<s>
In	O
mathematics	O
and	O
signal	O
processing	O
,	O
the	O
Z-transform	B-Algorithm
converts	O
a	O
discrete-time	O
signal	O
,	O
which	O
is	O
a	O
sequence	O
of	O
real	O
or	O
complex	O
numbers	O
,	O
into	O
a	O
complex	O
frequency-domain	O
(	O
z-domain	B-Algorithm
or	O
z-plane	O
)	O
representation	O
.	O
</s>
<s>
Whereas	O
the	O
continuous-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
is	O
evaluated	O
on	O
the	O
Laplace	O
s-domain	O
'	O
s	O
imaginary	O
line	O
,	O
the	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
evaluated	O
over	O
the	O
unit	O
circle	O
of	O
the	O
z-domain	B-Algorithm
.	O
</s>
<s>
What	O
is	O
roughly	O
the	O
s-domain	O
'	O
s	O
left	O
half-plane	O
,	O
is	O
now	O
the	O
inside	O
of	O
the	O
complex	O
unit	O
circle	O
;	O
what	O
is	O
the	O
z-domain	B-Algorithm
'	O
s	O
outside	O
of	O
the	O
unit	O
circle	O
,	O
roughly	O
corresponds	O
to	O
the	O
right	O
half-plane	O
of	O
the	O
s-domain	O
.	O
</s>
<s>
One	O
of	O
the	O
means	O
of	O
designing	O
digital	O
filters	O
is	O
to	O
take	O
analog	O
designs	O
,	O
subject	O
them	O
to	O
a	O
bilinear	O
transform	O
which	O
maps	O
them	O
from	O
the	O
s-domain	O
to	O
the	O
z-domain	B-Algorithm
,	O
and	O
then	O
produce	O
the	O
digital	O
filter	O
by	O
inspection	O
,	O
manipulation	O
,	O
or	O
numerical	O
approximation	O
.	O
</s>
<s>
The	O
basic	O
idea	O
now	O
known	O
as	O
the	O
Z-transform	B-Algorithm
was	O
known	O
to	O
Laplace	O
,	O
and	O
it	O
was	O
re-introduced	O
in	O
1947	O
by	O
W	O
.	O
Hurewicz	O
and	O
others	O
as	O
a	O
way	O
to	O
treat	O
sampled-data	O
control	O
systems	O
used	O
with	O
radar	O
.	O
</s>
<s>
It	O
was	O
later	O
dubbed	O
"	O
the	O
z-transform	B-Algorithm
"	O
by	O
Ragazzini	O
and	O
Zadeh	O
in	O
the	O
sampled-data	O
control	O
group	O
at	O
Columbia	O
University	O
in	O
1952	O
.	O
</s>
<s>
The	O
modified	O
or	O
advanced	B-Algorithm
Z-transform	I-Algorithm
was	O
later	O
developed	O
and	O
popularized	O
by	O
E	O
.	O
I	O
.	O
</s>
<s>
The	O
idea	O
contained	O
within	O
the	O
Z-transform	B-Algorithm
is	O
also	O
known	O
in	O
mathematical	O
literature	O
as	O
the	O
method	O
of	O
generating	O
functions	O
which	O
can	O
be	O
traced	O
back	O
as	O
early	O
as	O
1730	O
when	O
it	O
was	O
introduced	O
by	O
de	O
Moivre	O
in	O
conjunction	O
with	O
probability	O
theory	O
.	O
</s>
<s>
From	O
a	O
mathematical	O
view	O
the	O
Z-transform	B-Algorithm
can	O
also	O
be	O
viewed	O
as	O
a	O
Laurent	O
series	O
where	O
one	O
views	O
the	O
sequence	O
of	O
numbers	O
under	O
consideration	O
as	O
the	O
(	O
Laurent	O
)	O
expansion	O
of	O
an	O
analytic	O
function	O
.	O
</s>
<s>
The	O
Z-transform	B-Algorithm
can	O
be	O
defined	O
as	O
either	O
a	O
one-sided	O
or	O
two-sided	O
transform	O
.	O
</s>
<s>
(	O
Just	O
like	O
we	O
have	O
the	O
one-sided	O
Laplace	O
