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R2653 epfl
2_intro.tex
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\documentclass
[aspectratio=169]
{
beamer
}
\usepackage
{
../styles/com303
}
%\setbeameroption{show only notes}\def\logoEPFL{}
\setbeameroption
{
show notes
}
\begin
{
document
}
\intertitle
{
(digital) signal processing for communications
}
%\begin{frame} \frametitle{The three main chapters}
%
% \begin{enumerate}
% \item \textbf{signal}: analyze the information contained in things we call ``signals''
% \item \textbf{processing}: modify the information
% \item \textbf{communications}: craft signals to carry the information we want
% \end{enumerate}
%
% \vspace{2em}
% \centering
%
% let's start with a demo for each of the three chapters
%\end{frame}
%
%
%\intertitle{(digital) signal processing for communications}
%
%\begin{frame} \frametitle{Signal}
% \begin{center}
% Description of the evolution of a physical phenomenon
% \end{center}
%
% \pause
% \begin{itemize}
% \item temperature (weather)
% \item pressure (sound)
% \item magnetic deviation (recorded sound)
% \item gray level on paper (photograph)
% \item ...
% \end{itemize}
%\end{frame}
%
%
%
%\begin{frame} \frametitle{Processing}
% \begin{center}
% \textbf{Analysis}: \textit{understanding} the information carried by the signal
%
% \vspace{2em}
%
% \textbf{Synthesis}: \textit{creating} a signal to contain the given information
% \end{center}
%\end{frame}
%
%
%
%\begin{frame} \frametitle{Communications}
% \begin{center}
% \textbf{Reception}: \textit{analysis} of an incoming signal
%
% \vspace{2em}
%
% \textbf{Transmission}: \textit{syntesis} of an outgoing signal
% \end{center}
%\end{frame}
%
%
%
%\begin{frame} \frametitle{Signal Models}
% \begin{center}
% \only<1>{Description of the evolution of a physical phenomenon}
%
% \only<2,4>{
% \includegraphics[height=4cm]{rc.eps}
%
% \vspace{0.5cm}
%
% $v(t) = V_0(1-e^{-\frac{t}{RC}})$ \\
% \vspace{1ex}
% }
% \only<3>{$f : \mathbb{R} \rightarrow \mathbb{R}$}
% \only<4>{only 2 degrees of freedom: $R, C$}
% \end{center}
%\end{frame}
%
%
%%\begin{frame} \frametitle{Signal Models}
%% \begin{center}
%% \only<1>{Description of the evolution of a physical phenomenon}
%%
%% \only<2,4>{
%% \psset{xunit=1.5cm,yunit=0.8cm}
%% \begin{pspicture}(-.5,-.5)(4.5,3.5)
%% \psline{->}(0,-.1)(0,3.4)
%% \psline{->}(-.1,0)(4.4,0)
%% \psplot[linecolor=blue,plotpoints=50]{0}{4}{x 2 sub dup mul -1 mul 4 add 0.7 mul}
%% \psplot[linecolor=blue,plotstyle=dots,plotpoints=10]{0}{4}{x 2 sub dup mul -1 mul 4 add 0.7 mul}
%% \end{pspicture}
%%
%% \vspace{0.5cm}
%%
%% $\vec{x}(t) = \vec{v}_0 t + (1/2)\vec{g}\,t^2$ \\
%% \vspace{1ex}
%% }
%% \only<2>{Galileo, 1638}
%% \only<3>{$f : \mathbb{R} \rightarrow \mathbb{R}^2$}
%% \only<4>{only 2 degrees of freedom: $\vec{v}_0$}
%% \end{center}
%%\end{frame}
%
%
%\begin{frame} \frametitle{What about ``interesting'' signals?}
% \begin{center}
% \includegraphics[width=8.4cm,height=2.8cm]{bbc.eps}
%
% \vspace{2em}
% {\Large $f(t) = ? $}
% \end{center}
%\end{frame}
%
%
%
%\begin{frame} \frametitle{The Digital Model}
% Key ingredients:\\
% \begin{itemize}
% \item discrete time
% \item discrete amplitude
% \end{itemize}
%\end{frame}
%
%
%\begin{frame} \frametitle{From analog...}
% \begin{center}
% \includegraphics[width=8.4cm,height=2.8cm]{bbc.eps}
% \end{center}
%\end{frame}
%
%
%\begin{frame} \frametitle{... to digital}
% \begin{center}
% \begin{minipage}{0.6\textwidth}
% \ttfamily
% ... 74 31 -66 9 -123 33 159 -26
% 102 148 86 -136 -179 70 72 -84 -113
% -42 -88 88 8 -180 -7 -133 8 164
% -4 108 35 -82 74 -49 52 32 -31 ...
