Digital Signal Processing II Lecture 5: Filter Banks Preliminaries

1
Digital Signal Processing IILecture 5 Filter
Banks - Preliminaries

• Marc Moonen
• Dept. E.E./ESAT, K.U.Leuven
• marc.moonen_at_esat.kuleuven.be
• homes.esat.kuleuven.be/moonen/

2
Part-II Filter Banks

• 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
• 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
3
Filter Banks Introduction

• What we have in mind is this
• - Signals split into frequency
  channels/subbands
• - Per-channel/subband processing
• - Reconstruction synthesis of processed
  signal
• - Applications see below (audio coding
  etc.)
• - In practice, this is implemented as a
  multi-rate structure for higher
• efficiency (see next slides)

4
Filter Banks Introduction

• Step-1 Analysis filter bank
• - collection of M filters (analysis
  filters, decimation filters) with a
• common input signal
• - ideal (but non-practical) frequency
  responses ideal bandpass filters
• - typical frequency responses
  (overlapping, non-overlapping,)

5
Filter Banks Introduction

• Step-2 Decimators (downsamplers)
• - To increase efficiency, subband sampling
  rate is reduced by factor N
• ( Nyquist (bandpass) sampling
  theorem, see Lecture-6)
• - Maximally decimated filter banks
  (critically downsampled)
• subband samples fullband
  samples
• this sounds like maximum
  efficiency, but aliasing (see below)!
• - Oversampled filter banks
  (non-critically downsampled)
• subband samplesgt fullband
  samples

NM
NltM
M4
N3
3
H1(z)
3
IN
H2(z)
3
H3(z)
3
H4(z)
6
Filter Banks Introduction

• Step-3 Subband processing
• - Example
• coding (compression) (transmission
  or storage) decoding
• - Filter bank design mostly assumes subband
  processing has unit
• transfer function (output signalsinput
  signals), i.e. mostly ignores
• presence of subband processing

M4
N3
subband processing
H1(z)
subband processing
IN
H2(z)
subband processing
H3(z)
subband processing
H4(z)
7
Filter Banks Introduction

• Step-4 Expanders (upsamplers)
• - restore original fullband sampling rate by
  N-fold upsampling
• (insert N-1 zeros in between every two
  samples)

8
Filter Banks Introduction

• Step-5 Synthesis filter bank
• - upsampling has to be followed by
  (interpolation) filtering (see below)
• - collection of M synthesis
  (interpolation) filters, with a
• common (summed) output signal
• - frequency responses preferably
  matched to frequency responses of
• the analysis filters, e.g., to provide
  perfect reconstruction (see below)

9
Aliasing versus Perfect Reconstruction

• - Assume subband processing does not modify
  subband signals
• (e.g. lossless coding/decoding)
• -The overall aim would be to have
  ykuk-d, i.e. that the output signal
• is equal to the input signal up to a certain
  delay
• -But downsampling introduces ALIASING,
  especially so in maximally
• decimated (but also in non-maximally
  decimated) filter banks
• (see also Lecture-6)

10
Aliasing versus Perfect Reconstruction

• Question
• Can ykuk-d be achieved in the presence
  of aliasing ?
• Answer
• YES !! PERFECT RECONSTRUCTION banks with
• synthesis bank designed to remove aliasing
  effects !

11
Filter Banks Applications

• Subband coding
• Coding Fullband signal split into subbands
  downsampled
• subband signals separately
  encoded
• (e.g. subband with smaller
  energy content encoded with fewer bits)
• Decoding reconstruction of subband
  signals, then fullband
• signal synthesis
  (expanders synthesis filters)
• Example Image coding (e.g. wavelet
  filter banks)
• Example Audio coding
• e.g. digital compact
  cassette (DCC), MiniDisc, MPEG, ...
• Filter bandwidths and
  bit allocations chosen to further
• exploit perceptual
  properties of human hearing
• (perceptual coding,
  masking, etc.)

12
Filter Banks Applications

• Subband adaptive filtering
• - See Part III
• - Example Acoustic echo cancellation
• Adaptive filter models (time-varying)
  acoustic echo path and produces
• a copy of the echo, which is then
  subtracted from microphone signal.
• difficult problem !
• long acoustic impulse responses
• time-varying

13
Filter Banks Applications

• Subband adaptive filtering (continued)
• - Subband filtering M (simpler?) subband
  modeling problems instead
• of one (more complicated?) fullband
  modeling problem
• - Perfect reconstruction guarantees
  distortion-free desired near-end
• speech signal

