<s>
For	O
complementary	B-Algorithm
sequences	I-Algorithm
in	O
biology	O
,	O
see	O
complementarity	O
(	O
molecular	O
biology	O
)	O
.	O
</s>
<s>
In	O
applied	O
mathematics	O
,	O
complementary	B-Algorithm
sequences	I-Algorithm
(	O
CS	O
)	O
are	O
pairs	O
of	O
sequences	O
with	O
the	O
useful	O
property	O
that	O
their	O
out-of-phase	O
aperiodic	O
autocorrelation	O
coefficients	O
sum	O
to	O
zero	O
.	O
</s>
<s>
Binary	O
complementary	B-Algorithm
sequences	I-Algorithm
were	O
first	O
introduced	O
by	O
Marcel	O
J	O
.	O
E	O
.	O
Golay	O
in	O
1949	O
.	O
</s>
<s>
In	O
1961	O
–	O
1962	O
Golay	O
gave	O
several	O
methods	O
for	O
constructing	O
sequences	O
of	O
length	O
2N	O
and	O
gave	O
examples	O
of	O
complementary	B-Algorithm
sequences	I-Algorithm
of	O
lengths	O
10	O
and	O
26	O
.	O
</s>
<s>
Later	O
the	O
theory	O
of	O
complementary	B-Algorithm
sequences	I-Algorithm
was	O
generalized	O
by	O
other	O
authors	O
to	O
polyphase	O
complementary	B-Algorithm
sequences	I-Algorithm
,	O
multilevel	O
complementary	B-Algorithm
sequences	I-Algorithm
,	O
and	O
arbitrary	O
complex	O
complementary	B-Algorithm
sequences	I-Algorithm
.	O
</s>
<s>
So	O
we	O
can	O
say	O
that	O
the	O
sum	O
of	O
autocorrelation	O
functions	O
of	O
complementary	B-Algorithm
sequences	I-Algorithm
is	O
a	O
delta	O
function	O
,	O
which	O
is	O
an	O
ideal	O
autocorrelation	O
for	O
many	O
applications	O
like	O
radar	B-Application
pulse	B-Algorithm
compression	I-Algorithm
and	O
spread	B-Architecture
spectrum	I-Architecture
telecommunications	O
.	O
</s>
<s>
Complementary	B-Algorithm
sequences	I-Algorithm
have	O
complementary	O
spectra	O
.	O
</s>
<s>
As	O
the	O
autocorrelation	O
function	O
and	O
the	O
power	O
spectra	O
form	O
a	O
Fourier	O
pair	O
,	O
complementary	B-Algorithm
sequences	I-Algorithm
also	O
have	O
complementary	O
spectra	O
.	O
</s>
<s>
Sa	O
and	O
Sb	O
are	O
defined	O
as	O
a	O
squared	O
magnitude	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
sequences	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
a	O
direct	O
DFT	O
of	O
the	O
sequences	O
,	O
it	O
can	O
be	O
a	O
DFT	O
of	O
zero	O
padded	O
sequences	O
or	O
it	O
can	O
be	O
a	O
continuous	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
of	O
the	O
sequences	O
which	O
is	O
equivalent	O
to	O
the	O
Z	B-Algorithm
transform	I-Algorithm
for	O
.	O
</s>
<s>
A	O
new	O
pair	O
of	O
complementary	B-Algorithm
sequences	I-Algorithm
can	O
be	O
formed	O
as	O
[	O
a	O
b ]	O
and	O
 [ a−b ] 	O
where	O
 [ .. ] 	O
denotes	O
concatenation	O
and	O
a	O
and	O
b	O
are	O
a	O
pair	O
of	O
CS	O
;	O
</s>
<s>
Examples	O
include	O
the	O
Shapiro	O
polynomials	O
,	O
which	O
give	O
rise	O
to	O
complementary	B-Algorithm
sequences	I-Algorithm
of	O
length	O
a	O
power	O
of	O
two	O
.	O
</s>
<s>
coded	O
aperture	O
masks	O
are	O
designed	O
using	O
a	O
2-dimensional	O
generalization	O
of	O
complementary	B-Algorithm
sequences	I-Algorithm
.	O
</s>
