1  Text Conventions

f 1.

Definitions are indicated by $\triangleq$.

f 2.

When discussing digital modulation, $N$ is the number of dimensions (e.g., $N=2$ for QAM), $M$ is the number of constellation symbols, $S$ is the number of samples per symbol (also called oversampling factor) and $b={\text{log}<!--\; FUNCTION\; APPLICATION\; -->}_{2}M$ is the number of bits per symbol.

f 3.

Random variables are upper-case mathsf in Latex, e.g. $X,Y$ and $Z$.

f 4.

Random signals are lower-case mathsf in Latex, e.g. $x[n]$

f 5.

Random processes use cal in Latex, e.g., $X[n]$ and $X(t)$

f 6.

Sets also use cal in Latex, e.g., $C=A\cup B$. The context should be enough to disambiguate random processes and sets.

f 7.

We call D/C (discrete/continuous) and C/D (continuous/discrete) the theoretical models used as stages of the practical D/A and A/D conversions, respectively.

f 8.

Analog (unquantized continuous-time) signals $x(t)$. A sampled signal is indicated with a subscript $s$, e.g., $x_s(t)$. Discrete-time $x[n]$ and digital $x^q[n]$, sampled and quantized $x_s^q(t)$

f 9.

Distinguish the signal $x(t)$ (or x[n]) from a specific sample $x(t_0)$. At least in the beginning of the text.

f 10.

Units are taken from the international system: length in meters, time in seconds, $x(t)$ and $x[n]$ in Volts, $X(f)$ in Volts/Hz and PSDs in Watts/Hz.

f 11.

$𝔼$ is the expected value and Var is the variance.

f 12.

The superscripts *, $T$ and $H$ denote complex conjugate, transpose and Hermitian, respectively. For matrices, ${\mathbf{\text{A}}}^H={({\mathbf{\text{A}}}^{\ast })}^T$.

f 13.

Two Latin sentences will be used: i.e. (id est, which means “that is”) and e.g. (exempli gratia, which means “for example”)

f 14.

Whenever possible, constants and variables will be represented by upper and lower case letters, respectively

f 15.

Vectors are bold lower-case (e.g., $\mathbf{\text{x}}$) and matrices are bold upper-case (e.g., $\mathbf{\text{X}}$). When dealing with transforms, a bold upper-case letter is also used. For example, the vector $\mathbf{\text{X}}$ is the transform of $\mathbf{\text{x}}$. The context should be enough to disambiguate transforms and matrices.

f 16.

As in Matlab/Octave, the column $m$ of a $M\times N$ matrix $A$ is represented by $A(:,m)$, with $m=0,\dots ,M-1$. Similarly, $A(n,:)$ denotes the $n$-th row, with $n=0,\dots ,N-1$. Note that the first index in Matlab/Octaveis 1.

f 17.

Lower case can eventually denote frequency-domain variables

f 18.

$\text{|⋅|}$ magnitude or absolute value of a complex scalar.

f 19.

ang$\text{(⋅)}$ angle of a complex scalar

f 20.

$\parallel \mathbf{\text{x}}\parallel$ is the norm of vector $\mathbf{\text{x}}$

f 21.

Estimates are indicated by a hat over the symbol (e.g., $\widehat{x}$)

f 22.

diag is the main diagonal of a matrix

f 23.

A matrix $\left[\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right]$ can be denoted using Matlab/Octave syntax as $[a,b;c,d]$.

f 24.

The $(i,j)$-th element of a matrix $\mathbf{\text{A}}$, at row $i$ and column $j$, is represented by $a_{i,j}$.

f 25.

When the value of a variable $x$ is given in dB it will be denoted with a subscript $x_{\mathrm{\text{dB}}}$.

f 26.

In multicarrier systems, subscripts denote the user $n$ while the superscripts denote the tone $k$