<s>
The	O
Hann	B-Algorithm
function	I-Algorithm
is	O
named	O
after	O
the	O
Austrian	O
meteorologist	O
Julius	O
von	O
Hann	O
.	O
</s>
<s>
It	O
is	O
a	O
window	B-Language
function	I-Language
used	O
to	O
perform	O
Hann	O
smoothing	O
.	O
</s>
<s>
For	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
,	O
the	O
function	O
is	O
sampled	O
symmetrically	O
(	O
with	O
spacing	O
and	O
amplitude	O
)	O
:	O
</s>
<s>
(	O
see	O
)	O
It	O
is	O
also	O
known	O
as	O
the	O
raised	O
cosine	O
window	O
,	O
Hann	O
filter	O
,	O
von	O
Hann	B-Algorithm
window	I-Algorithm
,	O
etc	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
is	O
given	O
by	O
:	O
</s>
<s>
The	O
Discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
DTFT	O
)	O
of	O
the	O
length	O
,	O
time-shifted	O
sequence	O
is	O
defined	O
by	O
a	O
Fourier	O
series	O
,	O
which	O
also	O
has	O
a	O
3-term	O
equivalent	O
that	O
is	O
derived	O
similarly	O
to	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
derivation	O
:	O
</s>
<s>
The	O
truncated	O
sequence	O
is	O
a	O
DFT-even	O
(	O
aka	O
periodic	O
)	O
Hann	B-Algorithm
window	I-Algorithm
.	O
</s>
<s>
An	O
N-length	O
DFT	O
of	O
the	O
window	B-Language
function	I-Language
samples	O
the	O
DTFT	O
at	O
frequencies	O
for	O
integer	O
values	O
of	O
From	O
the	O
expression	O
immediately	O
above	O
,	O
it	O
is	O
easy	O
to	O
see	O
that	O
only	O
3	O
of	O
the	O
N	O
DFT	O
coefficients	O
are	O
non-zero	O
.	O
</s>
<s>
However	O
,	O
the	O
term	O
Hanning	B-Algorithm
function	I-Algorithm
is	O
also	O
conventionally	O
used	O
,	O
derived	O
from	O
the	O
paper	O
in	O
which	O
the	O
term	O
hanning	O
a	O
signal	O
was	O
used	O
to	O
mean	O
applying	O
the	O
Hann	B-Algorithm
window	I-Algorithm
to	O
it	O
.	O
</s>
