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diff --git a/chapter04/content_ch04.tex b/chapter04/content_ch04.tex index e3a90ac..e93f287 100644 --- a/chapter04/content_ch04.tex +++ b/chapter04/content_ch04.tex @@ -1061,7 +1061,7 @@ The \ac{DFT} \eqref{eq:ch04:dft} can be expressed as a linear system of equation \cmplxvect{X} = \underline{\mat{F}} \cdot \cmplxvect{x} \end{equation} -The $N \times N$ transformation matrix $\underline{\mat{F}}$ is the \index{DFTmatrx} \textbf{ac{DFT} matrix} with the elements: +The $N \times N$ transformation matrix $\underline{\mat{F}}$ is the \index{DFT matrx} \textbf{\ac{DFT} matrix} with the elements: \begin{equation} \underline{F}_{pq} = \underline{w}^{p \cdot q} \end{equation} @@ -1135,7 +1135,8 @@ A sequence of length $N$ is taken out of $\underline{x}[n]$: \end{equation} The act of extracting $N$ subsequent samples out of $\underline{x}[n]$ is called \index{windowing} \textbf{windowing}. To illustrate this, imagine that you watch a sequence moving to the left through a window. The movement to the left is the time advance. Because of the window, you see a certain extract of $\underline{x}[n]$ only at one time. -\todo{Window illustration} +%TODO +%\todo{Window illustration} This sequence is repeated indefinitely (\emph{periodic continuation}): \begin{equation} @@ -1169,22 +1170,31 @@ Windowing and periodic continuation is, in fact, a sampling of the \ac{DTFT} (se The widowing of $\underline{x}[n]$ to obtain $\underline{x}_N[n]$ as discussed before did not change the values of $\underline{x}[n]$. This can be expressed as a multiplication of the original sequence with the rectangular function: \begin{equation} - \underline{x}_W[n] = \underline{x}_N[n] \cdot \underbrace{w_{rect,N}[n]}_{\text{Rectangular window}} + \underline{x}_W[n] = \underline{x}_N[n] \cdot \underbrace{w_{rect,M}[n]}_{\text{Rectangular window}} \end{equation} -where +where $w_{rect,M}[n]$ is the rectangular function of the length $M$: \begin{equation} - w_{rect,N}[n] = \begin{cases} - 1 &\quad \text{if } \; 0 \geq n < N, \\ + w_{rect,M}[n] = \begin{cases} + 1 &\quad \text{if } \; -\frac{M-1}{2} \leq n \leq \frac{M-1}{2}, \\ 0 &\quad \text{else}. \end{cases} \end{equation} +$M$ shall be an odd number. -$w[n]$ is called \index{window function} \textbf{window function}. +$w_M[n]$ is called \index{window function} \textbf{window function}. \begin{definition}{Window function} - A \index{window function} \textbf{window function} $w[n]$ is applied to the periodically continued sequence $\underline{x}_N[n]$ by multiplication: + A \index{window function} \textbf{window function} $w_M[n]$ is applied to the periodically continued sequence $\underline{x}_N[n]$ by multiplication: \begin{equation} - \underline{x}_N[n] \equiv \underline{x}[n] \cdot w[n] + \underline{x}_N[n] \equiv \underline{x}[n] \cdot w_M[n] \end{equation} + + Window functions are $w_M[n]$: + \begin{itemize} + \item limited to a length of $M$ + \item where $M$ is an odd number, + \item always symmetric (even functions), and + \item real-valued. + \end{itemize} \end{definition} \subsubsection{Spectral Leakage} @@ -1207,6 +1217,16 @@ This has some implications: \end{itemize} This effect is called \index{spectral leakage} \textbf{spectral leakage}. +\begin{fact} + The window changes only the amplitude spectrum, but not the phase of the sampled signal. +\end{fact} +\begin{itemize} + \item The window function is an even, real-valued function. + \item Its Fourier transform is therefore always real-valued, too. + \item The purely real-valued ($\arg{\cdot} = 0 \text{ or } \pm \pi$) Fourier transform cannot change the phase of the signal. +\end{itemize} + + \begin{figure}[H] \centering @@ -1242,6 +1262,11 @@ This effect is called \index{spectral leakage} \textbf{spectral leakage}. %ytick={0}, ] \addplot[red, thick, smooth, domain=0:4, samples=50] plot(\x, {cos(deg(pi*\x))}); + + \pgfplotsinvokeforeach{0,0.25,...,4}{ + \addplot[blue] coordinates {(#1,0) (#1,{cos(deg(pi*#1))})}; + \addplot[blue, only marks, mark=o] coordinates {(#1,{cos(deg(pi*#1))})}; + } \end{axis} \end{tikzpicture} } @@ -1254,7 +1279,7 @@ This effect is called \index{spectral leakage} \textbf{spectral leakage}. width=0.35\linewidth, scale only axis, xlabel={$n$}, - ylabel={$w[n]$}, + ylabel={$w_9[n]$}, %grid style={line width=.6pt, color=lightgray}, %grid=both, grid=none, @@ -1269,7 +1294,7 @@ This effect is called \index{spectral leakage} \textbf{spectral leakage}. at={(ticklabel* cs:1.05)}, anchor=east, }, - xmin=-0.5, + xmin=-8.5, xmax=8.5, ymin=0, ymax=1.2, @@ -1277,12 +1302,23 @@ This effect is called \index{spectral leakage} \textbf{spectral leakage}. %xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$}, %ytick={0}, ] - \addplot[blue, thick] coordinates {(-0.5,0) (0,0)}; - \addplot[blue, thick] coordinates {(0,1) (8,1)}; - \addplot[blue, thick] coordinates {(8,0) (8.5,0)}; - - \addplot[blue, thick, dashed] coordinates {(0,0) (0,1)}; - \addplot[blue, thick, dashed] coordinates {(8,1) (8,0)}; + \pgfplotsinvokeforeach{-4,-3,...,4}{ + \addplot[blue] coordinates {(#1,0) (#1,1)}; + \addplot[blue, only marks, mark=o] coordinates {(#1,1)}; + } + \pgfplotsinvokeforeach{-8,-7,...,-5}{ + \addplot[blue, only marks, mark=o] coordinates {(#1,0)}; + } + \pgfplotsinvokeforeach{5,6,...,8}{ + \addplot[blue, only marks, mark=o] coordinates {(#1,0)}; + } + +% \addplot[blue, thick] coordinates {(-0.5,0) (0,0)}; +% \addplot[blue, thick] coordinates {(0,1) (8,1)}; +% \addplot[blue, thick] coordinates {(8,0) (8.5,0)}; +% +% \addplot[blue, thick, dashed] coordinates {(0,0) (0,1)}; +% \addplot[blue, thick, dashed] coordinates {(8,1) (8,0)}; \end{axis} \end{tikzpicture} } @@ -1310,15 +1346,26 @@ This effect is called \index{spectral leakage} \textbf{spectral leakage}. at={(ticklabel* cs:1.05)}, anchor=east, }, - xmin=0, - xmax=4.5, - ymin=-1.2, + xmin=-8.5, + xmax=8.5, + 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}, ] - \addplot[red, thick, smooth, domain=0:4, samples=50] plot(\x, {cos(deg(pi*\x))}); + \addplot[red] coordinates {(2,0) (2,1)}; + \addplot[red] coordinates {(-2,0) (-2,1)}; + \addplot[red, only marks, mark=o] coordinates {(-2,1) (2,1)}; + \pgfplotsinvokeforeach{-8,-7,...,-3}{ + \addplot[red, only marks, mark=o] coordinates {(#1,0)}; + } + \pgfplotsinvokeforeach{-1,0,1}{ + \addplot[red, only marks, mark=o] coordinates {(#1,0)}; + } + \pgfplotsinvokeforeach{3,4,...,8}{ + \addplot[red, only marks, mark=o] coordinates {(#1,0)}; + } \end{axis} \end{tikzpicture} } @@ -1346,20 +1393,65 @@ This effect is called \index{spectral leakage} \textbf{spectral leakage}. at={(ticklabel* cs:1.05)}, anchor=east, }, - xmin=-4.5, - xmax=4.5, + xmin=-8.5, + xmax=8.5, ymin=0, - ymax=1.2, + ymax=9.5, %xtick={-1, -0.5, 0, 0.5, 1}, %xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$}, %ytick={0}, ] - \addplot[blue, thick] coordinates {(-0.5,0) (0,0)}; - \addplot[blue, thick] coordinates {(0,1) (8,1)}; - \addplot[blue, thick] coordinates {(8,0) (8.5,0)}; - - \addplot[blue, thick, dashed] coordinates {(0,0) (0,1)}; - \addplot[blue, thick, dashed] coordinates {(8,1) (8,0)}; + \addplot[blue, dashed, smooth, domain=-8:8, samples=50] plot(\x, {9*abs(asinc((pi*\x/8), 9))}); + \pgfplotsinvokeforeach{-8,-7,...,8}{ + \addplot[red] coordinates {(#1,0) (#1,{9*abs(asinc((pi*#1/8), 9))})}; + \addplot[red, only marks, mark=o] coordinates {(#1,{9*abs(asinc((pi*#1/8), 9))})}; + } + + \draw[blue, dashed] (axis cs:1.5,7) -- (axis cs:2.5,7) node[right,align=left,color=blue]{$\underline{W}\left(e^{j\phi}\right)$\\ $\phi=2\pi\frac{k}{N}$}; + \draw (axis cs:-1,7) node[left,align=right,color=blue]{Main lobe}; + \draw (axis cs:-2,3) node[left,align=right,color=blue]{Side lobes}; + \end{axis} + \end{tikzpicture} + } + + + \subfloat[Amplitude spectrum of the windowed signal, showing spectral leakage]{ + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.7\linewidth, + scale only axis, + xlabel={$k$}, + ylabel={$\underline{X}_W[k]$}, + %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=-8.5, + xmax=8.5, + ymin=0, + ymax=12.5, + %xtick={-1, -0.5, 0, 0.5, 1}, + %xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$}, + %ytick={0}, + ] + \addplot[blue, dashed, smooth, domain=-8:8, samples=50] plot(\x, {9*abs(asinc((pi*(\x+2)/8), 9)) + 9*abs(asinc((pi*(\x-2)/8), 9))}); + \pgfplotsinvokeforeach{-8,-7,...,8}{ + \addplot[red] coordinates {(#1,0) (#1,{9*abs(asinc((pi*(#1+2)/8), 9)) + 9*abs(asinc((pi*(#1-2)/8), 9))})}; + \addplot[red, only marks, mark=o] coordinates {(#1,{9*abs(asinc((pi*(#1+2)/8), 9)) + 9*abs(asinc((pi*(#1-2)/8), 9))})}; + } + \draw[blue, dashed] (axis cs:3,9) -- (axis cs:3.7,9) node[right,align=left,color=blue]{$\left.\underline{X}_W\left(e^{j\phi}\right)\right|_{\phi=2\pi\frac{k}{N}}$}; \end{axis} \end{tikzpicture} } @@ -1367,8 +1459,54 @@ This effect is called \index{spectral leakage} \textbf{spectral leakage}. \caption{Spectral leakage of a mono-chromatic signal} \end{figure} +In the above example: +\begin{itemize} + \item The monochromatic signal has two ideal pulses in its amplitude spectrum. + \item The rectangular window has a sinc-like amplitude spectrum. + \item Due to the convolution, the ideal pulses of the monochromatic signal are ``leak'' across the frequency axis and take a sinc-like pattern, too. +\end{itemize} + +The selection of the window function has implications on the spectral leakage: +\begin{itemize} + \item The width of the \emph{main lobe}, defines how much the sampled signal is spread across the frequency axis. + \item The \emph{main lobe} should be narrow, in order to retain the original shape of the signal. + \item The amplitudes of the \emph{side lobes} define the noise, which emerges. + \item The \emph{side lobes} should be as weak as possible regarding their amplitudes, in order to reduce the noise and improve the \ac{SNR}. +\end{itemize} + +\begin{fact} + Generally, window functions with narrow main lobes have strong side lobes, and vice versa. So, there is always a trade-off. +\end{fact} + \begin{landscape} \subsubsection{Some Window Functions} + + \begin{longtable}[H]{|l|l|l|l|l|} + \hline + Name & Function $w[n]$ & Peak side lobe & Main lobe width & Time domain and frequency response \\ + \hline + \hline + Rectangular & $w[n] = \begin{cases}1, &\quad 0 \leq n \leq M,\\ 0, &\quad \text{otherwise}.\end{cases}$ & \SI{-13}{dB} & $\frac{4\pi}{M+1}$ & \includegraphics[width=5cm]{svg/ch04_win_rect.pdf} \\ + \hline + Hamming & $w[n] = \begin{cases}0.54 - 0.46 \cos\left(\frac{2 \pi n}{M}\right), &\quad 0 \leq n \leq M,\\ 0, &\quad \text{otherwise}.