DSP-CIS
Chapter 10: Cosine-Modulated Filter Banks
& Special Topics
Marc MoonenDept. E.E./ESAT-STADIUS, KU Leuven
www.esat.kuleuven.be/stadius/
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 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-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 3
Cosine-Modulated Filter Banks
Motivation :
Cosine-modulated FBs offer an alternative to DFT-modulated FBs…
• Similar to DFT-modulated FBs, cosine-modulated FBs offer economy in design- and implementation complexity
• Unlike DFT-modulated FBs, cosine-modulated FBs can be PR/FIR/paraunitary under maximal decimation (with design flexibility).
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 4
Cosine-Modulated Filter Banks
• Uniform DFT-modulated filter banks: Ho(z) is prototype lowpass filter, cutoff at for N filters
• Cosine-modulated filter banks :
Po(z) is prototype lowpass filter, cutoff at for N filters
Then...
etc...
N/
N2/
H0 H3H2H1
2N/
).(.).(.)()5.0(
0*0
)5.0(
000N
jN
jezPezPzH
P0
2
2
2
N2/
N/H1
Ho).(.).(.)(
)5.01(
0*1
)5.01(
011N
jN
jezPezPzH
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 5
Cosine-Modulated Filter Banks
• Cosine-modulated filter banks : - if Po(z) is prototype FIR lowpass filter with real coefficients po[k], k=0,1,…,L then
i.e. `cosine modulation’ (with real coefficients) instead of `exponential modulation’ (with complex coeffs, see DFT-modulated FBs Chapter-9)
- if Po(z) = `good’ lowpass filter, then Hk(z)’s = `good’ bandpass filters
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 6
Cosine-Modulated Filter Banks
Realization based on polyphase decomposition (analysis):
- if Po(z) has 2N-fold polyphase expansion (ps: 2N-fold for N filters!!!)
then...
NNT 2
u[k]
)( 20
NzE
)( 21
NzE
)( 212
NN zE
)().(0 zUzH
)().(1 zUzH
)().(1 zUzH N
: :
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 7
Cosine-Modulated Filter Banks
Realization based on polyphase decomposition (continued): - if Po(z) has L+1=m.2N taps, and m is even (similar formulas for m odd) (i.e. `m’ is the number of taps in each polyphase component) then...
With
00...1
:::
01...0
10...0
,
1...00
:::
0...10
0...01
)()(... 22
JI
JIJICNT NNNN
})5.0(cos{
})5.01(cos{
})5.0(cos{
...00
:::
0...0
0...0
mN
m
m
)}5.0).(5.0.(cos{2
}{ , qpNN
C qp
ign
ore
all
det
ails
h
ere
!!!!!
!!!!!!
!!!!
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DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 8
Cosine-Modulated Filter Banks
Realization based on polyphase decomposition (continued): - Note that C (the only dense matrix here) is NxN DCT-matrix (`Type 4’)
hence fast implementation (=fast matrix-vector product) based on fast discrete cosine transform (DCT) procedure, with complexity O(N.logN)
Modulated filter bank gives economy in * design (only prototype Po(z) ) * implementation (prototype + modulation (DCT))
Similar structure for synthesis bank
)}5.0).(5.0.(cos{2
}{ , qpNN
C qp
NNT 2
u[k]
)( 20
NzE
)( 21
NzE
)( 212
NN zE
)(0 zH
)(1 zH
)(1 zH N
: :Skip this slid
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DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 9
Cosine-Modulated Filter Banks
Maximally decimated cosine modulated (analysis) bank :
NNT 2
u[k]
)( 20
NzE
)( 21
NzE
)( 212
NN zE
:
N
N
N
NNT 2
u[k]
)( 20 zE
)( 21 zE
)( 212 zE N
:
N
N
N=
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 10
Cosine-Modulated Filter Banks
Question: How do we obtain Maximal Decimation + PR/FIR/Paraunitariness?
