1. DSP-CIS Chapter-7: Filter Banks -Preliminaries
Marc Moonen
Dept. E.E./ESAT-STADIUS, KU Leuven

2. Part-III : Filter Banks
Chapter-7
: Preliminaries
Filter bank (FB) set-up and applications
Perfect reconstruction (PR) problem
Multi-rate systems review (=10 slides)
Example: Ideal filter bank (=10 figures)
: Maximally Decimated PR-FBs
Example: DFT/IDFT filter bank
Perfect reconstruction theory
Paraunitary PR-FBs
: DFT-Modulated FBs
Maximally decimated DFT-modulated FBs
Oversampled DFT-modulated FBs
: Special Topics
Cosine-modulated FBs
Non-uniform FBs & Wavelets
Frequency domain filtering
Chapter-8
Chapter-9
Chapter-10

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)
subband processing
H1(z)
H2(z)
H3(z)
H4(z) IN +
OUT H1 H4 H3 H2

4. Filter Banks Introduction
Step-1: Analysis filter bank
- collection of N 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,…) H1 H4 H3 H2
H1(z)
H2(z)
H3(z)
H4(z) IN
N=4

5. Filter Banks Introduction
Step-2: Decimators (downsamplers)
- To increase efficiency, subband sampling rate is reduced by factor D
(= Nyquist (bandpass) sampling theorem)
- 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 samples> # fullband samples
D=N
D<N
N=4
D=3
H1(z) 3 IN
H2(z) 3
H3(z) 3
PS: 3-fold downsampling
= ‘remove every 2nd and 3rd sample’ (see p.32)
H4(z) 3

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 signals=input signals), i.e. mostly ignores
presence of subband processing
N=4
D=3
H1(z) 3
subband processing IN
H2(z)
subband processing
H3(z)
subband processing
H4(z)
subband processing

7. Filter Banks Introduction
Step-4: Expanders (upsamplers)
- restore original fullband sampling rate by D-fold upsampling
PS: 3-fold upsampling
= ‘insert 2 zeros in between every two samples’ (see p.30)
subband processing 3
H1(z)
H2(z)
H3(z)
H4(z) IN
N=4
D=3

8. Filter Banks Introduction
Step-5: Synthesis filter bank
- upsampling has to be followed by (interpolation) filtering (see p.22)
- collection of N `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) G1 G4 G3 G2
G1(z)
G2(z)
G3(z)
G4(z) +
OUT
subband processing 3
H1(z)
H2(z)
H3(z)
H4(z) IN
N=4
D=3

9. Aliasing versus Perfect Reconstruction
- Assume subband processing does not modify subband signals
(e.g. lossless coding/decoding)
-The overall aim would then be to have y[k]=u[k-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 p.32-33)
G1(z)
G2(z)
G3(z)
G4(z) +
output = input 3
H1(z)
H2(z)
H3(z)
H4(z)
N=4
D=3
u[k]
y[k]=u[k-d]?

10. Aliasing versus Perfect Reconstruction
Question :
Can y[k]=u[k-d] be achieved in the presence of aliasing ?
Answer :
YES !! PERFECT RECONSTRUCTION banks with
synthesis bank designed to remove aliasing effects !
G1(z)
G2(z)
G3(z)
G4(z) +
output = input 3
H1(z)
H2(z)
H3(z)
H4(z)
N=4
D=3
u[k]
y[k]=u[k-d]?

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 IV
- 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 filtering = N (simpler?) subband modeling problems instead
of one (more complicated?) fullband modeling problem
- Perfect reconstruction guarantees distortion-free desired near-end
speech signal 3
H1(z)
H2(z)
H3(z)
H4(z) +
G1(z)
G2(z)
G3(z)
G4(z)
OUT
ad.filter

14. Filter Banks Applications
Transmultiplexers :
Frequency Division Multiplexing (FDM) in digital communications
- N different source signals multiplexed into 1 transmit signal by
expanders & synthesis filters (ps: here interpolation factor )
- Received signal decomposed into N source signals by analysis filters
& decimators
- Again ideal filters = ideal bandpass filters
D>=N 5
G1(z)
G2(z)
G3(z)
G4(z) +
H1(z)
H2(z)
H3(z)
H4(z)
transmission
channel
signal-1
signal-2
signal-3
signal-4

