# Don't understand what is meant by signal dimension

I don't understand the concept of dimension of a signal. I ran into it in an explanation of Shannon Capacity, and in a paper on spread spectrum. I was hoping somebody could explain with an example. Does it apply to analog as well as digital? For instance is AM two dimensional considering time and amplitude? Or is QPSK two dimensional because it is a combination of a sin and cosine term? Or does multilevel signaling like Pulse Amplitude Modulation have as many dimensions as discrete levels it can take, or would dimensionality refer to the number of possible symbols it can represent in a pulse?

• Please quote some paragraphs so that we can be sure what you are asking about. – AlexTP Mar 9 '18 at 9:16

In digital communication systems, the dimension of a modulation scheme refers to the number of basis function, i.e. independent/orthogonal signals (in this case $\sin$ and $\cos$ function) used to represent the symbols. In these systems $k=\log_2 M$ binary digits are mapped into analog waveforms below $$\left\{s_m(t), m = 1, 2, \ldots, M \ \big\vert \ M = 2^k\right\}$$ Take a PAM signal for instance, this has the form \begin{align} s_m(t)&=\Re\left[A_mg(t)\exp\left(j2\pi f_c t\right)\right]\\ &= A_mg(t)\cos\left(2\pi f_c t\right) \end{align} One basis function (one axis), the in-phase component (here $x$-axis) is used to represent the signal. The signal is one dimensional. Now consider PSK modulation, the signal waveforms are represented as
\begin{align} s_m(t)&=\Re\bigg[g(t)\exp\left(j2\pi\frac{m - 1}{M}\right)\exp\left(j2\pi f_c t\right)\bigg]\\ &= g(t)\cos\left(2\pi f_c t + 2\pi\frac{m - 1}{M}\right) \end{align} Which can be decomposed down into sine and cosine components. This is a two-dimensional signal. The same goes for QAM signals.

A common misconception is taking the dimension for the number of bits per symbol. In brief, the dimension is the number of orthogonal basis functions used in the linear combination representing the waveforms $s_m (t)$. Note that despite the digital part of it, what is not always aparent is that the modulator takes in charge the mapping of the binary info into the symbols in the analog domain ready for transmission.

Having said that, there is also a notion of multi-dimensional signals where the time and frequency intervals are subdivided into short intervals for transmission. More on this and my explanation above see $[1]$.

$[1]$, J. G. Proakis, Digital Communications, 4th edition, McGraw Hill, chapter 4.

• Good expatiation with the number of axis', but I'm still trying to understand. Is M or K the number of dimensions? In the PAM example m can take on any value, if M had, for example, 4 discrete values then M=4 and k= 2. But it's still a one dimensional signal? – Frank Mar 16 '18 at 19:50
• That's right, in the PAM despite the $M = 4$ i.e. ($2^k$) and $k=2$, you have $M$ possible amplitudes for $k$-blocks of symbols; so neither $M$ nor $k$ represents the dimension. The dimensionality is seen on the number of basis functions. In my first equation it's seen in the $\cos$ multiplication factor. – Gilles Mar 17 '18 at 11:17

i'm not sure, but it might mean a couple of things:

1. not likely, but it could mean the class of units (but not a specific unit in that class) used to describe the physical quantity that the signal represents. like instantaneous current or voltage or pressure or position. whatever goes into the transducer that the signal comes out of.

2. some signals are in vectors. (for instance, a stereo audio signal is a 2-dim vector, a color video signal is a 3-dim vector) maybe it's about the number of dimensions of a vector signal: $$\mathbf{x}(t) = \begin{bmatrix} x_1(t) \\ x_2(t) \\ \vdots \\ x_N(t) \end{bmatrix}$$ or perhaps about one of the specific elements of the vector; $x_n(t)$.

and if Shannon Information Theory is the context, perhaps it is about the independence of the elements of the vector. e.g. a 6-dimensional vector signal might have an element that is a linear combination of the other 5. from an information (and communications) POV, perhaps Shannon thinks that just a 5-dimensional vector.