<s>
A	O
beta	B-Algorithm
encoder	I-Algorithm
is	O
an	O
analog-to-digital	O
conversion	O
(	O
A/D	O
)	O
system	O
in	O
which	O
a	O
real	O
number	O
in	O
the	O
unit	O
interval	O
is	O
represented	O
by	O
a	O
finite	O
representation	O
of	O
a	O
sequence	O
in	O
base	O
beta	O
,	O
with	O
beta	O
being	O
a	O
real	O
number	O
between	O
1	O
and	O
2	O
.	O
</s>
<s>
Beta	B-Algorithm
encoders	I-Algorithm
are	O
an	O
alternative	O
to	O
traditional	O
approaches	O
to	O
pulse-code	B-Algorithm
modulation	I-Algorithm
.	O
</s>
<s>
As	O
a	O
form	O
of	O
non-integer	O
representation	O
,	O
beta	O
encoding	O
contrasts	O
with	O
traditional	O
approaches	O
to	O
binary	B-Algorithm
quantization	I-Algorithm
,	O
in	O
which	O
each	O
value	O
is	O
mapped	O
to	O
the	O
first	O
N	O
bits	O
of	O
its	O
base-2	O
expansion	O
.	O
</s>
<s>
Rather	O
than	O
using	O
base	O
2	O
,	O
beta	B-Algorithm
encoders	I-Algorithm
use	O
base	O
beta	O
as	O
a	O
beta-expansion	O
.	O
</s>
<s>
In	O
practice	O
,	O
beta	B-Algorithm
encoders	I-Algorithm
have	O
attempted	O
to	O
exploit	O
the	O
redundancy	O
provided	O
by	O
the	O
non-uniqueness	O
of	O
the	O
expansion	O
in	O
base	O
beta	O
to	O
produce	O
more	O
robust	O
results	O
.	O
</s>
<s>
An	O
early	O
beta	B-Algorithm
encoder	I-Algorithm
,	O
the	O
Golden	B-Algorithm
ratio	I-Algorithm
encoder	I-Algorithm
used	O
the	O
golden	O
ratio	O
base	O
for	O
its	O
value	O
of	O
beta	O
,	O
but	O
was	O
susceptible	O
to	O
hardware	O
errors	O
.	O
</s>
<s>
Although	O
integrator	O
leaks	O
in	O
hardware	O
elements	O
make	O
some	O
beta	B-Algorithm
encoders	I-Algorithm
imprecise	O
,	O
specific	O
algorithms	O
can	O
be	O
used	O
to	O
provide	O
exponentially	O
accurate	O
approximations	O
for	O
the	O
value	O
of	O
beta	O
,	O
despite	O
the	O
imprecise	O
results	O
provided	O
by	O
some	O
circuit	O
components	O
.	O
</s>
<s>
An	O
alternative	O
design	O
called	O
the	O
negative	O
beta	B-Algorithm
encoder	I-Algorithm
(	O
called	O
so	O
due	O
to	O
the	O
negative	O
eigenvalue	O
of	O
the	O
transition	B-Algorithm
probability	I-Algorithm
matrix	I-Algorithm
)	O
has	O
been	O
proposed	O
to	O
further	O
reduce	O
the	O
quantization	B-Algorithm
error	O
.	O
</s>
