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Diffstat (limited to 'chapter02')
| -rw-r--r-- | chapter02/Rainbow.jpg | bin | 0 -> 5197 bytes | |||
| -rw-r--r-- | chapter02/content_ch02.tex | 269 |
2 files changed, 242 insertions, 27 deletions
diff --git a/chapter02/Rainbow.jpg b/chapter02/Rainbow.jpg Binary files differnew file mode 100644 index 0000000..7c84756 --- /dev/null +++ b/chapter02/Rainbow.jpg diff --git a/chapter02/content_ch02.tex b/chapter02/content_ch02.tex index 071400c..9dc7df2 100644 --- a/chapter02/content_ch02.tex +++ b/chapter02/content_ch02.tex @@ -389,7 +389,7 @@ If the signal $\underline{x_p}(t) = x_p(t)$ is real-valued, i.e., $\Im\left\{\un \begin{itemize} \item The amplitude spectrum $|\underline{c}_n|$ is an \underline{even function}. It is symmetric with respect to the $y$-axis. \item The phase spectrum $\arg\left(\underline{c}_n\right)$ is an \underline{odd function}. It is symmetric with respect to the origin. - \item As a consequence, the phase of $\arg\left(\underline{c}_0\right)$ at $n = 0$ must be either $0$ or $+\pi$. Note that, $+\pi$ is identical to $-\pi$ in the complex plane. Thus, $-\pi$ is valid, too, but not distinct from $+\pi$. This phase is the sign of the \ac{DC} bias: $\arg\left(\underline{c}_0\right) = 0$ means positive \ac{DC} bias and $\arg\left(\underline{c}_0\right) = \pi$ means negative \ac{DC} bias. + \item As a consequence, the phase of $\arg\left(\underline{c}_0\right)$ at $n = 0$ must be either $0$ or $\pm \pi$. Note that, $+\pi$ is identical to $-\pi$ in the complex plane. The phase is the sign of the \ac{DC} bias: $\arg\left(\underline{c}_0\right) = 0$ means positive \ac{DC} bias and $\arg\left(\underline{c}_0\right) = \pi$ means negative \ac{DC} bias. \end{itemize} \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 $t \in \mathbb{R}$. @@ -416,15 +416,15 @@ The symmetry rules do \underline{not} apply for complex-valued signals $\underli ymax=3, xtick={-3, -2, ..., 3}, ytick={0, 0.5, ..., 2.5}, - axis y line=middle,
+ axis y line=middle, axis x line=middle, - every axis x label/.style={
- at={(ticklabel* cs:1.05)},
- anchor=north,
- },
- every axis y label/.style={
- at={(ticklabel* cs:1.05)},
- anchor=east,
+ every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, } ] \addplot[red, thick] coordinates {(-3, 0) (-3, 2.0)}; @@ -460,17 +460,18 @@ The symmetry rules do \underline{not} apply for complex-valued signals $\underli ymin=-4, ymax=4, xtick={-3, -2, ..., 3}, - ytick={-3.14159, -1.5708,
1.5708, 3.14159}, - yticklabels={$-\pi\hspace{0.30cm}$, $-\frac{\pi}{2}$,
$\frac{\pi}{2}$, $\pi\hspace{0.10cm}$}, - axis y line=middle,
+ ytick={-3.14159, -1.5708, 1.5708, 3.14159}, + yticklabels={$-\pi\hspace{0.30cm}$, $-\frac{\pi}{2}$, +$\frac{\pi}{2}$, $\pi\hspace{0.10cm}$}, + axis y line=middle, axis x line=middle, - every axis x label/.style={
- at={(ticklabel* cs:1.05)},
- anchor=north,
- },
- every axis y label/.style={
- at={(ticklabel* cs:1.05)},
- anchor=east,
+ every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, } ] \addplot[red, thick] coordinates {(-3, 0) (-3, 3.14159)}; @@ -488,9 +489,61 @@ The symmetry rules do \underline{not} apply for complex-valued signals $\underli \label{fig:ch02:FSeries_Phase_Spectrum} \end{figure} -\section{Non-Periodic Signals and Fourier Transform} +\begin{excursus}{Spectra in the nature} + The spectrum is no abstract, mathematical theory. You can see spectra with your eye: + \begin{figure}[H] + \centering + \includegraphics[scale=1]{../chapter02/Rainbow.jpg} + \caption[A rainbow showing the spectrum of the sunlight]{A rainbow showing the spectrum of the sunlight: The white sunlight is composed of mono-chromatic, electromagnetic waves of all frequencies which are optically visible for humans. When