# Fourier Transform

We have already seen what a sine wave looks like in time and space – it’s the most basic form of wave.

The *Fourier transform* lets us to
see that same sine wave in a different way. It is a common
mathematical tool used by physicists and engineers that changes the
representation of spatial or temporal data into *frequency* data.
Among other applications, the Fourier transform is useful when we want
to understand how digital sound, such as that from a CD, becomes the
analog (continuous) sound that we hear, and whether or not it
faithfully reproduces the originally recorded sound.

Sound, of course, is one-dimensional; it changes with time. The Fourier transform can also be used on two-dimensional data such as images. A process very similar to the Fourier transform forms the basis of the JPEG image compression standard widely used on the Internet, for example.

## The Continuous Fourier Transform

A sine wave consists of only a single frequency, so its representation in the frequency space is very simple: just a single number representing that frequency. For example, the common musical note A above middle C on a piano has a frequency of 440 hertz, so its Fourier transform is just the number 440. We can represent this graphically just as a single vertical bar at the appropriate place:

More complex signals have more complex transforms, but any signal can be broken down into a combination of sine waves and cosine waves (which, of course, are just sine waves shifted by a quarter of a cycle). For our purposes here, we only need to learn about simple signals, but we encourage you to learn as much as you can about the Fourier transform.

## The Discrete Fourier Transform

Our abstract example above is the *continuous* Fourier transform. But
normally, when we are working with computers, we have *discrete* data,
a series of *samples* of the actual signal. We can also calculate the
original signal itself using any digital process. Consider the
following sequence of bits:

Obviously the pattern repeats itself once every four bits. We would expect that a digital transform of this would give us the number 4.

The result is actually somewhat more complicated than that. Not only does the pattern repeat every four bits, it also repeats every 8 bits, every 12 bits, and so on. Thus, after a little thought, we might instead expect that the transform would give us the sequence and this is, in fact, what we get:

A graph like this is said to be the *frequency domain representation*
of the signal, rather than in the more common *temporal domain
representation* (for a signal that varies in time, like sound) or the
*spatial domain representation* (e.g., for an image).

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