1. DSP-CIS Chapter 9: Filter Banks - Special Topics
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
Time-frequency analysis & Wavelets
Frequency domain filtering
Chapter-7
Chapter-8
Chapter-9

3. Topic-1: 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).

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... H0 H3 H2 H1 P0 H1 Ho

5. Cosine-Modulated Filter Banks
- if Po(z) is prototype FIR lowpass filter with real coefficients
po[n], n=0,1,…,L then
i.e. `cosine modulation’ (with real coefficients) instead of `exponential
modulation’ (with complex coeffs, see DFT-modulated FBs Chapter-8)
- if Po(z) = `good’ lowpass filter, then Hk(z)’s = `good’ bandpass filters

6. Cosine-Modulated Filter Banks
Skip this slide
Realization based on polyphase decomposition (analysis):
- if Po(z) has 2N-fold polyphase expansion (ps: 2N-fold for N filters!!!)
then...
u[k] : :

7. Cosine-Modulated Filter Banks
Skip this slide
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
ignore all details here !!!!!!!!!!!!!!!

8. Cosine-Modulated Filter Banks
Skip this slide
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
u[k] :

9. Cosine-Modulated Filter Banks
Skip this slide
Maximally decimated cosine modulated (analysis) bank :
u[k] : N
u[k] : N =

10. Cosine-Modulated Filter Banks
Skip this slide
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[n]=po[L-n] (linear phase),
with polyphase components Ek(z), k=0,1,…2N-1,
-then the (FIR) cosine-modulated analysis bank is PARAUNITARY
if and only if (for all k) are power complementary,
i.e. form a lossless 1 input/2 output system
And then FIR synthesis bank (for PR) can be obtained
by paraconjugation !!! = great result…
..this is the hard part…

11. Cosine-Modulated Filter Banks
Skip this slide
Perfect Reconstruction (continued)
Design procedure:
Parameterize lossless systems for k=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 k=0,1..,N-1,
-> lattice structure (see Part-I), where parameters are rotation angles

12. Cosine-Modulated Filter Banks
Skip this slide
lossless 1-in/2-out
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.
u[k]
p.9 = N : : N :

13. Cosine-Modulated Filter Banks
Skip this slide
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.
provides flexibility for FIR-design
no FIR-design flexibility

14. Topic2: 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

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)

16. Time-Frequency Analysis & Wavelets
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 result is discrete-time FT for all n
In following slides, will provide a filter bank version of STFT, also leading to simple inversion formula

17. Time-Frequency Analysis & Wavelets
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

18. Time-Frequency Analysis & Wavelets
Efficient implementation based on polyphase decomposition of prototype Ho + DFT-modulation
Often window length=N, hence
1-tap polyphase components
window length/N
u[k]
freq.resolution N
u[k]

19. Time-Frequency Analysis & Wavelets
If maximally decimated (M=N, decimation=`window shift’),
decimated DFT-modulated analysis bank corresponds to
xk[n] = 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

20. Time-Frequency Analysis & Wavelets
With a corresponding (PR) synthesis filter bank (see Chapter 7)
Ex:
the reconstruction/synthesis formula (=inverse STFT) is
..to be compared to page 14
PS: can also do oversampled versions
H2(z)
H3(z) 4
F2(z)
F3(z)
y[k]
H0(z)
H1(z)
u[k]
F0(z)
F1(z) +

21. 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 H0 H3 H2 H1
H2(z)
H3(z) 4 2
H0(z)
H1(z) 8
u[k]

22. Time-Frequency Analysis & Wavelets
This can be built as a tree-structure, based on a
2-channel filter bank with
u[k] 2 H0 H1 H2 H3

23. 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)

24. 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?) 2 +

25. Time-Frequency Analysis & 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 14 & 20

26. Time-Frequency Analysis & 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’

27. 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) ?

28. Time-Frequency Analysis & Wavelets
Skip this slide
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

29. Time-Frequency Analysis & Wavelets
Skip this slide
Not treated here :
Theory
- multiresolution analysis
- wavelet packets
- 2D transforms
- etc …
Applications :
- audio: de-noising, …
- communications : wavelet modulation
- image : image coding

30. Topic-3: Frequency Domain Filtering
See DSP-I : cheap FIR 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)

31. 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)

32. Frequency Domain Filtering
Starting point is this (see Chapter-7) :
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 +
at an N-fold lower rate (N=4)
N*N filters, L/N taps each

33. Frequency Domain Filtering
Now some matrix manipulation… :

34. 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 8, p.30 !).
The synthesis FB is matched to the analysis bank, for PR:
u[k] 4 +
y[k]
N/2-fold complexity reduction
2N filters, L/N taps each
at an N-fold lower rate ||

35. 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
`block’
`previous block’
`save’
`throw away’

36. Frequency Domain Filtering
`Overlap-add’ can be similarly derived :

37. 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
`block’
`zero padding’
`add’
`overlap’

38. 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.

39. 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 !!

40. Conclusions
Great (=FIR/paraunitary) perfect reconstruction FB designs based on `modulation’:
Oversampled DFT-modulated FBs (Chapter-8)
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