light passes through a dispersive medium (glass prism, raindrop, etc.), it is refracted. Each mono-chromatic component has a different refraction index. The light components are separated by its frequency and become individually visible. An example, is a rainbow as depicted above. \licensequote{\cite{Arz2007}}{``Arz''}{\href{https://creativecommons.org/licenses/by-sa/3.0/deed.en}{CC-BY-SA 3.0}}} + \end{figure} + The rainbow is a natural example of an visible spectrum of the sunlight. +\end{excursus} -\subsection{Derivation of The Fourier Transform} +\section{Non-Periodic Signals and The Continuous Fourier Transform} + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.6\linewidth, + scale only axis, + xlabel={$t$}, + ylabel={$x_{np}(t)$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-3, + xmax=3, + ymin=0, + ymax=2, + xtick={-2, -0.5, ..., 2}, + ytick={0, 0.5, ..., 1.5} + ] + \addplot[blue, thick] coordinates {(-2, 0) (-0.5, 0)}; + \addplot[blue, dashed] coordinates {(-0.5, 0) (-0.5, 1)}; + \addplot[blue, thick] coordinates {(-0.5, 1) (0.5, 1)}; + \addplot[blue, dashed] coordinates {(0.5, 1) (0.5, 0)}; + \addplot[blue, thick] coordinates {(0.5, 0) (2, 0)}; + \end{axis} + \end{tikzpicture} + \caption{The rectangular function $\mathrm{rect}$ as an example for a non-period signal} + \label{fig:ch02:rect_function} +\end{figure} +\index{rectangular function} + +\subsection{Derivation of The Continuous Fourier Transform} Non-periodic signals have no repeating pattern. Consequently, there is no period $T_0$. Mathematically, the period is indefinite $T_0 \rightarrow \infty$. @@ -529,15 +582,15 @@ The outer sum is a Rieman sum. $\frac{1}{T_0}$ is substituted by $\frac{\Delta \ \underline{x_{np}}(t) = \underbrace{\frac{1}{2 \pi} \int\limits_{\omega = -\infty}^{\infty} \underbrace{\left( \int\limits_{t' = -\infty}^{\infty} \underline{x_{np}}(t') \cdot e^{-j \omega t'} \, \mathrm{d} t' \right)}_{\text{Fourier transform}} \cdot e^{j \omega t} \, \mathrm{d} \omega}_{\text{Inverse Fourier transform}} \end{equation} -The inner integral is the \index{Fourier transform} \textbf{Fourier transform}. +The inner integral is the \textbf{continuous Fourier transform}, also called only \index{Fourier transform} \emph{Fourier transform}. \begin{definition}{Fourier Transform} - The \index{Fourier transform} \textbf{Fourier transform} of the function $\underline{x}(t)$ is: + The \index{continuous Fourier transform} \textbf{continuous Fourier transform} of the function $\underline{x}(t)$ is: \begin{equation} \underline{X}(j \omega) = \mathcal{F} \left\{\underline{x}(t)\right\} = \int\limits_{t = -\infty}^{\infty} \underline{x}(t) \cdot e^{-j \omega t} \, \mathrm{d} t \end{equation} - The \index{inverse Fourier transform} \textbf{inverse Fourier transform} is: + The \index{inverse Fourier transform} \index{inverse continuous Fourier transform} \textbf{inverse (continuous) Fourier transform} is: \begin{equation} \underline{x}(t) = \mathcal{F}^{-1} \left\{\underline{X}(j \omega)\right\} = \frac{1}{2 \pi} \int\limits_{\omega = -\infty}^{\infty} \underline{X}(j \omega) \cdot e^{+j \omega t} \, \mathrm{d} \omega \end{equation} @@ -545,7 +598,7 @@ The inner integral is the \index{Fourier transform} \textbf{Fourier transform}. \subsection{Amplitude and Phase Spectra} -The value-continuous complex frequency variable $j \omega$ in the Fourier transforms replaced the value-discrete index $n$ of the Fourier series. Due to their similarity, the constraints for all signals and the \index{spectrum!symmetry rules} \textbf{symmetry rules} for real-valued signals apply analogously. +The value-continuous complex frequency variable $j \omega$ in the continuous Fourier transforms replaced the value-discrete index $n$ of the Fourier series. Due