14
Filter Banks Applications Transmuxs

• Transmultiplexers
• Frequency Division Multiplexing (FDM) in
  digital communications
• - M different source signals multiplexed
  into 1 transmit signal by
• expanders synthesis filters (ps here
  interpolation factor )
• - Received signal decomposed into M source
  signals by analysis filters
• decimators
• - Again ideal filters ideal bandpass
  filters

NgtM
15
Filter Banks Applications Transmuxs

• Transmultiplexers (continued)
• - Non-ideal synthesis analysis filters
  result in aliasing and
• distortion, as well as CROSS-TALK
  between channels,
• i.e. each reconstructed signal contains
  unwanted
• contributions from other signals
• - Analysis synthesis are reversed here, but
  similar perfect
• reconstruction theory (try it!) (where
  analysis bank removes
• cross-talk introduced by synthesis bank, if
  transmission
• channel distortion free)

16
Filter Banks Applications Transmuxs

• Transmultiplexers (continued)
• PS special case is Time Division Multiplexing
  (TDM),
• if synthesis and analysis filters are
  replaced by delay
• operators (and NM)

u1k,u2k,u3k,u4k,u1k1,u2k1...
4
4
4
u2k-1,u2k
transmission channel
4
4
u3k-1,u3k
4
4
u4k-1,u4k
0,0,0,u4k,0,0,0,u4k1...
17
Filter Banks Applications Transmuxs
Skip this slide

• Transmultiplexers (continued)
• PS special case is Code Division Multiple
  Access (CDMA),
• where filter coefficients(orthogonal
  ) user codes
• CDMA basics (see digital coms courses)
  
• -Each user (i) is assigned a length-N
  pseudo-random code sequence
• -Transmission For each symbol
  (k-th symbol for user-i), a
• chip sequence is transmitted
• -Mostly binary codes (
  ) with BPSK/QPSK symbols
• -Multiple access based on
  code-orthogonality (see below)

18
Filter Banks Applications Transmuxs
Skip this slide

• CDMA basics
• -Reception If (x) received signal
  transmitted chip sequence (i.e. no
• channel effect, no noise), multiply
  chips with (synchronized) code
• sequence sum.
• -Example (user i)
• transmitted symbols 1 -1.
  -1 1
• code sequence 1,1,-1,-1
• transmitted chips 1,1,-1,-1
  -1,-1,1,1 -1,-1,1,1 1,1,-1,-1
• received chips 1,1,-1,-1
  -1,-1,1,1 -1,-1,1,1 1,1,-1,-1
• 
  1,1,-1,-1 (mult. with code sum)
• received symbols (1/4) 1
  -1...-11
• (x) PS real-world CDMA is considerably more
  complicated (different channels for different
  users channel dispersion, asynchronous users,
  scrambling codes, etc.)

19
Filter Banks Applications Transmuxs
Skip this slide

• CDMA Transmission/reception block scheme
• -transmitter code-multiplication may be
  viewed as filtering operation,
• with FIR transmit filter
• -receiver code-multiplication summation
  may be viewed as filtering
• operation, with receive filter
• -PR for flat channel H(z)1 and if
  codes are orthogonal
• (prove it!)

4
4
4
4
u2k1,u2k
Base station
transmission channel
User-2 terminal
4
4
4
4
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PR-FB Example DFT/IDFT Filter Bank

• Fundamental question is..
• Downsampling introduces ALIASING, then how
  can
• PERFECT RECONSTRUCTION (PR) (i.e.
  ykuk-d)
• be achieved ?
• Next slides provide simple PR-FB examples, to
• demonstrate that PR can indeed (easily) be
  obtained
• Discover the magic of aliasing-compensation.

21
DFT/IDFT Filter Bank

• First attempt to design a perfect reconstruction
  filter bank
• - Starting point is this
• convince yourself that ykuk-3

22
DFT/IDFT Filter Bank

• - An equivalent representation is ...
• As ykuk-d, this can already be viewed as
  a perfect
• reconstruction filter bank (with
  aliasing in the subbands!)
• All analysis/synthesis filters are seen to
  be pure delays,
• hence are not frequency selective (i.e.
  far from ideal
• case with ideal bandpass filters.)
• ps transmux version see p.17

23
DFT/IDFT Filter Bank

• -now insert DFT-matrix (discrete Fourier
  transform)
• and its inverse (I-DFT)...
• as this clearly does not
  change the input-output relation (hence perfect
  reconstruction property preserved)

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DFT/IDFT Filter Bank

• - and reverse order of decimators/expanders
  and DFT-matrices (not done in an efficient
  implementation!)
• analysis filter bank
  synthesis filter bank
• This is the DFT/IDFT filter bank. It is a
  first example of a maximally decimated perfect
  reconstruction filter bank !

uk
25
DFT/IDFT Filter Bank

• What do analysis filters look like?
• This is seen (known) to represent a collection
  of filters Ho(z),H1(z),...,
• each of which is a frequency shifted version
  of Ho(z)
• i.e. the Hi are obtained by uniformly
  shifting the prototype Ho over the
• frequency axis.

