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diff --git a/chapter04/content_ch04.tex b/chapter04/content_ch04.tex index c40e76c..9362134 100644 --- a/chapter04/content_ch04.tex +++ b/chapter04/content_ch04.tex @@ -109,6 +109,8 @@ Some corollaries can be deducted from these two points: \end{itemize} \end{itemize} +\subsubsection{Dirac comb} + We know already indefinitely narrow pulses. They are Dirac delta functions $\delta\left(t - n T_S\right)$. Taking out \underline{exactly one} sample out of $\underline{x}(t)$ is a multiplication of $\underline{x}(t)$ with $\delta\left(t - n T_S\right)$. @@ -181,49 +183,63 @@ The sum of Dirac delta functions \item is called \index{Dirac comb} \textbf{Dirac comb} $\Sha_{T_S}(t)$ or \index{impulse train} \textbf{impulse train}. \end{itemize} -\begin{equation} - \Sha_{T_S}(t) = \sum\limits_{n = -\infty}^{\infty} \delta\left(t - n T_S\right) -\end{equation} -\begin{figure}[H] - \centering - \begin{tikzpicture} - \begin{axis}[ - height={0.15\textheight}, - width=0.9\linewidth, - scale only axis, - xlabel={$t$}, - ylabel={$\Sha_{T_S}(t)$}, - %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=-5.5, - xmax=5.5, - ymin=0, - ymax=1.2, - xtick={-5, -4, ..., 5}, - xticklabels={$-5 T_S$, $-4 T_S$, $-3 T_S$, $-2 T_S$, $- T_S$, $0$, $T_S$, $2 T_S$, $3 T_S$, $4 T_S$, $5 T_S$}, - ytick={0}, - ] - \pgfplotsinvokeforeach{-5, -4, ..., 5}{ - \draw[-latex, blue, very thick] (axis cs:#1,0) -- (axis cs:#1,1); - %\addplot[blue, very thick] coordinates {(#1, 0) (#1, 1)}; - %\addplot[only marks, blue, thick, mark=triangle] coordinates {(#1, 1)}; - } - \end{axis} - \end{tikzpicture} - \caption{Dirac comb} -\end{figure} +\begin{definition}{Dirac comb} + The \index{Dirac comb} \textbf{Dirac comb} $\Sha_{T}(t)$ or \index{impulse train} \textbf{impulse train} is: + \begin{equation} + \Sha_{T}(t) = \sum\limits_{n = -\infty}^{\infty} \delta\left(t - n T\right) + \label{eq:ch04:dirac_comb} + \end{equation} + $T$ is the period of the equidistant Dirac pulses. + + It is a periodic signal and can be decomposed using the Fourier analysis: + \begin{equation} + \Sha_{T}(t) = \frac{1}{T} \sum\limits_{n = -\infty}^{\infty} e^{j n \frac{2 \pi}{T} t} + \label{eq:ch04:dirac_comb_fourier_series} + \end{equation} + + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$t$}, + ylabel={$\Sha_{T_S}(t)$}, + %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=-5.5, + xmax=5.5, + ymin=0, + ymax=1.2, + xtick={-5, -4, ..., 5}, + xticklabels={$-5 T_S$, $-4 T_S$, $-3 T_S$, $-2 T_S$, $- T_S$, $0$, $T_S$, $2 T_S$, $3 T_S$, $4 T_S$, $5 T_S$}, + ytick={0}, + ] + \pgfplotsinvokeforeach{-5, -4, ..., 5}{ + \draw[-latex, blue, very thick] (axis cs:#1,0) -- (axis cs:#1,1); + %\addplot[blue, very thick] coordinates {(#1, 0) (#1, 1)}; + %\addplot[only marks, blue, thick, mark=triangle] coordinates {(#1, 1)}; + } + \end{axis} + \end{tikzpicture} + \caption{Dirac comb} + \end{figure} +\end{definition} + +\subsubsection{Ideal Sampler} A \index{sampler} \textbf{sampler} is a system which \begin{itemize} @@ -231,14 +247,18 @@ A \index{sampler} \textbf{sampler} is a system which \item to a time-continuous signal $\underline{x}(t)$ (multiplication) and \item outputs a series of equidistant pulses $\underline{x}_S(t)$. \end{itemize} -\begin{equation} - \begin{split} - \underline{x}_S(t) &= \underline{x}(t) \cdot \Sha_{T_S}(t) \\ - &= \sum\limits_{n = -\infty}^{\infty} \underline{x}\left(t\right) \delta\left(t - n T_S\right) \\ - &= \sum\limits_{n = -\infty}^{\infty} \underline{x}\left(n T_S\right) \delta\left(t - n T_S\right) - \end{split} - \label{eq:ch04:ideal_sampling} -\end{equation} + +\begin{definition}{Ideally sampled signal} + An ideally \index{sampled signal} sampled signal is: + \begin{equation} + \begin{split} + \underline{x}_S(t) &= \underline{x}(t) \cdot \Sha_{T_S}(t) \\ + &= \sum\limits_{n = -\infty}^{\infty} \underline{x}\left(t\right) \delta\left(t - n T_S\right) \\ + &= \sum\limits_{n = -\infty}^{\infty} \underline{x}\left(n T_S\right) \delta\left(t - n T_S\right) + \end{split} + \label{eq:ch04:ideal_sampling} + \end{equation} +\end{definition} The sampled signal $\underline{x}_S(t)$ is a chain of pulses (red signal in Figure \ref{fig:ch04:sampling_of_signal}). The chain of pulses can then be reinterpreted as a time-discrete signal $\underline{x}[n]$. The value of $\underline{x}[n]$ is: \begin{equation} @@ -265,6 +285,8 @@ The sampled signal $\underline{x}_S(t)$ is a chain of pulses (red signal in Figu \caption{An abstract view on sampling} \end{figure} +\subsubsection{Irreversibility of Sampling} + \begin{fact} The act of sampling is irreversible. \end{fact} @@ -280,8 +302,467 @@ But there is generally no way back to reconstruct the original signal. $\mathrm{ Sampling is always lossy in general. -\subsection{Sampling Theorem and Aliasing} +\subsection{Sampling Theorem, Aliasing and Reconstruction} + +\subsubsection{Frequency Domain Representation} + +\begin{excursus}{Fourier transform of the Dirac comb} + The Fourier transform of the Dirac comb is again a Dirac comb: + \begin{equation} + \begin{split} + \Sha_{T}(t) \TransformHoriz \mathcal{F}\left\{\Sha_{T}(t)\right\} &= \frac{2 \pi}{T} \Sha_{\frac{2 \pi}{T}}(\omega) \\ + &= \frac{2 \pi}{T} \sum\limits_{k = -\infty}^{\infty} \delta\left(\omega - k \frac{2 \pi}{T}\right) \\ + &= \sum\limits_{k = -\infty}^{\infty} e^{- j \omega k T} + \end{split} + \label{eq:ch04:dirac_comb_fourier_tranform} + \end{equation} +\end{excursus} + +\eqref{eq:ch04:ideal_sampling} pointed out, that the sampled signal $\underline{x}_S(t)$ is the multiplication of the original time-domain signal $\underline{x}(t)$ and the Dirac comb $\Sha_{T_S}(t)$ with a periodicity of the sampling period $T_S$. +\begin{equation} + \begin{split} + \underline{x}_S(t) &= \underline{x}(t) \cdot \Sha_{T_S}(t) \\ + &= \underline{x}(t) \cdot \frac{1}{T_S} \sum\limits_{n = -\infty}^{\infty} e^{j n \frac{2 \pi}{T_S} t} \\ + &= \frac{1}{T_S} \sum\limits_{n = -\infty}^{\infty} \underbrace{\underline{x}(t) e^{j n \frac{2 \pi}{T_S} t}}_{\text{Frequency shift by } n \frac{2 \pi}{T_S}} \\ + \end{split} +\end{equation} + +The ideally sampled signal $\underline{x}_S(t)$ can be expressed as a sum of \emph{frequency shifts} of the original signal $\underline{x}(t)$. Its Fourier transform is: +\begin{equation} + \begin{split} + \underline{X}_S\left(j \omega\right) &= \mathcal{F}\left\{\frac{1}{T_S} \sum\limits_{k = -\infty}^{\infty} \underline{x}(t) e^{j k \frac{2 \pi}{T_S} t}\right\} \\ + & \qquad \text{Using the linearity