to their similarity, the constraints for all signals and the \index{spectrum!symmetry rules} \textbf{symmetry rules} for real-valued signals apply analogously. \begin{itemize} \item The Fourier transform $\underline{X}(j \omega) \in \mathbb{C}$ is always complex-valued, for both real-valued $\underline{x}(t) = x(t) \in \mathbb{R}$ and complex-valued $\underline{x}(t) \in \mathbb{C}$ signals. @@ -555,13 +608,175 @@ The value-continuous complex frequency variable $j \omega$ in the Fourier transf \begin{itemize} \item The amplitude spectrum $|\underline{X}(j \omega)|$ is an \underline{even function}. It is symmetric with respect to the $y$-axis. \item The phase spectrum $\arg\left(\underline{X}(j \omega)\right)$ is an \underline{odd function}. It is symmetric with respect to the origin. - \item As a consequence, the phase of $\arg\left(\underline{X}(0)\right)$ at $j \omega = 0$ must be either $0$ or $+\pi$. Note that, $+\pi$ is identical to $-\pi$ in the complex plane. Thus, $-\pi$ is valid, too, but not distinct from $+\pi$. This phase is the sign of the \ac{DC} bias: $\arg\left(\underline{X}(0)\right) = 0$ means positive \ac{DC} bias and $\arg\left(\underline{X}(0)\right) = \pi$ means negative \ac{DC} bias. + \item As a consequence, the phase of $\arg\left(\underline{X}(0)\right)$ at $j \omega = 0$ must be either $0$ or $\pm \pi$. Note that, $+\pi$ is identical to $-\pi$ in the complex plane. The phase is the sign of the \ac{DC} bias: $\arg\left(\underline{X}(0)\right) = 0$ means positive \ac{DC} bias and $\arg\left(\underline{X}(0)\right) = \pi$ means negative \ac{DC} bias. \end{itemize} \end{itemize} +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} + 0 & \qquad \text{if } \; |t| > \frac{1}{2}, \\ + 1 & \qquad \text{if } \; |t| < \frac{1}{2} + \end{cases} +\end{equation} +The function is undefined for $t = \pm \frac{1}{2}$. The function is now transformed, i.e., $\underline{x}(t) = \mathrm{rect}(t)$. + +\begin{equation} + \underline{X}\left(j \omega\right) = \int\limits_{t = -\infty}^{\infty} \mathrm{rect}(t) \cdot e^{-j \omega t} \, \mathrm{d} t = \mathrm{sinc}\left(\frac{\omega}{2 \pi}\right) +\end{equation} +where $\mathrm{sinc}(t)$ is the \emph{normalized} sinc function. + +\begin{attention} + Mathematics and engineering use a slightly different definition of the sinc function. + + In mathematics, it is \index{sinc function!unnormalized} \textbf{\textit{unnormalized} sinc function}: + \begin{equation*} + \mathrm{sinc}(t) = \frac{\sin\left(t\right)}{t} + \end{equation*} + + In the context of signal processing and information theory, it is the \index{sinc function!normalized} \textbf{\textit{normalized} sinc function}: + \begin{equation*} + \mathrm{sinc}(t) = \frac{\sin\left(\pi t\right)}{\pi t} + \end{equation*} + + In either case, the value at $t = 0$ is defined to: + \begin{equation*} + \mathrm{sinc}(t = 0) = \lim\limits_{t \rightarrow 0} \frac{\sin\left(t\right)}{t} = 1 + \end{equation*} +\end{attention} + +The resulting spectra of $\underline{X}\left(j \omega\right)$ can now be drawn. The rectangular function is special. The imaginary part $\Im\left\{\underline{X}\left(j \omega\right)\right\} = 0$ is zero. Thus, the phase can only be $0$ or $\pm \pi$. However, this is a special property of the sinc function, but not of every function. + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.6\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$|\underline{X}\left(j \omega\right)|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-52, + xmax=52, + ymin=0, + ymax=1.2, + xtick={-50, -40, ..., 50}, + ytick={0, 0.25, ..., 1.0} + ] + \addplot[red, thick, smooth, domain=-50:50, samples=200] plot (\x,{abs(sinc((1/(2*pi))*\x))}); + \end{axis} + \end{tikzpicture} + \caption{Amplitude spectrum of