# Ambiguity in the term 'dimension'?

We used to classify signals as 1D and 2D etc ie one dimensional and two dimensional. For example a periodic square wave signal is 1D and an image is a 2D signal etc (reference - Signals and systems by Simon Haykin and Barry Van Veen, 2nd edition , page 2).

But the same periodic square wave signal can be decomposed using fourier series to infinite sinusoids with different frequencies. In the linear algebra terms these infinite orthogonal sinusoids forms the basis and the the dimension of a periodic square wave is infinite.

So actually which is the actual dimension or what is dimension?

Dimension in mathematics is defined as the number of independent components of a structure (with some generalizations, like fractal dimensions). Now which number of dimensions you assign to e.g a discrete image (as in picture) depends on what aspect of the image you are describing. As a vector space where each pixel is a component, you get as many dimensions as pixels. The same for a discrete signal, the number of samples is your dimension of the vector space.

However, sometimes you're rather interested in the manifold the object is a function on. The real line is a 1-dimensional real manifold, and a real function on it would be a signal. So you can call the signal "one dimensional", explicitly referring to this property. Sometimes your function may map to the complex numbers or to R^2 (still from the real line!) instead, and you might want to call that a "two dimensional signal", and that's also correct. However, a two dimensional signal may also just be a function on a two dimensional manifold, like a plane.

So there's a lot ambiguity in verbal descriptions, which is why to be clear you should use exact mathematical notation. Opposite to common belief it was not just invented to torture students.

When you say dimension you're referring to the axis of the signal space, e.g., a 2D signal with time and amplitude axis. When you want to refer to transforms such as Fourier or Laplace you should say domain instead since what you're doing is a transform of the signal from one domain to another. For example, when applying the Fourier transform you go from the time domain to the frequency domain.

http://en.wikipedia.org/wiki/Time_domain

http://en.wikipedia.org/wiki/Frequency_domain