dsp-cis chapter-7: introduction to multi-rate systems & filter banks marc moonen dept....
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DSP-CIS
Chapter-7: Introduction to Multi-rate Systems & Filter Banks
Marc MoonenDept. E.E./ESAT, KU Leuven
www.esat.kuleuven.be/scd/
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 2
• Filter banks introduction Introduction Applications Aliasing & perfect reconstruction
• Review of multi-rate systems
• DFT/IDFT filter bank• Maximally decimated DFT-modulated filter banks• Other…
Overview
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 3
Filter Banks Introduction
What we have in mind is this… :
- Signals split into frequency channels/subbands - Per-channel/subband processing - Reconstruction : synthesis of processed signal - Applications : see below (audio coding etc.) - In practice, this is implemented as a multi-rate structure for higher efficiency (see next slides)
subband processing
subband processing
subband processing
subband processing
H1(z)
H2(z)
H3(z)
H4(z)
IN+
OUT
H1 H4H3H2
2
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 4
Filter Banks Introduction
Step-1: Analysis filter bank - collection of N filters (`analysis filters’, `decimation filters’) with a common input signal - ideal (but non-practical) frequency responses = ideal bandpass filters
- typical frequency responses (overlapping, non-overlapping,…)
2
H1 H4H3H2
H1 H4H3H2
H1 H4H3H2
2
2
H1(z)
H2(z)
H3(z)
H4(z)
IN
N=4
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 5
Filter Banks Introduction
Step-2: Decimators (downsamplers) - To increase efficiency, subband sampling rate is reduced by factor D
(= Nyquist (bandpass) sampling theorem)
- Maximally decimated filter banks (=critically downsampled):
# subband samples= # fullband samples this sounds like maximum efficiency, but aliasing (see below)!
- Oversampled filter banks (=non-critically downsampled):
# subband samples> # fullband samples
D=N
D<N
H1(z)
H2(z)
H3(z)
3
3
3
3H4(z)
IN
N=4 D=3
PS: 3-fold downsampling
= ‘remove every 2nd and 3rd sample’ (see below)
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 6
Filter Banks Introduction
Step-3: Subband processing
- Example :
coding (=compression) + (transmission or storage) + decoding
- Filter bank design mostly assumes subband processing has `unit
transfer function’ (output signals=input signals), i.e. mostly ignores
presence of subband processing
subband processingH1(z)
subband processingH2(z)
subband processingH3(z)
3
3
3
3 subband processingH4(z)
IN
N=4 D=3
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 7
Filter Banks Introduction
Step-4: Expanders (upsamplers)
- restore original fullband sampling rate by D-fold upsampling
subband processing 3H1(z)
subband processing 3H2(z)
subband processing 3H3(z)
3
3
3
3 subband processing 3H4(z)
IN
N=4 D=3 D=3
PS: 3-fold upsampling
= ‘insert 2 zeros in between every two samples’ (see below)
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 8
Filter Banks Introduction
Step-5: Synthesis filter bank - upsampling has to be followed by (interpolation) filtering (see p.22)
- collection of N `synthesis’ (`interpolation’) filters, with a `common’ (summed) output signal - frequency responses : preferably `matched’ to frequency responses of the analysis filters, e.g., to provide perfect reconstruction (see below)
G1(z)
G2(z)
G3(z)
G4(z)
+OUT
subband processing 3H1(z)
subband processing 3H2(z)
subband processing 3H3(z)
3
3
3
3 subband processing 3H4(z)
IN
N=4 D=3 D=3
G1 G4G3G2
2
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 9
Aliasing versus Perfect Reconstruction
- Assume subband processing does not modify subband signals
(e.g. lossless coding/decoding)
-The overall aim would then be to have y[k]=u[k-d], i.e. that the output signal is equal to the input signal up to a certain delay
-But: downsampling introduces ALIASING, especially so in maximally
decimated (but also in non-maximally decimated) filter banks
G1(z)
G2(z)
G3(z)
G4(z)
+
output = input 3H1(z)
output = input 3H2(z)
output = input 3H3(z)
3
3
3
3 output = input 3H4(z)
N=4 D=3 D=3
u[k]
y[k]=u[k-d]?
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 10
Aliasing versus Perfect Reconstruction
Question :
Can y[k]=u[k-d] be achieved in the presence of aliasing ?
Answer :
YES !! PERFECT RECONSTRUCTION banks with
synthesis bank designed to remove aliasing effects !
G1(z)
G2(z)
G3(z)
G4(z)
+
output = input 3H1(z)
output = input 3H2(z)
output = input 3H3(z)
3
3
3
3 output = input 3H4(z)
N=4 D=3 D=3
u[k]
y[k]=u[k-d]?
