From a7c67a1838333228a647a8d783bd6acfd8ae7f23 Mon Sep 17 00:00:00 2001 From: Philipp Le Date: Mon, 4 May 2020 01:22:22 +0200 Subject: Finishing chapter 1 --- chapter02/content_ch02.tex | 23 ++++++++++++++++++----- 1 file changed, 18 insertions(+), 5 deletions(-) (limited to 'chapter02') diff --git a/chapter02/content_ch02.tex b/chapter02/content_ch02.tex index 4ab3dea..d0cc535 100644 --- a/chapter02/content_ch02.tex +++ b/chapter02/content_ch02.tex @@ -1,4 +1,4 @@ -\chapter{Signals and Systems} +\chapter{Time-Continuous Signals and Systems} \begin{refsection} @@ -322,10 +322,14 @@ A special case is the coefficient $\tilde{a}_0$. \end{equation} $\cos\left(n \omega_0 t\right)$ is $1$ for $n = 0$. $\tilde{a}_0$ is the \index{DC offset} \textbf{\ac{DC} offset} of the signal. The above formula is known as the calculation of the signal mean in electrical engineering. -The composition of a series of mono-chromatic signals as shown in \eqref{eq:ch02:fourier_series} is called \index{Fourier series} \textbf{Fourier series}. -\begin{equation*} - x_p(t) = \sum\limits_{n=1}^{\infty} a_n \cos\left(n \omega_0 t\right) + \sum\limits_{m=1}^{\infty} b_m \sin\left(m \omega_0 t\right) -\end{equation*} +\begin{definition}{Fourier series} + The composition of a series of mono-chromatic signals as shown in \eqref{eq:ch02:fourier_series} is called \index{Fourier series} \textbf{Fourier series}. + \begin{equation*} + x_p(t) = \sum\limits_{n=1}^{\infty} a_n \cos\left(n \omega_0 t\right) + \sum\limits_{m=1}^{\infty} b_m \sin\left(m \omega_0 t\right) + \end{equation*} + + The coefficients can be calculated using \eqref{eq_ch02_fourier_series_coeff_an} and \eqref{eq_ch02_fourier_series_coeff_bm}. +\end{definition} \subsection{Complex-Valued Fourier Series} @@ -347,6 +351,15 @@ It is based on the orthogonality relation: \label{eq:ch02:orth_rel_exp} \end{equation} +\begin{definition}{Complex-Valued Fourier series} + A complex-valued, periodic signal $\underline{x_p}(t)$ can be decomposed into a series complex-valued mono-chromatic signals \eqref{eq:ch02:fourier_series_cmplx} -- the \index{Fourier series!complex-valued} \textbf{complex-valued Fourier series}. + \begin{equation*} + \underline{x_p}(t) = \sum\limits_{n = -\infty}^{\infty} \underline{c}_n \cdot e^{j n \omega_0 t} \qquad \forall \; n \in \mathbb{Z} + \end{equation*} + + The coefficients can be calculated using \eqref{eq_ch02_fourier_series_coeff_cn}. +\end{definition} + \subsection{Amplitude and Phase Spectra} \section{Non-Periodic Signals and Fourier Transform} -- cgit v1.1