Theorem: (proof omitted)
-If prototype Po(z) is a real-coefficient (L+1)-taps FIR filter, (L+1)=2N.m for integer m and po[k]=po[L-k] (linear phase), with polyphase components En(z), n=0,1,…2N-1, -then the (FIR) cosine-modulated analysis bank is PARAUNITARY if and only if (n=0..N-1) are power complementary, i.e. form a lossless 1 input/2 output system
FIR synthesis bank (for PR) can then be obtained by paraconjugation !!! = great result…
..th
is is
th
e h
ard
par
t…
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DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 11
Cosine-Modulated Filter Banks
Perfect Reconstruction (continued)
Design procedure: Parameterize lossless systems for n=0,1..,N-1 Optimize all parameters in this parametrization so that the prototype Po(z) based on these polyphase components is a linear-phase lowpass filter that satisfies the given specifications
Example parameterization: Parameterize lossless systems for n=0,1..,N-1, -> lattice structure (see Part-II), where parameters are rotation angles
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DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 12
Cosine-Modulated Filter Banks
PS: Linear phase property for po[n] implies that only half of the power
complementary pairs have to be designed. The other pairs are then
defined by symmetry properties.
NNT 2
u[k]
:
N
Np.9 = )( 20 zE
)( 2zEN
)( 21 zEN
)( 212 zE N
:
:
lossless 1-in/2-out
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 13
Cosine-Modulated Filter Banks
PS: Cosine versus DFT modulation In a maximally decimated cosine-modulated (analysis) filter bank 2 polyphase components of the prototype filter, ,
actually take the place of only 1 polyphase component in the DFT- modulated case. For paraunitariness (hence FIR-PR) in a cosine-modulated bank, each such pair of polyphase filters should form a power complementary pair, i.e. represent a lossless system.
In the DFT-modulated case, imposing paraunitariness is equivalent to imposing losslessness for each polyphase component separately, i.e. each polyphase component should be an `allpass’ transfer function. Allpass functions are always IIR, except for trivial cases (pure delays). Hence all FIR paraunitary DFT-modulated banks (with maximal decimation) are trivial modifications of the DFT bank.
no FIR-design flexibility
provides flexibility for FIR-design
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 14
Time-Frequency Analysis & Wavelets
Starting point is discrete-time Fourier transform:
= infinitely long sequence u[k] is evaluated at infinitely many frequencies
Inversion/reconstruction/synthesis (=filter bank jargon) is..
= sequence u[k] is represented as weighted sum of basis functions
20 , ].[)(
k
kjj ekueU
2
0
).(2
1][ deeUku kjj
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 15
Time-Frequency Analysis & Wavelets
• `uncertainty principle’ says that if u[k] has a narrow support
(i.e. is localized), then U(.) has a wide support (i.e. is non-
localized), and vice versa• Hence notion of `frequency that varies with time’ not
accommodated (e.g. `short lived sine’ will correspond to
non-localized spectrum)
20 , ].[)(
k
kjj ekueU
2
0
).(2
1][ deeUku kjj
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 16
Tool to fill this need is `short-time Fourier transform’(STFT)
where w[k] is your favorite window function (typically with `compact support’ (=FIR) )
• Window slides past the data. For each window position k, compute discrete-time Fourier transform
PS: If w[k]=1 (all k) then result is goodold discrete-time FT for all window positions
• In following slides, will provide a filter bank version of STFT, also leading to simple inversion formula
Time-Frequency Analysis & Wavelets
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 17
Time-Frequency Analysis & Wavelets
Rewrite STFT formula as…
or similarly…(swap k and k-bar)
• If we forget about the fase factor up front (=modulate window i.o input),
then this corresponds to performing a convolution of u[k] with a filter
(see Chapter-2, p.19)
In practice, will compute this for
a discrete set of (N) frequencies…
leading to a set of filters…
• This is a DFT-modulated analysis bank, prototype filter = window function
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 18
Time-Frequency Analysis & Wavelets
• If window length>N, can use efficient implementation
