<s>
Digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
(	O
DSP	O
)	O
is	O
the	O
use	O
of	O
digital	B-General_Concept
processing	I-General_Concept
,	O
such	O
as	O
by	O
computers	O
or	O
more	O
specialized	O
digital	B-Architecture
signal	I-Architecture
processors	I-Architecture
,	O
to	O
perform	O
a	O
wide	O
variety	O
of	O
signal	O
processing	O
operations	O
.	O
</s>
<s>
The	O
digital	O
signals	O
processed	O
in	O
this	O
manner	O
are	O
a	O
sequence	O
of	O
numbers	O
that	O
represent	O
samples	B-Algorithm
of	O
a	O
continuous	O
variable	O
in	O
a	O
domain	O
such	O
as	O
time	O
,	O
space	O
,	O
or	O
frequency	O
.	O
</s>
<s>
In	O
digital	O
electronics	O
,	O
a	O
digital	O
signal	O
is	O
represented	O
as	O
a	O
pulse	O
train	O
,	O
which	O
is	O
typically	O
generated	O
by	O
the	O
switching	O
of	O
a	O
transistor	B-Application
.	O
</s>
<s>
Digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
and	O
analog	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
are	O
subfields	O
of	O
signal	O
processing	O
.	O
</s>
<s>
DSP	O
applications	O
include	O
audio	B-Algorithm
and	O
speech	B-Algorithm
processing	I-Algorithm
,	O
sonar	B-Application
,	O
radar	B-Application
and	O
other	O
sensor	B-Algorithm
array	I-Algorithm
processing	O
,	O
spectral	O
density	O
estimation	O
,	O
statistical	O
signal	O
processing	O
,	O
digital	B-Algorithm
image	I-Algorithm
processing	I-Algorithm
,	O
data	B-General_Concept
compression	I-General_Concept
,	O
video	B-Algorithm
coding	I-Algorithm
,	O
audio	B-Algorithm
coding	O
,	O
image	B-General_Concept
compression	I-General_Concept
,	O
signal	O
processing	O
for	O
telecommunications	O
,	O
control	O
systems	O
,	O
biomedical	O
engineering	O
,	O
and	O
seismology	O
,	O
among	O
others	O
.	O
</s>
<s>
Nonlinear	O
signal	O
processing	O
is	O
closely	O
related	O
to	O
nonlinear	O
system	O
identification	O
and	O
can	O
be	O
implemented	O
in	O
the	O
time	O
,	O
frequency	O
,	O
and	O
spatio-temporal	B-Protocol
domains	I-Protocol
.	O
</s>
<s>
The	O
application	O
of	O
digital	O
computation	O
to	O
signal	O
processing	O
allows	O
for	O
many	O
advantages	O
over	O
analog	O
processing	O
in	O
many	O
applications	O
,	O
such	O
as	O
error	B-Error_Name
detection	I-Error_Name
and	I-Error_Name
correction	I-Error_Name
in	O
transmission	O
as	O
well	O
as	O
data	B-General_Concept
compression	I-General_Concept
.	O
</s>
<s>
Digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
is	O
also	O
fundamental	O
to	O
digital	O
technology	O
,	O
such	O
as	O
digital	O
telecommunication	O
and	O
wireless	O
communications	O
.	O
</s>
<s>
DSP	O
is	O
applicable	O
to	O
both	O
streaming	B-Operating_System
data	I-Operating_System
and	O
static	O
(	O
stored	O
)	O
data	O
.	O
</s>
<s>
Sampling	B-Algorithm
is	O
usually	O
carried	O
out	O
in	O
two	O
stages	O
,	O
discretization	B-Algorithm
and	O
quantization	B-Algorithm
.	O
</s>
<s>
Discretization	B-Algorithm
means	O
that	O
the	O
signal	O
is	O
divided	O
into	O
equal	O
intervals	O
of	O
time	O
,	O
and	O
each	O
interval	O
is	O
represented	O
by	O
a	O
single	O
measurement	O
of	O
amplitude	O
.	O
</s>
<s>
