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diff --git a/chapter04/content_ch04.tex b/chapter04/content_ch04.tex index 9362134..63def3d 100644 --- a/chapter04/content_ch04.tex +++ b/chapter04/content_ch04.tex @@ -778,16 +778,88 @@ The Fourier transform of the sampled signal $\underline{x}_S(t)$ is: \underline{X}_S \left(j \omega\right) &= \mathcal{F} \left\{\underline{x}_S(t)\right\} \\ &= \mathcal{F} \left\{\sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot \delta(t - n T_S)\right\} \\ &= \int\limits_{t = -\infty}^{\infty} \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot \delta(t - n T_S) \cdot e^{-j \omega t} \, \mathrm{d} t \\ - &= \sum\limits_{n = -\infty}^{\infty} \int\limits_{t = -\infty}^{\infty} \underline{x}[n] \cdot \delta(t - n T_S) \cdot e^{-j \omega t} \, \mathrm{d} t \\ + &= \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \int\limits_{t = -\infty}^{\infty} \cdot \delta(t - n T_S) \cdot e^{-j \omega t} \, \mathrm{d} t \\ + &\qquad \text{Using the Dirac measure:} \\ &= \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot e^{-j \omega n T_S} \end{split} \end{equation} Redefining $\phi = T_S \omega$: \begin{equation} - \underline{X}_S \left(j \omega\right) = \underline{X} \left(e^{j \phi}\right) = \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot e^{-j \phi n} + \underline{X}_S \left(j \omega\right) = \underline{X}_{2 \pi} \left(e^{j \phi}\right) = \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot e^{-j \phi n} \end{equation} +$\underline{X}_{2 \pi} \left(e^{j \phi}\right)$ is the discrete-time Fourier transform of the time-discrete, sampled signal $\underline{x}[n]$. +\begin{itemize} + \item The spectrum of the sampled signal $\underline{x}[n]$ is $\omega_S$-periodic. + \item The real-valued frequency-continuous variable $\omega$ is replaced by the complex-valued frequency-continuous variable $e^{j \phi}$ representing the periodicity of the spectrum. + \item $\phi = T_S \omega$ + \item The $\omega_S$-periodicity is equivalent to to a full $2\pi$-rotation on the unit circle $e^{j \phi}$ in the complex plane. +\end{itemize} + +\begin{definition}{Discrete-time Fourier transform} + The \index{discrete-time Fourier transform} \textbf{\acf{DTFT}} of a time-discrete signal $\underline{x}[n]$ with the sampling period $T_S$ is: + \begin{itemize} + \item Using the $T$-periodicity: + \begin{equation} + \underline{X}_{\frac{1}{T}} \left(e^{j T \omega}\right) = \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot e^{-j T \omega n} + \end{equation} + \item Using the $2 \pi$-periodicity: + \begin{equation} + \underline{X}_{2 \pi} \left(e^{j \phi}\right) = \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot e^{-j \phi n} + \end{equation} + \end{itemize} + Both expressions are equivalent. $\phi = T_S \omega$ + + The \index{discrete-time Fourier transform!inverse} \textbf{inverse discrete-time Fourier transform} of a time-discrete signal $\underline{x}[n]$ with the sampling period $T_S$ is: + \begin{itemize} + \item Using the $T$-periodicity: \todo{$T$ ???} + \begin{equation} + \underline{x}[n] = \frac{T}{2 \pi} \int\limits_{- \frac{\pi}{T}}^{+ \frac{\pi}{T}} \underline{X}_{\frac{1}{T}}(e^{j T \omega}) \cdot e^{+ j \omega T n} \, \mathrm{d} \omega + \end{equation} + \item Using the $2 \pi$-periodicity: + \begin{equation} + \underline{x}[n] = \frac{1}{2 \pi} \int\limits_{- \pi}^{+ \pi} \underline{X}_{2\pi}(e^{j \phi}) \cdot e^{+ j \phi n} \, \mathrm{d} \phi + \end{equation} + \end{itemize} + Both expressions are equivalent. +\end{definition} + +\begin{excursus}{z-Transform} + Analogous to the Fourier and Laplace transform, the \acf{DTFT} is a special case of the z-transform. The \index{z-transform} \textbf{z-transform} is: + \begin{equation} + \underline{X}\left(\underline{z}\right) = \mathcal{Z}\left\{\underline{x}[n]\right\} = \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot \underline{z}^{-n} + \end{equation} + $\underline{z}$ is the complex frequency variable, which can be decomposed into: + \begin{equation} + \underline{z} = A e^{j \phi} + \end{equation} + where $A$ represents the gain and $e^{j \phi}$ the frequency. + \begin{figure}[H] + \centering + \begin{tikzpicture} + \draw[->] (-2.2,0) -- (2.2,0) node[below, align=left]{$\Re\left\{\underline{z}\right\}$}; + \draw[->] (0,-2.2) -- (0,2.2) node[left, align=right]{$\Im\left\{\underline{z}\right\}$}; + %\draw (0:1) arc(0:360:1); + \draw (1,0.2) -- (1,-0.2) node[below]{$1$}; + \draw (-1,0.2) -- (-1,-0.2) node[below]{$-1$}; + \draw (0.2,1) -- (-0.2,1) node[left]{$1$}; + \draw (0.2,-1) -- (-0.2,-1) node[left]{$-1$}; + + \draw[thick, red] (0:1) arc(0:360:1); + \draw[dashed, red] (60:1) -- (45:1.5) node[right, align=left, color=red]{$e^{j \phi}$}; + \end{tikzpicture} + \caption{Complex plane of the complex frequency variable $\underline{z}$} + \label{fig:ch04:ztrafo_z_cmplx_plane} + \end{figure} + + In the \acf{DTFT}, $A = 1$ as a special case. The remainig $e^{j \phi}$ describes the unit circle in the complex plane. Like the Fourier transform, it assumes a steady-state, whereas the z-transform delivers a complete description of a time-discrete system. The z-transform is preferred for transient analysis of a time-discrete system. Its zeros $\underline{z}_0$ and poles $\underline{z}_\infty$ determine the stability of the system. + + \vspace{0.5em} + + Figure \ref{fig:ch04:ztrafo_z_cmplx_plane} makes evident the $2 \pi$-periodicity of both the \ac{DTFT} and z-transform. The frequency $e^{j \phi}$ repeats every $2 \pi$. +\end{excursus} + \subsection{Discrete Fourier Transform} \section{Analogies Of Time-Continuous and Time-Discrete Signals and Systems} |