<s>
Analog	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
is	O
a	O
type	O
of	O
signal	O
processing	O
conducted	O
on	O
continuous	O
analog	O
signals	O
by	O
some	O
analog	O
means	O
(	O
as	O
opposed	O
to	O
the	O
discrete	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
where	O
the	O
signal	O
processing	O
is	O
carried	O
out	O
by	O
a	O
digital	O
process	O
)	O
.	O
</s>
<s>
Examples	O
of	O
analog	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
include	O
crossover	O
filters	O
in	O
loudspeakers	O
,	O
"	O
bass	O
"	O
,	O
"	O
treble	O
"	O
and	O
"	O
volume	O
"	O
controls	O
on	O
stereos	O
,	O
and	O
"	O
tint	O
"	O
controls	O
on	O
TVs	O
.	O
</s>
<s>
Common	O
analog	O
processing	O
elements	O
include	O
capacitors	O
,	O
resistors	O
and	O
inductors	O
(	O
as	O
the	O
passive	O
elements	O
)	O
and	O
transistors	B-Application
or	O
opamps	O
(	O
as	O
the	O
active	O
elements	O
)	O
.	O
</s>
<s>
Convolution	B-Language
is	O
the	O
basic	O
concept	O
in	O
signal	O
processing	O
that	O
states	O
an	O
input	O
signal	O
can	O
be	O
combined	O
with	O
the	O
system	O
's	O
function	O
to	O
find	O
the	O
output	O
signal	O
.	O
</s>
<s>
It	O
is	O
the	O
integral	O
of	O
the	O
product	O
of	O
two	O
waveforms	O
after	O
one	O
has	O
reversed	O
and	O
shifted	O
;	O
the	O
symbol	O
for	O
convolution	B-Language
is	O
*	O
.	O
</s>
<s>
That	O
is	O
the	O
convolution	B-Language
integral	O
and	O
is	O
used	O
to	O
find	O
the	O
convolution	B-Language
of	O
a	O
signal	O
and	O
a	O
system	O
;	O
typically	O
a	O
=	O
-	O
∞	O
and	O
b	O
=	O
+∞	O
.	O
</s>
<s>
Consider	O
two	O
waveforms	O
f	O
and	O
g	O
.	O
By	O
calculating	O
the	O
convolution	B-Language
,	O
we	O
determine	O
how	O
much	O
a	O
reversed	O
function	O
g	O
must	O
be	O
shifted	O
along	O
the	O
x-axis	O
to	O
become	O
identical	O
to	O
function	O
f	O
.	O
The	O
convolution	B-Language
function	O
essentially	O
reverses	O
and	O
slides	O
function	O
g	O
along	O
the	O
axis	O
,	O
and	O
calculates	O
the	O
integral	O
of	O
their	O
(	O
f	O
and	O
the	O
reversed	O
and	O
shifted	O
g	O
)	O
product	O
for	O
each	O
possible	O
amount	O
of	O
sliding	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
a	O
function	O
that	O
transforms	O
a	O
signal	O
or	O
system	O
in	O
the	O
time	O
domain	O
into	O
the	O
frequency	O
domain	O
,	O
but	O
it	O
only	O
works	O
for	O
certain	O
functions	O
.	O
</s>
<s>
The	O
constraint	O
on	O
which	O
systems	O
or	O
signals	O
can	O
be	O
transformed	O
by	O
the	O
Fourier	B-Algorithm
Transform	I-Algorithm
is	O
that	O
:	O
</s>
<s>
This	O
is	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
integral	O
:	O
</s>
<s>
Usually	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
integral	O
is	O
n't	O
used	O
to	O
determine	O
the	O
transform	O
;	O
instead	O
,	O
a	O
table	O
of	O
transform	O
pairs	O
is	O
used	O
to	O
find	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
signal	O
or	O
system	O
.	O
</s>
<s>
The	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
used	O
to	O
go	O
from	O
frequency	O
domain	O
to	O
time	O
domain	O
:	O
</s>
<s>
Each	O
signal	O
or	O
system	O
that	O
can	O
be	O
transformed	O
has	O
a	O
unique	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
Laplace	O
transform	O
is	O
a	O
generalized	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
It	O
allows	O
a	O
transform	O
of	O
any	O
system	O
or	O
signal	O
because	O
it	O
is	O
a	O
transform	O
into	O
the	O
complex	O
plane	O
instead	O
of	O
just	O
the	O
jω	O
line	O
like	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
If	O
the	O
region	O
of	O
convergence	O
includes	O
the	O
jω	O
axis	O
,	O
jω	O
can	O
be	O
substituted	O
into	O
the	O
Laplace	O
transform	O
for	O
s	O
and	O
it	O
's	O
the	O
same	O
as	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Bode	B-Algorithm
plots	I-Algorithm
are	O
plots	O
of	O
magnitude	O
vs.	O
frequency	O
and	O
phase	O
vs.	O
frequency	O
for	O
a	O
system	O
.	O
</s>
<s>
These	O
can	O
be	O
found	O
by	O
taking	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
time	O
signal	O
and	O
are	O
plotted	O
similarly	O
to	O
a	O
bode	B-Algorithm
plot	I-Algorithm
.	O
</s>
<s>
While	O
any	O
signal	O
can	O
be	O
used	O
in	O
analog	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
,	O
there	O
are	O
many	O
types	O
of	O
signals	O
that	O
are	O
used	O
very	O
frequently	O
.	O
</s>
<s>
Sinusoids	O
are	O
the	O
building	O
block	O
of	O
analog	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
.	O
</s>
<s>
An	O
example	O
of	O
a	O
linear	O
system	O
is	O
a	O
first	O
order	O
low-pass	B-Algorithm
or	O
high-pass	O
filter	O
.	O
</s>
<s>
Linearity	O
and	O
time-invariance	O
are	O
important	O
because	O
they	O
are	O
the	O
only	O
types	O
of	O
systems	O
that	O
can	O
be	O
easily	O
solved	O
using	O
conventional	O
analog	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
methods	O
.	O
</s>
