<s>
In	O
signal	O
processing	O
,	O
undersampling	B-Algorithm
or	O
bandpass	B-Algorithm
sampling	I-Algorithm
is	O
a	O
technique	O
where	O
one	O
samples	B-Algorithm
a	O
bandpass-filtered	O
signal	O
at	O
a	O
sample	O
rate	O
below	O
its	O
Nyquist	O
rate	O
(	O
twice	O
the	O
upper	O
cutoff	O
frequency	O
)	O
,	O
but	O
is	O
still	O
able	O
to	O
reconstruct	O
the	O
signal	O
.	O
</s>
<s>
When	O
one	O
undersamples	O
a	O
bandpass	O
signal	O
,	O
the	O
samples	B-Algorithm
are	O
indistinguishable	O
from	O
the	O
samples	B-Algorithm
of	O
a	O
low-frequency	O
alias	B-Error_Name
of	O
the	O
high-frequency	O
signal	O
.	O
</s>
<s>
Such	O
sampling	O
is	O
also	O
known	O
as	O
bandpass	B-Algorithm
sampling	I-Algorithm
,	O
harmonic	O
sampling	O
,	O
IF	O
sampling	O
,	O
and	O
direct	O
IF-to-digital	O
conversion	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transforms	I-Algorithm
of	O
real-valued	O
functions	O
are	O
symmetrical	O
around	O
the	O
0	O
Hz	O
axis	O
.	O
</s>
<s>
After	O
sampling	O
,	O
only	O
a	O
periodic	B-Algorithm
summation	I-Algorithm
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
called	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
)	O
is	O
still	O
available	O
.	O
</s>
<s>
When	O
the	O
aliases	O
are	O
mutually	O
exclusive	O
(	O
spectrally	O
)	O
,	O
the	O
original	O
transform	O
and	O
the	O
original	O
continuous	O
function	O
,	O
or	O
a	O
frequency-shifted	O
version	O
of	O
it	O
(	O
if	O
desired	O
)	O
,	O
can	O
be	O
recovered	O
from	O
the	O
samples	B-Algorithm
.	O
</s>
<s>
The	O
first	O
and	O
third	O
graphs	O
of	O
Figure	O
1	O
depict	O
a	O
baseband	B-Algorithm
spectrum	O
before	O
and	O
after	O
being	O
sampled	O
at	O
a	O
rate	O
that	O
completely	O
separates	O
the	O
aliases	O
.	O
</s>
<s>
The	O
fourth	O
graph	O
depicts	O
the	O
spectral	O
result	O
of	O
sampling	O
at	O
the	O
same	O
rate	O
as	O
the	O
baseband	B-Algorithm
function	O
.	O
</s>
<s>
The	O
rate	O
was	O
chosen	O
by	O
finding	O
the	O
lowest	O
rate	O
that	O
is	O
an	O
integer	O
sub-multiple	O
of	O
A	O
and	O
also	O
satisfies	O
the	O
baseband	B-Algorithm
Nyquist	O
criterion	O
:	O
fs>2B	O
.	O
</s>
<s>
Consequently	O
,	O
the	O
bandpass	O
function	O
has	O
effectively	O
been	O
converted	O
to	O
baseband	B-Algorithm
.	O
</s>
<s>
If	O
n	O
>	O
1	O
,	O
then	O
the	O
conditions	O
result	O
in	O
what	O
is	O
sometimes	O
referred	O
to	O
as	O
undersampling	B-Algorithm
,	O
bandpass	B-Algorithm
sampling	I-Algorithm
,	O
or	O
using	O
a	O
sampling	O
rate	O
less	O
than	O
the	O
Nyquist	O
rate	O
(	O
2fH	O
)	O
.	O
</s>
<s>
For	O
the	O
case	O
of	O
a	O
given	O
sampling	B-Algorithm
frequency	I-Algorithm
,	O
simpler	O
formulae	O
for	O
the	O
constraints	O
on	O
the	O
signal	O
's	O
spectral	O
band	O
are	O
given	O
below	O
.	O
</s>
<s>
Example	O
:	O
Consider	O
FM	O
radio	O
to	O
illustrate	O
the	O
idea	O
of	O
undersampling	B-Algorithm
.	O
</s>
<s>
The	O
value	O
n	O
=	O
5	O
gives	O
the	O
lowest	O
sampling	B-Algorithm
frequencies	I-Algorithm
interval	O
and	O
this	O
is	O
a	O
scenario	O
of	O
undersampling	B-Algorithm
.	O
</s>
<s>
When	O
undersampling	B-Algorithm
a	O
real-world	O
signal	O
,	O
the	O
sampling	O
circuit	O
must	O
be	O
fast	O
enough	O
to	O
capture	O
the	O
highest	O
signal	O
frequency	O
of	O
interest	O
.	O
</s>
<s>
Thus	O
,	O
the	O
sampling	B-Algorithm
frequency	I-Algorithm
may	O
be	O
only	O
a	O
little	O
bit	O
greater	O
than	O
43.2	O
MHz	O
,	O
but	O
the	O
input	O
bandwidth	O
of	O
the	O
system	O
must	O
be	O
at	O
least	O
108	O
MHz	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
accuracy	O
of	O
the	O
sampling	O
timing	O
,	O
or	O
aperture	O
uncertainty	O
of	O
the	O
sampler	B-Algorithm
,	O
frequently	O
the	O
analog-to-digital	O
converter	O
,	O
must	O
be	O
appropriate	O
for	O
the	O
frequencies	O
being	O
sampled	O
108MHz	O
,	O
not	O
the	O
lower	O
sample	O
rate	O
.	O
</s>
<s>
Note	O
that	O
if	O
a	O
band	O
is	O
sampled	O
with	O
n	O
>	O
1	O
,	O
then	O
a	O
band-pass	O
filter	O
is	O
required	O
for	O
the	O
anti-aliasing	O
filter	O
,	O
instead	O
of	O
a	O
lowpass	O
filter	O
.	O
</s>
<s>
As	O
we	O
have	O
seen	O
,	O
the	O
normal	O
baseband	B-Algorithm
condition	O
for	O
reversible	O
sampling	O
is	O
that	O
X(f )	O
=	O
0	O
outside	O
the	O
interval	O
:	O
</s>
<s>
which	O
includes	O
the	O
normal	O
baseband	B-Algorithm
condition	O
as	O
case	O
n	O
=	O
1	O
(	O
except	O
that	O
where	O
the	O
intervals	O
come	O
together	O
at	O
0	O
frequency	O
,	O
they	O
can	O
be	O
closed	O
)	O
.	O
</s>
<s>
Rather	O
,	O
the	O
sample	O
sequence	O
can	O
be	O
treated	O
as	O
ordinary	O
samples	B-Algorithm
of	O
the	O
signal	O
frequency-shifted	O
to	O
near	O
baseband	B-Algorithm
,	O
and	O
digital	O
demodulation	O
can	O
proceed	O
on	O
that	O
basis	O
,	O
recognizing	O
the	O
spectrum	O
mirroring	O
when	O
n	O
is	O
even	O
.	O
</s>
<s>
Further	O
generalizations	O
of	O
undersampling	B-Algorithm
for	O
the	O
case	O
of	O
signals	O
with	O
multiple	O
bands	O
are	O
possible	O
,	O
and	O
signals	O
over	O
multidimensional	O
domains	O
(	O
space	O
or	O
space-time	O
)	O
and	O
have	O
been	O
worked	O
out	O
in	O
detail	O
by	O
Igor	O
Kluvánek	O
.	O
</s>
