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diff --git a/chapter06/content_ch06.tex b/chapter06/content_ch06.tex index e50260c..8bb25d6 100644 --- a/chapter06/content_ch06.tex +++ b/chapter06/content_ch06.tex @@ -320,6 +320,8 @@ There is another simple explanation for the \ac{BIBO} stability. \item The impulse response has a finite length in the time-domain. \end{itemize} +This leads to the definition of \ac{FIR} filters: + \begin{definition}{\ac{FIR} filters} Digital filters without a feed-back branch will always have a finite-length impulse response. They are called \index{finite impulse response filter} \textbf{\acf{FIR} filters}. \ac{FIR} filters are always \ac{BIBO} stable. \end{definition} @@ -364,6 +366,178 @@ Both \ac{IIR} and \ac{FIR} are causal. Their impulse response is $\underline{h}[ \section{Digital Mixer} +A digital mixer implements a frequency shift in the digital domain. It has the same purpose as the analogue mixer. However, there are some differences: +\begin{itemize} + \item There are usually an \ac{I} and \ac{Q} component in the digital domain. \textbf{The digital signal is complex-valued.} + \item The signal can be frequency-shifted as a whole, because both input and output are complex-valued. +\end{itemize} + +\begin{remark} + A signal is practically never mixed to exactly \SI{0}{Hz}. All ADC have a DC bias. The sampled signal is superimposed by a DC voltage at \SI{0}{Hz} in the time-domain. This adds an error. Therefore, the signal shifted some \si{kHz} away from DC. It can be digitally shifted to \SI{0}{Hz}, without adding any errors. +\end{remark} + +In the frequency-domain\footnote{The \ac{DTFT} is used because of the time-discrete signals.}, the frequency shift by $\omega_0$ is: +\begin{equation} + \underline{Y}_{\frac{2\pi}{T_S}}\left(e^{j \omega T_S}\right) = \underline{X}_{\frac{2\pi}{T_S}}\left(e^{j\left(\omega - \omega_0\right)T_S}\right) +\end{equation} +The input signal $\underline{X}_{\frac{2\pi}{T_S}}\left(e^{j \omega T_S}\right)$ is shifted as a whole block. $\underline{Y}_{\frac{2\pi}{T_S}}\left(e^{j \omega T_S}\right)$ is the mixer output signal. + +In the time-domain, the frequency shift is: +\begin{equation} + \begin{split} + \underline{y}[n] &= \mathcal{F}_{\mathrm{DTFT}}^{-1}\left\{\underline{X}_{\frac{2\pi}{T_S}}\left(e^{j\left(\omega - \omega_0\right)T_S}\right)\right\} \\ + &= \underbrace{\underline{x}[n]}_{\text{Mixer input}} \cdot \underbrace{e^{j \omega_c T_S n}}_{\text{\acs{LO} signal}} \\ + &= \left(\Re\left\{\underline{x}[n]\right\} + j \cdot \Im\left\{\underline{x}[n]\right\}\right) \left(\cos\left(\omega_c T_S n\right) + j \cdot \sin\left(\omega_c T_S n\right)\right) \\ + &= \left( \Re\left\{\underline{x}[n]\right\} \cdot \cos\left(\omega_c T_S n\right) - \Im\left\{\underline{x}[n]\right\} \cdot \sin\left(\omega_c T_S n\right) \right) \\ &\quad + j \left( \Im\left\{\underline{x}[n]\right\} \cdot \cos\left(\omega_c T_S n\right) + \Re\left\{\underline{x}[n]\right\} \cdot \sin\left(\omega_c T_S n\right) \right) + \end{split} + \label{eq:ch06:digi_mix} +\end{equation} + +Note the following: +\begin{itemize} + \item The mixer input is complex-valued in the time-domain. + \item The \ac{LO} signal is complex-valued in the time-domain. + \item The mixer output is complex-valued in the