diff options
Diffstat (limited to 'chapter02/content_ch02.tex')
| -rw-r--r-- | chapter02/content_ch02.tex | 45 |
1 files changed, 42 insertions, 3 deletions
diff --git a/chapter02/content_ch02.tex b/chapter02/content_ch02.tex index 8a9ec88..4ab3dea 100644 --- a/chapter02/content_ch02.tex +++ b/chapter02/content_ch02.tex @@ -109,9 +109,50 @@ When a signal passes through a \ac{LTI} system, the amplitude, the phase or both \end{itemize} remain. Both are absorbed by the complex-valued \index{phasor} \textbf{phasor} $\underline{X}$, which uniquely describes a mono-chromatic signal. \begin{equation} - \underline{X} = \hat{X} \cdot e^{-j \varphi_0} + \underline{X} = \hat{X} \cdot e^{-j \varphi_0} = \hat{X} \angle -\varphi_0 \end{equation} +\begin{excursus}{Complex numbers} + $j$ is the \index{imaginary unit} \textbf{imaginary unit}. It satisfies the equation + \begin{equation} + j^2 = -1 + \end{equation} + There is no real number $j \notin \mathbb{R}$ which satisfies the above solution. $j$ spans the set of complex numbers $\mathbb{C}$. + + In mathematics, the imaginary unit is noted as $i$. In engineering context, $j$ is used instead, because $i$ is the symbol of the electric current. + + A complex number $\underline{c} \in \mathbb{C}$ can be noted in \index{cartesian form} \textbf{cartesian form}: + \begin{equation} + \underline{c} = a + j b + \end{equation} + $a \in \mathbb{R}$ is the \index{real part} \textbf{real part} of $\underline{c}$. $b \in \mathbb{R}$ is the \index{imaginary part} \textbf{imaginary part} $\underline{c}$. + \begin{subequations} + \begin{align} + a &= \Re\{\underline{c}\} \\ + b &= \Im\{\underline{c}\} + \end{align} + \end{subequations} + Complex numbers $\underline{c}$ always carry an underline in this lecture to distinguish them from real numbers. However, this is not mandatory. + + Another notation is the \index{polar form} \textbf{polar form}: + \begin{equation} + \underline{c} = r \cdot e^{j \varphi} + \end{equation} + with + \begin{subequations} + \begin{align} + r &= |\underline{c}| = \sqrt{\Re\{\underline{c}\}^2 + \Im\{\underline{c}\}^2} \\ + \varphi &= \mathrm{atan2} \left(\Im\{\underline{c}\}, \Re\{\underline{c}\}\right) \\ + e^{j \varphi} &= \cos \varphi + j \sin \varphi + \end{align} + \end{subequations} + The polar form can be written in \index{angle notation} \textbf{angle notation}: + \begin{equation} + \underline{c} = r \angle \varphi + \end{equation} + $r \in \mathbb{R}$ and $\varphi \in \mathbb{R}$ are the \index{polar coordinates} \textbf{polar coordinates}. +\end{excursus} + The phasor $\underline{X} \in \mathbb{C}$ is a complex number, which is mostly represented in polar coordinates (see Figure \ref{fig:ch02:cmplxplane_phasor}). \begin{figure}[H] @@ -146,8 +187,6 @@ The real-valued function can be obtained by extracting the real part of the comp x_{mc}(t) = \Re\left\{\underline{x_{mc}}(t)\right\} \end{equation} -% Exercise: Is a sine wave with DC bias mono-chromatic -> no - \section{Periodic Signals and Fourier Series} Periodic signals $x_p(t)$ comprises a class of signals which indefinitely repeat at constant time intervals $T_0$. |