of the Fourier transform:} \\ + &= \frac{1}{T_S} \sum\limits_{k = -\infty}^{\infty} \mathcal{F}\left\{\underline{x}(t) e^{j k \frac{2 \pi}{T_S} t}\right\} \\ + & \qquad \text{Using the frequency shift theorem of the Fourier transform:} \\ + &= \frac{1}{T_S} \sum\limits_{k = -\infty}^{\infty} \underline{X}\left(j \left(\omega - k \frac{2 \pi}{T_S} \right)\right) + \end{split} +\end{equation} + +\begin{proof}{} + An alternative way is using the Fourier transform of this multiplication in the time-domain is a convolution in the frequency domain: + \begin{equation} + \begin{split} + \underline{X}_S\left(j \omega\right) &= \mathcal{F}\left\{\underline{x}(t) \cdot \Sha_{T_S}(t)\right\} \\ + &= \frac{1}{2 \pi} \underline{X}\left(j \omega\right) * \left(\frac{2 \pi}{T_S} \Sha_{\frac{2 \pi}{T_S}}(\omega)\right) \\ + &= \frac{1}{T_S} \int\limits_{-\infty}^{\infty} \underline{X}\left(j \left(\omega - \zeta\right)\right) \Sha_{\frac{2 \pi}{T_S}}\left(\zeta\right) \, \mathrm{d} \zeta \\ + &= \frac{1}{T_S} \int\limits_{-\infty}^{\infty} \underline{X}\left(j \left(\omega - \zeta\right)\right) \sum\limits_{k = -\infty}^{\infty} \delta\left(\zeta - k \frac{2 \pi}{T_S}\right) \, \mathrm{d} \zeta \\ + &= \frac{1}{T_S} \sum\limits_{k = -\infty}^{\infty} \int\limits_{-\infty}^{\infty} \underline{X}\left(j \left(\omega - \zeta\right)\right) \delta\left(\zeta - k \frac{2 \pi}{T_S}\right) \, \mathrm{d} \zeta \\ + & \qquad \text{Using the Dirac measure:} \\ + &= \frac{1}{T_S} \sum\limits_{k = -\infty}^{\infty} \underline{X}\left(j \left(\omega - k \frac{2 \pi}{T_S}\right)\right) \\ + \end{split} + \end{equation} +\end{proof} + +\textbf{Conclusion:} The spectrum of the sampled signal $\underline{X}_S\left(j \omega\right)$ +\begin{itemize} + \item consists of superimposed, frequency-shifted copies of the spectra of the original signal $\underline{X}\left(j\omega\right)$ and + \item the periodicity of the superimposed, frequency-shifted copies is the sampling angular frequency $\omega_S = \frac{2 \pi}{T_S}$ or sampling frequency $f_S$, respectively, + \item each frequency-shifted copy starts at $k \omega_S - \frac{\omega_S}{2}$ and ends at $k \omega_S + \frac{\omega_S}{2}$. +\end{itemize} + +\begin{figure}[H] + \subfloat[Original signal $\underline{X}\left(j\omega\right)$] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.9\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=-3.5, + xmax=3.5, + ymin=0, + ymax=1.2, + xtick={-1, -0.5, 0, 0.5, 1}, + xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$}, + ytick={0}, + ] + \draw[green, thick] (axis cs:-0.4,0) -- (axis cs:0,0.7); + \draw[red, thick] (axis cs:0,0.7) -- (axis cs:0.4,0); + \end{axis} + \end{tikzpicture} + } + \subfloat[Spectrum of the Dirac comb $\frac{2 \pi}{T} \Sha_{\frac{2 \pi}{T}}(\omega)$] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$t$}, + ylabel={$|\frac{2 \pi}{T} \Sha_{\frac{2 \pi}{T}}(\omega)|$}, + %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=-3.5, + xmax=3.5, + ymin=0, + ymax=1.2, + xtick={-3, -2, ..., 3}, + xticklabels={$-3 \omega_S$, $-2 \omega_S$, $- \omega_S$, $0$, $\omega_S$, $2 \omega_S$, $3 \omega_S$}, + ytick={0}, + ] + \pgfplotsinvokeforeach{-3, -2, ..., 3}{ + \draw[-latex, blue, very thick] (axis cs:#1,0) -- (axis