Digital Signal Processing Reference
In-Depth Information
3.2 THEORY OF SAMPLING
Let us first choose a continuous-time (analog) function
x
a
(t)
that can be repre-
sented by its Fourier transform
X
a
(j )
1
∞
x
a
(t)e
−
jt
dt
X
a
(j )
=
(3.1)
−∞
whereas the inverse Fourier transform of
X
a
(j )
is given by
2
∞
1
2
π
X
a
(j )e
jt
d
x
a
(t)
=
(3.2)
−∞
Now we generate a discrete-time sequence
x(nT )
by sampling
x
a
(t)
with a
sampling period
T
.Sowehave
x(nT )
=
x
a
(t)
|
t
=
nT
, and substituting
t
=
nT
in
(3.2), we can write
∞
1
2
π
X
a
(j )e
jnT
d
x
a
(nT )
=
x(nT )
=
(3.3)
−∞
The
z
transform of this discrete-time sequence is
3
∞
x(nT )z
−
n
X(z)
=
(3.4)
n
=−∞
e
jωT
,weget
and evaluating it on the unit circle in the
z
plane; thus, when
z
=
∞
X(e
jωT
)
x(nT )e
−
jωnT
=
(3.5)
n
=−∞
Next we consider
h(nT )
as the unit impulse response of a linear, time-
invariant, discrete-time system and the input
x(nT )
to the system as
e
jωnT
.Then
the output
y(nT )
is obtained by convolution as follows:
∞
e
jω(nT
−
kT )
h(kT )
y(nT )
=
k
=−∞
∞
∞
e
jωnT
e
−
jωkT
h(kT )
e
jωnT
h(kT )e
−
jωkT
=
=
(3.6)
k
=−∞
k
=−∞
1
The material in this section is adapted from a section with the same heading, in the author's book
Magnitude and Delay Approximation of 1-D and 2-D Digital Filters
[1], with permission from the
publisher, Springer-Verlag.
2
We have chosen
(measured in radians per second) to denote the frequency variable of an analog
function in this section and will choose the same symbol to represent the frequency to represent the
frequency response of a lowpass, normalized, prototype analog filter in Chapter 5.
3
Here we have used the bilateral
z
transform of the DT sequence, since we have assumed that it
is defined for
−∞
<n<
∞
in general. But the theory of bilateral
z
transform is not discussed in
this topic.
Search WWH ::
Custom Search