based on polyphase decomposition
of prototype Ho + DFT-modulation
• Often window length=N, hence
1-tap polyphase components u[k]
)(0 zH
)(1 zH
)(2 zH
)(3 zH
*NNF
fre
q.re
solu
tion
N
*NNF
u[k]
)( 40 zE
)( 41 zE
)( 42 zE
)( 43 zE
)(0 zH
)(1 zH
)(2 zH
)(3 zH
window length/N
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 19
Time-Frequency Analysis & Wavelets
In practice, will consider ‘window shift’ >1 (decimation D>1)• If maximally decimated (D=N), then analysis FB operation is
xn[k] = decimated subband signals
= `STFT-coefficients’
= infinitely long sequence u[k] is
evaluated at N frequencies,
infinitely many times (i.e. for
infinitely many window positions)
...to be compared to page 14
u[k]
*NNF 4
4
4
4
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 20
Time-Frequency Analysis & Wavelets
• Corresponding (PR) synthesis filter bank (also DFT-modulated) is
i.e. synthesis prototype filter fo[k]
= ‘synthesis window’
=
• Reconstruction/synthesis formula
(=inverse STFT) is
..to be compared to page 14
FRFE .)( .)( 1* ii wdiagzwdiagz
4
4
4
4
+
+
+
F
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 21
Time-Frequency Analysis & Wavelets
• If oversampled (D=N/d), then analysis FB operation is
Example: d=2 (‘window shift’= N/2)
u[k]
*NNF 2
2
2
2
u[k]
*NNF
2
2
B(z)
=
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 22
Time-Frequency Analysis & Wavelets
• Corresponding (PR) synthesis filter bank (also DFT-modulated)
with PR condition for synthesis window v_k
• Can easily generalize this for other oversampling factors d, leading to
PR condition explain…
(and try examples with
Hann windows)
• This is referred to as Weighted OverLap-Add (WOLA)
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 23
Time-Frequency Analysis & Wavelets
Now, for some applications (e.g. audio) would like to have
a non-uniform filter bank, hence also with non-uniform
(maximum) decimation, for instance
• non-uniform filters = low frequency resolution at high frequencies, high frequency resolution at low frequencies (as human hearing)
• non-uniform decimation = high time resolution at high frequencies, low time resolution at low frequencies
H2(z)
H3(z)
4
2
H0(z)
H1(z)
8
8u[k] H0 H3H2H1
2
8
4
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 24
Time-Frequency Analysis & Wavelets
This can be built as a tree-structure, based on a
2-channel filter bank with
H0 H3H2H1
2
8
4
)(zHLP )(zHHP
u[k]2
2)(zHHP
)(zHLP
2
2)(zHHP
)(zHLP
2
2)(zHHP
)(zHLP
)().().()(
)().().()(
)().()(
)()(
240
241
22
3
zHzHzHzH
zHzHzHzH
zHzHzH
zHzH
LPLPLP
LPLPHP
LPHP
HP
)(),( zHzH HPLP
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 25
Time-Frequency Analysis & Wavelets
Note that may be viewed as a prototype filter,
from which a series of filters is derived
The lowpass filters are then needed to turn these
multi-band filters into bandpass filters (i.e. remove images)
)()(1 zHzH HPN
)()( )2( 1
k
zHzH HPkN
2
8
4
)(zHHP
)( 4zHHP
)( 2zHHP
)().().()(
)().().()(
)().()(
)()(
240
241
22
3
zHzHzHzH
zHzHzHzH
zHzHzH
zHzH
LPLPLP
LPLPHP
LPHP
HP
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 26
Time-Frequency Analysis & Wavelets
Similar synthesis bank can be constructed with
• If and form a PR FB, then the complete analysis/synthesis structure is PR (why?)
)(),( zFzF HPLP
2
2 +
2
2 + 2
2 + )(zFLP
)(zFHP
)(zFLP
)(zFLP)(zFHP
)(zFHP
)(),( zFzF HPLP)(),( zHzH HPLP
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 27
Time-Frequency Analysis & Wavelets
• Analysis bank corresponds to `discrete-time wavelet transform’ (DTWT) (xk[n] = `DTWT-coefficients’)
• With a corresponding (PR) synthesis filter bank, the reconstruction/synthesis formula (inverse DTWT) is
..to be compared to page 14 & 20
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 28
Time-Frequency Analysis & Wavelets
• Reconstruction formula may be viewed as an expansion of u[k], using a set of basis functions (infinitely many)
• If the 2-channel filter bank is paraunitary, then this basis is orthonormal (which is a desirable property) :
=`orthonormal wavelet basis’
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 29
Time-Frequency Analysis & Wavelets
• Example : `Haar’ wavelet (after Alfred Haar)
• Compare to 2-channel DFT/IDFT bank• Derive formulas for Ho, H1, H2, H3, …
Derive formulas for Fo, F1, F2, F3, …
Paraunitary FB (orthonormal wavelet basis) ?
