Digital Signal Processing Reference
In-Depth Information
9.4.1 Local Interpolation
The simplest interpolation schemes create a continuous-time function
x
(
t
)
from a discrete-time sequence
x
[
n
]
, by setting
x
(
t
)
to be equal to
x
[
n
]
for
l
g
r
,
y
i
d
.
,
©
,
L
s
t
to be some linear combination of neighboring
sequence values when
t
lies in between interpolation instants. In general,
the local interpolation schemes can be expressed by the following formula:
=
nT
s
and by setting
x
(
t
)
I
t
∞
−
nT
s
T
s
x
(
t
)=
x
[
n
]
(9.11)
n
=
−∞
where
I
(
t
)
is called the interpolation function (for linear functions the no-
tation
I
N
is used and the subscript
N
indicates how many discrete-time
samples, besides the current one, enter into the computation of the inter-
polated values for
x
(
t
)
). The interpolation function must satisfy the funda-
mental
interpolation properties
:
I
(
t
)
(
0
)=
1
(9.12)
I
(
k
)=
0 r
k
∈
\{
0
}
where the second requirement implies that, no matter what the support of
I
is, its values should not affect other interpolation instants. By changing
the function
I
(
t
)
, we can change the type of interpolation and the properties
of the interpolated signal
x
(
t
)
.
Note that (9.11) can be interpreted either simply as a linear combina-
tion of shifted interpolation functions or, more interestingly, as a “mixed do-
main” convolution product, where we are convolving a discrete-time signal
x
(
t
)
[
n
]
with a continuous-time “impulse response”
I
(
t
)
scaled in time by the
interpolation period
T
s
.
Zero-Order Hold.
The simplest approach for the interpolating function
is the piecewise-constant interpolation; here the continuous-time signal is
kept constant between discrete sample values, yielding
for
n
T
s
n
T
s
1
2
1
2
x
(
t
)=
x
[
n
]
,
−
≤
t
<
+
and an example is shown in Figure 9.1; it is apparent that the resulting func-
tion is far from smooth since the interpolated function is discontinuous.
The interpolation function is simply:
I
0
(
t
)=
rect
(
t
)
and the values of
x
(
t
)
depend only on the current discrete-time sample value.