15. Filter Banks Applications
- Non-ideal synthesis & analysis filters result in aliasing 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
PS: special case is Time Division Multiplexing (TDM),
if synthesis and analysis filters are replaced by delay
operators (and N=D)
u1[k],u2[k],u3[k],u4[k],u1[k+1],u2[k+1]...
u1[k],u1[k+1]
u2[k],u2[k+1]
u3[k],u3[k+1]
u4[k],u4[k+1] 4 4
u1[k-1],u1[k] 4 4
u2[k-1],u2[k]
transmission
channel + 4 4
u3[k-1],u3[k] 4 4
u4[k-1],u4[k]
0,0,0,u4[k],0,0,0,u4[k+1]...
PS: another special case is OFDM… explain! What are Gi(z) & Hi(z) ?

17. Filter Banks Applications
PS : special case is Code Division Multiple Access (CDMA),
where filter coefficients=(orthogonal) user codes
CDMA basics: (see digital coms courses)……
-Each user (n=1..N) is assigned a length-N pseudo-random code
sequence
-Transmission : For each symbol (k-th symbol for user-n), a
`chip’ sequence is transmitted
-Mostly binary codes ( ) with BPSK/QPSK symbols
-Multiple access based on code-orthogonality (see below)
Skip this slide

18. Filter Banks Applications
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……… ……… ……… ………
code sequence ,+1,-1,-1
transmitted chips ,+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 (mult. with code & sum)
received symbols (*1/4) ………… -1…….…..-1…………+1
(x) PS: real-world CDMA is considerably more complicated (different channels for different users + channel dispersion, asynchronous users, scrambling codes, etc.)
Skip this slide

19. Filter Banks Applications
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!)
Skip this slide
u1[k],u1[k+1]
u2[k],u2[k+1]
u3[k],u3[k+1]
u4[k],u4[k+1] 4 4
Base station 4 4
u2[k+1],u2[k]
transmission
channel +
User-2 terminal 4 4 4 4

20. Review of Multi-rate Systems 1/10
Decimation : decimator (downsampler)
example : u[k]: 1,2,3,4,5,6,7,8,9,…
2-fold downsampling: 1,3,5,7,9,...
Interpolation : expander (upsampler)
2-fold upsampling: 1,0,2,0,3,0,4,0,5,0... D
u[0], u[N], u[2D]...
u[0],u[1],u[2]... D
u[0],0,..0,u[1],0,…,0,u[2]...
u[0], u[1], u[2],...

21. Review of Multi-rate Systems 2/10
Z-transform (frequency domain) analysis of expander
`expansion in time domain ~ compression in frequency domain’ D
u[0],0,..0,u[1],0,…,0,u[2]...
u[0], u[1], u[2],... D
`images’ 3

22. 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 p.8 & Chapter-8) D
u[0],0,..0,u[1],0,…,0,u[2]...
u[0], u[1], u[2],... 3
`images’ LP

23. Review of Multi-rate Systems 3/10
Z-transform (frequency domain) analysis of decimator
`compression in time domain ~ expansion in frequency domain’
PS: Note that is periodic with period while is periodic with period
The summation with d=0…D-1 restores the periodicity with period !
u[0],u[1],u[2]... D
u[0], u[D], u[2D]... D
d=2
d=0
d=1 3

24. Review of Multi-rate Systems 4/10
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-pass (see p.36 & Chapter-8)
u[0],u[1],u[2]... D
u[0], u[N], u[2D]...
d=2
d=0
d=1 LP 3

25. Review of Multi-rate Systems 5/10
Interconnection of multi-rate building blocks :
identities also hold if all decimators are replaced by expanders D x a = D x a
u1[k] D +
u2[k]
u1[k] + D =
u2[k]
u1[k] x D =
u1[k] D x
u2[k]
u2[k] D

26. Review of Multi-rate Systems 6/10
`Noble identities’ (I) : (only for rational functions)
Example : D=2
h[0],h[1],0,0,0,… = D
u[k]
y[k]

27. Review of Multi-rate Systems 7/10
`Noble identities’ (II) : (only for rational functions)
Example : D=2
h[0],h[1],0,0,0,…
u[k]
y[k]
u[k]
y[k] D = D

