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diff --git a/chapter06/content_ch06.tex b/chapter06/content_ch06.tex index 8bb25d6..dc2ea07 100644 --- a/chapter06/content_ch06.tex +++ b/chapter06/content_ch06.tex @@ -540,6 +540,8 @@ The \ac{LO} generates both a cos-signal and a \SI{90}{\degree}-phase-shifted sin \section{Resampling} +\index{resampling} \textbf{Resampling} refers to the change of the sampling rate in a \index{multi-rate system} \textbf{multi-rate system}. + \begin{figure}[H] \centering \begin{tikzpicture} @@ -560,7 +562,7 @@ The \ac{LO} generates both a cos-signal and a \SI{90}{\degree}-phase-shifted sin \draw ([shift={(-4cm,1cm)}] Data.west) node[left,align=right]{Input} to[adc,>,o-] ++(2cm,0) to[twoport,t=$\downarrow N$,>] ([yshift=1cm] Data.west) node[inputarrow]{}; \draw ([yshift=-1cm] Data.west) to[twoport,t=$\uparrow M$,>] ++(-2cm,0) to[dac,>] ++(-2cm,0) node[inputarrow,rotate=180]{} node[left,align=right]{Output}; \end{circuitikz} - \caption{A system with a down-sampler (decimation factor $N$) and up-sampler (interpolation factor $M$)} + \caption{A muli-rate system with a down-sampler (decimation factor $N$) and up-sampler (interpolation factor $M$)} \end{figure}% \nomenclature[Bd]{\begin{circuitikz}[baseline={(current bounding box.center)}]\draw (0,0) to[twoport,t=$\downarrow N$,>] (2,0);\end{circuitikz}}{Down-sampler (decimation factor $N$)}% \nomenclature[Bu]{\begin{circuitikz}[baseline={(current bounding box.center)}]\draw (0,0) to[twoport,t=$\uparrow M$,>] (2,0);\end{circuitikz}}{Up-sampler (interpolation factor $M$)}% @@ -573,11 +575,353 @@ The \ac{LO} generates both a cos-signal and a \SI{90}{\degree}-phase-shifted sin \subsection{Down-sampling} -\todo{Downsampling, Decimation} +\begin{definition}{Down-sampling} + \index{down-sampling} \textbf{Down-sampling} is the process of reducing the sampling rate. + + \begin{figure}[H] + \centering + \begin{circuitikz} + \draw (0,0) node[left,align=right]{Input $\underline{x}_i[n]$\\ Sample rate: $T_{S,i}$} to[lowpass,>,o-] ++(2,0) to[twoport,t=$\downarrow N$,>] ++(2,0) node[inputarrow,rotate=0]{} node[right,align=left]{Output $\underline{x}_o[n]$\\ Sample rate: $T_{S,o}$}; + \end{circuitikz} + \caption{A down-sampler with a decimation factor of $N$} + \end{figure} + + The ratio between input and output sampling rate is the \index{decimation factor} \textbf{decimation factor} $N$. + \begin{equation} + N = \frac{T_{S,i}}{T_{S,o}} \qquad, N \in \mathbb{N} + \end{equation} + The decimation factor $N$ must be a positive integer. + + \vspace{0.5em} + + \index{decimation} \textbf{Decimation} can be used synonymous for down-sampling. +\end{definition} + +Down-sampling is in fact the sampling of an already sampled time-discrete signal with a lower sampling rate. The term \emph{resampling} is derived from this. +\begin{itemize} + \item Prior to down-sampling, an \index{anti-aliasing filter} \textbf{anti-aliasing filter} is required. + \item The actual down-sampling is: + \begin{itemize} + \item Take every $n$-th sample out of the input signal. + \item Discard all samples in between. + \end{itemize} +\end{itemize} + +\begin{figure}[H] + \centering + + \subfloat[Input signal]{ + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.35\linewidth, + scale only axis, + xlabel={$n$}, + ylabel={$x_i[n]$}, + %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=0, + xmax=16.8, + ymin=-1.2, + ymax=1.2, + xtick={0,4,...,16}, + ytick={0}, + ] + \pgfplotsinvokeforeach{0,0.125,...,2}{ + \addplot[red, thick] coordinates {({#1*8},0) ({#1*8}, {cos(deg(2*pi*1*#1))})}; + \addplot[red, only marks, mark=o] coordinates {({#1*8}, {cos(deg(2*pi*1*#1))})}; + } + \end{axis} + \end{tikzpicture} + } + \hfill + \subfloat[Decimated output signal]{ + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.35\linewidth, + scale only axis, + xlabel={$n$}, + ylabel={$x_o[n]$}, + %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=0, + xmax=4.2, + ymin=-1.2, + ymax=1.2, + xtick={0,1,...,4}, + ytick={0}, + ] + \pgfplotsinvokeforeach{0,0.5,...,2}{ + \addplot[red, thick] coordinates {({#1*2},0) ({#1*2}, {cos(deg(2*pi*1*#1))})}; + \addplot[red, only marks, mark=o] coordinates {({#1*2}, {cos(deg(2*pi*1*#1))})}; + } + \end{axis} + \end{tikzpicture} + } + + \caption{Down-sampling by $N = 4$} +\end{figure} + +\subsubsection{Aliasing in Down-Sampling} + +Down-sampling means that a time-discrete signal is sampled again with a lower sampling rate. + +This has effects on the spectrum of the output signal: +\begin{itemize} + \item The spectrum of the input signal is band-limited by $\omega_i \in [-\frac{\pi}{T_{S,i}}, \frac{\pi}{T_{S,i}}]$. + \item The spectrum of the input signal repeats at $\frac{2\pi}{T_{S,i}}$. + \item The spectrum of the output signal is band-limited by $\omega_o \in [-\frac{\pi}{T_{S,o}}, \frac{\pi}{T_{So}}] = [-\frac{\pi}{N T_{S,i}}, \frac{\pi}{N T_{S,i}}]$. + \item The spectrum of the output signal repeats at $\frac{2\pi}{T_{S,o}} = \frac{2\pi}{N T_{S,i}}$. + \item The ``repetition frequency'' of the output signal spectrum is divided by $N$. +\end{itemize} + +\begin{fact} + The \index{Shannon-Nyquist sampling theorem} \emph{Shannon-Nyquist sampling theorem} applies to the down-sampling, too. The input signal must be band-limited to $[-\frac{\pi}{T_{S,o}}, \frac{\pi}{T_{So}}]$. Otherwise, the output signal will show \index{aliasing} \emph{aliasing}. +\end{fact} + +Therefore, a low-pass filter is applied as an \emph{anti-aliasing filter}. The anti-aliasing filter is implemented as an \ac{IIR} or \ac{FIR} filter. + +\begin{figure}[H] + \centering + + \subfloat[Spectrum of the input signal] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$|\underline{X}_i\left(j\omega\right)|$}, + %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=-2.5, + xmax=2.5, + ymin=0, + ymax=1.2, + xtick={-2, -1, -0.5, 0, 0.5, 1, 2}, + xticklabels={$-2 \omega_{S,i}$, $- \omega_{S,i}$, $- \frac{\omega_{S,i}}{2}$, $0$, $\frac{\omega_{S,i}}{2}$, $\omega_{S,i}$, $2 \omega_{S,i}$}, + ytick={0}, + ] + \draw[latex-latex] (axis cs:0,0.8) -- node[midway,above,align=center]{$\omega_{S,i}$-periodic} (axis cs:1,0.8); + + \pgfplotsinvokeforeach{-2, -1, ..., 2}{ + \draw[green, thick] (axis cs:{#1-0.25},0) -- (axis cs:#1,0.7); + \draw[red, thick] (axis cs:#1,0.7) -- (axis cs:{#1+0.25},0); + } + \end{axis} + \end{tikzpicture} + } + + \subfloat[Spectrum of the decimated output signal (decimation by 2)] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$|\underline{X}_o\left(j\omega\right)|$}, + %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=-2.5, + xmax=2.5, + ymin=0, + ymax=1.2, + xtick={-2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, 2}, + xticklabels={$-4 \omega_{S,o}$, $-3 \omega_{S,o}$, $-2 \omega_{S,o}$, $- \omega_{S,o}$, $0$, $\omega_{S,o}$, $2 \omega_{S,o}$, $3 \omega_{S,o}$, $4 \omega_{S,o}$}, + ytick={0}, + ] + \draw[latex-latex] (axis