Learning Networks (0): Open Course "Networks: Friend, Money and Bytes"

I'm gonna jump into Networks area. To broaden my horizons, I decide first to take the course "Networks: Friend, Money and Bytes", taught by Mung Chiang from Princeton University. The course attracts me by its interesting 20 Questions. Its goal is to teach students some important models and methods used in different areas.

Q1: What makes CDMA work for my smartphone

This question is not so interesting to me, perhaps because I don't use smartphone. The methods used to solve the problem is familiar to me. We will see what these methods are soon.

Problem Statement

The problem is CDMA performs frequency reuse, but it is non-orthogonal so that it suffers from interference. The goal is to achieve a target signal quality for each user.

Model

First to evaluate the signal quality, we define Signal-to-Interference-noise-Ratio (SIR) associated with $i$-th user:

\[SIR_i = \frac{P_iG_{ii}}{\sum_{j \neq i}{P_jG_{ij}} + n_i},\]

where $P_i$ denotes power at the transmitter side $i$, $G_{ij}$ represents the channel gain from the transmitter of the logical transceiver pair $j$ to the receiver of logical transceiver pair $i$, and $n_i$ is the noise at the $i$-th receiver.

The above formula is quite straightforward, $P_iG_{ii}$ is the signal power at the $i$-th receiver side. The denominator contains signal interference from other transmitters plus the noise. In this course, $G_{ij}$ and $n_i$ is considered as constants. We only need to decide the value of $P_i$ to achieve target SIR.

Methods

View1: Optimization

\[\min\sum_{i=1}^{N}P_i\]

\[s.t. STR_i(\vec{p}) \geq \beta_i, i = 1, 2, \ldots, N\]

where $\beta_i$ is the target SIR. After simple conversion of the constrains, we can find it is a linear programming.

About linear programming, I just know how to write down a LP formula, but don't know how to analyze its convexity and how to solve it. These questions left for future study.

View2: Game Theory
In my point of view, if optimization considers the problem from the perspective of overall performance, game theory is from the perspective of individual benefit when we think players are competing with each other. In the lecture, the professor talked about dominant action and Nash equilibrium, which reminds me of the course I took from coursera this spring. The professor gave the conclusion without proof that is the best strategy is Distributed power control (DPC).

DPC
This simple algorithm goes as follows:
\[P_i[t+1] = \frac{\beta_i}{STR_i[t]}P_i[t]\]

where $P_i[t]$ is current power, $STR_i[t]$ is the feedback.

Assignment