28. Review of Multi-rate Systems 8/10
Application of `noble identities : efficient multi-rate realizations of FIR filters through…
Polyphase decomposition:
example : (2-fold decomposition)
example : (3-fold decomposition)
general: (D-fold decomposition)

29. Review of Multi-rate Systems 9/10
Polyphase decomposition:
Example : efficient implementation of an FIR decimation filter
i.e. all filter operations performed at the lowest rate
H(z)
u[k] + 2
u[k] = 2 2 +

30. Review of Multi-rate Systems 10/10
Polyphase decomposition:
Example : efficient implementation of an FIR interpolation filter
i.e. all filter operations performed at the lowest rate
u[k]
H(z) 2 + =
u[k] 2 2 +

31. Example: Ideal Filter Bank
With ideal analysis/synthesis filters, FB operates as follows (1)
H1(z)
H2(z)
H3(z)
H4(z) IN … …
subband processing 4
H1(z)
H2(z)
H3(z)
H4(z) IN
G1(z)
G2(z)
G3(z)
G4(z) +
OUT

32. Example: Ideal Filter Bank
With ideal analysis/synthesis filters, FB operates as follows (2)
H1(z)
H2(z)
H3(z)
H4(z) x1
PS: H1(z) analysis filter
≈ lowpass anti-alias filter … … x1
subband processing 4
H1(z)
H2(z)
H3(z)
H4(z) IN
G1(z)
G2(z)
G3(z)
G4(z) +
OUT

33. Example: Ideal Filter Bank
With ideal analysis/synthesis filters, FB operates as follows (3)
x’1 … …
(ideal subband processing)
x’1
x’1
subband processing 4
H1(z)
H2(z)
H3(z)
H4(z) IN
G1(z)
G2(z)
G3(z)
G4(z) +
OUT

34. Example: Ideal Filter Bank
With ideal analysis/synthesis filters, FB operates as follows (4)
x”1 … …
x”1
subband processing 4
H1(z)
H2(z)
H3(z)
H4(z) IN
G1(z)
G2(z)
G3(z)
G4(z) +
OUT

35. Example: Ideal Filter Bank
With ideal analysis/synthesis filters, FB operates as follows (5)
G1(z)
G2(z)
G3(z)
G4(z)
PS: G1(z) synthesis filter
≈ lowpass interpolation filter
x’’’1 … …
x’’’1
subband processing 4
H1(z)
H2(z)
H3(z)
H4(z) IN
G1(z)
G2(z)
G3(z)
G4(z) +
OUT

36. Example: Ideal Filter Bank
With ideal analysis/synthesis filters, FB operates as follows (6)
H1(z)
H2(z)
H3(z)
H4(z) x2
PS: H2(z) analysis filter
≈ bandpass anti-alias filter … …
subband processing 4
H1(z)
H2(z)
H3(z)
H4(z) IN
G1(z)
G2(z)
G3(z)
G4(z) +
OUT x2

37. Example: Ideal Filter Bank
With ideal analysis/synthesis filters, FB operates as follows (7)
x’2 … …
subband processing 4
H1(z)
H2(z)
H3(z)
H4(z) IN
G1(z)
G2(z)
G3(z)
G4(z) +
OUT
x’2
x’2

38. Example: Ideal Filter Bank
With ideal analysis/synthesis filters, FB operates as follows (8) IN
x”2 … …
subband processing 4
H1(z)
H2(z)
H3(z)
H4(z) IN
G1(z)
G2(z)
G3(z)
G4(z) +
OUT
x”2

39. Example: Ideal Filter Bank
With ideal analysis/synthesis filters, FB operates as follows (9)
G1(z)
G2(z)
G3(z)
G4(z)
x”’2
PS: G2(z) synthesis filter
≈ bandpass interpolation filter … …
subband processing 4
H1(z)
H2(z)
H3(z)
H4(z) IN
G1(z)
G2(z)
G3(z)
G4(z) +
OUT
x”’2

40. Example: Ideal Filter Bank
With ideal analysis/synthesis filters, FB operates as follows (10)
OUT=IN … …
H1(z) 4
subband processing 4
G1(z)
OUT IN
H2(z)
subband processing 4
G2(z) +
H3(z)
subband processing 4
G3(z)
H4(z)
subband processing 4
G4(z)
Now try this with non-ideal filters…