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Fourier Transform

In this article, learn about the Fourier transform and the temporal and frequency domain representations of a signal.
© Keio University

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:

figure of a single bar at 440.

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 (4, 8, 12,…) and this is, in fact, what we get:

figure of bars at 4, 8, 12, 16.

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).




音声は時間と共に変化する1次元の波です。 フーリエ変換は画像などの2次元データに対しても使用できます。 フーリエ変換に非常に類似したプロセスは、例えば、インターネット上で広く使用されているJPEG画像圧縮標準の基礎技術としても用いられています。



figure of a single bar at 440.

複雑な信号になればなるほどより複雑な変換をしますが、どの信号も正弦波と位相差 2/π の余弦波との組み合わせに分解することができます。この分野を学ぶに当たって単純な信号だけではなく、フーリエ変換についてできるだけ多くのことを学ぶことが大切です。





一見単純見えるかもしれませんが、結果は実際にはそれよりいくらか複雑です。パターンは4ビットごとに繰り返されるだけでなく、8ビットごと、12ビットごとにも繰り返されることになります。したがって変換も 4,8,12,… のシーケンスになると考えられ、実際そうなっています。

figure of bars at 4, 8, 12, 16.


© Keio University
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