Previously, polyphase
interpolation and decimation were derived from the Noble
identities and motivated for reasons of computational
efficiency. Here we present a different interpretation of the
(ideal) polyphase filter.

Assume that HzHz is an ideal
lowpass filter with gain L L,
cutoff π
L
L, and constant group delay of
dd:
Heiω={Le−(idω) if ω∈
−πL
πL
0 if ω∈
−π
−πL
∧
πL
π
H
ω
L
d
ω
ω
L
L
0
ω
L
L

Recall that the polyphase filters are defined as
∀p,p∈0…L−1:
h
p
k=hkL+p
p
p
0
…
L
1
h
p
k
h
k
L
p

In other words,
h
p
k
h
p
k
is an advanced
(by pp samples)
and downsampled (by factor LL)
version of
hn
h
n
(see Figure 1).

The DTFT of the
pth
pth
polyphase filter impulse response
is then

H
p
z=1L∑l=0L−1Ve(−i)2πLlz1L
H
p
z
1
L
l
0
L
1
V
2
L
l
z
1
L

(1)
where

Vz=Hzzp
Vz
H
z
z
p
H
p
z=1L∑l=0L−1e(−i)2πLlpzpLHe(−i)2πLlz1L
H
p
z
1
L
l
0
L
1
2
L
l
p
z
p
L
H
2
L
l
z
1
L

(2)
H
p
eiω=1L∑l=0L−1eiω−2πlLpHeiω−2πlL=∀ω,|ω|≤π:1L(eiωLpHeiωL)=∀ω,|ω|≤π:e(−i)d−pLω
H
p
ω
1
L
l
0
L
1
ω
2
l
L
p
H
ω
2
l
L
ω
ω
1
L
ω
L
p
H
ω
L
ω
ω
d
p
L
ω

(3)
Thus, the ideal
pthpth
polyphase filter has a constant magnitude response of one and a
constant group delay of
d−pL
d
p
L
samples. The implication is that if the input to the
pthpth
polyphase filter is the unaliased
TT-sampled representation
xn=
x
c
nT
x
n
x
c
n
T
, then the output of the filter would be the unaliased
T T-sampled representation
y
p
n=
x
c
(n−d−pL)T
y
p
n
x
c
n
d
p
L
T
(see Figure 2).

Figure 3 shows the role of polyphase
interpolation filters assume zero-delay
(
d=0
d
0
) processing. Essentially, the
pthpth
filter interpolates the waveform
pL
p
L
-way
between consecutive input samples. The
LL polyphase outputs are then
interleaved to create the output stream. Assuming that
x
c
t
x
c
t
is bandlimited to
12THz
1
2
T
Hz
, perfect polyphase filtering yields a perfectly
interpolated output. In practice, we use the casual FIR
approximations of the polyphase filters
h
p
k
h
p
k
(which which correspond to some casual FIR
approximation of the master filter
hn
h
n
).