Digital Signal Processing Reference
In-Depth Information
Here, we have used the change of variable
m
k
.Wecannowinvoke
Euler's infinite product expansion for the sine function
=
n
−
1
k
2
∞
2
−
τ
l
g
r
,
y
i
d
.
,
©
,
L
s
sin
(
πτ
)=(
πτ
)
k
=
1
(whose derivation is in the appendix) to finally obtain
sinc
t
−
nT
s
T
s
L
(
N
)
(
)=
lim
N
t
(9.17)
n
→∞
The convergence of the Lagrange polynomial
L
(
N
)
0
to the sinc function is
illustrated in Figure 9.6. Note that, now, as the number of points becomes
infinite, the Lagrange polynomials converge to shifts of the
same
prototype
function, i.e. the sinc; therefore, the interpolation formula can be expressed
as in (9.11) with
I
(
t
)
(
t
)=
sinc
(
t
)
; indeed, if we consider an infinite sequence
x
[
n
]
and apply the Lagrange interpolation formula (9.15), we obtain:
sinc
t
∞
−
nT
s
T
s
(
)=
[
]
x
t
x
n
(9.18)
n
=
−∞
1.0
0.5
0
-10
-5
0
5
10
Figure 9.6
A portion of the sinc function and its Lagrange approximation
L
(
100
)
0
(
t
)
(light gray).
Spectral Properties of the Sinc Interpolation.
The sinc interpolation
of a discrete-time sequence gives rise to a strictly bandlimited continuous-
time function. If the DTFT
X
e
j
ω
)
of the discrete-time sequence exists, the
spectrum of the interpolated function
X
(
(
j
Ω)
can be obtained as follows:
∞
sinc
t
e
−j
Ω
t
dt
∞
−
nT
s
T
s
X
(
j
Ω)=
x
[
n
]
−∞
n
=
−∞
∞
sinc
t
e
−j
Ω
t
dt
∞
−
nT
s
T
s
=
x
[
n
]
n
=
−∞
−∞