Quantization	B-Algorithm
means	O
each	O
amplitude	O
measurement	O
is	O
approximated	O
by	O
a	O
value	O
from	O
a	O
finite	O
set	O
.	O
</s>
<s>
The	O
Nyquist	O
–	O
Shannon	O
sampling	B-Algorithm
theorem	O
states	O
that	O
a	O
signal	O
can	O
be	O
exactly	O
reconstructed	O
from	O
its	O
samples	B-Algorithm
if	O
the	O
sampling	B-Algorithm
frequency	I-Algorithm
is	O
greater	O
than	O
twice	O
the	O
highest	O
frequency	O
component	O
in	O
the	O
signal	O
.	O
</s>
<s>
In	O
practice	O
,	O
the	O
sampling	B-Algorithm
frequency	I-Algorithm
is	O
often	O
significantly	O
higher	O
than	O
this	O
.	O
</s>
<s>
It	O
is	O
common	O
to	O
use	O
an	O
anti-aliasing	O
filter	O
to	O
limit	O
the	O
signal	O
bandwidth	O
to	O
comply	O
with	O
the	O
sampling	B-Algorithm
theorem	O
,	O
however	O
careful	O
selection	O
of	O
this	O
filter	O
is	O
required	O
because	O
the	O
reconstructed	O
signal	O
will	O
be	O
the	O
filtered	O
signal	O
plus	O
residual	O
aliasing	B-Error_Name
from	O
imperfect	O
stop	O
band	O
rejection	O
instead	O
of	O
the	O
original	O
(	O
unfiltered	O
)	O
signal	O
.	O
</s>
<s>
Theoretical	O
DSP	O
analyses	O
and	O
derivations	O
are	O
typically	O
performed	O
on	O
discrete-time	O
signal	O
models	O
with	O
no	O
amplitude	O
inaccuracies	O
(	O
quantization	B-Algorithm
error	O
)	O
,	O
"	O
created	O
"	O
by	O
the	O
abstract	O
process	O
of	O
sampling	B-Algorithm
.	O
</s>
<s>
Numerical	B-General_Concept
methods	I-General_Concept
require	O
a	O
quantized	B-Algorithm
signal	I-Algorithm
,	O
such	O
as	O
those	O
produced	O
by	O
an	O
ADC	O
.	O
</s>
<s>
But	O
often	O
it	O
is	O
another	O
quantized	B-Algorithm
signal	I-Algorithm
that	O
is	O
converted	O
back	O
to	O
analog	O
form	O
by	O
a	O
digital-to-analog	O
converter	O
(	O
DAC	O
)	O
.	O
</s>
<s>
A	O
sequence	O
of	O
samples	B-Algorithm
from	O
a	O
measuring	O
device	O
produces	O
a	O
temporal	O
or	O
spatial	O
domain	O
representation	O
,	O
whereas	O
a	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
produces	O
the	O
frequency	O
domain	O
representation	O
.	O
</s>
<s>
Similarly	O
,	O
space	O
domain	O
refers	O
to	O
the	O
analysis	O
of	O
signals	O
with	O
respect	O
to	O
position	O
,	O
e.g.	O
,	O
pixel	O
location	O
for	O
the	O
case	O
of	O
image	B-Algorithm
processing	I-Algorithm
.	O
</s>
<s>
Digital	O
filtering	O
generally	O
consists	O
of	O
some	O
linear	O
transformation	O
of	O
a	O
number	O
of	O
surrounding	O
samples	B-Algorithm
around	O
the	O
current	O
sample	O
of	O
the	O
input	O
or	O
output	O
signal	O
.	O
</s>
<s>
The	O
surrounding	O
samples	B-Algorithm
may	O
be	O
identified	O
with	O
respect	O
to	O
time	O
or	O
space	O
.	O
</s>
<s>
The	O
output	O
of	O
a	O
linear	O
digital	O
filter	O
to	O
any	O
given	O
input	O
may	O
be	O
calculated	O
by	O
convolving	B-Language
the	O
input	O
signal	O
with	O
an	O
impulse	O
response	O
.	O
</s>
<s>
Signals	O
are	O
converted	O
from	O
time	O
or	O
space	O
domain	O
to	O
the	O
frequency	O
domain	O
usually	O
through	O
use	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
converts	O
the	O
time	O
or	O
space	O
information	O
to	O
a	O
magnitude	O
and	O
phase	O
component	O
of	O
each	O
