<s>
In	O
telecommunication	O
,	O
a	O
convolutional	B-Error_Name
code	I-Error_Name
is	O
a	O
type	O
of	O
error-correcting	B-Error_Name
code	I-Error_Name
that	O
generates	O
parity	O
symbols	O
via	O
the	O
sliding	O
application	O
of	O
a	O
boolean	B-Application
polynomial	I-Application
function	O
to	O
a	O
data	O
stream	O
.	O
</s>
<s>
The	O
sliding	O
application	O
represents	O
the	O
'	O
convolution	B-Language
 '	O
of	O
the	O
encoder	O
over	O
the	O
data	O
,	O
which	O
gives	O
rise	O
to	O
the	O
term	O
'	O
convolutional	B-Error_Name
coding	I-Error_Name
 '	O
.	O
</s>
<s>
The	O
sliding	O
nature	O
of	O
the	O
convolutional	B-Error_Name
codes	I-Error_Name
facilitates	O
trellis	O
decoding	O
using	O
a	O
time-invariant	O
trellis	O
.	O
</s>
<s>
Time	O
invariant	O
trellis	O
decoding	O
allows	O
convolutional	B-Error_Name
codes	I-Error_Name
to	O
be	O
maximum-likelihood	O
soft-decision	O
decoded	O
with	O
reasonable	O
complexity	O
.	O
</s>
<s>
The	O
ability	O
to	O
perform	O
economical	O
maximum	O
likelihood	O
soft	O
decision	O
decoding	O
is	O
one	O
of	O
the	O
major	O
benefits	O
of	O
convolutional	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
Convolutional	B-Error_Name
codes	I-Error_Name
are	O
often	O
characterized	O
by	O
the	O
base	O
code	O
rate	O
and	O
the	O
depth	O
(	O
or	O
memory	O
)	O
of	O
the	O
encoder	O
.	O
</s>
<s>
The	O
memory	O
is	O
often	O
called	O
the	O
"	O
constraint	B-Error_Name
length	I-Error_Name
"	O
,	O
where	O
the	O
output	O
is	O
a	O
function	O
of	O
the	O
current	O
input	O
as	O
well	O
as	O
the	O
previous	O
inputs	O
.	O
</s>
<s>
Convolutional	B-Error_Name
codes	I-Error_Name
are	O
often	O
described	O
as	O
continuous	O
.	O
</s>
<s>
However	O
,	O
it	O
may	O
also	O
be	O
said	O
that	O
convolutional	B-Error_Name
codes	I-Error_Name
have	O
arbitrary	O
block	O
length	O
,	O
rather	O
than	O
being	O
continuous	O
,	O
since	O
most	O
real-world	O
convolutional	O
encoding	O
is	O
performed	O
on	O
blocks	O
of	O
data	O
.	O
</s>
<s>
The	O
arbitrary	O
block	O
length	O
of	O
convolutional	B-Error_Name
codes	I-Error_Name
can	O
also	O
be	O
contrasted	O
to	O
classic	O
block	O
codes	O
,	O
which	O
generally	O
have	O
fixed	O
block	O
lengths	O
that	O
are	O
determined	O
by	O
algebraic	O
properties	O
.	O
</s>
<s>
The	O
code	O
rate	O
of	O
a	O
convolutional	B-Error_Name
code	I-Error_Name
is	O
commonly	O
modified	O
via	O
symbol	O
puncturing	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
convolutional	B-Error_Name
code	I-Error_Name
with	O
a	O
'	O
mother	O
 '	O
code	O
rate	O
may	O
be	O
punctured	O
to	O
a	O
higher	O
rate	O
of	O
,	O
for	O
example	O
,	O
simply	O
by	O
not	O
transmitting	O
a	O
portion	O
of	O
code	O
symbols	O
.	O
</s>
<s>
The	O
performance	O
of	O
a	O
punctured	O
