrete-time Fourier Transform are called Fast Fourier Transform (or FFT) algoritms.

Reference:
Oppenheim AV, Schafer RW (1989) Discrete-time signal processing Prentice-Hall, Englewood Cliffs, NJ

Links:
Fourier Transform from Eric Weisstein's World of Mathematics
An Introduction to Fourier Theory by Forrest Hoffman
Mathematics of the Discrete Fourier Transform (DFT) by Julius O. Smith III

(PC 07-09-1999)