% \end{minipage}
% \end{center}
%\end{frame}
%
%
%%% Temperature chart %%
%\def\tempFun{x 31 div 180 mul dup 1.1 mul cos 1 add 2 div exch 4 mul sin 0.1 mul add 10 mul x 4 div add}
%\begin{frame}[plain]
% \begin{columns}[c]
% \begin{column}{\paperwidth}
% \begin{overpic}[width=\paperwidth,height=\paperheight]{weather.eps}
% \only<2->{
% \put(90,50){%
% \begin{dspPlot}[yticks=5,xticks=none,sidegap=1,width=10cm]{1,31}{0,16}
% \moocStyle
% \psclip{\psframe[linewidth=0](0,0)(32,16)}
% \psgrid[gridlabels=0,xunit=5,yunit=5](0,0)(0,0)(32, 16)
% \psplot[showpoints=true, dotstyle=triangle*,plotstyle=dots, dotsize=\dspDotSize,%
% plotpoints=31]{1}{31}{\tempFun}
% \uput[180]{0}(2,14){$ ^{\circ}$C}
% \endpsclip
% \end{dspPlot}}}
% \end{overpic}
% \par
% \end{column}
% \end{columns}
%\end{frame}
%
%
%
%\begin{frame} \frametitle{Discretizing Time}
% \centering
% What is time?
%
%% \vspace{2em}
%% \only<2>{
%% \includegraphics[width=13em]{kant.eps}
%%
%% Immanuel Kant
%% }
%\end{frame}
%
%
%
%\begin{frame} \frametitle{A very old philosophical problem}
% \centering
% \includegraphics[width=15em]{zeno.eps}
%
% Zeno of Elea
%
%\end{frame}
%
%
%\setlength{\unitlength}{10mm}
%\def\zenotick#1#2{\psline(#1,0.8)(#1,1.2)\rput[t]{0}(#1,0.7){{\color{red} \bf #2}}}
%
%\begin{frame} \frametitle{The Dichotomy Paradox}
% \begin{pspicture}
% \put(2,0){
% \psset{unit=1cm}
% \psset{linecolor=red,linewidth=2pt}
% \psline(1,1)(10,1)
% \zenotick{1}{A}
% \zenotick{10}{B}
% \only<2->{\zenotick{5.5}{C}}
% \only<3->{\zenotick{7.75}{D}}
% \only<4->{\zenotick{8.875}{E}}
% \only<5->{\zenotick{9.4375}{F}\zenotick{9.71875}{G}}
% }
% \only<6->{
% \put(6.5,-2){\Huge $\displaystyle \sum_{n=1}^{\infty}\frac{1}{2^n} = 1$}
% }
% \end{pspicture}
%\end{frame}
%
%
%
%%% Trajectory of a ball & velocity
%\begin{frame}
% \frametitle{Calculus: from experiment to idealized abstraction}
% \begin{center}
% \vspace{3em}
% \psset{xunit=1.5cm,yunit=0.8cm}
% \begin{pspicture}(-.5,-.5)(4.5,3.5)
% \psline{->}(0,-.1)(0,3.4)
% \psline{->}(-.1,0)(4.4,0)
% \only<1-2>{\psplot[linecolor=blue,plotstyle=dots,plotpoints=10]{0}{4}{x 2 sub dup mul -1 mul 4 add 0.7 mul}}
% \only<2-3>{\psplot[linecolor=blue,plotpoints=50]{0}{4}{x 2 sub dup mul -1 mul 4 add 0.7 mul}}
% \end{pspicture}
%
% \vspace{0.5cm}
%
% \uncover<3->{
% $\vec{x}(t) = \vec{v}_0 t + (1/2)\vec{g}\,t^2$\\
% }
% \end{center}
%\end{frame}
\def\smooth
{
10 div 360 mul dup sin exch dup 2 mul sin exch dup mul 360 div sin add add
}
\def\interpolant
#1
{
\dspSinc
{
#1
}{
1
}
#1
\smooth
mul
}
\begin
{
frame
}
\frametitle
{
From continuous time to discrete time
}
\begin
{
center
}
\begin
{
figure
}
\begin
{
dspPlot
}
[yticks=none,xticks=none,sidegap=0]
{
0,10
}{
-3,3
}
\moocStyle
\only
<1-2>
{
\dspFunc
[linecolor=blue!50]
{
x
\smooth
}}
\only
<2->
{
\dspSignal
{
x
\smooth
}}
\end
{
dspPlot
}
\end
{
figure
}
\only
<1>
{
$
x
(
t
)
$
}
\only
<3>
{
$
x
[
n
]
$
}
\vphantom
{
$
x
[
n
]
x
(
t
)
$
}
\end
{
center
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
The two ``languages'' of Signal Processing
}
\begin
{
center
}
\Large
\begin
{
tabular
}{
cc
}
continuous time
&
$
x :
\mathbb
{R}
\rightarrow
\mathbb
{R}
$
\\
[3em]
discrete time
&
$
x :
\mathbb
{Z}
\rightarrow
\mathbb
{R}
$
\end
{
tabular
}
\end
{
center
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Discrete-time signals are just arrays of numbers
}
\begin
{
center
}
{
\Large
$
x
[
n
]
=
\ldots
,
1
.