)1(2
1
)1(2
1
1
1
zH
zHH
LP
HPHaar
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 30
Time-Frequency Analysis & Wavelets
Not treated here :• `continuous wavelet transform’ (CWT) of a continuous-time function u(t)
h(t)=prototype p,q are real-valued continuous variables p introduces `dilation’ of prototype, q introduces `shift’ of prototype • `discrete wavelet transform’ (DWT) is CWT with discretized p,q
T is sampling interval p-bar, q-bar are real-valued integer variables mostly a=2
dt
p
tqhtu
pqpxCWT )().(
1),(
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DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 31
Time-Frequency Analysis & Wavelets
Not treated here :• Theory
- multiresolution analysis
- wavelet packets
- 2D transforms
- etc …• Applications :
- audio: de-noising, …
- communications : wavelet modulation
- image : image coding Skip this slid
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DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 32
Frequency Domain Filtering
• See DSP-I : cheap FIR realization based on frequency domain processing (`time domain convolution equivalent to component-wise multiplication in the frequency domain’), cfr. `overlap-add’ & `overlap-save’ procedures
• This can be cast in the subband processing setting, as a non-critically downsampled (2-fold oversampled) DFT-modulated filter bank based operation!
• Leads to more general approach to performance/delay trade-off
PS: formulae given for N=4, for conciseness (but without loss of generality)
Chapter 6!
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 33
Frequency Domain Filtering
Have to know a theorem from linear algebra here: • A `circulant’ matrix is a matrix where each row is obtained from the previous row using a right-shift (by 1 position), the rightmost element which spills over is circulated back to become the leftmost element • The eigenvalue decomposition of a `circulant’ matrix is trivial…. example (4x4):
with F the NxN DFT-matrix, this means that the eigenvectors are equal to the column-vectors of the IDFT-matrix, and that then eigenvalues are obtained as the DFT of the first column of the circulant matrix (proof by Matlab)
d
c
b
a
F
D
C
B
A
F
D
C
B
A
F
abcd
dabc
cdab
bcda
. with ,.
000
000
000
000
.1
abcd
dabc
cdab
bcda
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 34
Frequency Domain Filtering
Starting point is this (see Chapter-8) :
meaning that a filtering with
can be realized in a multirate structure, based on a
pseudo-circulant matrix
T(z)*u[k-3]1z2z3z
1
u[k] 444
4 4444
+1z
2z
3z
1
)(zT
)()(.)(.)(.
)()()(.)(.
)()()()(.
)()()()(
)(
031
21
11
1031
21
21031
3210
zpzpzzpzzpz
zpzpzpzzpz
zpzpzpzpz
zpzpzpzp
zT
)()()()()( 43
342
241
140 zpzzpzzpzzpzT
N*N
filt
ers,
L/N
tap
s ea
ch
at a
n N
-fol
d lo
wer
rat
e (N
=4)
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 35
Frequency Domain Filtering
Now some matrix manipulation… :
)()(
)(
44.1
44
7
6
5
4
3
2
1
0
144
44.1
44
3210
3210
3210
3210
0321
1032
2103
3210
44
..
)(0000000
0)(000000
00)(00000
000)(0000
0000)(000
00000)(00
000000)(0
0000000)(
..0
.
0)()()()(000
00)()()()(00
000)()()()(0
0000)()()()(
)(0000)()()(
)()(0000)()(
)()()(0000)(
)()()()(0000
.0
zz
z
x
xx
x
xx
Iz
IF
zP
zP
zP
zP
zP
zP
zP
zP
FI
Iz
I
zpzpzpzp
zpzpzpzp
zpzpzpzp
zpzpzpzp
zpzpzpzp
zpzpzpzp
zpzpzpzp
zpzpzpzp
I
ER
T
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 36
Frequency Domain Filtering
• An (8-channel) filter bank representation of this is...