the rectangular function} + \label{fig:ch02:rect_function_ampl_spectrum} +\end{figure} + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.6\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$\arg\left(\underline{X}(j \omega)\right)$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-52, + xmax=52, + ymin=-4, + ymax=4, + xtick={-50, -40, ..., 50}, + ytick={-3.14159, -1.5708, 1.5708, 3.14159}, + yticklabels={$-\pi\hspace{0.30cm}$, $-\frac{\pi}{2}$, + $\frac{\pi}{2}$, $\pi\hspace{0.10cm}$}, + ] + \addplot[red, thick] coordinates {(-50, 0) (-39.48, 0)}; + \addplot[red, dashed] coordinates {(-39.48, 0)(-39.48, -3.14159)}; + \addplot[red, thick] coordinates {(-39.48, -3.14159) (-19.74, -3.14159)}; + \addplot[red, dashed] coordinates {(-19.74, -3.14159) (-19.74, 0)}; + \addplot[red, thick] coordinates {(-19.74, 0) (19.74, 0)}; + \addplot[red, dashed] coordinates {(19.74, 0) (19.74, 3.14159)}; + \addplot[red, thick] coordinates {(19.74, 3.14159) (39.48, 3.14159)}; + \addplot[red, dashed] coordinates {(39.48, 3.14159)(39.48, 0)}; + \addplot[red, thick] coordinates {(39.48, 0)(50, 0)}; + \end{axis} + \end{tikzpicture} + \caption[Phase spectrum of the rectangular function]{Phase spectrum of the rectangular function. Please note that $- \pi$ is equivalent to $+ \pi$.} + \label{fig:ch02:rect_function_phase_spectrum} +\end{figure} + \subsection{Time Domain and Frequency Domain} -\section{Properties of The Fourier Transform} +You have learnt two representations of a signal, so far. +\begin{itemize} + \item \index{time domain} \textbf{Time domain} -- A signal is a function $\underline{x}(t)$ of the time. + \item \index{frequency domain} \textbf{Frequency domain} -- A signal is a function $\underline{X}(j \omega)$ of the frequency. +\end{itemize} +Both $\underline{x}(t)$ and $\underline{X}(j \omega)$ refer to the same signal. + +The frequency domain is obtained from the time domain by a transform. For time-continuous signals, these transforms one of: +\begin{itemize} + \item Fourier series + \item continuous Fourier transform +\end{itemize} +The time domain is obtained by the respective inverse transform. + +\begin{definition}{Transform operator} + The operation of a transform between time and frequency domain is written as: + \begin{equation} + \underline{x}(t) \TransformHoriz \underline{X}(j \omega) + \end{equation} + for the transform from time to frequency domain, and vice versa: + \begin{equation} + \underline{X}(j \omega) \InversTransformHoriz \underline{x}(t) + \end{equation} +\end{definition} + +\textbf{But what is the purpose of the transforms?} + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \node[align=center, minimum width=2.5cm, minimum height=1.5cm] (ProbTD) {\textbf{Problem}\\ in time domain}; + \node[align=center, minimum width=2.5cm, minimum height=1.5cm, right=5cm of ProbTD] (ProbFD) {\textbf{Problem}\\ in frequency domain}; + \node[align=center, minimum width=2.5cm, minimum height=1.5cm, below=3cm of ProbTD] (SolTD) {\textbf{Solution}\\ in time domain}; + \node[align=center, minimum width=2.5cm, minimum height=1.5cm, below=3cm of ProbFD] (SolFD) {\textbf{Solution}\\ in frequency domain}; + + \draw[-latex, thick] (ProbTD.south) -- node[midway, left, align=right]{Hard to solve} (SolTD.north); + \draw[-latex, thick] (ProbTD.east) -- node[midway, above, align=center]{Transform} (ProbFD.west); + \draw[-latex, thick] (ProbFD.south) -- node[midway, right, align=left]{Easy to solve} (SolFD.north); + \draw[-latex, thick] (SolFD.west) -- node[midway, above, align=center]{Inverse Transform} (SolTD.east); + \end{tikzpicture} + \caption{Explanation of the purpose of transforms} +\end{figure} + +\section{Properties of The Continuous Fourier Transform} \subsection{Energy Signals and Power Signals} |