cs:#1,1); + } + \end{axis} + \end{tikzpicture} + } + + \subfloat[Sampled signal $\underline{X}_S\left(j\omega\right)$] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$|\underline{X}_S\left(j\omega\right)|$}, + %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=-3.5, + xmax=3.5, + ymin=0, + ymax=1.2, + xtick={-3, -2, -1, -0.5, 0, 0.5, 1, 2, 3}, + xticklabels={$-3 \omega_S$, $-2 \omega_S$, $- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$, $2 \omega_S$, $3 \omega_S$}, + ytick={0}, + ] + \pgfplotsinvokeforeach{-3, -2, ..., 3}{ + \draw[-latex, blue, dashed, very thick] (axis cs:#1,0) -- (axis cs:#1,1); + \draw[green, thick] (axis cs:{#1-0.4},0) -- (axis cs:#1,0.7); + \draw[red, thick] (axis cs:#1,0.7) -- (axis cs:{#1+0.4},0); + } + \end{axis} + \end{tikzpicture} + } + + \caption{Spectrum of the sampled signal} +\end{figure} + +\begin{attention} + The spectrum of the original signal $\underline{X}\left(j\omega\right)$ has both negative and positive frequencies. Remember that the symmetry rules apply \underline{only} for real-valued time-domain signals. +\end{attention} + +\subsubsection{Aliasing} + +The original signal in the previous example was limited to $- \frac{\omega_S}{2} \leq \omega \leq \frac{\omega_S}{2}$. The spectrum sampled signal consists of the frequency-shifted copies of the original signal's spectrum. Although they are superimposed, they do not overlap. + +A problem arises when the original signal is \underline{not} limited to $- \frac{\omega_S}{2} \leq \omega \leq \frac{\omega_S}{2}$. The original signal's spectrum will overlap. + +\begin{figure}[H] + \subfloat[Original signal $\underline{X}\left(j\omega\right)$ violating the band-limitation $- \frac{\omega_S}{2} \leq \omega \leq \frac{\omega_S}{2}$. The original signal's spectrum will overlap. + ] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.9\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=-3.5, + xmax=3.5, + ymin=0, + ymax=1.2, + xtick={-3, -2, -1, -0.5, 0, 0.5, 1, 2, 3}, + xticklabels={$-3 \omega_S$, $-2 \omega_S$, $- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$, $2 \omega_S$, $3 \omega_S$}, + ytick={0}, + ] + \draw[green, thick] (axis cs:-0.7,0) -- (axis cs:0,0.7); + \draw[red, thick] (axis cs:0,0.7) -- (axis cs:0.8,0); + \draw[olive, thick] (axis cs:2.1,0) -- (axis cs:2.1,0.5) -- (axis cs:2.3,0.5) -- (axis cs:2.3,0); + \end{axis} + \end{tikzpicture} + } + + \subfloat[Spectrum of the Dirac comb $\frac{2 \pi}{T} \Sha_{\frac{2 \pi}{T}}(\omega)$] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$t$}, + ylabel={$|\frac{2 \pi}{T} \Sha_{\frac{2 \pi}{T}}(\omega)|$}, + %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=-3.5, + xmax=3.5, + ymin=0, + ymax=1.2, + xtick={-3, -2, ..., 3}, + xticklabels={$-3 \omega_S$, $-2 \omega_S$, $- \omega_S$, $0$, $\omega_S$, $2 \omega_S$, $3 \omega_S$}, + ytick={0}, + ] + \pgfplotsinvokeforeach{-3, -2, ..., 3}{ + \draw[-latex, blue, very thick] (axis cs:#1,0) -- (axis cs:#1,1); + } + \end{axis} + \end{tikzpicture} + } + + \subfloat[Sampled signal $\underline{X}_S\left(j\omega\right)$ showing aliasing] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$|\underline{X}_S\left(j\omega\right)|$}, + %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=-3.5, + xmax=3.5, + ymin=0, + ymax=1.2, + xtick={-3, -2, -1, -0.5, 