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1_analogworld.tex
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1_analogworld.tex
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\documentclass
[aspectratio=169]
{
beamer
}
\def\stylepath
{
../styles
}
\usepackage
{
\stylepath
/com303
}
\begin
{
document
}
\begin
{
frame
}
\frametitle
{
Two views of the world
}
\begin
{
columns
}
\begin
{
column
}{
0.5
\paperwidth
}
\centering
\includegraphics
[width=0.4\paperwidth]
{
Picture
_
world
_
analog.eps
}
\end
{
column
}
\begin
{
column
}{
0.5
\paperwidth
}
\centering
\includegraphics
[width=0.4\paperwidth]
{
Picture
_
world
_
digital.eps
}
\end
{
column
}
\end
{
columns
}
\vspace
{
2em
}
\centering
Analog/continuous versus discrete/digital
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Two views of the world
}
\begin
{
columns
}
\begin
{
column
}{
0.3
\paperwidth
}
digital worldview:
\vspace
{
1em
}
\begin
{
itemize
}
\item
arithmetic
\item
combinatorics
\item
computer science
\item
DSP
\end
{
itemize
}
\end
{
column
}
\begin
{
column
}{
0.3
\paperwidth
}
analog worldview:
\vspace
{
1em
}
\begin
{
itemize
}
\item
calculus
\item
distributions
\item
system theory
\item
electronics
\end
{
itemize
}
\end
{
column
}
\end
{
columns
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Two views of the world
}
\begin
{
columns
}
\begin
{
column
}{
0.4
\paperwidth
}
digital worldview:
\vspace
{
1em
}
\begin
{
itemize
}
\item
countable integer index
$
n
$
\item
sequences
$
x
[
n
]
\in
\ell
_
2
(
\mathbb
{Z}
)
$
\item
frequency
$
\omega
\in
[-
\pi
,
\pi
]
$
\item
DTFT:
$
\ell
_
2
(
\mathbb
{Z}
)
\mapsto
L_
2
([-
\pi
,
\pi
])
$
\end
{
itemize
}
\end
{
column
}
\begin
{
column
}{
0.4
\paperwidth
}
analog worldview:
\vspace
{
1em
}
\begin
{
itemize
}
\item
real-valued time
$
t
$
(sec)
\item
functions
$
x
(
t
)
\in
L_
2
(
\mathbb
{R}
)
$
\item
frequency
$
\Omega
\in
\mathbb
{R}
$
(rad/sec)
\item
FT:
$
L_
2
(
\mathbb
{R}
)
\mapsto
L_
2
(
\mathbb
{R}
)
$
\end
{
itemize
}
\end
{
column
}
\end
{
columns
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Bridging the gap
}
\setbeamercovered
{
invisible
}
\begin
{
center
}
\begin
{
figure
}
\begin
{
dspBlocks
}{
2
}{
1
}
$
x
[
n
]
~~
$
&
\BDfilter
{
sound card
}
&
\raisebox
{
-1.2em
}{
\includegraphics
[height=6em]
{
speaker.eps
}}
\\
&
[mnode=circle]
$
T_s
$
&
\hspace
{
-5cm
}
{
\em
system clock
}
\\
\ncline
{
->
}{
2,2
}{
1,2
}
\ncline
{
->
}{
1,1
}{
1,2
}
\ncline
{
->
}{
1,2
}{
1,3
}
\end
{
dspBlocks
}
\end
{
figure
}
\end
{
center
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Bridging the gap
}
\setbeamercovered
{
invisible
}
\begin
{
center
}
\begin
{
figure
}
\begin
{
dspBlocks
}{
2
}{
1
}
\raisebox
{
-1.5em
}{
\includegraphics
[height=5em]
{
mike.eps
}}
&
\BDfilter
{
sound card
}
&
~~
$
x
[
n
]
$
\\
&
[mnode=circle]
$
T_s
$
&
\hspace
{
-3cm
}
{
\em
system clock
}
\\
\ncline
{
->
}{
2,2
}{
1,2
}
\ncline
{
->
}{
1,1
}{
1,2
}
\ncline
{
->
}{
1,2
}{
1,3
}
\end
{
dspBlocks
}
\end
{
figure
}
\end
{
center
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Bridging the gap
}
\begin
{
figure
}
\center
\psset
{
unit=3.5cm
}
%
\begin
{
pspicture
}
(-1,-1)(1,1)
%
\uput
[0]
(0.6,0)
{
$
x
[
n
]
$
}
\uput
[90]
(0,0.6)
{
\color
{
darkred
}
sampling
}
\uput
[180]
(-0.6,0)
{
$
x
(
t
)
$
}
\uput
[270]
(0,-0.6)
{
\color
{
darkred
}
interpolation
}
\psarc
[linewidth=3pt,linecolor=gray]
{
<-
}
(0,0)
{
0.73
}{
10
}{
65
}
\psarc
[linewidth=3pt,linecolor=gray]
{