\end{cases}$ & \SI{-41}{dB} & $\frac{8\pi}{M}$ & \includegraphics[width=5cm]{svg/ch04_win_hamming.pdf} \\ + \hline + Hann & $w[n] = \begin{cases}0.5 - 0.5 \cos\left(\frac{2 \pi n}{M}\right), &\quad 0 \leq n \leq M,\\ 0, &\quad \text{otherwise}.\end{cases}$ & \SI{-31}{dB} & $\frac{8\pi}{M}$ & \includegraphics[width=5cm]{svg/ch04_win_hann.pdf} \\ + \hline + Blackman & $w[n] = \begin{cases}0.42 - 0.5 \cos\left(\frac{2 \pi n}{M}\right) & \\ \quad + 0.08 \cos\left(\frac{4 \pi n}{M}\right), &\quad 0 \leq n \leq M,\\ 0, &\quad \text{otherwise}.\end{cases}$ & \SI{-57}{dB} & $\frac{12\pi}{M}$ & \includegraphics[width=5cm]{svg/ch04_win_blackman.pdf} \\ + \hline + Bartlett & $w[n] = \begin{cases}2n/M, &\quad 0 \leq n \leq M/2,\\ 2 - 2n/M, &\quad M/2 < n \leq M,\\ 0, &\quad \text{otherwise}.\end{cases}$ & \SI{-25}{dB} & $\frac{8\pi}{M}$ & \includegraphics[width=5cm]{svg/ch04_win_tri.pdf} \\ + \hline + Gauss & $w[n] = \begin{cases}\exp\left(- \frac{1}{2} \left(\frac{n - M/2}{\sigma M / 2}\right)^2\right), &\quad 0 \leq n \leq M,\\ 0, &\quad \text{otherwise}.\end{cases}$ & \multicolumn{2}{l|}{Adjustable by $\sigma$} & \includegraphics[width=5cm]{svg/ch04_win_gauss.pdf} \\ + \hline + \end{longtable} + + \textit{Picture credits:} \licensequote{\cite{Niemitalo2013}}{Olli Niemitalo}{\href{https://creativecommons.org/publicdomain/zero/1.0/deed.en}{CC0 1.0}} + + \begin{itemize} + \item Now, you see the trade-off which must be made between narrow main lobe and weak side lobes. + \item The rectangular window has a narrow main lobe but strong side lobes. + \item The Blackman window has weak side lobes but a wide main lobe. + \end{itemize} \end{landscape} @@ -1597,26 +1735,36 @@ All considerations apply for ergodic or \ac{WSS} processes only: \end{equation} \end{itemize} -\subsection{Energy Spectral Density} - -%TODO +\subsubsection{Energy Spectral Density} Parseval's theorem for discrete systems: \begin{equation} \sum\limits_{n=0}^{N-1} \left|\underline{x}[n]\right|^2 = \frac{1}{N} \sum\limits_{k=0}^{N-1} \left|\underline{X}[k]\right|^2 \end{equation} +The signal energy is defined to: \begin{equation} - E = \sum\limits_{n=0}^{N-1} \left|\underline{x}[n]\right|^2 + E = T_S^2 \sum\limits_{n=0}^{N-1} \left|\underline{x}[n]\right|^2 \stackrel{!}{=} \sum\limits_{n=0}^{N-1} \underline{\mathrm{S}}_{E,xx}[k] \end{equation} +The sampling period $T_S$ (spacing between the samples) must be kept, because the physical energy depends on the duration of the samples. +The energy spectral density is: \begin{equation} - \underline{\mathrm{S}}_{E,xx}[k] = \left|\underline{X}[k]\right|^2 + \underline{\mathrm{S}}_{E,xx}[k] = \frac{T_S^2}{N} \left|\underline{X}[k]\right|^2 \end{equation} -\todo{PSD} +\subsubsection{Power Spectral Density} -%\subsection{Noise} +The signal power is: +\begin{equation} + P = \frac{T_S^2}{T} \sum\limits_{n=0}^{N-1} \left|\underline{x}[n]\right|^2 \stackrel{!}{=} \sum\limits_{n=0}^{N-1} \underline{\mathrm{S}}_{xx}[k] +\end{equation} +Where $T = N T_S$ is the duration of the signal. + +Using the Parseval's theorem: +\begin{equation} + \underline{\mathrm{S}}_{xx}[k] = \underline{\mathrm{S}}_{P,xx}[k] = \frac{T_S}{N} \sum\limits_{n=0}^{N-1} \left|\underline{X}[k]\right|^2 +\end{equation} \section{Digital Signals and Systems} @@ -1928,7 +2076,8 @@ For example if the \ac{PCM} value is $\left(10010100\right)_2$, its real value i The \index{least significant bit} \textbf{\ac{LSB}} is the digit with the exponent $2^0$. \end{excursus} -\todo{Plot error} +%TODO +%\todo{Plot error} The quantization error superimposes the original signal and is therefore noise -- the \index{quantization noise} \textbf{quantization noise}. |