<s>
A	O
Butler	B-Algorithm
matrix	I-Algorithm
is	O
a	O
beamforming	B-Algorithm
network	O
used	O
to	O
feed	O
a	O
phased	O
array	O
of	O
antenna	O
elements	O
.	O
</s>
<s>
It	O
consists	O
of	O
an	O
matrix	O
(	O
some	O
power	O
of	O
two	O
)	O
with	O
hybrid	O
couplers	O
and	O
fixed-value	O
phase	B-Algorithm
shifters	I-Algorithm
at	O
the	O
junctions	O
.	O
</s>
<s>
The	O
Butler	B-Algorithm
matrix	I-Algorithm
feeds	O
power	O
to	O
the	O
elements	O
with	O
a	O
progressive	O
phase	O
difference	O
between	O
elements	O
such	O
that	O
the	O
beam	O
of	O
radio	O
transmission	O
is	O
in	O
the	O
desired	O
direction	O
.	O
</s>
<s>
Its	O
advantage	O
over	O
other	O
methods	O
of	O
angular	O
beamforming	B-Algorithm
is	O
the	O
simplicity	O
of	O
the	O
hardware	O
.	O
</s>
<s>
It	O
requires	O
far	O
fewer	O
phase	B-Algorithm
shifters	I-Algorithm
than	O
other	O
methods	O
and	O
can	O
be	O
implemented	O
in	O
microstrip	O
on	O
a	O
low-cost	O
printed	O
circuit	O
board	O
.	O
</s>
<s>
The	O
antenna	O
elements	O
fed	O
by	O
a	O
Butler	B-Algorithm
matrix	I-Algorithm
are	O
typically	O
horn	O
antennae	O
at	O
the	O
microwave	O
frequencies	O
at	O
which	O
Butler	O
matrices	O
are	O
usually	O
used	O
.	O
</s>
<s>
Horns	O
have	O
limited	O
bandwidth	B-Algorithm
and	O
more	O
complex	O
antennae	O
may	O
be	O
used	O
if	O
more	O
than	O
an	O
octave	B-Language
is	O
required	O
.	O
</s>
<s>
A	O
Butler	B-Algorithm
matrix	I-Algorithm
can	O
also	O
feed	O
a	O
circular	O
array	O
giving	O
360°	O
coverage	O
.	O
</s>
<s>
A	O
further	O
application	O
with	O
a	O
circular	O
antenna	O
array	O
is	O
to	O
produce	O
omnidirectional	O
beams	O
with	O
orthogonal	B-Application
phase-modes	O
so	O
that	O
multiple	O
mobile	O
stations	O
can	O
all	O
simultaneously	O
use	O
the	O
same	O
frequency	O
,	O
each	O
using	O
a	O
different	O
phase-mode	O
.	O
</s>
<s>
The	O
essential	O
components	O
needed	O
to	O
build	O
a	O
Butler	B-Algorithm
matrix	I-Algorithm
are	O
hybrid	O
couplers	O
and	O
fixed-value	O
phase	B-Algorithm
shifters	I-Algorithm
.	O
</s>
<s>
Additionally	O
,	O
fine	O
control	O
of	O
the	O
beam	O
direction	O
can	O
be	O
provided	O
with	O
variable	O
phase	B-Algorithm
shifters	I-Algorithm
in	O
addition	O
to	O
the	O
fixed	O
phase	B-Algorithm
shifters	I-Algorithm
.	O
</s>
<s>
By	O
using	O
the	O
variable	O
phase	B-Algorithm
shifters	I-Algorithm
in	O
combination	O
with	O
switching	O
the	O
power	O
to	O
the	O
beam	O
ports	O
,	O
a	O
continuous	O
sweep	O
of	O
the	O
beam	O
can	O
be	O
produced	O
.	O
</s>
<s>
An	O
alternative	O
which	O
allows	O
the	O
Butler	B-Algorithm
matrix	I-Algorithm
to	O
be	O
implemented	O
entirely	O
in	O
printed	O
circuit	O
form	O
,	O
and	O
thus	O
more	O
economically	O
,	O
is	O
a	O
crossover	O
in	O
the	O
form	O
of	O
a	O
branch-line	O
coupler	O
.	O
</s>
<s>
This	O
will	O
add	O
an	O
additional	O
90°	O
phase	O
shift	O
to	O
the	O
lines	O
being	O
crossed	O
,	O
but	O
this	O
can	O
be	O
compensated	O
for	O
by	O
adding	O
an	O
equivalent	O
amount	O
to	O
the	O
phase	B-Algorithm
shifters	I-Algorithm
in	O
lines	O
not	O
being	O
crossed	O
.	O
</s>
<s>
In	O
this	O
kind	O
of	O
implementation	O
,	O
the	O
phase	B-Algorithm
shifters	I-Algorithm
are	O
constructed	O
as	O
delay	O
lines	O
of	O
the	O
appropriate	O
length	O
.	O
</s>
<s>
When	O
there	O
are	O
a	O
large	O
number	O
of	O
antenna	O
elements	O
,	O
the	O
path	O
through	O
the	O
Butler	B-Algorithm
matrix	I-Algorithm
goes	O
through	O
a	O
large	O
number	O
of	O
hybrids	O
and	O
phase	B-Algorithm
shifters	I-Algorithm
.	O
</s>
<s>
Linear	O
antenna	O
arrays	O
driven	O
by	O
Butler	O
matrices	O
,	O
or	O
some	O
other	O
beam-forming	O
network	O
,	O
to	O
produce	O
a	O
scanning	O
beam	O
are	O
used	O
in	O
direction	B-Device
finding	I-Device
applications	O
.	O
</s>
<s>
These	O
need	O
to	O
allow	O
settling	B-Algorithm
time	I-Algorithm
for	O
the	O
servos	B-Algorithm
.	O
</s>
<s>
thumb|upright	O
=	O
1.5	O
|A	O
2.4	O
GHz	O
4×4	O
Butler	B-Algorithm
matrix	I-Algorithm
in	O
microstripInnok	O
et	O
al.	O
,	O
pp	O
.	O
</s>
<s>
Both	O
the	O
Blass	O
matrix	O
and	O
Butler	B-Algorithm
matrix	I-Algorithm
suffer	O
from	O
beam	O
squint	O
and	O
the	O
effect	O
limits	O
the	O
bandwidth	B-Algorithm
that	O
can	O
be	O
achieved	O
.	O
</s>
<s>
fixed	O
phase	B-Algorithm
shifters	I-Algorithm
.	O
</s>
<s>
Since	O
is	O
always	O
a	O
power	O
of	O
2	O
,	O
we	O
can	O
let	O
,	O
then	O
the	O
required	O
number	O
of	O
hybrids	O
is	O
and	O
phase	B-Algorithm
shifters	I-Algorithm
.	O
</s>
<s>
To	O
be	O
orthogonal	B-Application
(	O
that	O
is	O
,	O
not	O
interfere	O
with	O
each	O
other	O
)	O
the	O
beam	O
shapes	O
must	O
meet	O
the	O
Nyquist	O
ISI	O
criterion	O
,	O
but	O
with	O
distance	O
as	O
the	O
independent	O
variable	O
rather	O
than	O
time	O
.	O
</s>
<s>
Assuming	O
a	O
sinc	B-Algorithm
function	I-Algorithm
beam	O
shape	O
,	O
the	O
beams	O
must	O
be	O
spaced	O
so	O
that	O
their	O
crossovers	O
occur	O
at	O
of	O
their	O
peak	O
value	O
(	O
about	O
down	O
)	O
.	O
</s>