frequency	O
.	O
</s>
<s>
Where	O
phase	O
is	O
unimportant	O
,	O
often	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
converted	O
to	O
the	O
power	O
spectrum	O
,	O
which	O
is	O
the	O
magnitude	O
of	O
each	O
frequency	O
component	O
squared	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
cepstrum	B-Algorithm
converts	O
a	O
signal	O
to	O
the	O
frequency	O
domain	O
through	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
takes	O
the	O
logarithm	O
,	O
then	O
applies	O
another	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
Z-transform	B-Algorithm
provides	O
a	O
tool	O
for	O
analyzing	O
stability	O
issues	O
of	O
digital	O
IIR	O
filters	O
.	O
</s>
<s>
A	O
signal	O
is	O
represented	O
as	O
linear	O
combination	O
of	O
its	O
previous	O
samples	B-Algorithm
.	O
</s>
<s>
Coefficients	O
of	O
the	O
combination	O
are	O
called	O
autoregression	B-Algorithm
coefficients	O
.	O
</s>
<s>
This	O
method	O
has	O
higher	O
frequency	O
resolution	O
and	O
can	O
process	O
shorter	O
signals	O
compared	O
to	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Prony	B-Algorithm
's	I-Algorithm
method	I-Algorithm
can	O
be	O
used	O
to	O
estimate	O
phases	O
,	O
amplitudes	O
,	O
initial	O
phases	O
and	O
decays	O
of	O
the	O
components	O
of	O
signal	O
.	O
</s>
<s>
Linear	O
techniques	O
such	O
as	O
Short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
wavelet	B-Algorithm
transform	I-Algorithm
,	O
filter	O
bank	O
,	O
non-linear	O
(	O
e.g.	O
,	O
Wigner	O
–	O
Ville	O
transform	O
)	O
and	O
autoregressive	B-Algorithm
methods	O
(	O
e.g.	O
</s>
<s>
segmented	O
Prony	B-Algorithm
method	I-Algorithm
)	O
are	O
used	O
for	O
representation	O
of	O
signal	O
on	O
the	O
time-frequency	O
plane	O
.	O
</s>
<s>
Non-linear	O
and	O
segmented	O
Prony	B-Algorithm
methods	I-Algorithm
can	O
provide	O
higher	O
resolution	O
,	O
but	O
may	O
produce	O
undesirable	O
artifacts	O
.	O
</s>
<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
and	O
functional	B-Application
analysis	I-Application
,	O
a	O
discrete	O
wavelet	B-Algorithm
transform	I-Algorithm
is	O
any	O
wavelet	B-Algorithm
transform	I-Algorithm
for	O
which	O
the	O
wavelets	O
are	O
discretely	O
sampled	O
.	O
</s>
<s>
As	O
with	O
other	O
wavelet	B-Algorithm
transforms	I-Algorithm
,	O
a	O
key	O
advantage	O
it	O
has	O
over	O
Fourier	B-Algorithm
transforms	I-Algorithm
is	O
temporal	O
resolution	O
:	O
it	O
captures	O
both	O
frequency	O
and	O
location	O
information	O
.	O
</s>
<s>
DSP	O
algorithms	O
may	O
be	O
run	O
on	O
general-purpose	O
computers	O
and	O
digital	B-Architecture
signal	I-Architecture
processors	I-Architecture
.	O
</s>
<s>
Additional	O
technologies	O
for	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
include	O
more	O
powerful	O
general	O
purpose	O
microprocessors	B-Architecture
,	O
graphics	B-Architecture
processing	I-Architecture
units	I-Architecture
,	O
field-programmable	B-Architecture
gate	I-Architecture
arrays	I-Architecture
(	O
FPGAs	B-Architecture
)	O
,	O
digital	B-General_Concept
signal	I-General_Concept
controllers	I-General_Concept
(	O
mostly	O
for	O
