<s>
The	O
problem	O
of	O
reconstruction	B-Algorithm
from	I-Algorithm
zero	I-Algorithm
crossings	I-Algorithm
can	O
be	O
stated	O
as	O
:	O
given	O
the	O
zero	B-Algorithm
crossings	I-Algorithm
of	O
a	O
continuous	O
signal	O
,	O
is	O
it	O
possible	O
to	O
reconstruct	B-Algorithm
the	O
signal	O
(	O
to	O
within	O
a	O
constant	O
factor	O
)	O
?	O
</s>
<s>
Worded	O
differently	O
,	O
what	O
are	O
the	O
conditions	O
under	O
which	O
a	O
signal	O
can	O
be	O
reconstructed	O
from	O
its	O
zero	B-Algorithm
crossings	I-Algorithm
?	O
</s>
<s>
Firstly	O
,	O
proving	O
that	O
there	O
is	O
a	O
unique	O
reconstruction	O
of	O
the	O
signal	O
from	O
the	O
zero	B-Algorithm
crossings	I-Algorithm
,	O
and	O
secondly	O
,	O
how	O
to	O
actually	O
go	O
about	O
reconstructing	O
the	O
signal	O
.	O
</s>
<s>
According	O
to	O
Logan	O
,	O
a	O
signal	O
is	O
uniquely	O
reconstructible	O
from	O
its	O
zero	B-Algorithm
crossings	I-Algorithm
if	O
:	O
</s>
<s>
The	O
signal	O
x(t )	O
and	O
its	O
Hilbert	B-Algorithm
transform	I-Algorithm
xt	O
have	O
no	O
zeros	O
in	O
common	O
with	O
each	O
other	O
.	O
</s>
<s>
The	O
frequency-domain	O
representation	O
of	O
the	O
signal	O
is	O
at	O
most	O
1	O
octave	B-Language
long	O
,	O
in	O
other	O
words	O
,	O
it	O
is	O
bandpass-limited	O
between	O
some	O
frequencies	O
B	O
and	O
2B	O
.	O
</s>
