1. EEE4176 Applications of Digital Signal Processing
Fall 2011
Assistant Prof. Yangmo Yoo
Dept. of Electronic Engineering
Sogang University

2. Lecture 4: Filter Banks

3. Filter Banks Subband processing
Spliting the freq range into M subbands
Filter bank : Analysis filter bank
Process each subband independently : subband coding, etc.
Recombine the processed results
Synthesis filter bank
Perfect reconstruction
property
No aliasing error
No amplitude distortion
No phase distortion
Quantization error can not
be compensated.

4. Filter Banks
Uniform filter bank : All subbands have the same bandwidth
zeroth subband
Subbands are often overlapped.
m-th analysis and synthesis filters usually occupy the same frequency range for Perfect reconstruction.
Complex filter vs. real filter :
# of subbands are same.

5. Filter Banks Maximally decimated filter bank
Each subband has a bandwidth of (Uniform filter bank case)
Each subband signals can be decimated by a factor of
Even when the signals
are decimated, we still need
the perfect reconstruction
property.
That is, first aliasing error
must be eliminated.
Analysis and Synthesis filters are
to be properly designed to meet
the PR condition

6. Filter Banks
Two-channel Filter Bank

7. Filter Banks Two-channel Filter Bank Aliasing Term:
Alias free condition
: LPF
: HPF

8. Filter Banks Two-channel Filter Bank Alias free condition Note: If
: Perfect reconstruction
Amplitude distortion if
Phase distortion if is not linear

9. Filter Banks Quadrature Mirror Filter (QMF) Bank
Simple alias-free condition
Transfer function of a QMF bank
: is the mirror of around
that is, at the quadrature frequency (
: LPF
: HPF

10. Filter Banks Quadrature Mirror Filter (QMF) Bank Note
FIR QMF with simple alias-free condition
can’t have sharp response
If has linear phase, then no phase distortion produced.
should be odd. otherwise (severe amplitude distortion)
Amplitude distortion
By using proper IIR filters, we can eliminate aliasing error, amplitude distortion but not phase distortion.
: Linear phase
can be minimized using some filter design techniques (ex: Jonston's)

11. Filter Banks Quadrature Mirror Filter (QMF) Bank
Remarks:Linear phase QMF filter bank where
No ALD (Alias distortion
No PHD --> No AMD but trivial filter
So, try to minimize AMD using nontrivial filters
Johnston's technique to design filters optimally satisfying
Filter design using CAD technique
Objective function:
: Power symmetry property
If we don't stick to , we can achieve real PR FIR QMF

12. Perfect Reconstruction (PR) Filter Banks
Conjugate quadrature filters (CQF)
For Alias cancellation,
For Perfect reconstruction,
: Power symmetric condition

13. Perfect Reconstruction (PR) Filter Banks
Conjugate quadrature filters (CQF)
Note: How to design Power Symmetric Filters ?
Design : Half band filter
Spectral factorization to obtain from
for all

14. Perfect Reconstruction (PR) Filter Banks
Tree-structured filter banks
Two level maximally decimated tree-structured filter banks
If the QMF bank is alias free or PR systems, then so is the whole system.
Different QMF banks can be used at different levels.
Different tree-strictures (non-binary tree-structure) are also used.

15. Perfect Reconstruction (PR) Filter Banks
Tree-structured filter banks
Equivalent form

16. Perfect Reconstruction (PR) Filter Banks
Tree-structured filter banks

17. Perfect Reconstruction (PR) Filter Banks
Tree-structured filter banks

18. Perfect Reconstruction (PR) Filter Banks
A Tree-structured filter bank having
subbands
Draw a tree-structured QMF bank of which the equivalent analysis filter bank has the above frequency responses.
Subband coding for data compression
Each band signal is quantized differently to compress the data. Data loss should be limited to a certain tolerable level.
MPEG standard : audio compression with 32 subbands

19. Perfect Reconstruction (PR) Filter Banks
Octave-Band Filter banks
Redraw it so that it looks
more tree-structured !!

20. Uniform DFT filter bank
DFT matrix
: matrix with
: Conjugate of DFT matrix

21. Uniform DFT filter bank
Let's define a set of filters (analysis filters) :
: rectangular window

22. Uniform DFT filter bank
Properties of the uniform DFT filter bank
All filters , have the same length
are obtained by shifting the prototype filter
Right shift by
due to the use of conjugate DFT matrix
Mag response of is a sinc finction
Too high sidelobe: dB
unsatisfactory for subband processing
: complex filter
Spectrum analyzer (or moving DFT)

23. Uniform DFT filter bank
Properties of the uniform DFT filter bank
Spectrum analyzer (or moving DFT)
center frequency =
Frequency resolution is not good.
What if we use a better window to improve the frequency resolution ?
This is equivalent to Analysis filter bank / Conjugate DFT matrix cascade
Problems : Filter length limited to , RECTANGULAR WINDOW

24. Uniform DFT filter bank
Output of the DFT matrix in the synthesis bank
Since
We will get
Maximally decimated version: Decimated DFT filter bank
: Perfect reconstruction
Input vector to the
conjugate DFT matrix
Output:
- Same as before
- Perfect reconstruction

25. Uniform DFT filter bank
Polyphase implementation of DFT filter bank
DFT filter bank : Filter bank model
in the previous section
Consider an arbitrary length(N) filter with better characteristics
: Polyphase representation (length

26. Uniform DFT filter bank
Polyphase implementation of DFT filter bank
Decimated DFT filter bank
2008.9
EEE4176 Applications of Digital Signal Processing 26

27. Uniform DFT filter bank
Polyphase implementation of DFT filter bank
Decimated DFT filter bank
complex operations / unit sample time
In direct structure,
Can we make it have perfect reconstruction property ?