DSP-CIS
Chapter-9: Modulated Filter Banks
Marc MoonenDept. E.E./ESAT-STADIUS, KU Leuven
www.esat.kuleuven.be/stadius/
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 2
: 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
: Cosine-modulated FBs & Special topics• Cosine-modulated FBs• Time-frequency analysis & Wavelets• Frequency domain filtering
Part-II : Filter Banks
Chapter-7
Chapter-8
Chapter-9
Chapter-10
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 3
General `subband processing’ set-up (Chapter-7) :
PS: subband processing ignored in filter bank design
downsampling/decimation
Refresh (1)
subband processing 3H0(z)
subband processing 3H1(z)
subband processing 3H2(z)
3
3
3
3 subband processing 3H3(z)
IN
F0(z)
F1(z)
F2(z)
F3(z)
+
OUT
analysis bank synthesis bank
upsampling/expansion
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 4
Refresh (2)
Two design issues : - filter specifications, e.g. stopband attenuation, passband ripple, transition
band, etc. (for each (analysis) filter!)
- perfect reconstruction property (Chapter-8).
PS: still considering maximally decimated FB’s, i.e.
4444
+u[k-3]
1z
2z
3z
1
1z2z3z
1
u[k] 444
4)(zE )(zR
NIzzz )().( ER
PS: Equivalent perfect reconstruction condition for transmux’s ? Try it !
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 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)
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 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)
INH0 H3H2H1
H0 H3H2H1uniform
non-uniform
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 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
(N-fold polyphase decomposition)
then
i.e.
H0(z)
H1(z)
H2(z)
H3(z)
u[k]
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 8
Maximally Decimated DFT-Modulated FBs
where F is NxN DFT-matrix (and `*’ is complex conjugate)
This means that filtering with the Hn’s can be implemented by first filtering with polyphase components and then DFT
Nj
NN
N
N
N
N
NNN
N
N
N
eW
zU
zEz
zEz
zEz
zE
F
WWWW
WWWW
WWWW
WWWW
zU
zH
zH
zH
zH
/2
11
22
11
0
)1()1(2)1(0
)1(2420
)1(210
0000
1
2
1
0
)(.
)(.
:
)(.
)(.
)(
.
*
...
::::
...
...
...
)(.
)(
:
)(
)(
)(
2
i.e.
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 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 Hn(z)’s are
automatically `co-designed’ (same passband ripple, etc…) !
*F
u[k]
)( 40 zE
)( 41 zE
)( 42 zE
)( 43 zE
)(0 zH
)(1 zH
)(2 zH
)(3 zH
i.e.
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 10
Maximally Decimated DFT-Modulated FBs
• Special case: DFT-filter bank, if all En(z)=1
*F
u[k]
1 )(0 zH
)(1 zH
)(2 zH
)(3 zH
11
1
Ho(z) H1(z)
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 11
Maximally Decimated DFT-Modulated FBs
• PS: with F instead of F* (as in Chapter-6), only filter ordering is changed
Fu[k]
1 )(0 zH
)(1 zH
)(2 zH
)(3 zH11
1
Ho(z) H1(z)
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 12
Maximally Decimated DFT-Modulated FBs
• DFT-modulated analysis FB + maximal decimation
*F 4
4
4
4u[k]
)( 40 zE
)( 41 zE
)( 42 zE
)( 43 zE
4
4
4
4u[k]
*F)(0 zE
)(1 zE
)(2 zE
)(3 zE
= = efficient realization !
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 13
Maximally Decimated DFT-Modulated FBs
2. Synthesis FB
+
+
+
)(0 zF][0 ku
)(1 zF][1 ku
)(2 zF][2 ku
)(3 zF][3 kuy[k]
phase shift added
for convenience
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 14
Maximally Decimated DFT-Modulated FBs
where F is NxN DFT-matrix
Nj
NNNN
N
N
NNNNN
N
N
N
eW
zU
zU
zU
zU
F
WWWW
WWWW
WWWW
WWWW
zRzRzzRzzRz
zU
zU
zU
zU
zFzFzFzFzY
/2
1
2
1
0
)1()1(2)1(0
)1(2420
)1(210
0000
011
22
11
1
2
1
0
1210
)(
:
)(
)(
)(
.
...
::::
...
...
...
.)()(.)(....)(.
)(
:
)(
)(
)(
.)(...)()()()(
2
i.e.
