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 s1(t) and s2(t), any complex values α1, α2 and any time delay t1, t2, we can write:

images [1.2]

1.2.2. Scalar product

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

images [1.3]

For any continuous signals s1(t), s2(t), r1(t) and r2(t) and any complex values α1, α2, the following linearity properties hold:

images [1.4]

For vectors of size m × 1, s = [s0, s1, …, sm − 1]T and ...

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