**Baron Fourier** (1768-1830):

*Any continuous periodic signal can be represented as the sum of
carefully chosen sine waves.*

The *Fourier Transform* is used to express a function as a
continuous integral of sine waves. The *Fourier Series* (or
*Discrete Fourier Transform*) is used to express a discretely
sampled periodic function as a discrete sum of sine waves.

The DFT:

and its inverse:

An input series, *f*, of *N* real values fed through the
forward DFT will produce an output series, *F*, of *N*/2
complex values. Obviously, the highest frequency that can be
represented without aliasing given any sampling rate, *R*, is
*R*/2 Hz. One-zero-one-zero at 44100Hz is a square wave with a
frequency of 22050Hz. This lower frequency is called the Nyquist limit
or Nyquist frequency.

The first sample of the transformed series, *F _{0}*, is
the DC component (the average) of the input series.

N = number of samples R = sampling rate (samples per second) R/2 = Nyquist frequency (Hz) N/R = duration (seconds) |F[0]| = DC component Re F[n] = n-th real output Im F[n] = n-th imaginary output n*R/N = frequency (in Hz) corresponding to n-th output |F[n]| = amplitude of n*R/N Hz frequency Arg F[n] = phase of n*R/N Hz frequency

The complex output of a DFT is a polar representation of the present frequencies. The amplitude of any given frequency is the modulus of the corresponding complex output; the phase is the angle of the complex output:

nf kindly provided a 48Khz sample of him plucking the A string on his steel electric guitar. Here's a single period taken from it:

To get enough frequency resolution, I took a little more than two seconds
worth of samples and ran them through a DFT (example code:
dft.c - *slow*):

The most prominent frequency is at 107 point something Hz. The frequency of a perfectly tuned A-2 is 110Hz. The spectrum also shows diminishing overtones at integer multiples of the base frequency.

Graphs by gnuplot.[home]

copyright © 2004 Emil Mikulic

First try: 2001/09/25

$Date: 2004/10/09 08:56:10 $