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 11
Filter Banks Applications
• Subband coding : Coding = Fullband signal split into subbands & downsampled subband signals separately encoded (e.g. subband with smaller energy content encoded with fewer bits)
Decoding = reconstruction of subband signals, then fullband signal synthesis (expanders + synthesis filters) Example : Image coding (e.g. wavelet filter banks) Example : Audio coding e.g. digital compact cassette (DCC), MiniDisc, MPEG, ... Filter bandwidths and bit allocations chosen to further exploit perceptual properties of human hearing (perceptual coding, masking, etc.)
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 12
Filter Banks Applications
• Subband adaptive filtering :
- See Chapter-8
- Example : Acoustic echo cancellation
Adaptive filter models (time-varying) acoustic echo path and produces
a copy of the echo, which is then subtracted from microphone signal.
= difficult problem !
✪ long acoustic impulse responses
✪ time-varying
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 13
Filter Banks Applications
- Subband filtering = N (simpler?) subband modeling problems instead of one (more complicated?) fullband modeling problem
- Perfect reconstruction guarantees distortion-free desired near-end speech signal
3H1(z)
3H2(z)
3H3(z)
3H4(z)
3H1(z)
3H2(z)
3H3(z)
3H4(z) ++
++ 3 G1(z)
3 G2(z)
3 G3(z)
3 G4(z)
OUT+
ad.filter
ad.filter
ad.filter
ad.filter
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 14
Filter Banks Applications
• Transmultiplexers : Frequency Division Multiplexing (FDM) in digital communications - N different source signals multiplexed into 1 transmit signal by expanders & synthesis filters (ps: here interpolation
factor )
- Received signal decomposed into N source signals by analysis filters & decimators
- Again ideal filters = ideal bandpass filters
5 G1(z)
5 G2(z)
5 G3(z)
5 G4(z)
+
5H1(z)
5H2(z)
5H3(z)
5H4(z)
transmission
channel
signal-1
signal-2
signal-3
signal-4
signal-1
signal-2
signal-3
signal-4
D>=N
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 15
- Non-ideal synthesis & analysis filters result in aliasing as
well as CROSS-TALK between channels,
i.e. each reconstructed signal contains unwanted
contributions from other signals
- Analysis & synthesis are reversed here, but similar perfect
reconstruction theory (try it!) (where analysis bank removes cross-talk introduced by synthesis bank, if transmission channel = distortion free)
Filter Banks Applications
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 16
Filter Banks Applications
PS: special case is Time Division Multiplexing (TDM), if synthesis and analysis filters are replaced by delay operators (and N=D)
PS: special case is Code Division Multiplexing (CDMA)
4444
+444
transmission
channel
1z2z3z
1
1z
2z
3z
u1[k],u1[k+1]
u2[k],u2[k+1]
u3[k],u3[k+1]
u4[k],u4[k+1]
u1[k],u2[k],u3[k],u4[k],u1[k+1],u2[k+1]...
4 u1[k-1],u1[k]
u2[k-1],u2[k]
u3[k-1],u3[k]
u4[k-1],u4[k]
4z
0,0,0,u4[k],0,0,0,u4[k+1]...
PS: another special case is OFDM… explain! What are Gi(z) & Hi(z) ?
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 17
Review of Multi-rate Systems 1/10
• Decimation : decimator (downsampler)
example : u[k]: 1,2,3,4,5,6,7,8,9,…
2-fold downsampling: 1,3,5,7,9,...
• Interpolation : expander (upsampler)
example : u[k]: 1,2,3,4,5,6,7,8,9,… 2-fold upsampling: 1,0,2,0,3,0,4,0,5,0...
D u[0], u[N], u[2D]...u[0],u[1],u[2]...
D u[0],0,..0,u[1],0,…,0,u[2]...u[0], u[1], u[2],...
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 18
Review of Multi-rate Systems 2/10
• Z-transform (frequency domain) analysis of expander
`expansion in time domain ~ compression in frequency domain’
D u[0],0,..0,u[1],0,…,0,u[2]...u[0], u[1], u[2],...
D)(zU
jez
D
xHz
`images’
xHz
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 19
Review of Multi-rate Systems 2bis/10
• Z-transform (frequency domain) analysis of expander
expander mostly followed by `interpolation filter’ to remove images (and `interpolate the zeros’)
interpolation filter can be low-/band-/high-pass
D u[0],0,..0,u[1],0,…,0,u[2]...u[0], u[1], u[2],...
3
xHz
`images’
xHz
xHz
LP
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 20
Review of Multi-rate Systems 3/10
• Z-transform (frequency domain) analysis of decimator
`compression in time domain ~ expansion in frequency domain’ PS: Note that is periodic with period while is periodic with period
The summation with d=0…D-1 restores the periodicity with period !
