Digital Signal Processing Reference
In-Depth Information
where we have used
T
s
=
π/
Ω
N
and defined the sinc function as
⎧
⎨
⎩
sin
(
π
x
)
x
=
0
l
g
r
,
y
i
d
.
,
©
,
L
s
sinc
(
x
)=
π
x
1
x
=
0
The sinc function is plotted in Figure 9.6. Note the following:
•
The function is symmetric, sinc
(
x
)=
sinc
(
−
x
)
.
•
The sinc function is zero for all integer values of its argument, except
in zero. This feature is called the
interpolation property
of the sinc, as
we will shortly see more in detail.
•
The sinc function is square integrable (it has finite energy) but it is
not absolutely integrable (hence the discontinuity of its Fourier trans-
form).
•
The decay is slow, asymptotic to 1
/
x
.
•
The scaled sinc function represents the impulse response of an ideal,
continuous-time lowpass filter with cutoff frequency
Ω
N
.
9.4 Interpolation
Interpolation is a procedure whereby we convert a discrete-time sequence
x
[
]
(
)
. Since this can be done in an arbi-
trary number of ways, we have to start by formulating some requirements
on the resulting signal. At the heart of the interpolating procedure, as we
have mentioned, is the association of a physical time duration
T
s
to the in-
terval between the samples in the discrete-time sequence. An intuitive re-
quirement on the interpolated function is that its values at multiples of
T
s
be equal to the corresponding points of the discrete-time sequence, i.e.
n
to a continuous-time function
x
t
t
=
nT
s
=
x
(
t
)
x
[
n
]
The interpolation problem now reduces to “filling the gaps” between these
instants.