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 15
Maximally Decimated DFT-Modulated FBs
i.e.
y[k]
+
+
+)( 40 zR
)( 41 zR
)( 42 zR
)( 43 zR][0 ku
][1 ku
][2 ku
][3 ku
F
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 16
Maximally Decimated DFT-Modulated FBs
• Expansion + DFT-modulated synthesis FB :
y[k]
][0 ku
][1 ku
][2 ku
][3 ku
4
4
4
4
+
+
+)(0 zR
)(1 zR
)(2 zR
)(3 zR
F
y[k]
+
+
+
4
4
4
4 )( 40 zR
)( 41 zR
)( 42 zR
)( 43 zR][0 ku
][1 ku
][2 ku
][3 ku
F
= = efficient realization !
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 17
Maximally Decimated DFT-Modulated FBs
How to achieve Perfect Reconstruction (PR)
with maximally decimated DFT-modulated FBs?
polyphase components of synthesis bank prototype filter are obtained by inverting polyphase components of analysis bank prototype filter
y[k]
44
4
4
+
+
+)(0 zR
)(1 zR
)(2 zR
)(3 zR
F4
4
4
4u[k]
*F)(0 zE
)(1 zE
)(2 zE
)(3 zE
NIzzz )().( ER
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 18
Maximally Decimated DFT-Modulated FBs
Design Procedure : 1. Design prototype analysis filter Ho(z) (see Chapter-3).
2. This determines En(z) (=polyphase components).
3. Assuming all En(z) can be inverted (?), choose synthesis filters
y[k]
4444
+
+
+)(0 zR
)(1 zR
)(2 zR
)(3 zR
F444
4u[k]
*F)(0 zE
)(1 zE
)(2 zE
)(3 zE
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 19
Maximally Decimated DFT-Modulated FBs
• Will consider only FIR prototype analysis filter, leading to simple polyphase decomposition.
• However, FIR En(z)’s generally again lead to IIR Rn(z)’s, where stability is a concern…
• FIR unimodular E(Z)? ..such that Rn(z) are also FIR.
Only obtained with trivial choices for the En(z)’s, with
only 1 non-zero impulse response parameter,
i.e. En(z)=α or En(z)=α.z^{-d}.
Examples: next slide
all E(z)’s
FIR E(z)’s
FIR unimodular E(z)’s
E(z)=F*.diag{..}
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 20
Maximally Decimated DFT-Modulated FBs
• Simple example (1) is , which leads to
IDFT/DFT bank (Chapter-8)
i.e. Fn(z) has coefficients of Hn(z), but complex conjugated and in
reverse order (hence same magnitude response) (remember this?!)
• Simple example (2) is , where wn’s
are constants, which leads to `windowed’ IDFT/DFT bank, a.k.a. `short-time Fourier transform’ (see Chapter-10)
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 21
Maximally Decimated DFT-Modulated FBs
• FIR paraunitary E(Z)? ..such that Rn(z) are FIR + power complementary FB’s.
Only obtained when the En(z)’s are all-pass filters (and
FIR), i.e. En(z)=±1 or En(z)=±1.z^{-d}.
i.e. only trivial modifications
of DFT filter bank !
SIGH !all E(z)’s
FIR E(z)’s
FIR unimodular E(z)’s
E(z)=F*.diag{..}
FIR paraunitary E(z)’s
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 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-10)
– Oversampled PR FIR DFT-modulated FB’s (read on)
SIG
H!
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 23
Oversampled PR Filter Banks
• So far have considered maximal decimation (D=N), where aliasing makes PR design non-trivial.
• With downsampling factor (D) smaller than the number of channels (N), aliasing is expected to become a smaller problem, possibly negligible if D<<N.
• 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 (!)
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 24
Oversampled PR Filter Banks
• Starting point is (see Chapter-8):
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)…??
4444
+u[k-3]
1z
2z
3z
1
1z2z3z
1
u[k] 444
4
)(zE )(zR
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 25
Oversampled PR Filter Banks
• A more general case is :
where E(z) is now NxD (`tall-thin’) and R(z) is DxN (`short-fat’)
while still guarantees PR !
u[k-3]
4444
+1z
2z
3z
1
1z2z3z
1
u[k] 444
4
)(zE )(zR
N=
6 ch
ann
els
D=
4 d
ecim
atio
n
!
