It contains a simple diagram,alias-free equation derivation, polyphase representation, advantages, and disadvantages.

1. M - Channel QMF Filter Bank

2. The basic structure of M channel QMF filter bank is shown
below:-

3. I consider the generalization of the QMF bank to M channels. The figure given
previously illustrate the structure of an M channel QMF banks,where x(n) is the
input to the analysis section,𝑥𝑘
(
𝑎
)(n), 0 ≤ k ≤ M-1, are the outputs of th analysis
filters,𝑥𝑘
(
𝑠
)(n), 0 ≤ k ≤ M-1, are the inputs to the synthesis filters and X(n) is the
output of the synthesis section.
The M outputs from analysis filters may be expressed in the z-transform
domain as,
𝑥𝑘
(
𝑎
)
(z) =
1
𝑀
𝑚
=
0
𝑀
−
1 𝐻𝑘
(𝑧
1
/
𝑀
𝑊
𝑀
𝑚
)𝑥(𝑧
1
/
𝑀
𝑊
𝑀
𝑚
),
0 ≤ k ≤ M-1 ...(1)
where 𝑊
𝑀 = 𝑒
−
𝑗2𝜋
/
𝑀
. The output from the synthesis section is
X(z) = 𝑘
=
0
𝑀
−
1 𝑥𝑘
(
𝑠
)
(𝑧
𝑀
)𝐺𝑘(z) ...(2)
As in the case of the two channel QMF bank,we set 𝑥𝑘
(
𝑎
)
(z) = 𝑥𝑘
(
𝑠
)
(z). Then , if we
substitute (1) into (2) ,we obtain,

4. X(z) = 𝑘=0
𝑀−1
𝐺𝑘 (z)[
1
𝑀 𝑚=0
𝑀−1
𝐻𝑘(𝑧𝑊𝑀
𝑚
)𝑥(𝑧𝑊𝑀
𝑚
)]
= 𝑚=0
𝑀−1
[
1
𝑀 𝑘=0
𝑀−1
𝐺𝑘 (𝑧)𝐻𝑘(𝑧𝑊𝑀
𝑚
)]𝑥(𝑧𝑊𝑀
𝑚
) ....(3)
It is convenient to define the term in the bracket as,
Rm(z) =
1
𝑀 𝑘=0
𝑀−1
𝐺𝑘𝐻𝑘(z𝑊𝑀
𝑚
), 0 ≤ m ≤ M-1 ....(4)
Then (3) can be expressed as,
X(z) = 𝑚=0
𝑀−1
𝑅𝑚(𝑧)𝑥(𝑧𝑊𝑁
𝑚
)
= R0(z)𝑥(z)+ 𝑚=1
𝑀−1
𝑅𝑚(𝑧)(z𝑊𝑀
𝑚
) ....(5)
We note the first term in (5) is the alias-free component of the QMF
bank and the second term is the aliasing component.

5. • From (5), it is clear that aliasing is eliminating by forcing the condition,
Rm(z) = 0, 1 ≤ m ≤ M-1 ....(6)
With the elimination of the alias terms, the M-channel QMF bank becomes
a linear time-invariant system that satisfies the input-output relation,
X(z) = R0(z)𝑥(z) ....(7)
where
R0(z) =
1
𝑀 𝑘=0
𝑀−1
𝐻𝑘(𝑧)𝐺𝑘(z) ....(8)
Then, the condition for a perfect reconstruction M-channel QMF bank
becomes
R0(z) = Cz-k ....(9)
Where C and k are positive constants

6. Polyphase form of the M-channel QMF Bank
• An efficient implementation of the M-channel QMF bank is achieved by
employing polyphase filters. To obtain the polyphase form for the
analysis filter bank,the kth filter Hk(z) is represented as
• Hk(z) = 𝑚=0
𝑀−1
𝑧−𝑚
𝑃𝑘𝑚(𝑧), 1 ≤ k ≤ M-1 ....(10)
We may express the equation for the M polyphase filter in matrix form as
H(z) = P(zM)a(z) ....(11)
where
H(z) = [H0(z) H1(z) ... HM-1(z)]𝑡
a(z) = [1 z-1 z-2 ... z-(M-1)]𝑡 ....(12)

7. • P(z) =
𝑃00(𝑧) 𝑃01(𝑧) ⋯ 𝑃0𝑀(𝑧)
⋮ ⋱ ⋮
𝑃𝑀−1 0(𝑧) 𝑃𝑀−1 1(𝑧) ⋯ 𝑃𝑀−1 𝑀−1(𝑧)
....(13)
• The polyphase form of the analysis filter bank shown in figure
below (a) and after applying the first noble identity we obtain
the structure shown in figure (b)

8. • The synthesis section can be constructed in a similar manner.
Suppose we use a type II (transpose) for the polyphase
representation of the filter {Gk(z)}. Thus,
Gk(z) = 𝑚=0
𝑀−1
𝑧−(𝑀−1−𝑚)
𝑄𝑘𝑚(𝑧𝑀
), 1 ≤ k ≤ M-1 ....(14)
• When expressed in matrix form,(14) becomes
G(z) = 𝑧−(𝑀−1)
Q(zM)a(z-1) ....(15)
where a(z) is defined in (12) and
Q(z) =
𝑄00(𝑧) 𝑄(𝑧) ⋯ 𝑄0𝑀(𝑧)
⋮ ⋱ ⋮
𝑄𝑀−1 0(𝑧) 𝑄𝑀−1 1(𝑧) ⋯ 𝑄𝑀−1 𝑀−1(𝑧)
....(16)

9. • Therefore the synthesis of the M-channel QMF bank is realised
as shown in figure below.
(a)-Polyphase structure of the synthesis section of an M-channel
QMF bank and (b)-after applying the first noble identity.

10. • We obtain the polyphase structure of th complete M-channel QMF bank
shown in figure below,
• From the structure of the M-channel QMF bank in figure above, it is
observed that perfect reconstruction condition can be restated as
Q(z)P(z) = Cz-kI ....(17)

11. where I is the M x M marix. Hence, if the polyphase matrix P(z)
is known, then the polyphase synhtesis matrix Q(z) is
Q(z) = Cz-k[P(z)]-1 ....(18)

12. • It has significanty reduced computations.
• It has reduced storage.
• It has simplified filter design.
• It has reduced finite word length effects.

13. • It has increased control structure required to implement
the design.