time-domain. +\end{itemize} + +The \ac{LO} generates both a cos-signal and a \SI{90}{\degree}-phase-shifted sin-signal. The mixer signal is time-discrete. The digital oscillator is a \index{numerically-controlled oscillator} \textbf{\acf{NCO}} which emits the \ac{LO} real and imaginary values at the sample period $T_S$. The frequency can be configured. + +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=0.8} + \begin{tikzpicture} + \node[mixer](MixIcos) {}; + \node[mixer](MixIsin) at([shift={(3cm, -1cm)}]MixIcos) {}; + \node[mixer](MixQcos) at([shift={(0cm, -2cm)}]MixIcos) {}; + \node[mixer](MixQsin) at([shift={(3cm, -3cm)}]MixIcos) {}; + \node[adder](AddI) at([shift={(7cm, 0cm)}]MixIcos) {}; + \node[adder](AddQ) at([shift={(6cm, 0cm)}]MixQcos) {}; + \node[block,draw,minimum width=4cm](NCO) at([shift={(0cm, -6cm)}]MixIcos) {\acs{NCO}}; + + \draw ([xshift=-1.5cm]NCO.north) node[above left,align=right]{$\cos\left(\omega_c T_S n\right)$} -- ([shift={(-1.5cm, -1cm)}]MixIcos.south) -- ([shift={(0.147cm, 0.147cm)}]MixIcos.south west) node[inputarrow,rotate=45]{}; + \draw ([shift={(-1.5cm, -1cm)}]MixQcos.south) to[short,*-] ([shift={(0.147cm, 0.147cm)}]MixQcos.south west) node[inputarrow,rotate=45]{}; + \draw ([xshift=1.5cm]NCO.north) node[above right,align=left]{$\sin\left(\omega_c T_S n\right)$} -- ([shift={(-1.5cm, -1cm)}]MixIsin.south) -- ([shift={(0.147cm, 0.147cm)}]MixIsin.south west) node[inputarrow,rotate=45]{}; + \draw ([shift={(-1.5cm, -1cm)}]MixQsin.south) to[short,*-] ([shift={(0.147cm, 0.147cm)}]MixQsin.south west) node[inputarrow,rotate=45]{}; + + \draw ([shift={(-3cm, 0cm)}]MixIcos.west) node[left,align=right]{Input \ac{I} $\Re\left\{\underline{x}[n]\right\}$\\ (real part)} to[short,o-] (MixIcos.west) node[inputarrow]{}; + \draw ([shift={(-3cm, 0cm)}]MixQcos.west) node[left,align=right]{Input \ac{Q} $\Im\left\{\underline{x}[n]\right\}$\\ (imaginary part)} to[short,o-] (MixQcos.west) node[inputarrow]{}; + \draw ([shift={(-2cm, 0cm)}]MixIcos.west) to[short,*-] ++(0cm,-0.5cm) |- (MixIsin.west) node[inputarrow]{}; + \draw ([shift={(-2cm, 0cm)}]MixQcos.west) to[short,*-] ++(0cm,-0.5cm) |- (MixQsin.west) node[inputarrow]{}; + + \draw (MixIcos.east) to[short] (AddI.west) node[inputarrow]{}; + \draw (MixQcos.east) to[short] (AddQ.west) node[inputarrow]{}; + \draw (MixQsin.east) to[amp,>,l_=$-1$] ++(2cm,0) -| (AddI.south) node[inputarrow,rotate=90]{}; + \draw (MixIsin.east) to[short] ++(1cm,0) -| (AddQ.north) node[inputarrow,rotate=-90]{}; + + \draw (AddI.east) to[short] ++(1cm,0) node[inputarrow]{} node[right,align=left,xshift=5mm]{Output \ac{I} $\Re\left\{\underline{y}[n]\right\}$\\ (real part)}; + \draw (AddQ.east) to[short] ++(2cm,0) node[inputarrow]{} node[right,align=left,xshift=5mm]{Output \ac{Q} $\Im\left\{\underline{y}[n]\right\}$\\ (real part)}; + \end{tikzpicture} + \end{adjustbox} + \caption[Block diagram of a digital mixer]{Block diagram of a digital mixer, implementing \eqref{eq:ch06:digi_mix}} +\end{figure} + +\begin{figure}[H] + \centering + + \subfloat[Mixer input $\underline{X}_{\frac{2\pi}{T_S}}\left(e^{j \omega T_S}\right)$] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.10\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$|\underline{X}_{\frac{2\pi}{T_S}}\left(e^{j \omega