PS: Here E(z) has 6 rows (defining 6 analysis filters),
with four 4-fold polyphase components in each row
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 26
Oversampled PR Filter Banks
• The PR condition
appears to be a `milder’ requirement if D<N
for instance for D=N/2, we have (where Ei and Ri are DxD 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
)()( 111 zz ER
DxDDxN
NxD
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 27
Oversampled DFT-Modulated FBs
Example-1 : # channels N = 8 Ho(z),H1(z),…,H7(z)
decimation D = 4
prototype analysis filter Ho(z)
will consider N’-fold polyphase expansion, with
Sh
ou
ld n
ot
try
to u
nd
ers
tan
d t
his
…
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 28
Oversampled DFT-Modulated FBs
In general, it is proved that the N-channel DFT-modulated (analysis) filter bank can be realized based on an N-point DFT cascaded with an NxD `polyphase matrix’ B, which contains the (N’-fold) polyphase components of the prototype Ho(z)
Example-1 (continued):
*88xF
u[k]
)(0 zH
)(6 zH
)(7 zH
484 )( xzB
)(.000
0)(.00
00)(.0
000)(.
)(000
0)(00
00)(0
000)(
)(
87
4
86
4
85
4
84
4
83
82
81
80
4
zEz
zEz
zEz
zEz
zE
zE
zE
zE
zB
Convince yourself that this is indeed correct.. (or see next slide)N
=8
chan
nel
s
D=
4 d
ecim
atio
n
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 29
Oversampled DFT-Modulated FBs
Proof is simple:*
88xF
u[k]
)(0 zH
)(6 zH
)(7 zH
484 )( xzB
)(.
)(
)(
)(
)(
)(
)(
)(
)(
)(.
)(.
)(.
)(.
)(.
)(.
)(.
)(.
)(
....)(.
1
).(.
7
6
5
4
3
2
1
0
87
7
86
6
85
5
84
4
83
3
82
2
81
1
80
*
3
2
14* zU
zH
zH
zH
zH
zH
zH
zH
zH
zU
zEz
zEz
zEz
zEz
zEz
zEz
zEz
zE
FzU
z
z
zzBF
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 30
Oversampled DFT-Modulated FBs
-With 4-fold decimation, this is…
u[k]
*88xF48)( xzB
4
4
4
4
)(.)( * zBFz E
)(.000
0)(.00
00)(.0
000)(.
)(000
0)(00
00)(0
000)(
)(
27
1
26
1
25
1
24
1
23
22
21
20
zEz
zEz
zEz
zEz
zE
zE
zE
zE
zB
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 31
Oversampled DFT-Modulated FBs
- Similarly, synthesis FB is…
y[k]
4
4
4
4
+
+
+
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 32
Oversampled DFT-Modulated FBs
- Perfect Reconstruction (PR) ?
4
4
4
4
+u[k-3]
1z
2z
3z
1
1z2z3z
1
u[k] 4
4
4
4
)(zE )(zR
)(.)( * zBFz E
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 33
Oversampled DFT-Modulated FBs
- Perfect Reconstruction (PR) ?
4
4
4
4
+u[k-3]
1z
2z
3z
1
1z2z3z
1
u[k] 4
4
4
4
)(zE )(zR
)(.)( * zBFz E
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 34
Oversampled DFT-Modulated FBs
- FIR Perfect Reconstruction FB (unimodular-like)
Design Procedure :1. Design FIR prototype analysis filter Ho(z).
2. This determines En(z) (=polyphase components).
3. Compute pairs of FIR Ri(z)’s (Lr+1 coefficients each) from pairs of FIR Ei(z)’s (Le+1 coefficients each)
i.e. solve set of linear equations in Ri(z) coefficients :
(for sufficiently high synthesis prototype filter order, this
set of equations can be solved, except in special cases)
= EASY !
4
4
4
4
+u[k-3]
1z
2z
3z
1
1z2z3z
1
u[k] 4
4
4
4
)(zE )(zR
Lr+Le+1 equations in 2(Lr+1) unknowns, can (mostly) be solved if Le-1 ≤ Lr
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 35
Oversampled DFT-Modulated FBs
- FIR Paraunitary Perfect Reconstruction FB
– If E(z)=F*.B(z) is chosen to be paraunitary,
then PR is obtained with R(z)=B~(z).F – E(z) is paraunitary only if B(z) is paraunitary
So how can we make B(z) paraunitary ?