2390
,
-
0
.
7372
,
0
.
8987
,
0
.
1798
,
-
1
.
1501
,
-
0
.
2642
\ldots
$
}
\vspace
{
4em
}
\textit
{
(one could say that discrete-time signal processing is ``data-driven'')
}
\end
{
center
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Remember Galileo?
}
\begin
{
center
}
\vspace
{
3em
}
\psset
{
xunit=1.5cm,yunit=0.8cm
}
\begin
{
pspicture
}
(-.5,-.5)(4.5,3.5)
\psline
{
->
}
(0,-.1)(0,3.4)
\psline
{
->
}
(-.1,0)(4.4,0)
\psplot
[linecolor=blue,plotstyle=dots,plotpoints=10]
{
0
}{
4
}{
x 2 sub dup mul -1 mul 4 add 0.7 mul
}
\end
{
pspicture
}
\vspace
{
0.5cm
}
$
\vec
{x}
[
n
]
=
(
0
,
0
)
,
(
1
,
3
.
7
)
,
(
2
,
5
.
9
)
,
\ldots
$
\end
{
center
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Are we losing information?
}
\begin
{
center
}
\begin
{
dspBlocks
}{
2
}{
0.4
}
$
x
(
t
)
$
~
&
\raisebox
{
-1.4em
}{
\psframebox
[linewidth=1.5pt]
{
%
\psset
{
xunit=1em,yunit=1em,linewidth=1.8pt
}
%
\pspicture
(-3,-1.8)(2,1.8)
%
\psline
(-2.8,0)(-1.6,0)(1.2,1.4)
\psline
(1.1,0)(1.8,0)
\psarc
[linewidth=1pt]
{
<-
}
(-1.6,0)
{
2em
}{
-10
}{
55
}
\endpspicture
}}
&
$
x
[
n
]
$
\\
&
$
T_s
$
\psset
{
linewidth=1.5pt
}
\ncline
{
->
}{
1,1
}{
1,2
}
\ncline
{
->
}{
1,2
}{
1,3
}
\end
{
dspBlocks
}
\end
{
center
}
\end
{
frame
}
%\begin{frame} \frametitle{Are we burning bridges?}
% \begin{center}
% \includegraphics[height=4cm]{rc.eps}
%
% \vspace{1em}
%
% $x[n] = ?$
% \end{center}
%\end{frame}
\begin
{
frame
}
\frametitle
{
Translating between languages: the founding fathers
}
\begin
{
center
}
\includegraphics
[height=0.6\paperheight]
{
nyquistshannon.eps
}
\vspace
{
1ex
}
Harry Nyquist and Claude Shannon
\end
{
center
}
\end
{
frame
}
%% Sampling theorem
\begin
{
frame
}
\frametitle
{
The Sampling Theorem
}
\centering
Under appropriate ``slowness'' conditions for
$
x
(
t
)
$
we have:
\[
x
(
t
)
%= \sum_{n = -\infty}^{\infty} x[n] \frac{\sin(\pi(t - nT_s)/T_s)}{\pi(t-nT_s)/T_s} \\ \pause
=
\sum
_{n
=
-
\infty
}^{
\infty
} x
[
n
]
\sinc\left
(
\frac
{t
-
nT_s}{T_s}
\right
)
\]
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
The sinc function
}
\begin
{
center
}
\begin
{
figure
}
\begin
{
dspPlot
}
[yticks=none,xticks=none,sidegap=0]
{
0,10
}{
-3,3
}
\moocStyle
\dspFunc
[linecolor=darkgreen!60]
{
x
\dspSinc
{
5
}{
1
}
2 mul
}
\end
{
dspPlot
}
\end
{
figure
}
\end
{
center
}
\end
{
frame
}
\def\smooth
{
10 div 360 mul dup sin exch dup 2 mul sin exch dup mul 360 div sin add add
}
\def\interpolant
#1
{
\dspSinc
{
#1
}{
1
}
#1
\smooth
mul
}
\begin
{
frame
}
\frametitle
{
From continuous to discrete time and back!