convolutional	B-Error_Name
code	I-Error_Name
generally	O
scales	O
well	O
with	O
the	O
amount	O
of	O
parity	O
transmitted	O
.	O
</s>
<s>
The	O
ability	O
to	O
perform	O
economical	O
soft	O
decision	O
decoding	O
on	O
convolutional	B-Error_Name
codes	I-Error_Name
,	O
as	O
well	O
as	O
the	O
block	O
length	O
and	O
code	O
rate	O
flexibility	O
of	O
convolutional	B-Error_Name
codes	I-Error_Name
,	O
makes	O
them	O
very	O
popular	O
for	O
digital	O
communications	O
.	O
</s>
<s>
Convolutional	B-Error_Name
codes	I-Error_Name
were	O
introduced	O
in	O
1955	O
by	O
Peter	O
Elias	O
.	O
</s>
<s>
It	O
was	O
thought	O
that	O
convolutional	B-Error_Name
codes	I-Error_Name
could	O
be	O
decoded	O
with	O
arbitrary	O
quality	O
at	O
the	O
expense	O
of	O
computation	O
and	O
delay	O
.	O
</s>
<s>
In	O
1967	O
,	O
Andrew	O
Viterbi	O
determined	O
that	O
convolutional	B-Error_Name
codes	I-Error_Name
could	O
be	O
maximum-likelihood	O
decoded	O
with	O
reasonable	O
complexity	O
using	O
time	O
invariant	O
trellis	O
based	O
decoders	O
—	O
the	O
Viterbi	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Other	O
trellis-based	O
decoder	O
algorithms	O
were	O
later	O
developed	O
,	O
including	O
the	O
BCJR	B-Error_Name
decoding	O
algorithm	O
.	O
</s>
<s>
Recursive	B-Error_Name
systematic	I-Error_Name
convolutional	I-Error_Name
codes	I-Error_Name
were	O
invented	O
by	O
Claude	O
Berrou	O
around	O
1991	O
.	O
</s>
<s>
These	O
codes	O
proved	O
especially	O
useful	O
for	O
iterative	O
processing	O
including	O
the	O
processing	O
of	O
concatenated	O
codes	O
such	O
as	O
turbo	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
Using	O
the	O
"	O
convolutional	O
"	O
terminology	O
,	O
a	O
classic	O
convolutional	B-Error_Name
code	I-Error_Name
might	O
be	O
considered	O
a	O
Finite	O
impulse	O
response	O
(	O
FIR	O
)	O
filter	O
,	O
while	O
a	O
recursive	B-Error_Name
convolutional	I-Error_Name
code	I-Error_Name
might	O
be	O
considered	O
an	O
Infinite	O
impulse	O
response	O
(	O
IIR	O
)	O
filter	O
.	O
</s>
<s>
Convolutional	B-Error_Name
codes	I-Error_Name
are	O
used	O
extensively	O
to	O
achieve	O
reliable	O
data	O
transfer	O
in	O
numerous	O
applications	O
,	O
such	O
as	O
digital	O
video	O
,	O
radio	O
,	O
mobile	O
communications	O
(	O
e.g.	O
,	O
in	O
GSM	O
,	O
GPRS	B-General_Concept
,	O
EDGE	O
and	O
3G	O
networks	O
(	O
until	O
3GPP	O
Release	O
7	O
)	O
)	O
and	O
satellite	O
communications	O
.	O
</s>
<s>
These	O
codes	O
are	O
often	O
implemented	O
in	O
concatenation	O
with	O
a	O
hard-decision	O
code	O
,	O
particularly	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
.	O
</s>
<s>
Prior	O
to	O
turbo	B-Error_Name
codes	I-Error_Name
such	O
constructions	O
were	O
the	O
most	O
efficient	O
,	O