<-
}
(0,0)
{
0.73
}{
115
}{
170
}
\psarc
[linewidth=3pt,linecolor=gray]
{
<-
}
(0,0)
{
0.73
}{
190
}{
235
}
\psarc
[linewidth=3pt,linecolor=gray]
{
<-
}
(0,0)
{
0.73
}{
305
}{
350
}
\end
{
pspicture
}
\end
{
figure
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Digital processing of signals from/to the analog world
}
\begin
{
itemize
}
\item
input is continuous-time:
$
x
(
t
)
$
\item
output is continuous-time:
$
y
(
t
)
$
\item
processing is on sequences:
$
x
[
n
]
, y
[
n
]
$
\end
{
itemize
}
\vspace
{
1em
}
\centering
\begin
{
figure
}
\begin
{
dspBlocks
}{
1.5
}{
.5
}
analog world~~
&
\BDfilter
{
processing
}
&
~~analog world
\\
\ncline
{
->
}{
1,1
}{
1,2
}
\ncline
{
->
}{
1,2
}{
1,3
}
\end
{
dspBlocks
}
\end
{
figure
}
examples: MP3, digital photography
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Digital processing of signals to the analog world
}
\begin
{
itemize
}
\item
input is discrete-time:
$
x
[
n
]
$
\item
output is continuous-time:
$
y
(
t
)
$
\item
processing is on sequences:
$
x
[
n
]
, y
[
n
]
$
\end
{
itemize
}
\vspace
{
1em
}
\centering
\begin
{
figure
}
\begin
{
dspBlocks
}{
1.5
}{
.5
}
digital world~~
&
\BDfilterMulti
{
processing
\\
for synthesis
}
&
~~analog world
\\
\ncline
{
->
}{
1,1
}{
1,2
}
\ncline
{
->
}{
1,2
}{
1,3
}
\end
{
dspBlocks
}
\end
{
figure
}
examples: computer graphics, video games
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Digital processing of signals from the analog world
}
\begin
{
itemize
}
\item
input is continuous-time:
$
x
(
t
)
$
\item
output is discrete-time:
$
y
[
n
]
$
\item
processing is on sequences:
$
x
[
n
]
, y
[
n
]
$
\end
{
itemize
}
\vspace
{
1em
}
\centering
\begin
{
figure
}
\begin
{
dspBlocks
}{
1.5
}{
.5
}
analog world~~
&
\BDfilterMulti
{
processing
\\
for analysis
}
&
~~digital world
\\
\ncline
{
->
}{
1,1
}{
1,2
}
\ncline
{
->
}{
1,2
}{
1,3
}
\end
{
dspBlocks
}
\end
{
figure
}
examples: storage(compression), control systems, monitoring
\end
{
frame
}
\intertitle
{
continuous-time signal processing
}
\begin
{
frame
}
\frametitle
{
About continuous time
}
\begin
{
itemize
}
[<+->]
\item
time: real variable
$
t
$
\item
signal
$
x
(
t
)
$
: complex functions of a real variable
\item
finite energy:
$
x
(
t
)
\in
L_
2
(
\mathbb
{R}
)
$
\item
inner product in
$
L_
2
(
\mathbb
{R}
)
$
\[
\langle
x
(
t
)
, y
(
t
)
\rangle
=
\int
_{
-
\infty
}^{
\infty
} x^
*(
t
)
y
(
t
)
dt
\]
\item
energy:
$
||x
(
t
)
||^
2
=
\langle
x
(
t
)
, x
(
t
)
\rangle
$
\end
{
itemize
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Analog LTI filters
}
\begin
{
figure
}
\center
\begin
{
dspBlocks
}{
2.5
}{
1
}
$
x
(
t
)
$
~~
&
\rnode
{
F
}{
\BDfilter
{
$
\mathcal
{H}
$
}}
&
~~
$
y
(
t
)
$
\ncline
{
->
}{
1,1
}{
1,2
}
\ncline
{
->
}{
1,2
}{
1,3
}
\end
{
dspBlocks
}
\end
{
figure
}
\vspace
{
1em
}
\begin
{
align*
}
y(t)
&
= (x
\ast
h)(t)
\\
\pause
&
=
\int
_{
-
\infty
}^{
\infty
}
x(
\tau
)h(t-
\tau
) d
\tau
\\
\pause
&
=
\langle
h
^
*(t-
\tau
), x(
\tau
)
\rangle
\end
{
align*
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Fourier analysis
}
\begin
{
itemize
}
[<+->]
\item
in discrete time max angular frequency is
$
\pm\pi
$
\item
in continuous time no max frequency:
$
\Omega
\in
\mathbb
{R}
$
\item
concept is the same: similarity to sinusoidal components
\begin
{
align*
}
X(j
\Omega
)
&
=
\langle
e
^{
j
\Omega
t
}
, x(t)
\rangle
\\
&
=
\int
_{
-
\infty
}^{
\infty
}
x(t) e
^{
-j
\Omega
t
}
dt
\qquad
\leftarrow\mbox
{
\em
not periodic!