}
\begin
{
center
}
\begin
{
figure
}
\begin
{
dspPlot
}
[yticks=none,xticks=none,sidegap=0]
{
0,10
}{
-3,3
}
\moocStyle
\only
<1-2>
{
\dspFunc
[linecolor=blue!50]
{
x
\smooth
}}
\only
<2->
{
\dspSignal
{
x
\smooth
}}
\only
<4->
{
\dspFunc
[linecolor=darkgreen!60,linewidth=0.3pt]
{
x
\interpolant
{
2
}
}}
\only
<5->
{
\dspFunc
[linecolor=darkgreen!60,linewidth=0.3pt]
{
x
\interpolant
{
3
}
}}
\only
<6->
{
\dspFunc
[linecolor=darkgreen!60,linewidth=0.3pt]
{
x
\interpolant
{
4
}
}}
\only
<7->
{
\dspFunc
[linecolor=darkgreen!60,linewidth=0.3pt]
{
x
\interpolant
{
5
}
}}
\only
<8->
{
\dspFunc
[linecolor=darkgreen!60,linewidth=0.3pt]
{
x
\interpolant
{
0
}
}
\dspFunc
[linecolor=darkgreen!60,linewidth=0.3pt]
{
x
\interpolant
{
1
}
}
\dspFunc
[linecolor=darkgreen!60,linewidth=0.3pt]
{
x
\interpolant
{
6
}
}
\dspFunc
[linecolor=darkgreen!60,linewidth=0.3pt]
{
x
\interpolant
{
7
}
}
\dspFunc
[linecolor=darkgreen!60,linewidth=0.3pt]
{
x
\interpolant
{
8
}
}
\dspFunc
[linecolor=darkgreen!60,linewidth=0.3pt]
{
x
\interpolant
{
9
}
}
\dspFunc
[linecolor=darkgreen!60,linewidth=0.3pt]
{
x
\interpolant
{
10
}
}}
\only
<9->
{
\dspFunc
[linecolor=blue!50]
{
x
\smooth
}}
\end
{
dspPlot
}
\end
{
figure
}
\only
<1>
{
$
x
(
t
)
$
}
\only
<3>
{
$
x
[
n
]
$
}
\vphantom
{
$
x
[
n
]
x
(
t
)
$
}
\end
{
center
}
\end
{
frame
}
%\begin{frame} \frametitle{When can we do all this? Ask Fourier!}
% \begin{center}
% \includegraphics[height=0.6\paperheight]{fourier.eps}
% \end{center}
%\end{frame}
%% Mean value, CT & DT
\savedata
{
\smooth
}
[
1.0000 1.0000 1.1579 1.0011 1.3158 1.0130 1.4737 1.0484
1.6316 1.1201 1.7895 1.2408 1.9474 1.4232 2.1053 1.6777
2.2632 1.9852 2.4211 2.3076 2.5789 2.6068 2.7368 2.8445
2.8947 2.9824 3.0526 2.9828 3.2105 2.8418 3.3684 2.6079
3.5263 2.3338 3.6842 2.0725 3.8421 1.8769 4.0000 1.8000]
\begin
{
frame
}
\frametitle
{
By the way, discrete time is easy!
