# 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 Cf, 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.