coming	O
closest	O
to	O
the	O
Shannon	O
limit	O
.	O
</s>
<s>
To	O
convolutionally	O
encode	O
data	O
,	O
start	O
with	O
k	O
memory	B-General_Concept
registers	I-General_Concept
,	O
each	O
holding	O
one	O
input	O
bit	O
.	O
</s>
<s>
Unless	O
otherwise	O
specified	O
,	O
all	O
memory	B-General_Concept
registers	I-General_Concept
start	O
with	O
a	O
value	O
of	O
0	O
.	O
</s>
<s>
The	O
figure	O
below	O
is	O
a	O
rate	O
(	O
)	O
encoder	O
with	O
constraint	B-Error_Name
length	I-Error_Name
(	O
k	O
)	O
of	O
3	O
.	O
</s>
<s>
Convolutional	B-Error_Name
codes	I-Error_Name
can	O
be	O
systematic	O
and	O
non-systematic	O
:	O
</s>
<s>
Non-systematic	O
convolutional	B-Error_Name
codes	I-Error_Name
are	O
more	O
popular	O
due	O
to	O
better	O
noise	O
immunity	O
.	O
</s>
<s>
It	O
relates	O
to	O
the	O
free	O
distance	O
of	O
the	O
convolutional	B-Error_Name
code	I-Error_Name
.	O
</s>
<s>
Recursive	O
systematic	O
convolutional	O
(	O
RSC	O
)	O
codes	O
have	O
become	O
more	O
popular	O
due	O
to	O
their	O
use	O
in	O
Turbo	B-Error_Name
Codes	I-Error_Name
.	O
</s>
<s>
Useful	O
for	O
LDPC	B-Error_Name
code	I-Error_Name
implementation	O
and	O
as	O
inner	O
constituent	O
code	O
for	O
serial	B-Error_Name
concatenated	I-Error_Name
convolutional	I-Error_Name
codes	I-Error_Name
(	O
SCCC	O
's	O
)	O
.	O
</s>
<s>
Useful	O
for	O
SCCC	O
's	O
and	O
multidimensional	O
turbo	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
Useful	O
as	O
constituent	O
code	O
in	O
low	O
error	O
rate	O
turbo	B-Error_Name
codes	I-Error_Name
for	O
applications	O
such	O
as	O
satellite	O
links	O
.	O
</s>
<s>
A	O
convolutional	O
encoder	O
is	O
called	O
so	O
because	O
it	O
performs	O
a	O
convolution	B-Language
of	O
the	O
input	O
stream	O
with	O
the	O
encoder	O
's	O
impulse	O
responses	O
:	O
</s>
<s>
where	O
is	O
an	O
input	O
sequence	O
,	O
is	O
a	O
sequence	O
from	O
output	O
,	O
is	O
an	O
impulse	O
response	O
for	O
output	O
and	O
denotes	O
convolution	B-Language
.	O
</s>
<s>
An	O
impulse	O
response	O
is	O
connected	O
with	O
a	O
transfer	O
function	O
through	O
Z-transform	B-Algorithm
.	O
</s>
<s>
,	O
and	O
the	O
constraint	B-Error_Name
length	I-Error_Name
is	O
defined	O
as	O
.	O
</s>
<s>
For	O
instance	O
,	O
in	O
the	O
first	O
example	O
the	O
constraint	B-Error_Name
length	I-Error_Name
is	O
3	O
,	O
and	O
in	O
the	O
second	O
the	O
constraint	B-Error_Name
length	I-Error_Name
is	O
4	O
.	O
</s>
<s>
A	O
convolutional	O
encoder	O
is	O
a	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
.	O
</s>
<s>
The	O
correcting	O
capability	O
(	O
t	O
)	O
of	O
a	O
convolutional	B-Error_Name
code	I-Error_Name
is	O
the	O
number	O
of	O
errors	O
that	O
can	O
be	O
corrected	O
by	O
the	O
code	O
.	O
</s>
<s>
Since	O
a	O
convolutional	B-Error_Name
