Interpolation is the process of upsampling and filtering a signal to increase its effective sampling rate. To be more specific, say that
xm
x
m
is an (unaliased) TT-sampled version of
x
c
t
x
c
t
and
vn
v
n
is an LL-upsampled version version of
xm
x
m
.
If we filter
vn
v
n
with an ideal πLL-bandwidth lowpass filter (with DC gain LL) to obtain
yn
y
n
, then
yn
y
n
will be a TL TL -sampled version of
x
c
t
x
c
t
. This process is illustrated in Figure 1.

We justify our claims about interpolation using frequency-domain arguments.
From the sampling theorem, we know that TT-
sampling
x
c
t
x
c
t
to create
xn
x
n
yields

Xejω=1T∑k
X
c
jω−2πkT
X
ω
1
T
k
k
X
c
ω
2
k
T

(1)
After upsampling by factor

LL,

Equation 1 implies

Vejω=1T∑k
X
c
jωL−2πkT=1T∑k
X
c
jω−2πLkTL
V
ω
1
T
k
k
X
c
ω
L
2
k
T
1
T
k
k
X
c
ω
2
L
k
T
L
Lowpass filtering with cutoff

πL
L and gain

LL yields

Yejω=LT∑kL∈Z
X
c
jω−2πLkTL=LT∑l
X
c
jω−2πlTL
Y
ω
L
T
kL
k
L
X
c
ω
2
L
k
T
L
L
T
l
l
X
c
ω
2
l
T
L
since the spectral copies with indices other than

k=lL
klL
(for

l∈Z
l) are removed.
Clearly, this process yields a

TL
TL-shaped version of

x
c
t
x
c
t
.

Figure 2 illustrates these frequency-domain arguments for

L=2L2.