<s>
Staggered	B-Algorithm
tuning	I-Algorithm
is	O
a	O
technique	O
used	O
in	O
the	O
design	O
of	O
multi-stage	O
tuned	B-Algorithm
amplifiers	I-Algorithm
whereby	O
each	O
stage	O
is	O
tuned	O
to	O
a	O
slightly	O
different	O
frequency	O
.	O
</s>
<s>
In	O
comparison	O
to	O
synchronous	B-Algorithm
tuning	I-Algorithm
(	O
where	O
each	O
stage	O
is	O
tuned	O
identically	O
)	O
it	O
produces	O
a	O
wider	O
bandwidth	B-Algorithm
at	O
the	O
expense	O
of	O
reduced	O
gain	O
.	O
</s>
<s>
It	O
also	O
produces	O
a	O
sharper	O
transition	B-Algorithm
from	O
the	O
passband	B-Algorithm
to	O
the	O
stopband	O
.	O
</s>
<s>
Both	O
staggered	B-Algorithm
tuning	I-Algorithm
and	O
synchronous	B-Algorithm
tuning	I-Algorithm
circuits	O
are	O
easier	O
to	O
tune	O
and	O
manufacture	O
than	O
many	O
other	O
filter	O
types	O
.	O
</s>
<s>
Applications	O
include	O
television	O
IF	O
amplifiers	O
(	O
mostly	O
20th	O
century	O
receivers	O
)	O
and	O
wireless	B-Architecture
LAN	I-Architecture
.	O
</s>
<s>
Staggered	B-Algorithm
tuning	I-Algorithm
improves	O
the	O
bandwidth	B-Algorithm
of	O
a	O
multi-stage	O
tuned	B-Algorithm
amplifier	I-Algorithm
at	O
the	O
expense	O
of	O
the	O
overall	O
gain	O
.	O
</s>
<s>
Staggered	B-Algorithm
tuning	I-Algorithm
also	O
increases	O
the	O
steepness	O
of	O
passband	B-Algorithm
skirts	B-Algorithm
and	O
hence	O
improves	O
selectivity	O
.	O
</s>
<s>
The	O
value	O
of	O
staggered	B-Algorithm
tuning	I-Algorithm
is	O
best	O
explained	O
by	O
first	O
looking	O
at	O
the	O
shortcomings	O
of	O
tuning	O
every	O
stage	O
identically	O
.	O
</s>
<s>
This	O
method	O
is	O
called	O
synchronous	B-Algorithm
tuning	I-Algorithm
.	O
</s>
<s>
Each	O
stage	O
of	O
the	O
amplifier	O
will	O
reduce	O
the	O
bandwidth	B-Algorithm
.	O
</s>
<s>
Thus	O
the	O
bandwidth	B-Algorithm
becomes	O
progressively	O
narrower	O
with	O
each	O
additional	O
stage	O
.	O
</s>
<s>
The	O
fractional	O
bandwidth	B-Algorithm
of	O
an	O
LC	O
circuit	O
is	O
given	O
by	O
,	O
</s>
<s>
The	O
bandwidth	B-Algorithm
is	O
thus	O
reduced	O
by	O
a	O
factor	O
of	O
.	O
</s>
<s>
Thus	O
,	O
the	O
four	O
stage	O
synchronously	O
tuned	B-Algorithm
amplifier	I-Algorithm
will	O
have	O
a	O
bandwidth	B-Algorithm
of	O
only	O
19%	O
of	O
a	O
single	O
stage	O
.	O
</s>
<s>
Even	O
in	O
a	O
two-stage	O
amplifier	O
the	O
bandwidth	B-Algorithm
is	O
reduced	O
to	O
41%	O
of	O
the	O
original	O
.	O
</s>
<s>
Staggered	B-Algorithm
tuning	I-Algorithm
allows	O
the	O
bandwidth	B-Algorithm
to	O
be	O
widened	O
at	O
the	O
expense	O
of	O
overall	O
gain	O
.	O
</s>
<s>
The	O
overall	O
gain	O
is	O
reduced	O
because	O
when	O
any	O
one	O
stage	O
is	O
at	O
resonance	O
(	O
and	O
thus	O
maximum	O
gain	O
)	O
the	O
others	O
are	O
not	O
,	O
unlike	O
synchronous	B-Algorithm
tuning	I-Algorithm
where	O
all	O
stages	O
are	O
at	O
maximum	O
gain	O
at	O
