1. DSP-CIS Chapter 8: Modulated Filter Banks
Marc Moonen
Dept. E.E./ESAT, K.U.Leuven

2. Part-II : Filter Banks Chapter-6 Chapter-7 Chapter-8 Chapter-9
: Preliminaries
Filter bank set-up and applications
`Perfect reconstruction’ problem + 1st example (DFT/IDFT)
Multi-rate systems review (10 slides)
: Maximally decimated FBs
Perfect reconstruction filter banks (PR FBs)
Paraunitary PR FBs
: Modulated FBs
Maximally decimated DFT-modulated FBs
Oversampled DFT-modulated FBs
: Special Topics
Cosine-modulated FBs
Non-uniform FBs & Wavelets
Frequency domain filtering
Chapter-7
Chapter-8
Chapter-9

3. downsampling/decimation upsampling/expansion
Refresh (1)
General `subband processing’ set-up (Chapter-6) :
PS: subband processing ignored in filter bank design
analysis bank
synthesis bank
subband processing 3
H0(z)
H1(z)
H2(z)
H3(z) IN
F0(z)
F1(z)
F2(z)
F3(z) +
OUT
downsampling/decimation
upsampling/expansion

4. Refresh (2) Two design issues : 4 +
- filter specifications, e.g. stopband attenuation, passband ripple, transition band, etc. (for each (analysis) filter!)
- perfect reconstruction property (Chapter-6).
PS: we are now still considering maximally decimated FB’s, i.e. 4 +
u[k-3]
u[k]
PS: Equivalent perfect reconstruction condition for transmux’s ? Try it !

5. Introduction
-All design procedures so far involve monitoring of characteristics (passband ripple, stopband suppression,…) of all (analysis) filters, which may be tedious.
-Design complexity may be reduced through usage of
`uniform’ and `modulated’ filter banks.
DFT-modulated FBs (this Chapter)
Cosine-modulated FBs (next Chapter)

6. Introduction Uniform versus non-uniform (analysis) filter bank:
N-channel uniform FB:
i.e. frequency responses are uniformly shifted over the unit circle
Ho(z)= `prototype’ filter (=one and only filter that has to be designed)
Time domain equivalent is:
non-uniform = everything that is not uniform
e.g. for speech & audio applications (cfr. human hearing)
example: wavelet filter banks (next Chapter)
H0(z)
H1(z)
H2(z)
H3(z) IN H0 H3 H2 H1
uniform
non-uniform

7. Maximally Decimated DFT-Modulated FBs
Uniform filter banks can be realized cheaply based on
polyphase decompositions + DFT(FFT) (hence name `DFT-modulated FB)
1. Analysis FB If
(polyphase decomposition)
then
H0(z)
H1(z)
H2(z)
H3(z)
u[k]
i.e.

8. Maximally Decimated DFT-Modulated FBs
where F is NxN DFT-matrix (and `*’ is complex conjugate)
This means that filtering with the Hi’s can be implemented by first filtering with polyphase components and then DFT
i.e.

9. Maximally Decimated DFT-Modulated FBs
conclusion: economy in…
implementation complexity (for FIR filters):
N filters for the price of 1, plus DFT (=FFT) !
design complexity:
Design `prototype’ Ho(z), then other Hi(z)’s are
automatically `co-designed’ (same passband ripple, etc…) !
i.e.
u[k]

10. Maximally Decimated DFT-Modulated FBs
Special case: DFT-filter bank, if all Ei(z)=1
u[k]
Ho(z)
H1(z)

11. Maximally Decimated DFT-Modulated FBs
PS: with F instead of F* (as in Chapter-6), only filter
ordering is changed
u[k]
Ho(z)
H1(z)

12. Maximally Decimated DFT-Modulated FBs
DFT-modulated analysis FB + maximal decimation (M=N)
u[k] 4 4
u[k] =
= efficient realization !

