2. INTRODUCTION TO FILTER BANKS
TYPES OF FILTER BANKS
UNIFORM DFT FILTER BANK
TWO CHANNEL FILTER BANK
POLYPHASE TWO CHANNEL FILTER BANK
SUMMARY
REFERENCES
TOPICS
3. Array of BPF that seperates the input signal into
multiple components, each one carrying a single
sub-band of the original signal.
A Digital filter bank is a collection of filters having
a common input or output.
2 types of filter banks
Analysis filter bank
Synthesis filter bank
INTRODUCTION
4.
5. 1-D AND 2-D FILTER BANKS
1-D FILTER BANK 2-D FILTER BANK
7. DIRECTIONAL FILTER
BANK
•modulating the input signal and
using diamond-shaped filters.
•Advantages:
•offers perfect reconstruction.
•directional-selectivity and efficient
structure.
•use in 3-D to achieve the frequency
sectioning.
• Sparse image representation,
medical imageing[ ignal and image
processing
9. Decomposition performed by the filter bank is called analysis
Output of analysis - >subband signal
The Analysis filter may be
non-overlapping,
slightly overlapping
substantially overlapping.
APPLICATION OF ANALYSIS FILTER BANKS:
Spectrum analysis. (split the input signal into R different so
– called subband signals)
ANALYSIS FILTER BANKS
10. Synthesis (i.e. recombining the outputs of multiple receivers)
INVOLVES:
upsampling each one at a rate with the total bandwidth to be
created,
translating each channel to its new center frequency, and
summing the streams of samples.
Interpolation filter +Upsampling -> Synthesis filter.
The combination of several signal into a common output signal is
called a Synthesis filter bank.
The reconstruction process is called synthesis, meaning
reconstitution of a complete signal resulting from the filtering
process.
SYNTHESIS FILTER BANKS
11. Important characteristic of filter banks
Bandwidth and spacing of the center frequencies of the
filters.
FILTER BANKS
UNIFORM FILTER BANKS
same bandwidth and same sampling rates
NON UNIFORM FILTER BANKS
Different bandwidth and different sampling rates
Uniform, maximal decimation filter banks are often
preferred .
OTHER FILTER BANK TYPES
13. DFT filter bank :
If the rth band filter hr[n] is computed from the
“modulation” of a single prototype filter h[n]
A DFT filter interpolation R = Number of bands K
Applications Of Filter Banks
Sub-band Adaptive Filtering
Transmultiplexers
Graphic Equalizer
Signal Compression
Vocoder
UNIFORM DFT FILTER BANK
15. ℎ 𝑟 𝑛 = ℎ 𝑛 𝑊𝑅
𝑟𝑛
= ℎ[𝑛]𝑒−𝑗2𝜋𝑟𝑛/𝑅
An efficient implementation of the R channel filter bank can be
generated using polyphasedecomposition of the filter ℎ 𝑟[𝑛]
and the input signal 𝑧[𝑛].
Because each of these bandpass filter is critically sampled, we
use a decomposition with R polyphase signals according to
ℎ 𝑛 =
𝑘=0
𝑅−1
ℎ 𝑘 𝑛 ↔ ℎ 𝑘 𝑚 = ℎ[𝑚𝑅 − 𝑘]
𝑥 𝑛 =
𝑘=0
𝑅−1
𝑥 𝑘 𝑛 ↔ 𝑥 𝑘 𝑚 = 𝑥[𝑚𝑅 − 𝑘]
18. 𝑓 𝑟
𝑛 =
1
𝑅
𝑓 𝑛 𝑊𝑅
𝑟𝑛
= 𝑓[𝑛]𝑒 𝑗2𝜋𝑟𝑛/𝑅
If we now combine the analysis and synthesis filter banks,
we can see that the DFT and IDFT annihilate each other, and
perfect reconstruction occurs if the convolution of the
included polyphase filter gives a unit sample function, i.e.,
ℎ 𝑟 𝑛 × 𝑓𝑟 𝑛 =
1
0
𝑛 = 𝑑
𝑒𝑙𝑠𝑒
In other words, the two polyphase functions must be
inverse filters of each other, i.e.,
𝐻𝑟 𝑧 × 𝐹𝑟 𝑧 = 𝑧−𝑑
𝐹𝑟 𝑧 =
𝑧−𝑑
𝐻𝑟(𝑧)
Where we allow a delay d in order to have casual
(realizable) filters. These ideal conditions cannot be met
exactly by two FIR filters.
21. The construction rule is normally given by
ℎ 𝑛 = (−1) 𝑛 𝑔 𝑛 ⊶ 𝐻 𝑧 = 𝐺 −𝑧
For the synthesis use an expander (a sampling rate
increase of 2), and then two separate reconstruction
filters, 𝐺^ 𝑧 and 𝐻^ 𝑧 ,to reconstruct 𝑥[𝑛].
A perfectly reconstructed signal has the sample shape
as the original, up to a phase (time) shift.
CONTD…
24. 1. Run-length filter using short Winograd convolution
algorithms
2. Fast convolution using FFT
3. Using advanced arithmetic concepts such as
distribute arithmetic, reduced adder graph, or
residue number system
VARIOUS METHODS TO IMPLEMENT
L/2 FILTERS
25. Constructs fast and efficient two-channel filter banks
The basic idea is the use of cross-terms (called lifting
and dual lifting), as in a lattice filter, to construct a
longer filter from short filter, while preserving the
perfect reconstruction conditions
LIFTING
26. Thus we have discussed about Analysis and Synthesis
filter banks,Uniform DFT and Non-uniform filter
banks
Two channel filter banks /Polyphase two-channel filter
bank.
Lifting approach has been listed out briefly.
Summary
27. Digital Signal Processing With Field Programmable
Gate Arrays By Uwe Meyer Baese
https://en.wikipedia.org/wiki/Filter_bank
REFERENCES