diff options
| -rw-r--r-- | .gitignore | 1 | ||||
| -rw-r--r-- | doc/thesis/Fading.tex | 6 | ||||
| -rw-r--r-- | doc/thesis/Makefile | 14 | ||||
| -rw-r--r-- | doc/thesis/chapters/theory.tex | 15 | ||||
| -rw-r--r-- | doc/thesis/figures/data/multipath_frequency_response.dat | 300 | ||||
| -rw-r--r-- | doc/thesis/figures/data/multipath_frequency_response.py | 35 | ||||
| -rw-r--r-- | doc/thesis/figures/tikz/multipath-frequency-response-plots.tex | 70 | ||||
| -rw-r--r-- | references/books/mit_ocw_book_6.pdf | bin | 0 -> 653430 bytes |
8 files changed, 435 insertions, 6 deletions
@@ -1,3 +1,4 @@ +*.figlist # Created by https://www.toptal.com/developers/gitignore/api/windows,linux,latex,python # Edit at https://www.toptal.com/developers/gitignore?templates=windows,linux,latex,python diff --git a/doc/thesis/Fading.tex b/doc/thesis/Fading.tex index b80b9e7..1844ceb 100644 --- a/doc/thesis/Fading.tex +++ b/doc/thesis/Fading.tex @@ -30,14 +30,20 @@ \usepackage{tikz} % Pretty drawings \usepackage{tikz-3dplot} % More dimensions! +\pgfplotsset{compat=1.17} \usetikzlibrary{ + external, calc, positioning, backgrounds, decorations.pathreplacing, + decorations.markings, matrix, arrows, } +% \tikzexternalize[ + % mode=graphics if exists, + % figure list=true] \pgfdeclarelayer{background} \pgfdeclarelayer{foreground} \pgfsetlayers{background,main,foreground} diff --git a/doc/thesis/Makefile b/doc/thesis/Makefile index d5f02ab..167ad29 100644 --- a/doc/thesis/Makefile +++ b/doc/thesis/Makefile @@ -1,5 +1,5 @@ TEX := xelatex -TEXARGS := -halt-on-error -interaction=nonstopmode +TEXARGS := -halt-on-error -interaction=nonstopmode -shell-escape=1 SOURCES := \ Fading.tex \ @@ -20,7 +20,10 @@ SOURCES := \ figures/tikz/qam-modulator.tex \ figures/tikz/qam-constellation.tex \ figures/tikz/multipath-sketch.tex \ - figures/tikz/multipath-impulse-response.tex + figures/tikz/multipath-impulse-response.tex \ + figures/tikz/multipath-frequency-response-plots.tex + +DATA := figures/data/multipath_frequency_response.dat # Get the main file from the file MAIN := $(shell sed -ne 's/^.*\!TeX root =\(.*\)$$/\1/ p' $(SOURCES)) @@ -34,7 +37,10 @@ endif .PHONY: all all: $(PDF) -%.pdf: %.tex $(SOURCES) +%.dat: %.py + /usr/bin/env python3 $< + +%.pdf: %.tex $(SOURCES) $(DATA) $(TEX) $(TEXARGS) $< # TODO: makeindex biber $(basename $(MAIN)) @@ -42,7 +48,7 @@ all: $(PDF) .PHONY: clean cleanall clean: - @rm -vf *.aux *.aue *.lof *.log *.lot *.fls *.out *.toc *.fmt *.fot *.cb *.cb2 .*.lb *.dvi *.xdv *-converted-to.* *.bbl *.bcf *.blg *-blx.aux *-blx.bib *.run.xml *.fdb_latexmk *.synctex *.synctex\(busy\) *.synctex.gz *.synctex.gz\(busy\) *.pdfsync *.alg *.loa acs-*.bib *.thm *.nav *.pre *.snm *.vrb *.soc *.cut *.cpt *.spl *.ent *.lox *.mf *.acn *.acr *.glg *.glo *.gls *.glsdefs *.lzo *.lzs *.lol + @rm -vf *.aux *.aue *.lof *.log *.lot *.fls *.out *.toc *.fmt *.fot *.cb *.cb2 .*.lb *.dvi *.xdv *-converted-to.* *.bbl *.bcf *.blg *-blx.aux *-blx.bib *.run.xml *.fdb_latexmk *.synctex *.synctex\(busy\) *.synctex.gz *.synctex.gz\(busy\) *.pdfsync *.alg *.loa acs-*.bib *.thm *.nav *.pre *.snm *.vrb *.soc *.cut *.cpt *.spl *.ent *.lox *.mf *.acn *.acr *.glg *.glo *.gls *.glsdefs *.lzo *.lzs *.lol *.auxlock *.figlist @rm -vf tex/*.aux @rm -vf chapters/*.aux diff --git a/doc/thesis/chapters/theory.tex b/doc/thesis/chapters/theory.tex index 2c4687b..2506333 100644 --- a/doc/thesis/chapters/theory.tex +++ b/doc/thesis/chapters/theory.tex @@ -193,13 +193,24 @@ that describes the \emph{impulse response} of the channel. This function is depe \subsection{Spectrum of a multipath fading channel} -With a continuous time channel model we can now discuss the spectral properties of a fading channel, since the frequency response is the Fourier transform of the impulse response, mathematically \(H(f, t) = \fourier h(\tau, t)\). In this case however \(h(\tau, t)\) depends on two time variables, but that is actually not an issue, it just means that the frequency response is also changing with time. Hence we perform the Fourier transform with respect to the channel (convolution) time variable \(\tau\) to obtain +With a continuous time channel model we can now discuss the spectral properties of a fading channel since the frequency response is the Fourier transform of the impulse response, mathematically \(H(f, t) = \fourier h(\tau, t)\). In this case however \(h(\tau, t)\) depends on two time variables, but that is actually not an issue, it just means that the frequency response is also changing with time. Hence we perform the Fourier transform with respect to the channel (convolution) time variable \(\tau\) to obtain \begin{equation} \label{eqn:multipath-frequency-response} H(f, t) = \int_\mathbb{R} \sum_k c_k(t) \delta(\tau - \tau_k(t)) e^{-2\pi jf\tau} \, d\tau = \sum_k c_k(t) e^{-2\pi jf \tau_k(t)}. \end{equation} -Equation \eqref{eqn:multipath-frequency-response} indicates that the frequency response is has a periodic (complex) sinusoidal shape, which has some important implications. A series of plots of the magnitude of the frequency response is shown in figure \ref{fig:multipath-frequency-response-plots}. +Equation \eqref{eqn:multipath-frequency-response} indicates that the frequency response is a periodic complex exponential, which has some important implications. Notice that if there is only one tap (term), the magnitude of \(H(f, t)\) is a constant (with respect to \(f\)) since \(|e^{j\alpha f}| = 1\). This means that the channels attenuates all frequencies by the same amount, therefore it is said to be a \emph{frequency non-selective} channel. Whereas in the case when there is more than one tap, the taps interfere destructively at certain frequencies and the channel is called \emph{frequency selective}. Plots of the frequency response of a two tap channel model are shown in figure \ref{fig:multipath-frequency-response-plots}. On the left is the magnitude of \(H(f, t)\), which presents periodic ``dips'', and on the right complex loci for the two taps (red and blue), as well as their sum (magenta), over the frequency range near the first dip (2 to 2.5 MHz) are shown. + +\begin{figure} + \centering + \resizebox{\linewidth}{!}{ + \input{figures/tikz/multipath-frequency-response-plots} + } + \caption{ + Frequency response of a multipath fading channel. + \label{fig:multipath-frequency-response-plots} + } +\end{figure} \subsection{Discrete-time model} diff --git a/doc/thesis/figures/data/multipath_frequency_response.dat b/doc/thesis/figures/data/multipath_frequency_response.dat new file mode 100644 index 0000000..bb10780 --- /dev/null +++ b/doc/thesis/figures/data/multipath_frequency_response.dat @@ -0,0 +1,300 @@ +1.000000e+05 1.000000e-05 1.996053e-05 1.000000e+00 -1.133108e-15 -8.090170e-01 -5.877853e-01 1.909830e-01 -5.877853e-01 +1.031283e+05 9.696658e-06 1.935262e-05 9.999862e-01 5.253475e-03 -8.071602e-01 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-9.999801e-01 4.202648e-03 -3.532661e-05 +9.696658e+08 1.031283e-09 2.017258e-09 5.253475e-03 9.999862e-01 -3.152094e-03 -9.999950e-01 2.101381e-03 -8.831734e-06 +1.000000e+09 1.000000e-09 2.000000e-09 3.061617e-16 1.000000e+00 -1.836970e-16 -1.000000e+00 1.224647e-16 0.000000e+00 diff --git a/doc/thesis/figures/data/multipath_frequency_response.py b/doc/thesis/figures/data/multipath_frequency_response.py new file mode 100644 index 0000000..e5207ae --- /dev/null +++ b/doc/thesis/figures/data/multipath_frequency_response.py @@ -0,0 +1,35 @@ +import sys, os +import numpy as np + +nsamples = 300 + +np.seterr(over='raise') + +def