1. Digital Signal Processing II Lecture 8: Filter Banks - Special Topics
Marc Moonen
Dept. E.E./ESAT, K.U.Leuven
homes.esat.kuleuven.be/~moonen/

2. Part-II : Filter Banks Lecture-5 Lecture-6 Lecture-7 Lecture-8
: Preliminaries
Applications
Intro perfect reconstruction filter banks (PR FBs)
: Maximally decimated FBs
Multi-rate systems review
PR FBs
Paraunitary PR FBs
: Modulated FBs
DFT-modulated FBs
Cosine-modulated FBs
: Special Topics
Non-uniform FBs & Wavelets
Oversampled DFT-modulated FBs
Frequency domain filtering
Lecture-5
Lecture-6
Lecture-7
Lecture-8
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3. Topic-1: Non-Uniform FBs / Wavelets
Prelude
Starting point is discrete-time Fourier transform:
inversion/reconstruction/synthesis (=filter bank jargon) is..
`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
Notion of `frequency that varies with time’ not accommodated
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4. Non-Uniform FBs / Wavelets
Prelude
Tool to fill this need is `short-time Fourier transform’(STFT):
where w[n] is your favorite window function (typically with `compact support’ (=FIR) )
Window slides past the data. For each window position n, compute discrete-time Fourier transform.
PS: If w[n]=1 for all n, then STFT = discrete-time FT for all n
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5. Non-Uniform FBs / Wavelets
Prelude
Rewrite STFT formula as…
If we forget about the fase factor up front (meaning what?),
then this corresponds to performing a convolution with a filter
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
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6. Non-Uniform FBs / Wavelets
Prelude
Efficient implementation based on polyphase decomposition of prototype Ho + FFT-modulation
Often window length=N, hence
1-tap polyphase components
u[k]
u[k]
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7. Non-Uniform FBs / Wavelets
Prelude
If maximally decimated (M=N, decimation=`window shift’), this DFT-modulated analysis bank corresponds to a STFT
xk[n] = decimated subband signals = STFT-coefficients
With a corresponding (PR) synthesis filter bank (see Lecture 7)
Ex:
the reconstruction/synthesis formula (inverse STFT) is
..to be compared to page 3
H2(z)
H3(z) 4
F2(z)
F3(z)
y[k]
H0(z)
H1(z)
u[k]
F0(z)
F1(z) +
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8. Non-Uniform FBs / 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 H0 H3 H2 H1
H2(z)
H3(z) 4 2
H0(z)
H1(z) 8
u[k]
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9. Non-Uniform FBs / Wavelets
This can be built as a tree-structure, based on a
2-channel filter bank with
u[k] 2 H0 H1 H2 H3
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10. Non-Uniform FBs / 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)
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11. Non-Uniform FBs / Wavelets
Similar synthesis bank can be constructed with
If and form a PR FB, then the complete analysis/synthesis structure is PR (why?) 2 +
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12. Non-Uniform FBs / Wavelets
Analysis bank corresponds to `discrete-time wavelet transform’ (DTWT)
With a corresponding (PR) synthesis filter bank, the reconstruction/synthesis formula (inverse DTWT) is
..to be compared to page 3 & 7
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13. Non-Uniform FBs / Wavelets
Reconstruction formula may be viewed as an expansion of u[n], 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’
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14. Non-Uniform FBs / Wavelets
Example : `Haar’ wavelet (after Alfred Haar)
Formulas for Ho, H1, H2, H3, … ?
Formulas for Fo, F1, F2, F3, …?
Paraunitary FB (orthonormal wavelet basis) ?
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15. Non-Uniform FBs / Wavelets
ignore details…
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
k,n are real-valued integer variables mostly a=2
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16. Non-Uniform FBs / Wavelets
Not treated here :
Theory
- multiresolution analysis
- wavelet packets
- 2D transforms
- etc …
Applications :
- audio: de-noising, …
- communications : wavelet modulation
- image : image coding
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17. Topic-2: 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 PR FIR (!) DFT-modulated banks
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18. Oversampled PR Filter Banks
Starting point is (see Lecture-6):
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]
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19. 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
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20. 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
FIR PR-FBs, which can even (see below) be modulated and paraunitary
In the sequel, give 2 examples of oversampled DFT-modulated FBs
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21. Oversampled PR Filter Banks
Oversampled DFT-modulated PR FBs :
Example-1 : # channels M = Ho(z),H1(z),…,H7(z)
decimation N = 4
prototype filter Ho(z)
will consider N’-fold polyphase expansion, with
Should not really try
to understand this…
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22. Oversampled PR Filter Banks
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]
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Convince yourself that this is indeed correct..