}
\\
\\
x(t)
&
=
\frac
{
1
}{
2
\pi
}
\int
_{
-
\infty
}^{
\infty
}
X(j
\Omega
) e
^{
j
\Omega
t
}
d
\Omega
\end
{
align*
}
\end
{
itemize
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Real-world frequency
}
\begin
{
itemize
}
\item
$
\Omega
$
expressed in rad/s
\item
$
F
=
\displaystyle
\frac
{
\Omega
}{
2
\pi
},
$
expressed in Hertz (
$
1
/
s
$
)
\item
period
$
T
=
\displaystyle
\frac
{
1
}{F}
=
\frac
{
2
\pi
}{
\Omega
}
$
\end
{
itemize
}
\end
{
frame
}
\def\s
{
10
}
\begin
{
frame
}
\frametitle
{
Example
}
\begin
{
center
}
$
x
(
t
)
=
e^{
-
\frac
{t^
2
}{
2
\sigma
^
2
}}
$
\begin
{
dspPlot
}
[xticks=10,xout=true,sidegap=0,ylabel=
{
$
x
(
t
)
$
}
]
{
-50,50
}{
0,1.1
}
\moocStyle
\dspFunc
{
x dup mul neg 2
\s
\s
mul mul div 2.718281828459046 exch exp
}
\end
{
dspPlot
}
\end
{
center
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Example
}
\begin
{
center
}
$
X
(
j
\Omega
)
=
\sigma\sqrt
{
2
\pi
} e^{
-
\frac
{
\sigma
^
2
}{
2
}
\Omega
^
2
}
$
\begin
{
dspPlot
}
[xticks=10,xout=true,sidegap=0,yticks=custom,ylabel=
{
$
X
(
j
\Omega
)
$
}
]
{
-50,50
}{
0,26
}
\moocStyle
\dspFunc
{
x dup mul neg 2
\s
\s
mul mul 2 div div 2.718281828459046 exch exp
\s
2.506628274631 mul mul
}
\dspCustomTicks
[axis=y]
{
25
$
\sigma\sqrt
{
2
\pi
}
$
}
\end
{
dspPlot
}
\end
{
center
}
\end
{
frame
}
\def\s
{
10
}
\begin
{
frame
}
\frametitle
{
Example
}
\begin
{
center
}
$
x
(
t
)
=
\cos
(
2
\pi
f_
0
t
)
$
\begin
{
dspPlot
}
[xticks=10,xout=true,sidegap=0,ylabel=
{
$
x
(
t
)
$
}
]
{
-4,4
}{
-1.1,1.1
}
\moocStyle
\dspFunc
{
x 360 mul cos
}
\dspCustomTicks
[axis=x]
{
1
$
1
/
f_
0
$
2
$
2
/
f_
0
$
3
$
3
/
f_
0
$
-1
$
-
1
/
f_
0
$
}
\end
{
dspPlot
}
\end
{
center
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Example
}
\begin
{
center
}
$
X
(
j
\Omega
)
=
2
\pi\delta
(
\Omega
\pm
f_
0
/
2
\pi
)
$
\begin
{
dspPlot
}
[xout=true,xticks=10,yticks=10,sidegap=0,ylabel=
{
$
X
(
j
\Omega
)
$
}
]
{
-2,2
}{
0,1.1
}
\moocStyle
\dspDiracs
{
1 1 -1 1
}
\dspCustomTicks
[axis=x]
{
1
$
f_
0
/
2
\pi
$
-1
$
-
f_
0
/
2
\pi
$
}
\dspCustomTicks
[axis=y]
{
1
$
2
\pi
$
}
\end
{
dspPlot
}
\end
{
center
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Convolution theorem
}
\begin
{
figure
}
\center
\begin
{
dspBlocks
}{
1
}{
1
}
$
x
(
t
)
$
~~
&
\rnode
{
F
}{
\BDfilter
{
$
\mathcal
{H}
$
}}
&
~~
$
y
(
t
)
$
\ncline
{
->
}{
1,1
}{
1,2
}
\ncline
{
->
}{
1,2
}{
1,3
}
\end
{
dspBlocks
}
\end
{
figure
}
\vspace
{
2em
}
\[
Y
(
j
\Omega
)
=
X
(
j
\Omega
)
\,
H
(
j
\Omega
)
\]
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
A new concept: bandlimited functions
}
