Reading the literature, we often encounter sums of the form

for a point process Φ. This is an instance of what may be called “over-indexing”, since the subscript *k* is clearly not needed. It makes the notation unnecessarily cumbersome, but there is nothing mathematically wrong with it. If it is tacitly assumed that using a dummy variable of the form *x*_{k} somehow defines *k*, problems arise. For instance, what exactly is meant by this formula?

To make this concrete, let us focus on this simple form:

What is the result? It is not 14 but undefined, since *k* is undefined. This problem occurs fairly frequently in the form

where *h(k)* is used to denotes a fading random variable indexed by some (undefined) *k* and *f* is a path loss function. What is meant is usually

Alternatively,

Here the fading random variables are indexed by the points, i.e., they can be interpreted as marks associated with the points. This is the form I personally prefer, for it does not require a particular way of indexing the points and it is more compact.

### Like this:

Like Loading...

*Related*

Good one Martin, what do you think about the following notation:

$\sum_{i\in\Phi}h_{i}l(r_{i})$ in this case I believe the over-indexing issue can be avoided and also if instead of h we wanted to use ho for instance, i.e. gain between ith node and node placed at the origin it could be explicitly stated.

LikeLike

I am not a fan of using $i$ to represent a point of a point process; most people use $i$ for integers, so using it for an element of R^d may be confusing. While there is nothing mathematically wrong about using $i$ here, $\sum_{x\in\Phi}h_{x}l(r_{x}$ is better notation. Similarly, most people probably prefer to write (and read) $\int_{x=0}^10 f(x)dx$ for example, rather than $\int_{i=0}^10 f(i)di$.

Next, I am not sure how to connect $i\in\Phi$ with the “$i$-th node”. The latter would suggest that $i$ is an integer, but if $i\in\Phi$, this is definitely not the case. So that is an inconsistency.

LikeLike

Thank you, Martin, for the prompt and interesting response. I was just thinking back on the notation used for modeling dyadic fading where notation slightly gets complex and defining using $i$ could simplify life but I agree with your point about this is not a preferable option as connecting $i\in\Phi$ with the “$i$-th node” interpretation might be misleading. Thanks for your valuable feedback.

LikeLike