transform	O
and	O
the	O
two-sided	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
.	O
)	O
</s>
<s>
In	O
signal	O
processing	O
,	O
this	O
definition	O
can	O
be	O
used	O
to	O
evaluate	O
the	O
Z-transform	B-Algorithm
of	O
the	O
unit	O
impulse	O
response	O
of	O
a	O
discrete-time	O
causal	O
system	O
.	O
</s>
<s>
An	O
important	O
example	O
of	O
the	O
unilateral	O
Z-transform	B-Algorithm
is	O
the	O
probability-generating	O
function	O
,	O
where	O
the	O
component	O
is	O
the	O
probability	O
that	O
a	O
discrete	O
random	O
variable	O
takes	O
the	O
value	O
,	O
and	O
the	O
function	O
is	O
usually	O
written	O
as	O
in	O
terms	O
of	O
.	O
</s>
<s>
The	O
properties	O
of	O
Z-transforms	B-Algorithm
(	O
below	O
)	O
have	O
useful	O
interpretations	O
in	O
the	O
context	O
of	O
probability	O
theory	O
.	O
</s>
<s>
With	O
this	O
contour	O
,	O
the	O
inverse	O
Z-transform	B-Algorithm
simplifies	O
to	O
the	O
inverse	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
or	O
Fourier	O
series	O
,	O
of	O
the	O
periodic	O
values	O
of	O
the	O
Z-transform	B-Algorithm
around	O
the	O
unit	O
circle	O
:	O
</s>
<s>
The	O
Z-transform	B-Algorithm
with	O
a	O
finite	O
range	O
of	O
n	O
and	O
a	O
finite	O
number	O
of	O
uniformly	O
spaced	O
z	O
values	O
can	O
be	O
computed	O
efficiently	O
via	O
Bluestein	O
's	O
FFT	O
algorithm	O
.	O
</s>
<s>
The	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
DTFT	O
)	O
—	O
not	O
to	O
be	O
confused	O
with	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
DFT	O
)	O
—	O
is	O
a	O
special	O
case	O
of	O
such	O
a	O
Z-transform	B-Algorithm
obtained	O
by	O
restricting	O
z	O
to	O
lie	O
on	O
the	O
unit	O
circle	O
.	O
</s>
<s>
The	O
region	O
of	O
convergence	O
(	O
ROC	O
)	O
is	O
the	O
set	O
of	O
points	O
in	O
the	O
complex	O
plane	O
for	O
which	O
the	O
Z-transform	B-Algorithm
summation	O
converges	O
.	O
</s>
<s>
Examples	O
2	O
&	O
3	O
clearly	O
show	O
that	O
the	O
Z-transform	B-Algorithm
X(z )	O
of	O
x[n]	O
is	O
unique	O
when	O
and	O
only	O
when	O
specifying	O
the	O
ROC	O
.	O
</s>
<s>
Creating	O
the	O
pole	B-Algorithm
–	I-Algorithm
zero	I-Algorithm
plot	I-Algorithm
for	O
the	O
causal	O
and	O
anticausal	O
case	O
show	O
that	O
the	O
ROC	O
for	O
either	O
case	O
does	O
not	O
include	O
the	O
pole	O
that	O
is	O
at	O
0.5	O
.	O
</s>
<s>
Let	O
us	O
assume	O
we	O
are	O
provided	O
a	O
Z-transform	B-Algorithm
of	O
a	O
system	O
without	O
a	O
ROC	O
(	O
i.e.	O
,	O
an	O
ambiguous	O
x[n]	O
)	O
.	O
</s>
<s>
Bilateral	B-Algorithm
Z-transform	I-Algorithm
:	O
</s>
<s>
Unilateral	O
Z-transform	B-Algorithm
:	O
</s>
<s>
which	O
is	O
also	O
known	O
as	O
the	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
DTFT	O
)	O
of	O
the	O
sequence	O
.	O
</s>
<s>
This	O
2-periodic	O
function	O
is	O
the	O