\centering
$
\Omega
_N
$
-bandlimitedness:
\[
X
(
j
\Omega
)
=
0
\quad
\mbox
{for } |
\Omega
| >
\Omega
_N
\]
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
Prototypical bandlimited function
}
\center
\begin
{
figure
}
\begin
{
dspPlot
}
[xtype=freq,xticks=custom,yticks=custom]
{
-1.5,1.5
}{
0,1.4
}
\moocStyle
\dspFunc
{
x
\dspRect
{
0
}{
1
}}
\dspCustomTicks
[axis=x]
{
0 0 -0.5
$
-
\Omega
_N
$
0.5
$
\Omega
_N
$
}
\dspCustomTicks
[axis=y]
{
1
$
G
$
}
%{1 $\pi/\Omega_N$}
\end
{
dspPlot
}
\end
{
figure
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
The prototypical bandlimited function
}
\note
<1>
{
We will normalize so that a) area is
$
2
\pi
$
\\
or (equivalent) b) max time-domain
\\
value is 1
}
\[
\Phi
(
j
\Omega
)
=
G
\,
\rect\left
(
\frac
{
\Omega
}{
2
\Omega
_N}
\right
)
\]
\vspace
{
1em
}
\pause
\begin
{
align*
}
\varphi
(t)
&
=
\frac
{
1
}{
2
\pi
}
\int
_{
-
\infty
}^{
\infty
}
\Phi
(j
\Omega
) e
^{
j
\Omega
t
}
d
\Omega
\\
&
=
\ldots
\\
&
= G
\,\frac
{
\Omega
_
N
}{
\pi
}
\sinc\left
(
\frac
{
\Omega
_
N
}{
\pi
}
t
\right
)
\end
{
align*
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
The prototypical bandlimited function
}
\begin
{
itemize
}
\item
normalization:
$
G
=
\displaystyle
\frac
{
\pi
}{
\Omega
_N}
$
\item
total bandwidth:
$
\Omega
_B
=
2
\Omega
_N
$
\item
define
$
T_s
=
\displaystyle
\frac
{
2
\pi
}{
\Omega
_B}
=
\frac
{
\pi
}{
\Omega
_N}
$
\end
{
itemize
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
The prototypical bandlimited function
}
\begin
{
align*
}
\Phi
(j
\Omega
)
&
=
\frac
{
\pi
}{
\Omega
_
N
}
\mbox
{
rect
}
\left
(
\frac
{
\Omega
}{
2
\Omega
_
N
}
\right
)
\\
\\
\varphi
(t)
&
=
\mbox
{
sinc
}
\left
(
\frac
{
t
}{
T
_
s
}
\right
)
\end
{
align*
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
The prototypical bandlimited function
}
\center
\begin
{
figure
}
\begin
{
dspPlot
}
[xtype=freq,xticks=custom,yticks=custom]
{
-1.5,1.5
}{
0,1.4
}
\moocStyle
\dspFunc
{
x
\dspRect
{
0
}{
1
}}
\dspCustomTicks
[axis=x]
{
0 0 -0.5
$
-
\Omega
_N
$
0.5
$
\Omega
_N
$
}
\dspCustomTicks
[axis=y]
{
1
$
\pi
/
\Omega
_N
$
}
\end
{
dspPlot
}
\end
{
figure
}
\end
{
frame
}
\begin
{
frame
}
\frametitle
{
The prototypical bandlimited function
}
\center
\begin
{
figure
}
\begin
{
dspPlot
}
[xticks=custom,sidegap=0,xout=true]
{
-8,8
}{
-0.3,1.2
}
\moocStyle
\dspFunc
{
x
\dspSinc
{
0
}{
1
}}
\dspCustomTicks
[axis=x]
{
0 0 1
$
T_s
$
-1
$
-
T_s
$
2
$
2
T_s
$
3
$
3
T_s
$
4
$
4
T_s
$
}
\end
{
dspPlot
}
\end
{
figure
}
\end
{
frame
}
\end
{
document
}
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