)(.)( * zBFz E
4
4
4
4
+u[k-3]
1z
2z
3z
1
1z2z3z
1
u[k] 4
4
4
4
)(zE )(zR
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 36
Oversampled DFT-Modulated FBs
• B(z) is paraunitary if and only if
i.e. (n=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-5 on FIR lossless lattices realizations (!)…
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 37
Oversampled DFT-Modulated FBs
• Design Procedure: Optimize parameters (=angles) of 4 (=D) FIR lossless lattices (defining polyphase components of Ho(z) ) such that Ho(z) satisfies specifications.
p.30 =
*88F
u[k]
:
4
4 )( 20 zE
)( 24 zE
)( 23 zE
)( 27 zE
:
:
Lossless 1-in/2-out
= not-so-easy but DOABLE !
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 38
Oversampled DFT-Modulated FBs
• Result = oversampled DFT-modulated FB (N=8, D=4), that is PR/FIR/paraunitary !! All great properties combined in one design !!
• PS: With 2-fold oversampling (D=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 (D=N/d), paraunitary design will be based on 1-input/d-output lossless systems (see also Chapter-5 on multi-channel FIR lossless lattices). With maximal decimation (D=N), 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) !
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 39
Oversampled DFT-Modulated FBs
Example-2 (non-integer oversampling) :
# channels N = 6 Ho(z),H1(z),…,H5(z)
decimation D = 4
prototype analysis filter Ho(z)
will consider N’-fold polyphase expansion, with
Sh
ou
ld n
ot
try
to u
nd
erst
and
th
is…
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 40
Oversampled DFT-Modulated FBs
DFT modulated (analysis) filter bank can be realized based on an N-point IDFT cascaded with an NxD polyphase matrix B, which contains the (N’-fold) polyphase components of the prototype Ho(z)
)(.0)(.0
0)(.0)(.
)(0)(.0
0)(0)(.
)(.0)(0
0)(.0)(
)(
1211
8125
4
1210
8124
4
123
129
8
122
128
8
127
4121
126
4120
4
zEzzEz
zEzzEz
zEzEz
zEzEz
zEzzE
zEzzE
zB
*66xF
u[k]
)(0 zH
)(1 zH
)(2 zH
)(3 zH
)(4 zH
)(5 zH
464 )( xzB
Convince yourself that this is indeed correct.. (or see next slide)
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 41
Oversampled DFT-Modulated FBs
Proof is simple:
)(.
)(
)(
)(
)(
)(
)(
)(.
iondecomposit polyphase fold-6
)(.)(
)(.)(
)(.)(
)(.)(
)(.)(
)(.)(
....)(.
1
).(.
5
4
3
2
1
0
1211
11125
5
1210
10124
4
129
9123
3
128
8122
2
127
7121
1
126
6120
*
3
2
14* ZU
zH
zH
zH
zH
zH
zH
ZU
zEzzEz
zEzzEz
zEzzEz
zEzzEz
zEzzEz
zEzzE
FzU
z
z
zzBF
*66xF
u[k]
)(0 zH
)(1 zH
)(2 zH
)(3 zH
)(4 zH
)(5 zH
464 )( xzB
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 42
Oversampled DFT-Modulated FBs
-With 4-fold decimation, this is
-Similar synthesis FB (R(z)=C(z).F), and then PR conditions...
)(.)( * zBFz E
u[k]
*66xF46)( xzB4
44
4
)(.0)(.0
0)(.0)(.
)(0)(.0
0)(0)(.
)(.0)(0
0)(.0)(
)(
311
235
1
310
234
1
33
39
2
32
38
2
37
131
36
130
zEzzEz
zEzzEz
zEzEz
zEzEz
zEzzE
zEzzE
zB
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 43
Oversampled DFT-Modulated FBs
- FIR Perfect Reconstruction FB: try it..- FIR Paraunitary Perfect Reconstruction FB:
E(z) is paraunitary iff B(z) is 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!
)(.)(.
)()(.
)(.)(
and
)(.)(.
)()(.
)(.)(
311
235
1
33
39
2
37
131
310
234
1
32
38
2
36
130
zEzzEz
zEzEz
zEzzE
zEzzEz
zEzEz
zEzzE
= EASY !
= not-so-easy but DOABLE !
DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 44
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?