cs:0.5,0.8) -- node[midway,above,align=center]{$\omega_{S,o}$-periodic\\ (i.e. $\omega_{S,i}/N$-periodic)} (axis cs:1,0.8); + + \pgfplotsinvokeforeach{-2, -1.5, ..., 2}{ + \draw[green, thick] (axis cs:{#1-0.25},0) -- (axis cs:#1,0.7); + \draw[red, thick] (axis cs:#1,0.7) -- (axis cs:{#1+0.25},0); + } + \end{axis} + \end{tikzpicture} + } + + \caption[Effects of the down-sampling on the spectrum]{Effects of the down-sampling on the spectrum. If the input signal occupied a little more bandwidth and thereby violated the Shannon-Nyquist sampling theorem, the output signal would show aliasing.} +\end{figure} + +\begin{fact} + Changing the sampling rate changes to periodicity of the spectrum of the sampled signal. +\end{fact} + +\subsubsection{Processing Gain} + +Digitizing an analogue signal and then down-sampling it seems pointless. Why is the \ac{ADC} not configured to the desired sampling rate? + +An advantage of the down-sampling is its \index{processing gain!down-sampling} \textbf{processing gain}. + +Let's consider a signal with the power $P_i$ (linear scale, \si{mW}) or $L_{P,i}$ (logarithmic scale, \si{dBm}), respectively. The signal sampled and quantized with $B$ bits. The \ac{SQNR} is: +\begin{equation} + L_{\mathrm{SQNR},i} = \SI{1.761}{dB} + B \cdot \SI{6.02}{dB} +\end{equation} + +The quantization noise power is in the logarithmic scale (\si{dBm}): +\begin{equation} + L_{P,N,i} = L_{P,i} - L_{\mathrm{SQNR},i} +\end{equation} +or in the linear scale (\si{mW}): +\begin{equation} + \begin{split} + P_{N,i} &= \frac{P_i}{\mathrm{SQNR}_i} \\ + &= P_i \cdot 10^{\frac{\SI{1.761}{dB} + B \cdot \SI{6.02}{dB}}{\SI{10}{dB}}} + \end{split} +\end{equation} + +The quantization noise power is distributed equally over the frequency axis between $[-\frac{1}{2 T_{S,i}}, \frac{1}{2 T_{S,i}}]$, which is the band limit for the sampled input signal. The \index{noise bandwidth} \textbf{noise bandwidth} is therefore $\Delta f_{S,i} = \frac{1}{T_{S,i}}$. The quantization noise floor $S_{N,i}$, which is a \ac{PSD} (\si{mW/Hz}), is: +\begin{equation} + \begin{split} + S_{N,i} = \frac{P_{N,i}}{\Delta f_{S,i}} \\ + &= \frac{P_{N,i}}{\frac{1}{T_{S,i}}} \\ + &= P_{N,i} T_{S,i} + \end{split} +\end{equation} + +\begin{attention} + Please note the difference between noise power and noise floor (frequency distribution of the power). +\end{attention} + +\begin{itemize} + \item The quantization noise floor depends only on the number of bits of the quantizer $B$. + \item Prior to the down-sampling, a low-pass filter is applied. + \item The noise bandwidth is reduced to the filter bandwidth $\Delta f_{S,o} = \frac{1}{T_{S,o}} = \frac{1}{N T_{S,i}}$, which is determined by the Shannon-Nyquist sampling theorem. +\end{itemize} + +The quantization noise floor remains constant while the noise bandwidth is divided by $N$. That is, the quantization noise floor in the down-sampler input equals the quantization noise floor in the down-sampler output. +\begin{equation} + S_{N,o} = S_{N,i} +\end{equation} +The noise power in the output will be: +\begin{equation} + \begin{split} + P_{N,o} &= S_{N,o} \cdot \Delta f_{S,o} \\ + &= S_{N,i} \cdot \frac{1}{T_{S,o}} \\ + &= P_{N,i} T_{S,i} \cdot \frac{1}{N T_{S,i}} \\ + &= \frac{P_{N,i}}{N} + \end{split} +\end{equation} +In the logarithmic scale: +\begin{equation} + L_{P,N,o} = L_{P,N,i} - \SI{10}{dB} \cdot \log_{10} \left(N\right) +\end{equation} + +\begin{fact} + The