13. Maximally Decimated DFT-Modulated FBs
2. Synthesis FB +
y[k]
phase shift added
for convenience

14. Maximally Decimated DFT-Modulated FBs
where F is NxN DFT-matrix
i.e.

15. Maximally Decimated DFT-Modulated FBs
i.e. +
y[k]

16. Maximally Decimated DFT-Modulated FBs
Expansion (M=N) + DFT-modulated synthesis FB : 4 + + +
y[k] 4 + =
= efficient realization !
y[k]

17. Maximally Decimated DFT-Modulated FBs
How to achieve Perfect Reconstruction (PR)
with maximally decimated DFT-modulated FBs?
i.e. synthesis bank polyphase components are obtained by inverting analysis bank polyphase components
y[k] 4 +
u[k]

18. Maximally Decimated DFT-Modulated FBs
y[k] 4 +
u[k]
Design Procedure :
1. Design prototype analysis filter Ho(z) (see Chapter-3).
2. This determines Ei(z) (=polyphase components).
3. Assuming all Ei(z) can be inverted (?), choose synthesis filters

19. Maximally Decimated DFT-Modulated FBs
Will consider only FIR prototype analysis filters, leading to simple polyphase decompositions.
However, FIR Ei(z)’s generally still lead to IIR Ri(z)’s, where stability is a concern…
Ri(z) ’s are stable only if Ei(z)’s have stable zeros (`minimum-phase filters’).
Example: LPC lattice filters with all |ki|<1 (see Chapter-4).
The design of such minimum phase FIR filters is (significantly) more difficult..
FIR Ri(z)’s (=guaranteed stability) are only obtained with trivial choices for the Ei(z)’s, i.e. with only 1 non-zero impulse response parameter.
E(z) is then unimodular (see Chapter-7). Examples: see next slide.

20. Maximally Decimated DFT-Modulated FBs
Simple example (1) is , which leads to
IDFT/DFT bank (Chapter-6)
i.e. Fl(z) has coefficients of Hl(z), but complex conjugated and in
reverse order (hence same magnitude response) (remember this?!)
Simple example (2) is , where wi’s
are constants, which leads to `windowed’ IDFT/DFT bank, a.k.a. `short-time Fourier transform’ (see Chapter-9)

21. Maximally Decimated DFT-Modulated FBs
Question (try to answer):
Can we have paraunitary FBs here (=desirable property) ?
When is maximally decimated DFT-modulated FB
at the same time
- PR
- FIR (both analysis & synthesis)
- Paraunitary ?
Hint:
E(z) is paraunitary only if the Ei(z)’s are all-pass filters.
An FIR all-pass filter takes a trivial form, e.g. Ei(z)=1 or Ei(z)=z^{-d}

22. Maximally Decimated DFT-Modulated FBs
Bad news: It is seen that the maximally
decimated IDFT/DFT filter bank (or trivial modifications
thereof) is the only possible maximally decimated DFT-
modulated FB that is at the same time...
- PR
- FIR (all analysis+synthesis filters)
- Paraunitary
Good news:
Cosine-modulated PR FIR FB’s (Chapter-9)
Oversampled PR FIR DFT-modulated FB’s (read on)

23. Oversampled PR Filter Banks
So far have considered maximal decimation (M=N), where aliasing makes PR design non-trivial.
With downsampling factor (N) smaller than the number of channels (M), aliasing is expected to become a smaller problem, possibly negligible if N<<M.
Still, PR theory (with perfect alias cancellation) is not necessarily simpler !
Will not consider PR theory as such here, only give some examples of
oversampled DFT-modulated FBs that are
PR/FIR/paraunitary (!)

24. Oversampled PR Filter Banks
Starting point is (see Chapter-7):
(delta=0 for conciseness here)
where E(z) and R(z) are NxN matrices
(cfr maximal decimation)
What if we try other dimensions for E(z) and R(z)…?? 4 +
u[k-3]
u[k]

25. Oversampled PR Filter Banks!
A more general case is :
where E(z) is now MxN (`tall-thin’) and R(z) is NxM (`short-fat’)
while still guarantees PR ! 4 +
u[k]
u[k-3]
N=4 decimation
M=6 channels

26. Oversampled PR Filter Banks
The PR condition
appears to be a `milder’ requirement if M>N
for instance for M=2N, we have (where Ei and Ri are NxN matrices)
which does not necessarily imply that
meaning that inverses may be avoided, creating possibilities for (great)
DFT-modulated FBs, which can (see below) be PR/FIR/paraunitary
In the sequel, will give 2 examples of oversampled DFT-modulated FBs

27. Oversampled DFT-Modulated FBs
Example-1 : # channels M = Ho(z),H1(z),…,H7(z)
decimation N = 4
prototype analysis filter Ho(z)
will consider N’-fold polyphase expansion, with
to understand this…
Should not try