23. Oversampled PR Filter Banks
With N-fold (N=4) decimation, this is…
Perfect Reconstruction by paraunitariness ?
- If E(z )=F*.B(z) is paraunitary, then PR is obtained with R(z)=B~(z).F (=NxM)
- Analysis bank E(z) is paraunitary if B(z) (=MxN) is paraunitary
- When is B(z) paraunitary ?
u[k] 4 4 +
u[k-3]
u[k]
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24. Oversampled PR Filter Banks
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
Solution & design procedure: FIR lattices
In general : non-trivial FIR solutions exist for M>N (unlike for M=N, see
Lecture 7). This is great news : oversampled PR-FBs can be at the same
time DFT-modulated (`M filters for the price of 1’), FIR and paraunitary !
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25. Oversampled PR Filter Banks
Example-2 : # channels M = Ho(z),H1(z),…,H5(z)
decimation N = 4
prototype filter Ho(z)
will consider N’-fold polyphase expansion, with
Should not really try
to understand this…
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26. Oversampled PR Filter Banks
Remember (page 18) that the 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]
Version Lecture-8 Filter Banks – Special Topics
Convince yourself that this is indeed correct..

27. Oversampled PR Filter Banks
With N-fold (N=4) decimation, this is…
Perfect Reconstruction by paraunitariness ?
- E(z) paraunitary iff B(z) paraunitary. Etc….
- A bit more complicated here (hence skipped), but still leading to non-trivial solutions
Conclusion : Oversampling provides additional design flexibility, not
available in maximally decimated case.
Hence can have it all at once : FIR, PR, DFT-modulated, paraunitariness!
u[k] 4
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28. Topic-3 : Frequency Domain Filtering
See DSP-I : cheap filtering based on frequency domain realization (`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 operation!
Leads to more general approach to performance/delay trade-off
PS: formulae given for N=4, for conciseness (but without loss of generality)
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29. 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
example (4x4):
The eigenvalue decomposition of a `circulant’ matrix is trivial….
with F the NxN DFT-matrix, this means that the eigenvectors are
always equal to the column-vectors of the IDFT-matrix, and that the
eigenvalues are obtained as the DFT of the first column of the circulant
matrix (...proof by Matlab)
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30. Frequency Domain Filtering
Starting point is this (see Lecture-6) :
meaning that a filtering with
can be realized in a multirate structure, based on a pseudo-
circulant matrix
T(z)*u[k-3]
u[k] 4 +
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31. Frequency Domain Filtering
Now some matrix manipulation… :
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32. Frequency Domain Filtering
A subband processing (8-channel filter bank) representation of this is...
Analysis bank:
Synthesis bank:
Subband processing:
This is a 2N-channel filter bank, with N-fold downsampling. The analysis filter
bank is a 2N-channel uniform DFT filter bank. The synthesis bank is (different, but) matched to the analysis bank, for PR under 2-fold oversampling.
u[k] 4 +
y[k]
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33. 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’ operations
(see next slide)
`Overlap-add’ can be similarly derived (try it!)
`block’
`previous block’
`throw away’
`save’
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34. 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 N-tap filter T(z). 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 N-tap filter, with large N, this leads to a cheap realization based on FFT/IFFTs instead of DFT/IDFTs.
However, for large N, as 2N-point FFT/IFFTs are needed, this may also lead to an unacceptably large processing delay (latency) between filter input and output.
In the more general case, with higher-order polyphase components (hence N smaller than the filter length), a smaller complexity reduction is achieved, but the processing delay is also smaller. This provides an interesting trade-off between complexity reduction and latency.
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