industrial	O
applications	O
such	O
as	O
motor	O
control	O
)	O
,	O
and	O
stream	B-Application
processors	I-Application
.	O
</s>
<s>
For	O
systems	O
that	O
do	O
not	O
have	O
a	O
real-time	B-General_Concept
computing	I-General_Concept
requirement	O
and	O
the	O
signal	O
data	O
(	O
either	O
input	O
or	O
output	O
)	O
exists	O
in	O
data	O
files	O
,	O
processing	O
may	O
be	O
done	O
economically	O
with	O
a	O
general-purpose	O
computer	O
.	O
</s>
<s>
This	O
is	O
essentially	O
no	O
different	O
from	O
any	O
other	O
data	B-General_Concept
processing	I-General_Concept
,	O
except	O
DSP	O
mathematical	O
techniques	O
(	O
such	O
as	O
the	O
DCT	B-General_Concept
and	O
FFT	O
)	O
are	O
used	O
,	O
and	O
the	O
sampled	O
data	O
is	O
usually	O
assumed	O
to	O
be	O
uniformly	O
sampled	O
in	O
time	O
or	O
space	O
.	O
</s>
<s>
An	O
example	O
of	O
such	O
an	O
application	O
is	O
processing	O
digital	B-Application
photographs	I-Application
with	O
software	O
such	O
as	O
Photoshop	B-Operating_System
.	O
</s>
<s>
When	O
the	O
application	O
requirement	O
is	O
real-time	O
,	O
DSP	O
is	O
often	O
implemented	O
using	O
specialized	O
or	O
dedicated	O
processors	O
or	O
microprocessors	B-Architecture
,	O
sometimes	O
using	O
multiple	O
processors	O
or	O
multiple	O
processing	O
cores	O
.	O
</s>
<s>
For	O
more	O
demanding	O
applications	O
FPGAs	B-Architecture
may	O
be	O
used	O
.	O
</s>
<s>
Parallel	O
implementations	O
of	O
DSP	O
algorithms	O
,	O
utilising	O
multi-core	O
CPU	O
and	O
many-core	O
GPU	B-Architecture
architectures	O
,	O
are	O
developed	O
to	O
improve	O
the	O
performances	O
in	O
terms	O
of	O
latency	O
of	O
these	O
algorithms	O
.	O
</s>
<s>
is	O
done	O
by	O
the	O
computer	O
's	O
CPU	O
rather	O
than	O
by	O
DSP	O
or	O
outboard	O
processing	O
,	O
which	O
is	O
done	O
by	O
additional	O
third-party	O
DSP	B-Architecture
chips	I-Architecture
located	O
on	O
extension	O
cards	O
or	O
external	O
hardware	O
boxes	O
or	O
racks	O
.	O
</s>
<s>
Many	O
digital	B-General_Concept
audio	I-General_Concept
workstations	I-General_Concept
such	O
as	O
Logic	B-Application
Pro	I-Application
,	O
Cubase	B-Application
,	O
Digital	B-Operating_System
Performer	I-Operating_System
and	O
Pro	B-Application
Tools	I-Application
LE	O
use	O
native	O
processing	O
.	O
</s>
<s>
Others	O
,	O
such	O
as	O
Pro	B-Application
Tools	I-Application
HD	O
,	O
Universal	O
Audio	B-Algorithm
's	O
UAD-1	O
and	O
TC	O
Electronic	O
's	O
Powercore	O
use	O
DSP	O
processing	O
.	O
</s>
<s>
Audio	B-Algorithm
data	B-General_Concept
compression	I-General_Concept
e.g.	O
</s>
<s>
Specific	O
examples	O
include	O
speech	B-Application
coding	I-Application
and	O
transmission	O
in	O
digital	O
mobile	O
phones	O
,	O
room	O
correction	O
of	O
sound	O
in	O
hi-fi	O
and	O
sound	O
reinforcement	O
applications	O
,	O
analysis	O
and	O
control	O
of	O
industrial	O
processes	O
,	O
medical	B-Application
imaging	I-Application
such	O
as	O
CAT	O
scans	O
and	O
MRI	B-Algorithm
,	O
audio	B-Algorithm
crossovers	I-Algorithm
and	O
equalization	O
,	O
digital	O
synthesizers	O
,	O
and	O
audio	B-Algorithm
effects	O
units	O
.	O
</s>