Analysis bank:
Subband processing:
Synthesis bank:
This is a 2N-channel filter bank, with N-fold downsampling. The analysis FB is a 2N-channel uniform DFT filter bank (see Chapter 9, p.30 !). The synthesis FB is matched to the analysis bank, for PR:
..
.)(44
144
x
x
Iz
IFzE
144 .0)( FIz xR
1z2z3z
1
u[k] 444
4 4444
+y[k]
1z
2z
3z
1
)(zR)(zH)(zE
)}(),...(),({)( 710 zPzPzPdiagzH
441)().( xIzzz ER
2N
filte
rs,
L/N
tap
s ea
chat
an
N-f
old
low
er r
ate
||
N/2
-fol
d co
mpl
exity
red
uctio
n
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 37
Frequency Domain Filtering
• This is known as an `overlap-save’ realization :– Analysis bank: performs 2N-point DFT (FFT) of a block of (N=4)
samples, together with the previous block of (N) samples (hence `overlap’)
– Synthesis bank: performs 2N-point IDFT (IFFT), throws away the first half of the result, saves the second half
(hence `save’)
– Subband processing corresponds to `frequency domain’ operation
..
.)(44
144
x
x
Iz
IFzE
`block’
`previous block’
144 .0)( FIz xR
`save’`throw away’
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 38
Frequency Domain Filtering
`Overlap-add’ can be similarly derived :
)()(
)(
44
44
7
6
5
4
3
2
1
0
14444
.1
44
44
3210
3210
3210
3210
0321
1032
2103
3210
4444.1
0..
)(0000000
0)(000000
00)(00000
000)(0000
0000)(000
00000)(00
000000)(0
0000000)(
..
0.
0)()()()(000
00)()()()(00
000)()()()(0
0000)()()()(
)(0000)()()(
)()(0000)()(
)()()(0000)(
)()()()(0000
.
zz
z
x
xxx
x
xxx
IF
zP
zP
zP
zP
zP
zP
zP
zP
FIIz
I
zpzpzpzp
zpzpzpzp
zpzpzpzp
zpzpzpzp
zpzpzpzp
zpzpzpzp
zpzpzpzp
zpzpzpzp
IIz
ER
T
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 39
Frequency Domain Filtering
• This is known as an `overlap-add’ realization :– Analysis bank: performs 2N-point DFT (FFT) of a block of (N=4)
samples, padded with N zero samples
– Synthesis bank: performs 2N-point IDFT (IFFT), adds second half of the result to first half of previous IDFT (hence `add’)
– Subband processing corresponds to `frequency domain’ operation
.0
.)(44
44
x
xIFzE`block’
`zero padding’
14444
1 ..)( FIIzz xxR
`add’`overlap’
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 40
Frequency Domain Filtering
• Standard `Overlap-add’ and `overlap-save’ realizations are derived when 0th order poly-phase components are used in the above derivation, i.e. each poly-phase component represents 1 tap of an L-tap filter T(z). (N=L)
The corresponding 0th order subband processing (H) then corresponds to what is usually referred to as the `component-wise multiplication’ in the frequency domain.
Note that for an L-tap filter, with large L, this leads to a cheap realization based on FFT/IFFTs instead of DFT/IDFTs.
However, for large L, as 2L-point FFT/IFFTs are needed, this may also lead to an unacceptably large processing delay (latency) between filter input and output.
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 41
Frequency Domain Filtering
• In the more general case, with higher-order polyphase components (hence N smaller than the filter length L), a smaller complexity reduction is achieved, but the processing delay is also smaller.
• This provides an interesting trade-off between complexity reduction and latency !!
DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 p. 42
Conclusions
• Great (=FIR/paraunitary) perfect reconstruction FB designs based on `modulation’:– Oversampled DFT-modulated FBs (Chapter-9)– Maximally decimated (and oversampled (not treated here))
cosine-modulated FBs
• `Perfect reconstruction’ concept provides framework for time-frequency analysis of signals
• Filter bank concept provides framework for frequency domain realization of long FIR filters