tap(c, tau, f): + return np.exp(2j * np.pi * f * tau) + + +# attenuations for frequency plot +f_whole = np.logspace(5, 9, num=nsamples) +linear_whole = 1 / f_whole +multipath_whole = tap(c=.8, tau=500e-9, f=f_whole) + tap(c=.4, tau=300e-9, f=f_whole) +channel_whole = linear_whole * multipath_whole + +# coordinates for complex diagram +f_tap = np.linspace(2e6, 2.5e6, num=nsamples) +linear_tap = 1 / f_tap + +multipath_tap_1 = tap(c=.8, tau=500e-9, f=f_tap) +multipath_tap_2 = tap(c=.4, tau=300e-9, f=f_tap) +multipath_taps = multipath_tap_1 + multipath_tap_2 + +data = np.array(list(zip(f_whole, linear_whole, np.abs(channel_whole), + np.real(multipath_tap_1), np.imag(multipath_tap_1), + np.real(multipath_tap_2), np.imag(multipath_tap_2), + np.real(multipath_taps), np.imag(multipath_taps)))) + +# save to file +location = os.path.realpath(os.path.join(os.getcwd(), os.path.dirname(__file__))) +name, _ = os.path.splitext(os.path.basename(__file__)) +filename = os.path.join(location, name + ".dat") +np.savetxt(filename, data, fmt='%.6e') diff --git a/doc/thesis/figures/tikz/multipath-frequency-response-plots.tex b/doc/thesis/figures/tikz/multipath-frequency-response-plots.tex new file mode 100644 index 0000000..094e33a --- /dev/null +++ b/doc/thesis/figures/tikz/multipath-frequency-response-plots.tex @@ -0,0 +1,70 @@ +% vim: set ts=2 sw=2 noet: +\begin{tikzpicture} + \begin{loglogaxis}[ + width = .6\linewidth, height = 5cm, + ylabel = {Response \(|H(f, t)|\)}, + xlabel = {Frequency \(f\)/Hz}, + xlabel near ticks, + ylabel near ticks, + ytick = \empty, + smooth, + ] + + \addplot[solid, magenta] table[x index = 0, y index = 2] + {figures/data/multipath_frequency_response.dat}; + \addlegendentry{Multipath} + + \addplot[dashed, thick, black] table[x index = 0, y index = 1] + {figures/data/multipath_frequency_response.dat}; + \addlegendentry{Linear} + + \end{loglogaxis} +\end{tikzpicture} +\hskip 5mm +\begin{tikzpicture}[ + decorated/.style = { + solid, thick, + postaction={decorate}, + decoration={markings, + mark=at position 0.35 with {\arrow{stealth}}, + mark=at position 0.65 with {\arrow{stealth}}}, + }, + ] + \begin{axis}[ + width = 5cm, height = 5cm, + ylabel = {\(\Im{H(f,t)}\)}, + xlabel = {\(\Re{H(f,t)}\)}, + xlabel near ticks, + ylabel near ticks, + grid = major, + xmin = -1.2, xmax = 1.2, + ymin = -1.2, ymax = 1.2, + ] + + \addplot[decorated, red] table[x index = 3, y index = 4] + {figures/data/multipath_frequency_response.dat} + node[pos = 0, circle, fill = white, draw, inner sep = 1pt] {} + node[pos = .2, outer sep = 1pt, inner sep = 0pt] (A) {} + node[pos = 1, circle, fill = white, draw, inner sep = 1pt] {}; + + \addplot[decorated, blue] table[x index = 5, y index = 6] + {figures/data/multipath_frequency_response.dat} + node[pos = 0, circle, fill = white, draw, inner sep = 1pt] {} + node[pos = .2, outer sep = 1pt, inner sep = 0pt] (B) {} + node[pos = 1, circle, fill = white, draw, inner sep = 1pt] {}; + + \addplot[decorated, magenta] table[x index = 7, y index = 8] + {figures/data/multipath_frequency_response.dat} + node[pos = 0, circle, fill = white, draw, inner sep = 1pt] {} + node[pos = 0, below, font = \tiny] {2 MHz} + node[pos = .2, outer sep = 1pt, inner sep = 0pt] (C) {} + node[pos = 1, circle, fill = white, draw, inner sep = 1pt] {} + node[pos = 1, above left, font = \tiny] {2.5 MHz}; + + \node[outer sep = 2pt, inner sep = 0pt] (O) at (0,0) {}; + \draw[-latex, red!50!white] (O) -- (A); + \draw[-latex, blue!50!white] (O) -- (B); + \draw[-latex, magenta!50!white] (O) -- (C); + + \end{axis} +\end{tikzpicture} diff --git a/references/books/mit_ocw_book_6.pdf b/references/books/mit_ocw_book_6.pdf Binary files differnew file mode 100644 index 0000000..bfb6457 --- /dev/null +++ b/references/books/mit_ocw_book_6.pdf |