Digital Signal Processing Reference
In-Depth Information
Chapter 6
The Fourier Transform
M
any dierent elds, including medicine, optics, physics, and electrical engineer-
ing, use the Fourier Transform (FT) as a common analysis tool. In practice, the
standards by compression groups JPEG (Joint Photographic Experts Group) and
MPEG (Motion Picture Experts Group) use a modied form of the Fourier trans-
form. Essentially, it allows us to look at frequency information instead of time
information, which people nd more natural for some data. For example, many
stereo systems have rows of little lights that glow according to the strength of fre-
quency bands. The stronger the treble, for example, the more lights along the row
are lit, creating a light bar that rises and falls according to the music. This is the
type of information produced by the Fourier transform.
The Discrete Fourier Transform (DFT) is the version of this transform that
we will concentrate on, since it works on discrete data. Our data are discrete
in time, and we can assume that they are periodic. That is, if we took another
N samples they would just be a repeat of the data we already have, in terms of
the frequencies present. In this case, we use the DFT, which produces discrete
frequency information that we also assume is periodic. Below, in Figure 6.1, we see
the frequency magnitude response graph for the \ee" sound. We got this by applying
the discrete Fourier transform to the sound le. The gure shows the whole range
along the top, and a close-up view on the bottom.
The following graph, Figure 6.2, shows the frequency magnitude response for
another sound le: a brief recording of Adagio from Toccata and Fuge in C, written
by J.S. Bach. The top of this gure shows the entire frequency range, from 0 to
22,050 Hz, while the bottom part shows a close-up view of the rst 4500 frequencies.
An interesting thing to notice is that the spikes in magnitude are regularly spaced.
This happens often with real signals, especially music. For example, right before
1000 Hz, we see three spikes increasing in magnitude, corresponding to three dierent
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