}
\setbeamercovered
{
invisible
}
\begin
{
center
}
\vspace
{
2em
}
\psset
{
xunit=1.5cm,yunit=0.8cm
}
\begin
{
pspicture
}
(-.5,-.5)(5.5,3.5)
\psline
{
->
}
(0,-.1)(0,3.4)
\psline
{
->
}
(-.1,0)(5.4,0)
\rput
[t]
(1,-.3)
{
$
a
$
}
\rput
[t]
(4,-.3)
{
$
b
$
}
\only
<1,2,5->
{
\dataplot
[linecolor=blue,plotstyle=dots]
{
\smooth
}}
\only
<2-4>
{
\dataplot
[linecolor=blue,plotstyle=curve]
{
\smooth
}}
\onslide
<4,6>
{
\psline
[linecolor=red,linestyle=dotted]
{
-
}
(1,1.983)(4,1.983)
}
\end
{
pspicture
}
\vspace
{
1em
}
\onslide
<4>
{
$
\displaystyle
\bar
{x}
=
\frac
{
1
}{b
-
a}
\int
_a^b f
(
t
)
dt
$
}
\onslide
<6>
{
$
\displaystyle
\bar
{x}
=
\frac
{
1
}{N}
\sum
_{n
=
0
}^{N
-
1
} x
[
n
]
$
}
\end
{
center
}
\end
{
frame
}
\end
{
document
}
\begin
{
frame
}
\frametitle
{
Digital Signals
}
\begin
{
center
}
(digital) signal processing for communications
\end
{
center
}
\vspace
{
2em
}
Key ingredients:
\\
\begin
{
itemize
}
\item
discrete time
\item
{
\bf
discrete amplitude
}
\end
{
itemize
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
The Digital Model
}
\begin
{
center
}
\Large
$
x :
\mathbb
{Z}
\rightarrow
\mathbb
{Z}
$
\vspace
{
2em
}
$
x
[
n
]
=
\ldots
,
123
,
-
73
,
89
,
17
,
-
11
,
-
26
,
\ldots
$
\end
{
center
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Digital Signals
}
\begin
{
center
}
\begin
{
figure
}
\begin
{
dspPlot
}
[yticks=none,xticks=none,sidegap=2]
{
0,32
}{
-1.2,1.2
}
\moocStyle
\only
<1-2>
{
\dspFunc
[xmin=0,xmax=32]
{
x 32 div 360 mul sin
}}
\only
<2->
{
\multido
{
\n
=0+1
}{
33
}{
\psline
[linecolor=lightgray,linewidth=.4pt]
(
\n
,-1.2)(
\n
,1.2)
}}
\only
<3-4>
{
\dspSignal
{
x 32 div 360 mul sin
}}
\only
<4->
{
\multido
{
\n
=-1+0.25
}{
9
}{
\psline
[linecolor=lightgray,linewidth=.4pt]
(-2,
\n
)(34,
\n
)
}}
\only
<5>
{
\dspSignal
{
x 32 div 360 mul sin 4 mul cvi 4 div
}}
\end
{
dspPlot
}
\end
{
figure
}
\only
<1-2>
{
$
x
(
t
)
$
}
\only
<3-4>
{
$
x
[
n
]
$
}
\only
<5->
{
$
\hat
{x}
[
n
]
$
}
\end
{
center
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Digital amplitude
}
Why it is important:
\\
\begin
{
itemize
}
\item
storage
\item
processing
\item
transmission
\end
{
itemize
}
\end
{
frame
}
%% Storage
\begin
{
frame
}
\frametitle
{
Data storage
}
\begin
{
center
}
{
\color
{
red
}
Analog storage:
}
\\
\uncover
<2->
{
paper, wax cylinders, reel-to-reel, vinyl, compact cassette, VHS, Betamax, silver plates, Kodachrome, Super8, 8-Track, microfilm, ...
}
\vspace
{
2em
}
\uncover
<3->
{{
\color
{
red
}
Digital storage:
}}
\\
\uncover
<4->
{
\Large
\{
0, 1
\}
}
\end
{
center
}
\end
{
frame
}
%%
\begin
{
frame
}
[plain]
\begin
{
columns
}
[c]
\begin
{
column
}{
\paperwidth
}
\includegraphics
[width=\paperwidth,height=\paperheight]
{
25-years-of-storage.eps
}
\par
\end
{
column
}
\end
{
columns
}
\end
{
frame
}
%% Processing
\begin
{
frame
}
\frametitle
{
Processing
}
\begin
{
center
}
\includegraphics
[width=\textwidth,height=0.7\textheight]
{
processing.eps
}
\end
{
center
}
\end
{
frame
}
\end
{
document
}
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