28. Oversampled DFT-Modulated FBs
In general, it is proved that the M-channel DFT-modulated (analysis) filter
bank can be realized based on an M-point DFT cascaded with an
MxN `polyphase matrix’ B, which contains the (N’-fold) polyphase
components of the prototype Ho(z)
ps: note that if M=N, then N’=N, and then B is a diagonal matrix (cfr. supra)
Example-1 (continued):
u[k]
N=4 decimation
M=8 channels
Convince yourself that this is indeed correct.. (or see next slide)

29. Oversampled DFT-Modulated FBs
Proof is simple:
u[k]

30. Oversampled DFT-Modulated FBs
-With (N=) 4-fold decimation, this is…
u[k] 4

31. Oversampled DFT-Modulated FBs
Perfect Reconstruction (PR) can now be obtained
based on an E(z) that is FIR and paraunitary :
If E(z )=F*.B(z) is chosen to be paraunitary, then PR is obtained with R(z)=B~(z).F (=NxM) (=DFT-modulated synthesis bank).
E(z) is paraunitary only if B(z) is paraunitary.
So how can we make B(z) paraunitary ? 4 +
u[k-3]
u[k]

32. Oversampled DFT-Modulated FBs
Example 1 (continued) :
From the structure of B(z)
It follows that B(z) is paraunitary if and only if
(for k=0,1,2,3) are power complementary
i.e. form a lossless 1-input/2-output system (explain!)
For 1-input/2-output power complementary FIR systems,
see Chapter-6 on FIR lossless lattices realizations (!)…

33. Oversampled DFT-Modulated FBs
Lossless 1-in/2-out
Design Procedure: Optimize parameters (=angles) of (4)
FIR lossless lattices (defining polyphase components of Ho(z) )
such that Ho(z) satisfies specifications.
u[k] : 4
p.30 =

34. Oversampled DFT-Modulated FBs
Result = oversampled DFT-modulated FB (M=8, N=4), that
is PR/FIR/paraunitary !!
All great properties combined in one design !!
PS:
With 2-fold oversampling (M/N=2 in example-1), paraunitary design is
based on 1-input/2-output lossless systems (see page 32-33).
In general, with D-fold oversampling (for D=integer), paraunitary design
will be based on 1-input/D-output lossless systems (see also Chapter-3
on multi-channel FIR lossless lattices).
With maximal decimation (D=1), paraunitary design will then be based
on 1-input/1-output lossless systems, i.e. all-pass (polyphase) filters,
which in the FIR case can only take trivial forms (=page 21-22) !

35. Oversampled DFT-Modulated FBs
Example-2 (non-integer oversampling) :
# channels M = Ho(z),H1(z),…,H5(z)
decimation N = 4
prototype analysis filter Ho(z)
will consider N’-fold polyphase expansion, with
to understand this…
Should not try

36. Oversampled DFT-Modulated FBs
DFT modulated (analysis) filter bank can be realized based on an
M-point IDFT cascaded with an MxN polyphase matrix B, which contains
the (N’-fold) polyphase components of the prototype Ho(z)
u[k]
Convince yourself that this is indeed correct.. (or see next slide)

37. Oversampled DFT-Modulated FBs
Proof is simple:
u[k]

38. Oversampled DFT-Modulated FBs
-With (N=) 4-fold decimation, this is…
u[k] 4

39. Oversampled DFT-Modulated FBs
Perfect Reconstruction by paraunitariness?
- E(z) paraunitary iff B(z) paraunitary
- B(z) is paraunitary if and only if submatrices
are paraunitary (explain!)
Hence paraunitary design based on (two) 2-input/3-output
lossless systems. Such systems can again be FIR, then
parameterized and optimized. Details skipped, but doable!

40. PS: Equivalent PR theory for transmux’s? How does OFDM fit in?
Conclusions
Uniform DFT-modulated filter banks are great:
Economy in design- and implementation complexity
Maximally decimated DFT-modulated FBs:
Sounds great, but no PR/FIR design flexibility 
- Oversampled DFT-modulated FBs:
Oversampling provides additional design flexibility,
not available in maximally decimated case.
Hence can have it all at once : PR/FIR/paraunitary! 
PS: Equivalent PR theory for transmux’s? How does OFDM fit in?