D)(zU
D u[0], u[D], u[2D]...u[0],u[1],u[2]...
d=0d=2 d=1
3
xHz
3
xHz
)( jeU 2
2
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 21
Review of Multi-rate Systems 4/10
• Z-transform (frequency domain) analysis of decimator
decimation introduces ALIASING if input signal occupies frequency band larger than , hence mostly preceded by anti-aliasing (decimation) filter
anti-aliasing filter can be low-/band-/high-pass
D u[0], u[N], u[2D]...u[0],u[1],u[2]...
LP
d=0d=2 d=1
3
xHz
3
xHz
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 22
Review of Multi-rate Systems 5/10
• Interconnection of multi-rate building blocks :
identities also hold if all decimators are replaced by expanders
D x
aDx
a=
=
=
D+
u2[k]
Dx
u2[k]
u1[k]
u1[k]
D +
Du2[k]
u1[k]
D x
Du2[k]
u1[k]
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 23
Review of Multi-rate Systems 6/10
• `Noble identities’ (I) : (only for rational functions)
Example : D=2 h[0],h[1],0,0,0,…
=D D )(zHu[k] u[k]y[k] y[k]
]3[
]2[
]1[
]0[
.0100
0001.
)(
]1[0
]0[]1[
0]0[
...
]3[
]2[
]1[
]0[
.
)2(
]1[000
0]1[00
]0[0]1[0
0]0[0]1[
00]0[0
000]0[
.
010000
000100
000001
]2[
]1[
]0[
ngdownsampli fold-2
ngdownsampli fold-2
u
u
u
u
zH
h
hh
h
u
u
u
u
zH
h
h
hh
hh
h
h
y
y
y
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 24
Review of Multi-rate Systems 7/10
• `Noble identities’ (II) : (only for rational functions)
Example : D=2 h[0],h[1],0,0,0,…
=D D)(zHu[k] u[k]y[k] y[k]
]1[
]0[.
00
10
00
01
.
)2(
]1[000
0]1[00
]0[0]1[0
0]0[0]1[
00]0[0
000]0[
...]1[
]0[.
)(
]1[0
]0[]1[
0]0[
.
000
100
000
010
000
001
]5[
]4[
]3[
]2[
]1[
]0[
upsampling fold-2
upsampling fold-2
u
u
zH
h
h
hh
hh
h
h
u
u
zH
h
hh
h
y
y
y
y
y
y
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 25
Review of Multi-rate Systems 8/10
Application of `noble identities : efficient multi-rate realizations of FIR filters through…
• Polyphase decomposition: example : (2-fold decomposition)
example : (3-fold decomposition)
general: (D-fold decomposition)
)(
421
)(
642
654321
21
20
)].5[].3[]1[(.)].6[].4[].2[]0[(
].6[].5[].4[].3[].2[].1[]0[)(
zEzE
zhzhhzzhzhzhh
zhzhzhzhzhzhhzH
)(
32
)(
31
)(
63
654321
32
31
30
)].5[]2[(.)].4[]1[(.)].6[].3[]0[(
].6[].5[].4[].3[].2[].1[]0[)(
zEzEzE
zhhzzhhzzhzhh
zhzhzhzhzhzhhzH
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 26
Review of Multi-rate Systems 9/10
• Polyphase decomposition: Example : efficient implementation of an FIR decimation filter
i.e. all filter operations performed at the lowest rate
u[k]
2
)( 20 zE
)( 21 zE
1z+
H(z)
u[k]
2
1z
)(0 zE
)(1 zE +
= 2
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 27
Review of Multi-rate Systems 10/10
• Polyphase decomposition:
Example : efficient implementation of an FIR interpolation filter
i.e. all filter operations performed at the lowest rate
=
u[k]
2
)( 20 zE
)( 21 zE
1z
+
H(z)
u[k]
2
1z
+)(0 zE
)(1 zE
2
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 28
DFT/IDFT Filter Bank
• Basic question is..: Downsampling introduces ALIASING, then how can PERFECT RECONSTRUCTION (PR) (i.e. y[k]=u[k-d]) be achieved ?• Next slides provide simple PR-FB example, to
demonstrate that PR can indeed (easily) be obtained• Discover the magic of aliasing-compensation….
G1(z)
G2(z)
G3(z)
G4(z)
+
output = input 4H1(z)
output = input 4H2(z)
output = input 4H3(z)
4
4
4
4 output = input 4H4(z)
u[k]
y[k]=u[k-d]?
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 29
DFT/IDFT Filter Bank
First attempt to design a perfect reconstruction filter bank
- Starting point is this :
convince yourself that y[k]=u[k-3] …
4
4
4
4
u[k]
4
4
4
4
+
+
+u[k-3]
u[0],0,0,0,u[4],0,0,0,...
u[-1],u[0],0,0,u[3],u[4],0,0,...
u[-2],u[-1],u[0],0,u[2],u[3],u[4],0,...
u[-3],u[-2],u[-1],u[0],u[1],u[2],u[3],u[4],...
