Digital Signal Processing Reference
In-Depth Information
1
1
l
g
r
,
y
i
d
.
,
©
,
L
s
0
0
-1.5
-1.0
-0.5
0 .5 .0 .5
-1.5
-1.0
-0.5
0
0.5
1.0
1.5
Figure 9.3
Interpolation functions for the zero-order (left) and first-order interpo-
lators (right).
Higher-Order Interpolators.
The zero- and first-order interpolators are
widely used in practical circuits due to their extreme simplicity. Note that
the interpolating functions
I
0
are alsol knows as the B-spline
functions of order zero and one respectively. These schemes can be ex-
tended to higher order interpolation functions and, in general,
I
N
(
t
)
and
I
1
(
t
)
is a
N
-
th order polynomial in
t
. The advantage of the local interpolation schemes
is that, for small
N
, they can easily be implemented in practice as
causal
interpolation schemes (locality is akin to FIR filtering); their disadvantage
is that, because of the locality, their
N
-th derivative is discontinuous. This
discontinuity represents a lack of smoothness in the interpolated function;
from a spectral point of view this corresponds to a high frequency energy
content, which is usually undesirable.
(
t
)
9.4.2 Polynomial Interpolation
The lack of smoothness of local interpolations is easily eliminated when we
need to interpolate just a
finite
number of discrete-time samples. In fact, in
this case the task becomes a classic polynomial interpolation problem for
which the optimal solution has been known for a long time under the name
of
Lagrange interpolation
. Note that a polynomial interpolating a finite set
of samples is amaximally smooth function in the sense that it is continuous,
together with all its derivatives.
Consider a length
(
2
N
+
1
)
discrete-time signal
x
[
n
]
,with
n
=
−
N
,...,
N
.
Associate to each sample an abscissa
t
n
nT
s
; we know from basic algebra
that there is one and only one polynomial
P
=
(
t
)
of degree 2
N
which passes
]
and this polynomial is the Lagrange
interpolator. The coefficients of the polynomial could be found by solving
the set of 2
N
1pairs
t
n
,
x
through all the 2
N
+
[
n
+
1equations:
P
]
n
=
−N
,...,
N
(
)=
[
t
n
x
n
(9.13)