the	O
same	O
frequency	O
.	O
</s>
<s>
A	O
two-stage	O
stagger-tuned	B-Algorithm
amplifier	I-Algorithm
will	O
have	O
a	O
gain	O
less	O
than	O
a	O
synchronously	O
tuned	B-Algorithm
amplifier	I-Algorithm
.	O
</s>
<s>
Even	O
in	O
a	O
design	O
that	O
is	O
intended	O
to	O
be	O
synchronously	O
tuned	O
,	O
some	O
staggered	B-Algorithm
tuning	I-Algorithm
effect	O
is	O
inevitable	O
because	O
of	O
the	O
practical	O
impossibility	O
of	O
keeping	O
all	O
tuned	O
circuits	O
perfectly	O
in	O
step	O
and	O
because	O
of	O
feedback	O
effects	O
.	O
</s>
<s>
The	O
overall	O
gain	O
of	O
a	O
synchronously	O
tuned	B-Algorithm
amplifier	I-Algorithm
will	O
always	O
be	O
less	O
than	O
the	O
theoretical	O
maximum	O
because	O
of	O
this	O
.	O
</s>
<s>
Tuned	B-Algorithm
amplifiers	I-Algorithm
such	O
as	O
the	O
one	O
illustrated	O
at	O
the	O
beginning	O
of	O
this	O
article	O
can	O
be	O
more	O
generically	O
depicted	O
as	O
a	O
chain	O
of	O
transconductance	B-Algorithm
amplifiers	I-Algorithm
each	O
loaded	O
with	O
a	O
tuned	O
circuit	O
.	O
</s>
<s>
Solving	O
this	O
for	O
ωc	O
and	O
taking	O
the	O
difference	O
between	O
the	O
two	O
positive	O
solutions	O
finds	O
the	O
bandwidth	B-Algorithm
Δω	O
,	O
</s>
<s>
and	O
the	O
fractional	O
bandwidth	B-Algorithm
B	O
,	O
</s>
<s>
It	O
is	O
desirable	O
to	O
be	O
able	O
to	O
design	O
the	O
filter	O
from	O
a	O
standard	O
low-pass	B-Algorithm
prototype	O
filter	O
of	O
the	O
required	O
specification	O
.	O
</s>
<s>
A	O
popular	O
choice	O
for	O
a	O
polynomial	O
with	O
ripple	O
is	O
the	O
Chebyshev	O
response	O
for	O
its	O
steep	O
skirt	B-Algorithm
.	O
</s>
<s>
where	O
ω'''c	O
is	O
the	O
cutoff	O
frequency	O
of	O
the	O
low-pass	B-Algorithm
prototype	O
.	O
</s>
<s>
This	O
can	O
be	O
done	O
straightforwardly	O
for	O
the	O
complete	O
filter	O
in	O
the	O
case	O
of	O
synchronously	O
tuned	B-Algorithm
amplifiers	I-Algorithm
where	O
every	O
stage	O
has	O
the	O
same	O
ω0	O
but	O
for	O
a	O
stagger-tuned	B-Algorithm
amplifier	I-Algorithm
there	O
is	O
no	O
simple	O
analytical	O
solution	O
to	O
the	O
transform	O
.	O
</s>
<s>
Stagger-tuned	O
designs	O
can	O
be	O
approached	O
instead	O
by	O
calculating	O
the	O
poles	O
of	O
a	O
low-pass	B-Algorithm
prototype	O
of	O
the	O
desired	O
form	O
(	O
e.g.	O
</s>
<s>
From	O
the	O
band-pass	O
to	O
low-pass	B-Algorithm
transform	O
given	O
above	O
,	O
an	O
expression	O
can	O
be	O
found	O
for	O
the	O
poles	O
in	O
terms	O
of	O
the	O
poles	O
of	O
the	O
low-pass	B-Algorithm
prototype	O
,	O
qk	O
,	O
</s>
<s>
This	O
expression	O
is	O
greatly	O
simplified	O
if	O
the	O
cutoff	O
frequency	O
of	O
the	O
prototype	O
,	O
ω'c	O
,	O
is	O
set	O
to	O
the	O
final	O
filter	O
bandwidth	B-Algorithm
ω0B/Qeff	O
.	O
</s>
<s>
Staggered	B-Algorithm
tuning	I-Algorithm
is	O
of	O
most	O
benefit	O
in	O
wideband	O
applications	O
.	O
</s>