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 30
DFT/IDFT Filter Bank
- An equivalent representation is ...
As y[k]=u[k-d], this can already be viewed as a (1st) perfect reconstruction filter bank (with lots of aliasing in the subbands!) All analysis/synthesis filters are seen to be pure delays, hence are not frequency selective (i.e. far from ideal case with ideal bandpass filters….)
PS: Transmux version (TDM) see p.16
4444
+1z2z3z
1
u[k-3]444
1z
2z
3z4
1
u[k]
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 31
DFT/IDFT Filter Bank
-now insert DFT-matrix (discrete Fourier transform)
and its inverse (I-DFT)...
as this clearly does not change the input-output relation (hence perfect reconstruction property preserved)
4444
+u[k-3]
1z
2z
3z
1
1z2z3z
1
444
4u[k]
F 1F
IFF .1
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 32
DFT/IDFT Filter Bank
- …and reverse order of decimators/expanders and DFT-matrices (not done in an efficient implementation!) :
=analysis filter bank =synthesis filter bank
This is the `DFT/IDFT filter bank’. It is a first (or 2nd) example of a maximally decimated perfect reconstruction filter bank !
4444
1z2z3z
1
u[k] 444
4
F +u[k-3]
1z
2z
3z
1
1F
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 33
DFT/IDFT Filter Bank
What do analysis filters look like? (N-channel case)
This is seen (known) to represent a collection of filters Ho(z),H1(z),..., each of which is a frequency shifted version of Ho(z) :
i.e. the Hn are obtained by uniformly shifting the `prototype’ Ho over the frequency axis.
F
u[k]
Nj
N
F
NNN
N
N
N
eW
z
z
z
WWWW
WWWW
WWWW
WWWW
zH
zH
zH
zH
/2
1
2
1
)1()1(210
)1(2420
1210
0000
1
2
1
0
:
1
.
...
::::
...
...
...
)(
:
)(
)(
)(
2
1210 ...1)( NzzzzH
: :
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 34
DFT/IDFT Filter Bank
The prototype filter Ho(z) is a not-so-great
lowpass filter with significant sidelobes.
Ho(z) and Hi(z)’s are thus far from ideal
lowpass/bandpass filters.
Hence (maximal) decimation introduces
significant ALIASING in the decimated
subband signals
Still, we know this is a PERFECT RECONSTRUCTION filter bank (see construction previous slides), which means the synthesis filters can apparently restore the aliasing distortion. This is remarkable!
Other perfect reconstruction banks : read on..
Ho(z)H1(z)
N=4
H3(z)H2(z)
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 35
DFT/IDFT Filter Bank
Synthesis filters ?
synthesis filters are (roughly) equal to analysis filters…
PS: Efficient DFT/IDFT implementation based on FFT algorithm
(`Fast Fourier Transform’).
...
)(
:
)(
)(
)(
1
2
1
0
zF
zF
zF
zF
N
+1z
1
1F
*(1/N)N=4
::
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 36
Maximally Decimated DFT-Modulated FBs
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
H0(z)
H1(z)
H2(z)
H3(z)
INH0 H3H2H1
H0 H3H2H1uniform
non-uniform
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 37
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]
NM
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 38
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 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 39
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 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 40
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 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 41
Maximally Decimated DFT-Modulated FBs
• PS: with F instead of F* (see p.32), only filter ordering is changed
Fu[k]
1 )(0 zH
)(1 zH
)(2 zH
)(3 zH11
1
Ho(z) H1(z)
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 42
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 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 43
Maximally Decimated DFT-Modulated FBs
y[k]
+
+
+)( 40 zR
)( 41 zR
)( 42 zR
)( 43 zR][0 ku
][1 ku
][2 ku
][3 ku
F
2. Synthesis FB: similar..
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 44
Maximally Decimated DFT-Modulated FBs
• Expansion + DFT-modulated synthesis FB :
y[k]
4
4
4
4
+
+
+)(0 zR
)(1 zR
)(2 zR
)(3 zR][0 ku
][1 ku
][2 ku
][3 ku
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 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 45
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
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 46
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 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 47
Maximally Decimated DFT-Modulated FBs
• Will consider only FIR prototype analysis filters, leading to simple polyphase decompositions.
• However, FIR En(z)’s generally again lead to IIR Rn(z)’s, where stability is a concern…
• Hence need for other/better design procedures
DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 48
Other FBs…
• Para-unitary perfect reconstruction filter banks• Cosine-modulated perfect reconstruction filter banks• Non-critically downsampled DFT-modulated perfect
recinstruction filter banks• Non-uniform filter banks• Wavelet filter banks• Etc…