0, 0.5, 1, 2, 3}, + xticklabels={$-3 \omega_S$, $-2 \omega_S$, $- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$, $2 \omega_S$, $3 \omega_S$}, + ytick={0}, + ] + \pgfplotsinvokeforeach{-3, -2, ..., 3}{ + \draw[-latex, blue, dashed, very thick] (axis cs:#1,0) -- (axis cs:#1,1); + \draw[green, thick] (axis cs:{#1-0.7},0) -- (axis cs:#1,0.7); + \draw[red, thick] (axis cs:#1,0.7) -- (axis cs:{#1+0.8},0); + \draw[olive, thick] (axis cs:{#1+0.1},0) -- (axis cs:{#1+0.1},0.5) -- (axis cs:{#1+0.3},0.5) -- (axis cs:{#1+0.3},0); + } + \end{axis} + \end{tikzpicture} + } + + \caption{Aliasing} +\end{figure} + +The sampled signal $\underline{X}_S\left(j\omega\right)$ contains overlapping, frequency-shifted copies of the original signal's spectrum. This is not feasible for most applications. + +\begin{definition}{Anti-aliasing filter} + A signal $\underline{x}(t)$ must be band-limited by an \index{anti-aliasing filter} \textbf{anti-aliasing filter} to avoid aliasing. The anti-aliasing filter is a \ac{LPF} with the cut-off frequency $\omega_o = \frac{\omega_S}{2}$. + + \begin{figure}[H] + \centering + \begin{adjustbox}{scale=0.7} + \begin{circuitikz} + \node[draw, block] (Sampler) {Ideal sampler}; + \node[draw, block, right=3cm of Sampler] (ReInterp) {Reinterpret as\\ time-discrete signal}; + + \draw[<-o] (Sampler.west) to[lowpass] ++(-2.5cm, 0) -- ++(-0.7cm,0) node[above, align=center]{Time-continuous\\ signal $\underline{x}(t)$}; + \draw[->] (Sampler.east) -- (ReInterp.west) node[midway, above, align=center]{Series of pulses\\ $\underline{x}_S(t)$}; + \draw[<-] (Sampler.south) -- ++(0, -0.75cm) node[below, align=center]{Dirac comb\\ $\Sha_{T_S}(t)$}; + \draw[->] (ReInterp.east) -- ++(1.5cm, 0) node[above, align=center]{Time-discrete\\ signal $\underline{x}[n]$}; + + \draw[dashed] (ReInterp.north) -- ++(0, 2cm) node[below left, align=right]{Time-continuous\\ domain} node[below right, align=left]{Time-discrete\\ domain}; + \draw[dashed] (ReInterp.south) -- ++(0, -1cm); + \end{circuitikz} + \end{adjustbox} + \caption{An abstract view on sampling, including the anti-aliasing filter} + \end{figure} +\end{definition} + +The anti-aliasing filter's cut-off frequency must be half of the sampling frequency, because its bandwidth $\omega_S$ or $f_S$, respectively, must be distributed equally over the negative and positive part of the frequency axis. + +\subsubsection{Reconstruction} + +\textit{Remark:} Due to aliasing, there is no inverse function $\mathrm{Sampling}^{-1} \left(\underline{x}_S(t)\right)$ reversing the sampling process. + +However, the original signal $\underline{x}(t)$ can be reconstructed if it was band-limited to the sampling (angular) frequency $\omega_S$ or $f_S$, respectively, before sampling. + +\begin{definition}{Shannon-Nyquist sampling theorem} + According to the \index{Shannon-Nyquist sampling theorem} \textbf{Shannon-Nyquist sampling theorem}, the original signal $\underline{x}(t)$ can be reconstructed if the sample rate $T_S$ is at least twice the inverse of signal's highest (angular) frequency $\omega_B$ or $f_B$, respectively. + \begin{equation} + T_S \geq \frac{1}{2 f_B} = \frac{\pi}{\omega_B} + \end{equation} +\end{definition} + +The \index{reconstruction} \textbf{reconstruction} of a sampled signal is done by: +\begin{itemize} + \item Reinterpreting the time-discrete signal $\underline{x}[n]$ again as a time-continuous, sampled signal $\underline{x}_S(t)$. + \item Removing the copies of the original signal in the frequency domain, using a \ac{LPF} (\index{reconstruction filter} \textbf{reconstruction filter}) with the cut-off frequency $\omega_o = \frac{\omega_S}{2}$. +\end{itemize} + +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=0.7} + \begin{circuitikz} + \node[draw, block, right=3cm of Sampler] (ReInterp) {Reinterpret as\\ time-continuous signal}; + + \draw[<-o] (ReInterp.west) -- ++(-1.5cm, 0) node[above, align=center]{Time-discrete\\ signal $\underline{x}[n]$}; + \draw (ReInterp.east) -- ++(3cm,0) node[midway, above, align=center]{Series of pulses\\ $\underline{x}_S(t)$} + to[lowpass] ++(2.5cm,0 ) -- ++(0.7cm, 0) node[above, align=center]{Reconstructed\\ time-continuous\\ signal $\underline{\tilde{x}}(t)$}; + + \draw[dashed] (ReInterp.north) -- ++(0, 2cm) node[below left, align=right]{Time-discrete\\ domain} node[below right, align=left]{Time-continuous\\ domain}; + \draw[dashed] (ReInterp.south) -- ++(0, -1cm); + \end{circuitikz} + \end{adjustbox} + \caption{An abstract view on reconstruction} +\end{figure} + +The reconstructed signal $\underline{\tilde{x}}(t)$ equals the original signal $\underline{x}(t)$ only if the Shannon-Nyquist theorem is fulfilled. +\begin{equation} + \underline{\tilde{x}}(t) = \underline{x}(t) \qquad \text{if $\underline{x}(t)$ band-limtied to $-\frac{\omega_S}{2} \leq \omega \leq \frac{\omega_S}{2}$} +\end{equation} + +\begin{figure}[H] + + \subfloat[Sampled signal $\underline{X}_S\left(j\omega\right)$] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$|\underline{X}_S\left(j\omega\right)|$}, + %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=-3.5, + xmax=3.5, + ymin=0, + ymax=1.2, + xtick={-3, -2, -1, -0.5, 0, 0.5, 1, 2, 3}, + xticklabels={$-3 \omega_S$, $-2 \omega_S$, $- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$, $2 \omega_S$, $3 \omega_S$}, + ytick={0}, + ] + \pgfplotsinvokeforeach{-3, -2, ..., 3}{ + \draw[-latex, blue, dashed, very thick] (axis cs:#1,0) -- (axis cs:#1,1); + \draw[green, thick] (axis cs:{#1-0.4},0) -- (axis cs:#1,0.7); + \draw[red, thick] (axis cs:#1,0.7) -- (axis cs:{#1+0.4},0); + } + + \draw[black, thick, dashed] (axis cs:-0.5,0) -- (axis cs:-0.5,0.9) -- (axis cs:0.5,0.9) -- (axis cs:0.5,0); + \draw (axis cs:0.3,0.9) -- (axis cs:0.4,1.0) node[above right, align=left]{Reconstruction filter}; + \end{axis} + \end{tikzpicture} + } + + \subfloat[Reconstructed signal $\underline{\tilde{X}}\left(j\omega\right)$] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$|\underline{\tilde{X}}\left(j\omega\right)|$}, + %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=-3.5, + xmax=3.5, + ymin=0, + ymax=1.2, + xtick={-1, -0.5, 0, 0.5, 1}, + xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$}, + ytick={0}, + ] + \draw[green, thick] (axis cs:-0.4,0) -- (axis cs:0,0.7); + \draw[red, thick] (axis cs:0,0.7) -- (axis cs:0.4,0); + \end{axis} + \end{tikzpicture} + } + + \caption{Reconstruction of a sampled signal} +\end{figure} \subsection{Discrete-Time Fourier Transform} |