periodic	B-Algorithm
summation	I-Algorithm
of	O
a	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
which	O
makes	O
it	O
a	O
widely	O
used	O
analysis	O
tool	O
.	O
</s>
<s>
To	O
understand	O
this	O
,	O
let	O
be	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
any	O
function	O
,	O
,	O
whose	O
samples	O
at	O
some	O
interval	O
,	O
T	O
,	O
equal	O
the	O
x[n]	O
sequence	O
.	O
</s>
<s>
And	O
now	O
,	O
with	O
the	O
substitution	O
can	O
be	O
expressed	O
in	O
terms	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
X( •	O
)	O
:	O
</s>
<s>
Due	O
to	O
periodicity	O
,	O
there	O
are	O
only	O
a	O
finite	O
number	O
of	O
unique	O
amplitudes	O
,	O
which	O
are	O
readily	O
computed	O
by	O
the	O
much	O
simpler	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
DFT	O
)	O
.	O
</s>
<s>
The	O
bilinear	O
transform	O
can	O
be	O
used	O
to	O
convert	O
continuous-time	O
filters	O
(	O
represented	O
in	O
the	O
Laplace	O
domain	O
)	O
into	O
discrete-time	O
filters	O
(	O
represented	O
in	O
the	O
Z-domain	B-Algorithm
)	O
,	O
and	O
vice	O
versa	O
.	O
</s>
<s>
from	O
the	O
Z-domain	B-Algorithm
to	O
the	O
Laplace	O
domain	O
.	O
</s>
<s>
Through	O
the	O
bilinear	O
transformation	O
,	O
the	O
complex	O
s-plane	O
(	O
of	O
the	O
Laplace	O
transform	O
)	O
is	O
mapped	O
to	O
the	O
complex	O
z-plane	O
(	O
of	O
the	O
z-transform	B-Algorithm
)	O
.	O
</s>
<s>
As	O
such	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
which	O
is	O
the	O
Laplace	O
transform	O
evaluated	O
on	O
the	O
axis	O
)	O
becomes	O
the	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
This	O
assumes	O
that	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
exists	O
;	O
i.e.	O
,	O
that	O
the	O
axis	O
is	O
in	O
the	O
region	O
of	O
convergence	O
of	O
the	O
Laplace	O
transform	O
.	O
</s>
<s>
Given	O
a	O
one-sided	O
Z-transform	B-Algorithm
,	O
X(z )	O
,	O
of	O
a	O
time-sampled	O
function	O
,	O
the	O
corresponding	O
starred	B-Algorithm
transform	I-Algorithm
produces	O
a	O
Laplace	O
transform	O
and	O
restores	O
the	O
dependence	O
on	O
sampling	O
parameter	O
,	O
T	O
:	O
</s>
<s>
The	O
zeros	O
and	O
poles	O
are	O
commonly	O
complex	O
and	O
when	O
plotted	O
on	O
the	O
complex	O
plane	O
(	O
z-plane	O
)	O
it	O
is	O
called	O
the	O
pole	B-Algorithm
–	I-Algorithm
zero	I-Algorithm
plot	I-Algorithm
.	O
</s>
<s>
By	O
performing	O
partial	O
fraction	O
decomposition	O
on	O
Y(z )	O
and	O
then	O
taking	O
the	O
inverse	O
Z-transform	B-Algorithm
the	O
output	O
y[n]	O
can	O
be	O
found	O
.	O
</s>
<s>
In	O
practice	O
,	O
it	O
is	O
often	O
useful	O
to	O
fractionally	O
decompose	O
before	O
multiplying	O
that	O
quantity	O
by	O
z	O
to	O
generate	O
a	O
form	O
of	O
Y(z )	O
which	O
has	O
terms	O
with	O
easily	O
computable	O
inverse	O
Z-transforms	B-Algorithm
.	O
</s>
