<s>
In	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
,	O
downsampling	B-Algorithm
,	O
compression	O
,	O
and	O
decimation	B-Algorithm
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>
Both	O
downsampling	B-Algorithm
and	O
decimation	B-Algorithm
can	O
be	O
synonymous	O
with	O
compression	O
,	O
or	O
they	O
can	O
describe	O
an	O
entire	O
process	O
of	O
bandwidth	O
reduction	O
(	O
filtering	B-Algorithm
)	O
and	O
sample-rate	O
reduction	O
.	O
</s>
<s>
Decimation	B-Algorithm
is	O
a	O
term	O
that	O
historically	O
means	O
the	O
removal	O
of	O
every	O
tenth	O
one	O
.	O
</s>
<s>
But	O
in	O
signal	O
processing	O
,	O
decimation	B-Algorithm
by	O
a	O
factor	O
of	O
10	O
actually	O
means	O
keeping	O
only	O
every	O
tenth	O
sample	O
.	O
</s>
<s>
A	O
system	O
component	O
that	O
performs	O
decimation	B-Algorithm
is	O
called	O
a	O
decimator	O
.	O
</s>
<s>
Decimation	B-Algorithm
by	O
an	O
integer	O
factor	O
is	O
also	O
called	O
compression	O
.	O
</s>
<s>
Reduce	O
high-frequency	O
signal	O
components	O
with	O
a	O
digital	O
lowpass	B-Algorithm
filter	I-Algorithm
.	O
</s>
<s>
Step	O
2	O
alone	O
allows	O
high-frequency	O
signal	O
components	O
to	O
be	O
misinterpreted	O
by	O
subsequent	O
users	O
of	O
the	O
data	O
,	O
which	O
is	O
a	O
form	O
of	O
distortion	O
called	O
aliasing	B-Error_Name
.	O
</s>
<s>
Step	O
1	O
,	O
when	O
necessary	O
,	O
suppresses	O
aliasing	B-Error_Name
to	O
an	O
acceptable	O
level	O
.	O
</s>
<s>
In	O
this	O
application	O
,	O
the	O
filter	O
is	O
called	O
an	O
anti-aliasing	O
filter	O
,	O
and	O
its	O
design	O
is	O
discussed	O
below	O
.	O
</s>
<s>
Also	O
see	O
undersampling	B-Algorithm
for	O
information	O
about	O
decimating	O
bandpass	O
functions	O
and	O
signals	O
.	O
</s>
<s>
When	O
the	O
anti-aliasing	O
filter	O
is	O
an	O
IIR	O
design	O
,	O
it	O
relies	O
on	O
feedback	O
from	O
output	O
to	O
input	O
,	O
prior	O
to	O
the	O
second	O
step	O
.	O
</s>
<s>
With	O
FIR	O
filtering	B-Algorithm
,	O
it	O
is	O
an	O
easy	O
matter	O
to	O
compute	O
only	O
every	O
Mth	O
output	O
.	O
</s>
<s>
x[•]	O
represents	O
the	O
input	O
sequence	O
being	O
downsampled	B-Algorithm
.	O
</s>
<s>
In	O
the	O
case	O
M	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>
Furthermore	O
,	O
because	O
of	O
downsampling	B-Algorithm
by	O
M	O
,	O
the	O
stream	O
of	O
x[•]	O
samples	O
involved	O
in	O
any	O
one	O
of	O
the	O
M	O
dot	O
products	O
is	O
never	O
involved	O
in	O
the	O
other	O
dot	O
products	O
.	O
</s>
<s>
Thus	O
M	O
low-order	O
FIR	O
filters	O
are	O
each	O
filtering	B-Algorithm
one	O
of	O
M	O
multiplexed	O
phases	O
of	O
the	O
input	O
stream	O
,	O
and	O
the	O
M	O
outputs	O
are	O
being	O
summed	O
.	O
</s>
<s>
Let	O
X(f )	O
be	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
any	O
function	O
,	O
x(t )	O
,	O
whose	O
samples	O
at	O
some	O
interval	O
,	O
T	O
,	O
equal	O
the	O
x[n]	O
sequence	O
.	O
</s>
<s>
Then	O
the	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
DTFT	O
)	O
is	O
a	O
Fourier	O
series	O
representation	O
of	O
a	O
periodic	B-Algorithm
summation	I-Algorithm
of	O
X(f )	O
:	O
</s>
<s>
The	O
periodic	B-Algorithm
summation	I-Algorithm
has	O
been	O
reduced	O
in	O
amplitude	O
and	O
periodicity	O
by	O
a	O
factor	O
of	O
M	O
.	O
An	O
example	O
of	O
both	O
these	O
distributions	O
is	O
depicted	O
in	O
the	O
two	O
traces	O
of	O
Fig	O
1	O
.	O
</s>
<s>
Aliasing	B-Error_Name
occurs	O
when	O
adjacent	O
copies	O
of	O
X(f )	O
overlap	O
.	O
</s>
<s>
The	O
purpose	O
of	O
the	O
anti-aliasing	O
filter	O
is	O
to	O
ensure	O
that	O
the	O
reduced	O
periodicity	O
does	O
not	O
create	O
overlap	O
.	O
</s>
<s>
The	O
condition	O
that	O
ensures	O
the	O
copies	O
of	O
X(f )	O
do	O
not	O
overlap	O
each	O
other	O
is	O
:	O
so	O
that	O
is	O
the	O
maximum	O
cutoff	O
frequency	O
of	O
an	O
ideal	O
anti-aliasing	O
filter	O
.	O
</s>
<s>
Let	O
M/L	O
denote	O
the	O
decimation	B-Algorithm
factor	O
,	O
where	O
:	O
</s>
<s>
Increase	O
(	O
resample	O
)	O
the	O
sequence	O
by	O
a	O
factor	O
of	O
L	O
.	O
This	O
is	O
called	O
Upsampling	B-Algorithm
,	O
or	O
interpolation	O
.	O
</s>
<s>
Step	O
1	O
requires	O
a	O
lowpass	B-Algorithm
filter	I-Algorithm
after	O
increasing	O
(	O
expanding	O
)	O
the	O
data	O
rate	O
,	O
and	O
step	O
2	O
requires	O
a	O
lowpass	B-Algorithm
filter	I-Algorithm
before	O
decimation	B-Algorithm
.	O
</s>
<s>
For	O
the	O
M>L	O
case	O
,	O
the	O
anti-aliasing	O
filter	O
cutoff	O
,	O
cycles	O
per	O
intermediate	O
sample	O
,	O
is	O
the	O
lower	O
frequency	O
.	O
</s>