down-sampling (decimation) divides the quantization noise power by $N$. +\end{fact} + +Now, the \ac{SNR} of the output signal is: +\begin{equation} + \begin{split} + \mathrm{SNR}_o &= \frac{P_i}{P_{N,o}} + \end{split} +\end{equation} +The power of the input signal $P_i$ is not affected and remains the same. + +In the logarithmic scale: +\begin{equation} + \begin{split} + L_{\mathrm{SNR},o} &= L_{P,i} - L_{P,N,o} \\ + &= \underbrace{L_{P,i} - L_{P,N,i}}_{= L_{\mathrm{SQNR},i}} + \SI{10}{dB} \cdot \log_{10} \left(N\right) \\ + &= L_{\mathrm{SQNR},i} + \SI{10}{dB} \cdot \log_{10} \left(N\right) + \end{split} +\end{equation} -\todo{Sampling of digital signal, Shannon-Nyquist theorem, filtering} +\begin{definition}{Processing gain of down-sampling} + The \emph{down-sampling} by $N$ (or \emph{decimation} by $N$) reduces the quantization noise power by factor $N$. The \ac{SNR} is improved (increased) by factor $N$. + + \vspace{0.5em} + + In measures of decibel, the \ac{SNR} is improved by $+ \SI{10}{dB} \cdot \log_{10} \left(N\right)$. + + \vspace{0.5em} + + The \ac{SNR} improvement is achieved in the digital signal processing. It is therefore called \index{processing gain} \textbf{processing gain}. +\end{definition} -\todo{Processing Gain (Noise bandwidth)} +Increasing the \ac{SNR} is equivalent with increasing the number of bits: +\begin{equation} + \begin{split} + L_{\mathrm{SNR},o} &= \SI{1.761}{dB} + \tilde{B} \cdot \SI{6.02}{dB} \\ + \tilde{B} &= \frac{L_{\mathrm{SNR},o} - \SI{1.761}{dB}}{\SI{6.02}{dB}} \\ + \tilde{B} &= \frac{L_{\mathrm{SQNR},i} + \SI{10}{dB} \cdot \log_{10} \left(N\right) - \SI{1.761}{dB}}{\SI{6.02}{dB}} \\ + \tilde{B} &= \frac{\SI{1.761}{dB} + B \cdot \SI{6.02}{dB} + \SI{10}{dB} \cdot \log_{10} \left(N\right) - \SI{1.761}{dB}}{\SI{6.02}{dB}} \\ + \tilde{B} &= \frac{B \cdot \SI{6.02}{dB} + \SI{10}{dB} \cdot \log_{10} \left(N\right)}{\SI{6.02}{dB}} + \end{split} +\end{equation} +$\tilde{B}$ is the \index{effective number of bits!down-sampling} \textbf{\acf{ENOB}}, which would have been necessary if the analogue signal was directly sampled of the decimated sampling frequency $\frac{1}{T_{S,o}} = \frac{1}{N T_{S,i}}$. + +The processing gain adds $\Delta B$ bits in significance of the quantized values. +\begin{equation} + \Delta B = \tilde{B} - B = \frac{\SI{10}{dB} \cdot \log_{10} \left(N\right)}{\SI{6.02}{dB}} +\end{equation} +\begin{remark} + When such a system is implemented in real, the designer must be careful and really add the bits in the implementation. For example, if the input signal is stored at the \ac{ADC} bit size of \SI{8}{bit} and the processing gain adds \SI{3}{bit}, the output must be stored in the next word size that the machine supports, e.g. \SI{16}{bit}. When it is stored as a \SI{8}{bit} value again, the processing gain is immediately lost due to truncation. +\end{remark} + +The real number of bits of the \ac{ADC} is not changed. The new number of bits is solely the corollary of the processing. + +\begin{conclusion} + The processing gain, which is a result of the decimation, improves the \ac{SNR} and increases the \ac{ENOB}. + + \vspace{0.5em} + + Sampling the analogue signal at a much higher frequency than required by the Shannon-Nyquist theorem is called \index{oversampling} \textbf{oversampling}. Oversampling may be used to increase the \ac{ENOB} of the \ac{ADC} and improve the receiver sensitivity. +\end{conclusion} \subsection{Up-sampling} |