Discrete Fourier Transform of an Arbitrary (Finite) Energy Signal

Features

• Input data is supplied by the user using the copy/paste mechanism from a local text file
• DFT approximation for various definitions of the continuous Fourier transform
• Plot for real-valued time-domain signal
• Plots for complex-valued frequency-domain signal in either Real/Imaginary or Magnitude/Argument format
• Scale factors are reported so the coordinates are well quantified if the data have physical dimensions attached

Rules and Theories

There are many variations of continuous Fourier transform definitions. But, they all fall into one of the two categories:

Type 1:

Type 2:

where and typically .

This applet utilizes a discrete Fourier transform (DFT) via the popular FFT algorithm to approximate the Fourier transform. For a given definition and choice of C_f, the forward Fourier transform is performed on a REAL time-domain (finite) energy signal x(t) evenly sampled at a span of

where N is a number that is power of 2. The inverse transform is carried out using the inverse FFT algorithm.

The Applet and User's Guide

• Enter a sequence of time-domain sampled data arranged in one column into the input text area. Alternatively, user can open a local file containing appropriate data, first copy then paste (using Ctrl-V) into the text area
• Change the time-domain sampling rate (Delta t) and set the correct number of points
• Click the "Forward Transform" button to perform the forward transform. Results will be used to plot the signals in both time and frequency domains. The spectral sequence is automatically filled
• Click the "Inverse Transform" to perform an inverse transform

References

[1] Alexander D. Poularikas, "The Transform and Applications Handbook", CRC Press, Boca Raton, 1996.

[2] George Arfken, "Mathematical Methods for Physicists", Academic Press, San Diego, 1985.

[3] William H. Press, Saul A. Teukolsky, Willaim T. Vetterling and Brian P. Flannery, "Numerical Recipes in C", Cambridge University Press, Cambridge, 1995.