Ambiguity in the term 'dimension'?

[dexterdev]

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?

 

[hilbert2]

If you represent a "1D" signal as a function $f(t)$, which could give a value of electric current as a function of time, for example, the domain of the function $f$ is one-dimensional, but the set of all possible signal functions $f$ is infinite-dimensional.

 

[PSarkar]

I think when talking about 1D, 2D waves, the dimension refers to the dimension of the range space which is $\mathbb{R}$ and $\mathbb{R}^2$ respectively. These have dimensions 1 and 2 when talking about the usual vector space of $\mathbb{R}$ or $\mathbb{R}^2$ over $\mathbb{R}$ with the usual operations.

Now when you talk about the Fourier series of the square wave, the cosines and sines are indeed a basis but for a completely different vector space. Now you are looking at the vector space of the set of all continuous functions, not the range space which is $\mathbb{R}$ or $\mathbb{R}^2$.