We start by analyzing the kkth
filterbank branch, analyzed in Figure 2:

The first step is to reverse the modulation and filtering
operations. To do this, we define a "modulated filter"
H
k
z
H
k
z
:

vk
n=∑ihixn−iej2πMk(n−i)=
∑ihie(−j)2πMkixn−i
ej2πMkn=
∑i
hk
ixn−i
ej2πMkn
vk
n
i
h
i
x
n
i
j
2
M
k
n
i
i
h
i
j
2
M
k
i
x
n
i
j
2
M
k
n
i
hk
i
x
n
i
j
2
M
k
n

(1)
xn
xn
is convolved with the modulated filter and that the
filter output is modulated. This is illustrated in

Figure 3:

Notice that the only modulator outputs not discarded by the
downsampler are those with time index
n=mM
n
m
M
for
m∈Z
m
. For these outputs, the modulator has the value
ej2πMkmM=1
j
2
M
k
m
M
1
,
and thus it can be ignored. The resulting system is portrayed
by:

Next we would like to reverse the order of filtering and
downsampling. To apply the Noble identity, we must decompose
H
k
z
H
k
z
into a bank of upsampled polyphase filters. The technique
used to derive polyphase decimation can be employed here:

Hk
z=∑
n
=−∞∞
hk
nz−n=∑
l
=0M−1∑
m
=−∞∞
h
k
mM+lz−(mM+l)
Hk
z
n
hk
n
z
n
l
0
M
1
m
h
k
m
M
l
z
m
M
l

(2)
Noting the fact that the

llth
polyphase filter has impulse response:

h
k
mM+l=hmM+le-j2πM(k(mM+l))=hmM+le-j2πMkl=
p
l
me-j2πMkl
h
k
m
M
l
h
m
M
l
-j
2
M
k
m
M
l
h
m
M
l
-j
2
M
k
l
p
l
m
-j
2
M
k
l
where

p
l
m
p
l
m
is the

llth polyphase filter
defined by the original (unmodulated) lowpass filter

Hz
Hz
, we obtain

Hk
z=∑
l
=0M−1∑
m
=−∞∞
p
l
me-j2πMklz−(mM+l)=∑
l
=0M−1e-j2πMklz−l∑
m
=−∞∞
p
l
mzM−m=∑
l
=0M−1e-j2πMklz−l
P
l
zM
Hk
z
l
0
M
1
m
p
l
m
-j
2
M
k
l
z
m
M
l
l
0
M
1
-j
2
M
k
l
z
l
m
p
l
m
z
M
m
l
0
M
1
-j
2
M
k
l
z
l
P
l
z
M

(3) kkth filterbank branch (now
containing

MM polyphase branches)
is in

Figure 5:

Because it is a linear operator, the downsampler can be moved
through the adders and the (time-invariant) scalings
e(−j)2πMkl
j
2
M
k
l
. Finally, the Noble identity is employed to
exchange the filtering and downsampling. The
kkth filterbank branch becomes:

Observe that the polyphase outputs
v
l
m
v
l
m
l=0…M−1
v
l
m
l
0
…
M
1
v
l
m
are identical for each filterbank branch, while the scalings
e(−j)2πMkl
l=0…M−1
j
2
M
k
l
l
0
…
M
1
once. Using these outputs we can compute the branch
outputs via

y
k
m=∑
l
=0M−1
v
l
me(−j)2πMkl
y
k
m
l
0
M
1
v
l
m
j
2
M
k
l

(4)
From the previous equation it is clear that

y
k
m
y
k
m
corresponds to the

kkth DFT
output given the

MM-point input
sequence

v
l
m
l=0…M−1
v
l
m
l
0
…
M
1
. Thus the

MM
filterbank branches can be computed in parallel by taking an

MM-point DFT of the

MM polyphase outputs (see

Figure 7).

The polyphase/DFT synthesis bank can be derived in a similar
manner.