The Noble identities (illustrated in Figure 1 and Figure 2) describe
when it is possible to reverse the order of upsampling/downsampling and filtering.
We prove the Noble identities showing the equivalence of each pair of block diagrams.

The Noble identity for interpolation can be depicted as in Figure 1:

For the left side of the diagram, we have
Yz=HzL
V
1
z
Y
z
H
z
L
V
1
z
where
V
1
z=XzL
V
1
z
X
z
L
Yz=HzLXzL
Y
z
H
z
L
X
z
L
while for the right side,
Yz=
V
2
zL
Y
z
V
2
z
L
where
V
2
z=HzXz
V
2
z
H
z
X
z
Yz=HzLXzL
Y
z
H
z
L
X
z
L
Thus we have established the Noble identity for interpolation.

The Noble identity for decimation can be depicted as in Figure 2:

For the left side of the preceding diagram, we have
V
1
z=1M∑k=0M−1Xe(−j)2πMkz1M
V
1
z
1
M
k
0
M
1
X
2
M
k
z
1
M

Yz=Hz
V
1
z=Hz(1M∑k=0M−1Xe(−j)2πMkz1M)
Y
z
H
z
V
1
z
H
z
1
M
k
0
M
1
X
2
M
k
z
1
M

(1)
while for the right side,

Yz=1M∑k=0M−1
V
z
e(−j)2πMkz1M
Y
z
1
M
k
0
M
1
V
z
2
M
k
z
1
M

(2)
where

V
2
z=XzHzM
V
2
z
X
z
H
z
M
Yz=1M∑k=0M−1Xe(−j)2πMkz1MHe(−j)2πMkMzMM=Hz1M∑k=0M−1Xe(−j)2πMkz1M
Y
z
1
M
k
0
M
1
X
2
M
k
z
1
M
H
2
M
k
M
z
M
M
H
z
1
M
k
0
M
1
X
2
M
k
z
1
M

(3)
Thus we have established the Noble identity for decimation. Note
that the impulse response of

HzL
H
z
L
is the

LL-upsampled
impulse response of

Hz
H
z
.