code	I-Error_Name
does	O
n't	O
use	O
blocks	O
,	O
processing	O
instead	O
a	O
continuous	O
bitstream	O
,	O
the	O
value	O
of	O
t	O
applies	O
to	O
a	O
quantity	O
of	O
errors	O
located	O
relatively	O
near	O
to	O
each	O
other	O
.	O
</s>
<s>
The	O
fact	O
that	O
errors	O
appear	O
as	O
"	O
bursts	O
"	O
should	O
be	O
accounted	O
for	O
when	O
designing	O
a	O
concatenated	O
code	O
with	O
an	O
inner	O
convolutional	B-Error_Name
code	I-Error_Name
.	O
</s>
<s>
The	O
popular	O
solution	O
for	O
this	O
problem	O
is	O
to	O
interleave	O
data	O
before	O
convolutional	O
encoding	O
,	O
so	O
that	O
the	O
outer	O
block	O
(	O
usually	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
)	O
code	O
can	O
correct	O
most	O
of	O
the	O
errors	O
.	O
</s>
<s>
Several	O
algorithms	O
exist	O
for	O
decoding	O
convolutional	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
For	O
relatively	O
small	O
values	O
of	O
k	O
,	O
the	O
Viterbi	B-Algorithm
algorithm	I-Algorithm
is	O
universally	O
used	O
as	O
it	O
provides	O
maximum	O
likelihood	O
performance	O
and	O
is	O
highly	O
parallelizable	O
.	O
</s>
<s>
Viterbi	O
decoders	O
are	O
thus	O
easy	O
to	O
implement	O
in	O
VLSI	O
hardware	O
and	O
in	O
software	O
on	O
CPUs	O
with	O
SIMD	B-Device
instruction	O
sets	O
.	O
</s>
<s>
Longer	O
constraint	B-Error_Name
length	I-Error_Name
codes	O
are	O
more	O
practically	O
decoded	O
with	O
any	O
of	O
several	O
sequential	B-Error_Name
decoding	I-Error_Name
algorithms	O
,	O
of	O
which	O
the	O
Fano	B-Error_Name
algorithm	I-Error_Name
is	O
the	O
best	O
known	O
.	O
</s>
<s>
Unlike	O
Viterbi	O
decoding	O
,	O
sequential	B-Error_Name
decoding	I-Error_Name
is	O
not	O
maximum	O
likelihood	O
but	O
its	O
complexity	O
increases	O
only	O
slightly	O
with	O
constraint	B-Error_Name
length	I-Error_Name
,	O
allowing	O
the	O
use	O
of	O
strong	O
,	O
long-constraint-length	O
codes	O
.	O
</s>
<s>
Such	O
codes	O
were	O
used	O
in	O
the	O
Pioneer	O
program	O
of	O
the	O
early	O
1970s	O
to	O
Jupiter	O
and	O
Saturn	O
,	O
but	O
gave	O
way	O
to	O
shorter	O
,	O
Viterbi-decoded	O
codes	O
,	O
usually	O
concatenated	O
with	O
large	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
error	I-Error_Name
correction	I-Error_Name
codes	O
that	O
steepen	O
the	O
overall	O
bit-error-rate	O
curve	O
and	O
produce	O
extremely	O
low	O
residual	O
undetected	O
error	O
rates	O
.	O
</s>
<s>
Both	O
Viterbi	O
and	O
sequential	B-Error_Name
decoding	I-Error_Name
algorithms	O
return	O
hard	O
decisions	O
:	O
the	O
bits	O
that	O
form	O
the	O
most	O
likely	O
codeword	O
.	O
</s>
<s>
An	O
approximate	O
confidence	O
measure	O
can	O
be	O
added	O
to	O
each	O
bit	O
by	O
use	O
of	O
the	O
Soft	O
output	O
Viterbi	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
