<s>
Generalized	B-Algorithm
pencil-of-function	I-Algorithm
method	I-Algorithm
(	O
GPOF	B-Algorithm
)	O
,	O
also	O
known	O
as	O
matrix	B-Algorithm
pencil	I-Algorithm
method	I-Algorithm
,	O
is	O
a	O
signal	O
processing	O
technique	O
for	O
estimating	O
a	O
signal	O
or	O
extracting	O
information	O
with	O
complex	O
exponentials	O
.	O
</s>
<s>
Being	O
similar	O
to	O
Prony	B-Algorithm
and	O
original	O
pencil-of-function	O
methods	O
,	O
it	O
is	O
generally	O
preferred	O
to	O
those	O
for	O
its	O
robustness	O
and	O
computational	O
efficiency	O
.	O
</s>
<s>
The	O
method	O
was	O
originally	O
developed	O
by	O
Yingbo	O
Hua	O
and	O
Tapan	O
Sarkar	O
for	O
estimating	O
the	O
behaviour	O
of	O
electromagnetic	O
systems	O
by	O
its	O
transient	O
response	O
,	O
building	O
on	O
Sarkar	O
's	O
past	O
work	O
on	O
the	O
original	O
pencil-of-function	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
method	O
has	O
a	O
plethora	O
of	O
applications	O
in	O
electrical	O
engineering	O
,	O
particularly	O
related	O
to	O
problems	O
in	O
computational	O
electromagnetics	O
,	O
microwave	B-Algorithm
engineering	I-Algorithm
and	O
antenna	O
theory	O
.	O
</s>
<s>
is	O
the	O
signal	B-Algorithm
noise	I-Algorithm
,	O
</s>
<s>
by	O
the	O
identities	O
of	O
Z-transform	B-Algorithm
,	O
</s>
<s>
The	O
same	O
sequence	O
,	O
sampled	B-Algorithm
by	O
a	O
period	O
of	O
,	O
can	O
be	O
written	O
as	O
the	O
following	O
:	O
</s>
<s>
Generalized	B-Algorithm
pencil-of-function	I-Algorithm
estimates	O
the	O
optimal	O
and	O
'	O
s	O
.	O
</s>
<s>
and	O
are	O
diagonal	B-Algorithm
matrices	I-Algorithm
with	O
sequentially-placed	O
and	O
values	O
,	O
respectively	O
.	O
</s>
<s>
In	O
this	O
decomposition	O
,	O
and	O
are	O
unitary	B-Algorithm
matrices	I-Algorithm
with	O
respective	O
eigenvectors	O
and	O
and	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
with	O
singular	O
values	O
of	O
.	O
</s>
<s>
Superscript	O
denotes	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
.	O
</s>
<s>
and	O
are	O
the	O
maximum	O
singular	O
value	O
and	O
significant	B-Architecture
decimal	I-Architecture
digits	I-Architecture
,	O
respectively	O
.	O
</s>
<s>
For	O
a	O
data	O
with	O
significant	B-Architecture
digits	I-Architecture
accurate	O
up	O
to	O
,	O
singular	O
values	O
below	O
are	O
considered	O
noise	O
.	O
</s>
<s>
Band-pass	O
matrix	O
pencil	O
(	O
BPMP	O
)	O
method	O
is	O
a	O
modification	O
of	O
the	O
GPOF	B-Algorithm
method	I-Algorithm
via	O
FIR	O
or	O
IIR	O
band-pass	O
filters	O
.	O
</s>
<s>
GPOF	B-Algorithm
can	O
handle	O
up	O
to	O
25	O
dB	O
SNR	O
.	O
</s>
<s>
For	O
GPOF	B-Algorithm
,	O
as	O
well	O
as	O
for	O
BPMP	O
,	O
variace	O
of	O
the	O
estimates	O
approximately	O
reaches	O
Cramér	O
–	O
Rao	O
bound	O
.	O
</s>
<s>
Residues	O
of	O
the	O
complex	O
poles	O
are	O
obtained	O
through	O
the	O
least	B-Algorithm
squares	I-Algorithm
problem	I-Algorithm
:	O
</s>
<s>
The	O
method	O
is	O
generally	O
used	O
for	O
the	O
evaluation	O
of	O
Sommerfeld	O
integrals	O
in	O
discrete	O
complex	O
image	O
method	O
for	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
,	O
where	O
the	O
spectral	O
Green	O
's	O
function	O
is	O
approximated	O
as	O
a	O
sum	O
of	O
complex	O
exponentials	O
.	O
</s>
<s>
Additionally	O
,	O
the	O
method	O
is	O
used	O
in	O
antenna	O
analysis	O
,	O
S-parameter-estimation	O
in	O
microwave	B-Algorithm
integrated	I-Algorithm
circuits	I-Algorithm
,	O
wave	O
propagation	O
analysis	O
,	O
moving	O
target	O
indication	O
and	O
radar	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
.	O
</s>