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DFT/IDFT Filter Bank
Ho(z)
H1(z)

• The prototype filter Ho(z) is a not-so-great
• lowpass filter with first sidelobe only
• 13 dB below the main lobe. Ho(z) and
• Hi(z)s are thus far from ideal lowpass/
• bandpass filters.
• Hence (maximal) decimation introduces
• significant ALIASING in the decimated
  subband signals
• Still, we know this is a PERFECT RECONSTRUCTION
  filter bank (see construction p.23-26), which
  means the synthesis filters can apparently
  restore the aliasing distortion. This is
  remarkable!
• Other perfect reconstruction banks see
  Lecture-6

27
DFT/IDFT Filter Bank

• What do synthesis filters look like?
• synthesis filters are (roughly) equal to
  analysis filters
• (details omitted, see also Lecture-6)
• PS Efficient DFT/IDFT implementation based on
  FFT algorithm
• (Fast Fourier Transform).

(1/N)
28
Conclusions

• Seen the general subband processing set-up
  applications
• Filter bank system is multi-rate structure, with
  decimators and expanders, hence ALIASING is a
  major concern
• Seen a first (simple not-so-great) example of a
  PERFECT RECONSTRUCTION filter bank (DFT/IDFT)
• Sequel other (better) PR structures
• Lecture 6 Maximally decimated filter banks
• Lecture 7 Modulated filter banks
• Lecture 8 Oversampled filter banks, etc..
• Reference Multirate Systems Filter Banks ,
  P.P. Vaidyanathan
• Prentice Hall 1993.

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Review of Multi-rate Systems 1/10

• Decimation decimator (downsampler)
• example uk 1,2,3,4,5,6,7,8,9,
• 2-fold downsampling
  1,3,5,7,9,...
• Interpolation expander (upsampler)
• example uk 1,2,3,4,5,6,7,8,9,
• 2-fold upsampling
  1,0,2,0,3,0,4,0,5,0...

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Review of Multi-rate Systems 2/10

• Z-transform (frequency domain) analysis of
  expander
• expansion in time domain compression in
  frequency domain

images
31
Review of Multi-rate Systems 2bis/10

• Z-transform (frequency domain) analysis of
  expander
• expander mostly followed by interpolation
  filter to remove images
• (and interpolate the zeros)
• interpolation filter can be
  low-/band-/high-pass (see Lecture-6)

32
Review of Multi-rate Systems 3/10

• Z-transform (frequency domain) analysis of
  decimator
• compression in time domain expansion in
  frequency domain

N
u0, uN, u2N...
u0,u1,u2...
N
i0
i2
i1
3
33
Review of Multi-rate Systems 3bis/10

• Z-transform (frequency domain) analysis of
  decimator
• decimation introduces ALIASING if input signal
  occupies frequency band
• larger than , hence mostly
  preceded by anti-aliasing (decimation)
• filter
• anti-aliasing filter can be low-/band-/high-pas
  s (see Lecture-6)

N
u0, uN, u2N...
u0,u1,u2...
i0
i2
i1
3
34
Review of Multi-rate Systems 4/10

• Z-transform analysis of decimator (continued)
• - Note that is periodic with
  period
• while is periodic
  with period
• the summation with i0N-1 restores the
  periodicity with period !
• - Example

35
Review of Multi-rate Systems 5/10

• Interconnection of multi-rate building blocks
• identities also hold if all decimators are
  replaced by expanders

u1k

N

u2k
u1k

N
x
u1k
x
u2k
u2k
36
Review of Multi-rate Systems 6/10

• Noble identities (I) (only for rational
  functions)
• Example N2
• h0,h1,0,0,0,

37
Review of Multi-rate Systems 7/10

• Noble identities (II) (only for rational
  functions)
• Example N2
• h0,h1,0,0,0,

uk
uk
yk
yk

N
N
38
Review of Multi-rate Systems 8/10

• Application of noble identities efficient
  multi-rate filter implementations through
• Polyphase decomposition
• example (2-fold decomposition)
• example (3-fold decomposition)
• general (N-fold decomposition)

39
Review of Multi-rate Systems 9/10

• Polyphase decomposition
• Example efficient implementation of a
  decimation filter
• i.e. all filter operations
  performed at the lowest rate

H(z)
uk

uk


40
Review of Multi-rate Systems 10/10

• Polyphase decomposition
• Example efficient implementation of an
  interpolation filter
• i.e. all filter operations
  performed at the lowest rate

uk