(	O
MAP	O
)	O
soft	O
decisions	O
for	O
each	O
bit	O
can	O
be	O
obtained	O
by	O
use	O
of	O
the	O
BCJR	B-Error_Name
algorithm	I-Error_Name
.	O
</s>
<s>
In	O
fact	O
,	O
predefined	O
convolutional	B-Error_Name
codes	I-Error_Name
structures	O
obtained	O
during	O
scientific	O
researches	O
are	O
used	O
in	O
the	O
industry	O
.	O
</s>
<s>
This	O
relates	O
to	O
the	O
possibility	O
to	O
select	O
catastrophic	O
convolutional	B-Error_Name
codes	I-Error_Name
(	O
causes	O
larger	O
number	O
of	O
errors	O
)	O
.	O
</s>
<s>
An	O
especially	O
popular	O
Viterbi-decoded	O
convolutional	B-Error_Name
code	I-Error_Name
,	O
used	O
at	O
least	O
since	O
the	O
Voyager	O
program	O
has	O
a	O
constraint	B-Error_Name
length	I-Error_Name
of	O
7	O
and	O
a	O
rate	O
r	O
of	O
1/2	O
.	O
</s>
<s>
The	O
convolutional	B-Error_Name
code	I-Error_Name
with	O
a	O
constraint	B-Error_Name
length	I-Error_Name
of	O
2	O
and	O
a	O
rate	O
of	O
1/2	O
is	O
used	O
in	O
GSM	O
as	O
an	O
error	O
correction	O
technique	O
.	O
</s>
<s>
Convolutional	B-Error_Name
code	I-Error_Name
with	O
any	O
code	O
rate	O
can	O
be	O
designed	O
based	O
on	O
polynomial	O
selection	O
;	O
however	O
,	O
in	O
practice	O
,	O
a	O
puncturing	O
procedure	O
is	O
often	O
used	O
to	O
achieve	O
the	O
required	O
code	O
rate	O
.	O
</s>
<s>
Code	O
rate	O
Puncturing	O
matrix	O
Free	O
distance	O
(	O
for	O
NASA	O
standard	O
K	O
=	O
7	O
convolutional	B-Error_Name
code	I-Error_Name
)	O
1/2	O
( No	O
perf	O
.	O
)	O
</s>
<s>
Punctured	O
convolutional	B-Error_Name
codes	I-Error_Name
are	O
widely	O
used	O
in	O
the	O
satellite	O
communications	O
,	O
for	O
example	O
,	O
in	O
INTELSAT	O
systems	O
and	O
Digital	B-Application
Video	I-Application
Broadcasting	I-Application
.	O
</s>
<s>
Punctured	O
convolutional	B-Error_Name
codes	I-Error_Name
are	O
also	O
called	O
"	O
perforated	O
"	O
.	O
</s>
<s>
Simple	O
Viterbi-decoded	O
convolutional	B-Error_Name
codes	I-Error_Name
are	O
now	O
giving	O
way	O
to	O
turbo	B-Error_Name
codes	I-Error_Name
,	O
a	O
new	O
class	O
of	O
iterated	O
short	O
convolutional	B-Error_Name
codes	I-Error_Name
that	O
closely	O
approach	O
the	O
theoretical	O
limits	O
imposed	O
by	O
Shannon	O
's	O
theorem	O
with	O
much	O
less	O
decoding	O
complexity	O
than	O
the	O
Viterbi	B-Algorithm
algorithm	I-Algorithm
on	O
the	O
long	O
convolutional	B-Error_Name
codes	I-Error_Name
that	O
would	O
be	O
required	O
for	O
the	O
same	O
performance	O
.	O
</s>
<s>
Concatenation	O
with	O
an	O
outer	O
algebraic	O
code	O
(	O
e.g.	O
,	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
)	O
addresses	O
the	O
issue	O
of	O
error	B-Error_Name
floors	I-Error_Name
inherent	O
to	O
turbo	B-Error_Name
code	I-Error_Name
designs	O
.	O
</s>
