1  Text Conventions

f 1.

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

f 2.

Definitions are indicated by $\triangleq$.

f 3.

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 4.

Similarly, S/D denotes the conversion of a sampled signal into a discrete-time signal, while D/S is the reverse process.

f 5.

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

f 6.

A specific value of the signal $x(t)$ at time $t=t_0$ is denoted as $x(t_0)$. Similarly, $x[n_0]$ indicates a single sample of the signal $x[n]$ at time $n=n_0$.

f 7.

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 8.

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

f 9.

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

f 10.

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

f 11.

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 12.

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

f 13.

ang$\text{(⋅)}$ denotes the angle of a complex number.

f 14.

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

f 15.

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

f 16.

diag is the main diagonal of a matrix.

f 17.

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 18.

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

f 19.

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, while Python adopts 0.

f 20.

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

f 21.

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

f 22.

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

f 23.

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

f 24.

The value of a variable $x$ given in dB is denoted with a subscript $x_{\mathrm{\text{dB}}}$.