# 1Background

## 1.1. Introduction

This chapter provides some background necessary for the remainder of this book. The common operations and functions are presented first. The common transforms required for calculations are detailed in section 1.3. Then, some background on discrete and continuous probabilities is provided in section 1.4. Finally, some elements of digital signal processing are recalled in section 1.5.

## 1.2. Common operations and functions

### 1.2.1. *Convolution*

The convolution between two signals *s*(*t*) and *h*(*t*) is defined as:

We will write it as (*s* * *h*)(*t*) = *s*(*t*) * *h*(*t*) with a notational abuse.

The convolution is linear and invarious to time. Consequently, for any continuous signals *s*_{1}(*t*) and *s*_{2}(*t*), any complex values α_{1}, α_{2} and any time delay *t*_{1}, *t*_{2}, we can write:

### 1.2.2. *Scalar product*

The scalar product between two continuous signals *s*(*t*) and *r(t)* is defined as:

For any continuous signals *s*_{1}(*t*), *s*_{2}(*t*), *r*_{1}(*t*) and *r*_{2}(*t*) and any complex values α_{1}, α_{2}, the following linearity properties hold:

For vectors of size *m ×* 1, s = [s_{0}, s_{1}, …, *s _{m − }*

_{1}]

*and ...*

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