T_S}\right)|$}, + %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=-2.5, + xmax=2.8, + ymin=0, + ymax=1.2, + xtick={-2, -1, -0.5, 0, 0.5, 1, 2}, + xticklabels={$-2 \omega_S$, $- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$, $2 \omega_S$}, + ytick={0}, + ] + \draw[dashed] (axis cs:-0.5,0) -- (axis cs:-0.5,1) -- (axis cs:0.5,1) --(axis cs:0.5,0); + + \draw[green, thick] (axis cs:{0-0.2},0) -- (axis cs:0,0.7); + \draw[red, thick] (axis cs:0,0.7) -- (axis cs:{0+0.4},0); + + \pgfplotsinvokeforeach{-2, -1, 1, 2}{ + \draw[green, thick, dashed] (axis cs:{#1-0.2},0) -- (axis cs:#1,0.7); + \draw[red, thick, dashed] (axis cs:#1,0.7) -- (axis cs:{#1+0.4},0); + } + \end{axis} + \end{tikzpicture} + } + + \subfloat[Mixer output $\underline{Y}_{\frac{2\pi}{T_S}}\left(e^{j \omega T_S}\right)$] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.13\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$|\underline{Y}_{\frac{2\pi}{T_S}}\left(e^{j \omega T_S}\right)|$}, + %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=-2.5, + xmax=2.8, + ymin=-0.4, + ymax=1.2, + xtick={-2, -1, -0.5, 0, 0.5, 1, 2}, + xticklabels={$-2 \omega_S$, $- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$, $2 \omega_S$}, + ytick={0}, + ] + \draw[dashed] (axis cs:-0.5,0) -- (axis cs:-0.5,1) -- (axis cs:0.5,1) --(axis cs:0.5,0); + + \draw[latex-latex] (axis cs:0,-0.1) -- node[midway,below,align=center]{$\omega_0$} (axis cs:0.25,-0.1); + \draw[dotted] (axis cs:0.25,-0.2) -- (axis cs:0.25,0.8); + + \draw[green, thick, dashed] (axis cs:{-0.75-0.2},0) -- (axis cs:-0.75,0.7); + \draw[red, thick, dashed] (axis cs:-0.75,0.7) -- (axis cs:{-0.75+0.25},0.262); + \draw[red, thick] (axis cs:{-0.75+0.25},0.262) -- (axis cs:{-0.75+0.4},0); + + \draw[green, thick] (axis cs:{0.25-0.2},0) -- (axis cs:0.25,0.7); + \draw[red, thick] (axis cs:0.25,0.7) -- (axis cs:{0.25+0.25},0.262); + \draw[red, thick, dashed] (axis cs:{0.25+0.25},0.262) -- (axis cs:{0.25+0.4},0); + + \pgfplotsinvokeforeach{-1.75, 1.25, 2.25}{ + \draw[green, thick, dashed] (axis cs:{#1-0.2},0) -- (axis cs:#1,0.7); + \draw[red, thick, dashed] (axis cs:#1,0.7) -- (axis cs:{#1+0.4},0); + } + \end{axis} + \end{tikzpicture} + } + + \caption[Digital mixing in the frequency-domain]{Digital mixing in the frequency-domain. It must be noted that the spectra of sampled signals are periodic. But only the interval between $[- \frac{\omega_S}{2}, \frac{\omega_S}{2}]$ is visible. Therefore, it might appear that frequencies are moved from one end of the interval to the other end.} +\end{figure} + \section{Resampling} \begin{figure}[H] @@ -397,11 +571,23 @@ Both \ac{IIR} and \ac{FIR} are causal. Their impulse response is $\underline{h}[ \item The \ac{ADC} can be operated at maximum sampling rate. The signal is oversampled. Down-sampling provides processing gain and enhances the receiver performance. \end{itemize} +\subsection{Down-sampling} + \todo{Downsampling, Decimation} +\todo{Sampling of digital signal, Shannon-Nyquist theorem, filtering} + +\todo{Processing Gain (Noise bandwidth)} + +\subsection{Up-sampling} + \todo{Upsampling, Interpolation} -\todo{Processing Gain} +\todo{Inserting zeros, filtering} + +\begin{excursus}{\acs{CIC} filter} + The \index{cascaded integrator-comb} \textbf{\acf{CIC}} filter an optimized \ac{FIR} filter. +\end{excursus} \todo{CIC filter} |