## Basic Mathematical Tools

Now I realize that for performance analysis in wireless communication systems, we frequently use the following three mathematical tools:

I will explain how to use these mathematical tools by showing related simple examples.

## Theoretical approach vs. Experimental approach

When we want to solve simple mathematical problems, we usually try to find solutions exactly matched to the given problem, i.e., try to find closed form solutions. If we are interested only in a closed form solution, we will meet a number of fundamental limitations to solve highly complex problems.

- First, because of computational complexity it is not easy for the recent approach to find a solution for more than third order polynomial equations or for equations including non-linear functions.
- Second, by simplification in order to reduce the complexity of the given problems we will lose accuracy in its solution as well. For high level complexity problems, we simplify the given problems so as to be less order polynomials or to be linear equations. By doing that, the complex problems can be changed to solvable problems at the expense of solution accuracy.
- Third, we have also difficulty to solve multiple conditional problems without using additional mathematical tools when the number of conditions are more than one. We can solve the multiple conditional problems using, e.g., Lagrange multiplication, which has also computational limitation.

We can overcome these limitations by the computer simulation approach. The computer simulation can solve complex problems with high order polynomials, nonlinear equations and many conditions. However, traditional mathematicians do not tend to rely on the computer simulation results because they think it is unproved yet. It is obvious that the computer simulation may provide solutions associated only for a number of given situations because we can run the simulation for all situations. However, it is also wonder whether the closed form solution applicable for all cases without limitation.

Many Japan paper associations are interested in experimental results while western paper associations give less attention to the experimental results without theoretical analysis, i.e., the computer simulation. We now need to rethink which is a better way for the future research between the pure theoretical approach and the fusion experimental approach.