Recall the standard interpolation procedure illustrated in
Figure 1.

Note that this procedure is computationally inefficient
because the lowpass filter operates on a sequence that is
mostly composed of zeros. Through the use of the Noble
identities, it is possible to rearrange the preceding block
diagram so that operations on zero-valued samples are avoided.

In order to apply the Noble identity for interpolation, we must
transform
Hz
H
z
into its upsampled polyphase components
H
p
zL
H
p
z
L
,
p=0…L−1
p
0
…
L
1
.

Hz=∑nnhnz−n=∑kk∑p=0L−1hkL+pz−(kL+p)
H
z
n
n
h
n
z
n
k
k
p
0
L
1
h
k
L
p
z
k
L
p

(1)
via

k≔⌊nL⌋
≔
k
n
L
,

p≔nmodL
≔
p
n
L
Hz=∑p=0L−1∑kk
h
p
kz−(kL)z−p
H
z
p
0
L1
k
k
h
p
k
z
k
L
z
p

(2)
via

h
p
k≔hkL+p
≔
h
p
k
h
k
L
p
Hz=∑p=0L−1
H
p
zLz−p
H
z
p
0
L1
H
p
z
L
z
p

(3)
Above,

⌊·⌋·
denotes the floor operator and

·modM
·
M
the modulo-

MM operator. Note that the

pthpth
polyphase filter

h
p
k
h
p
k
is
constructed by downsampling the "master filter"

hn
h
n
at offset

pp. Using the
unsampled polyphase components, the

Figure 1 diagram can be redrawn as in

Figure 2.

Applying the Noble identity for interpolation to Figure 3 yields Figure 2. The ladder of upsamplers and delays on the
right below accomplishes a form of
parallel-to-serial conversion.