From 36a634d52563d6cb44f0bb8414fa20ca32f55531 Mon Sep 17 00:00:00 2001 From: Philipp Le Date: Sat, 16 May 2020 16:40:39 +0200 Subject: WIP: Chapter 3 - Spectral Density, Decibel --- chapter02/content_ch02.tex | 21 ++++++++++++++++++++- 1 file changed, 20 insertions(+), 1 deletion(-) (limited to 'chapter02') diff --git a/chapter02/content_ch02.tex b/chapter02/content_ch02.tex index 745c55d..39134f1 100644 --- a/chapter02/content_ch02.tex +++ b/chapter02/content_ch02.tex @@ -353,7 +353,7 @@ Using the orthogonality relations, the coefficients $\tilde{a}_n$ and $\tilde{b} \begin{subequations} \begin{align} \tilde{a}_n &= \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \tilde{x}_p(t) \cdot \cos\left(n \omega_0 t\right) \, \mathrm{d} t \label{eq_ch02_fourier_series_coeff_an} \\ - \tilde{b}_m &= \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \tilde{x}_p(t) \cdot \sin\left(n \omega_0 t\right) \, \mathrm{d} t \label{eq_ch02_fourier_series_coeff_bm} + \tilde{b}_m &= \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \tilde{x}_p(t) \cdot \sin\left(m \omega_0 t\right) \, \mathrm{d} t \label{eq_ch02_fourier_series_coeff_bm} \end{align} \end{subequations} @@ -476,6 +476,20 @@ If the signal $\underline{x_p}(t) = x_p(t)$ is real-valued, i.e., $\Im\left\{\un \end{itemize} These symmetry rules apply for \underline{all} real-valued signals $\underline{x_p}(t) = x_p(t) \in \mathbb{R}$. The symmetry rules ensure that the mono-chromatic components of the Fourier series \eqref{eq:ch02:fourier_series_cmplx} sum up to a real value at each time instance $t \in \mathbb{R}$. +\begin{definition}{Hermitian function} + A complex-valued function $\underline{f}(t)$ is a \index{Hermitian function} \textbf{Hermitian function} if + \begin{equation} + \overline{\underline{f}(t)} = \underline{f}(-t) + \label{eq:ch02:hermitian} + \end{equation}% + \nomenclature[Na]{$\overline{\left(\cdot\right)}$}{Complex conjugate of $\left(\cdot\right)$} + where $\overline{\left(\cdot\right)}$ denotes the complex conjugate. + + Hermitian function are \index{conjugate symmetric} \textbf{conjugate symmetric}. +\end{definition} + +$\underline{c}_n$ is Hermitian if and only if $\underline{x_p}(t) = x_p(t)$ is real-valued. + The symmetry rules do \underline{not} apply for complex-valued signals $\underline{x_p}(t) \in \mathbb{C}$. \begin{figure}[H] @@ -706,6 +720,11 @@ The value-continuous complex frequency variable $j \omega$ in the continuous Fou \end{itemize} \end{itemize} +Analogue to the Fourier series, the Fourier transform $\underline{X}(j \omega)$ is Hermitian \eqref{eq:ch02:hermitian} if and only if $\underline{x}(t) = x(t)$ is real-valued. +\begin{equation} + \overline{\underline{X}(j \omega)} = \underline{X}(-j \omega) \qquad \forall \; \mathcal{F}^{-1}\left\{\underline{X}(j \omega)\right\} \in \mathbb{R} +\end{equation} + Let's investigate the \index{rectangular function} rectangular function from Figure \ref{fig:ch02:rect_function}. It is defined as: \begin{equation} \mathrm{rect}(t) = \begin{cases} -- cgit v1.1