# What is signal dimension

Estimating Unknown Sparsity in Compressed Sensing is a paper about sparse signal. I am just learning the concepts. In the first paragraph, it says that when the number of observation data samples $n$ is less than the signal dimension $p$, then the desired signal $x$ is sparse. I have never come across he term signal dimension. Can somebody please explain what is signal dimension ?

The signal dimension $p$ refers to the dimension of the signal. The paper in question specifies that it is working in $R^p$, not ($R^1$) because the author wants to take into account multiple dimensions. For example, if your data is a geographic location, you have dimension $p=3$ because the complete description requires 3 numbers: longitude, latitude and altitude.

• That means it is the number of variables?Can you please explain with an example equation?Or is it like a vector auto regressive model? Commented Aug 6, 2014 at 2:05
• No that does not mean the number of variables. I gave you an example of geographic location. If that is your variable, you need 3 numbers to completely describe the geographic location of an object: longitude, latitude and altitude. For example, point A is located at co-ordinates (x,y,z). It has nothing to do with vector auto-regressive model. It is not a model at all, just a description. Commented Aug 6, 2014 at 2:21
• Are they the number of coefficients of a linear regression model, say auto regression model of order 2: AR(2) then is signal dimension p=2? Commented Aug 6, 2014 at 2:24
• No. It is not the number of coefficients of any model. It is just the complete description of a single variable. It has nothing to do with the dimension of a model. AR(2) tells you nothing about the dimension of the signal. Commented Aug 6, 2014 at 2:36
• Ok, so by looking at the signal is there a way to know what the dimension is? The thing is I am learning about sparse regression, and everywhere I get is that the signal dimension must be greater than number of samples observed. So, in this context how do I determine if the model is sparse or not if nothing is reflected in the functional form of the model equation?Shall be obliged for your insights into this matter. Commented Aug 6, 2014 at 2:41

Signal dimension is just the dimension of the random variable that is the signal. "Dimension $p$" is a slightly sloppy way of stating the dimension of the support of the random variable is $p$. That is, it's a multivariate signal with $p$ components.

• Can you please explain with an example of what a multivariate signal is?How it looks like in eQuation form? Commented Aug 6, 2014 at 2:06
• Color on a computer screen is a 3-dimensional signal of red, green, and blue. This would be represented as a vector $color = (red, green, blue) \in \mathbb{R}^3$. Commented Aug 6, 2014 at 2:36
• Both of your answers are different. Does dimensionality means multivariate signal where each variable = dimension?So, in linear regression I will have 3 equations if dimension = 3?Can you please explain with AR model? Commented Aug 6, 2014 at 2:46
• It's a matter of interpretation. You can think of "color" as one variable, or "red," "green," and "blue" as three separate variables. Commented Aug 6, 2014 at 2:51