<s>
In	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
,	O
upsampling	B-Algorithm
,	O
expansion	O
,	O
and	O
interpolation	O
are	O
terms	O
associated	O
with	O
the	O
process	O
of	O
resampling	O
in	O
a	O
multi-rate	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
system	O
.	O
</s>
<s>
Upsampling	B-Algorithm
can	O
be	O
synonymous	O
with	O
expansion	O
,	O
or	O
it	O
can	O
describe	O
an	O
entire	O
process	O
of	O
expansion	O
and	O
filtering	O
(	O
interpolation	O
)	O
.	O
</s>
<s>
When	O
upsampling	B-Algorithm
is	O
performed	O
on	O
a	O
sequence	O
of	O
samples	O
of	O
a	O
signal	O
or	O
other	O
continuous	O
function	O
,	O
it	O
produces	O
an	O
approximation	O
of	O
the	O
sequence	O
that	O
would	O
have	O
been	O
obtained	O
by	O
sampling	O
the	O
signal	O
at	O
a	O
higher	O
rate	O
(	O
or	O
density	O
,	O
as	O
in	O
the	O
case	O
of	O
a	O
photograph	O
)	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
compact	O
disc	O
audio	O
at	O
44,100	O
samples/second	O
is	O
upsampled	B-Algorithm
by	O
a	O
factor	O
of	O
5/4	O
,	O
the	O
resulting	O
sample-rate	O
is	O
55,125	O
.	O
</s>
<s>
Interpolation	O
:	O
Smooth	O
out	O
the	O
discontinuities	O
with	O
a	O
lowpass	B-Algorithm
filter	I-Algorithm
,	O
which	O
replaces	O
the	O
zeros	O
.	O
</s>
<s>
In	O
the	O
case	O
L	O
=	O
2	O
,	O
h[•]	O
can	O
be	O
designed	O
as	O
a	O
half-band	B-Algorithm
filter	I-Algorithm
,	O
where	O
almost	O
half	O
of	O
the	O
coefficients	O
are	O
zero	O
and	O
need	O
not	O
be	O
included	O
in	O
the	O
dot	O
products	O
.	O
</s>
<s>
Computing	O
L-1	O
zeros	O
between	O
the	O
useful	O
outputs	O
of	O
a	O
phase	O
and	O
adding	O
them	O
to	O
a	O
sum	O
is	O
effectively	O
decimation	B-Algorithm
.	O
</s>
<s>
Let	O
be	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
any	O
function	O
,	O
whose	O
samples	O
at	O
some	O
interval	O
,	O
equal	O
the	O
sequence	O
.	O
</s>
<s>
The	O
second	O
graph	O
depicts	O
a	O
lowpass	B-Algorithm
filter	I-Algorithm
and	O
resulting	O
in	O
the	O
desired	O
spectral	O
distribution	O
(	O
third	O
graph	O
)	O
.	O
</s>
<s>
Let	O
L/M	O
denote	O
the	O
upsampling	B-Algorithm
factor	O
,	O
where	O
L>M	O
.	O
</s>
<s>
Upsampling	B-Algorithm
requires	O
a	O
lowpass	B-Algorithm
filter	I-Algorithm
after	O
increasing	O
the	O
data	O
rate	O
,	O
and	O
downsampling	B-Algorithm
requires	O
a	O
lowpass	B-Algorithm
filter	I-Algorithm
before	O
decimation	B-Algorithm
.	O
</s>
