This document discusses filter banks in digital communication. It covers topics such as digital transmultiplexing, discrete multitone modulation, precoding for channel equalization, and equalization with fractionally spaced sampling. It provides mathematical formulations and examples related to optimizing filter banks for digital modulation techniques over noisy channels.

1. EE 5632
小波轉換與應用
Chapter 9
Filter Banks in Digital Communication
1.
2.
3.
4.
1
Digital transmultiplexing
Discrete multitone modulation
Precoding for channel equalization
Equalization with fractionally spaced
sampling

2. The Noisy Channel
2

3. 3

4. x(n) and e(n) are uncorrelated w.s.s. random processes with
power spectrum Sxx( e jω ) and See( e jω respectively.
)
4

5. See ( e jω )
Sqq ( e jω ) ≡
| C ( e jω ) |2
If C(z) has zeros close to the unit circle, then 1/C(z) has poles
near the unit circle and the noise gain can be large.
5

6. Water filling rule:
λ − Sqq ( e jω ) when ≥ 0
S xx ( e ) = 
0
otherwise

jω
6

7. M-fold Decimator, Expander and
Multiplexer
7

8. 8

9. The Digital Transmultiplexer
9

10. 10

11. Discrete Multitone Modulation (DMT)
11

12. Biorthogonality and Perfect DMT
Systems
12

13. 13
H k ( z ) Fm ( z ) |↓M = δ (k − m )

14. Gkm ( z ) ≡ H k ( z ) Fm ( z )
g km ( M n ) = 0
14

15. In a biorthogonal DMT system with zero-forcing equalizer,
yk ( n ) = xk ( n ) + qk (n )
15

16. Optimization of DMT Filter Banks
Let Pk = variance of xk(n) = average power of xk(n)
where xk(n) comes from bk–bit modulation constellation
Noise qk(n) is Gaussian with variance
2
σ qk
.
2
Then the probability of error in detecting xk(n) can be expressed
σ qk
in terms of Pk ,
, and bk .
This expression can be inverted to obtain the total power in the
symbols xk(n) :
M −1
M −1
k =0
k =0
2
P = ∑ Pk = ∑ β ( Pe ( k ), bk ) ×σ qk
16

17. Given the channel C(z) and the channel noise spectrum See
2
σ qis the
(z) , the only freedom we have in order to control
k
choice of the filters Hk(z) . But we have to control these filters
under the constraint that {Hk , Fm} is biorthogonal.
17
2
Since the scaled system {αk Hk , Fm / αk } is also biorthogonal,
σqk
it appears that the variances
can be made arbitrarily
small by making αk small. The catch is that the transmitting
filters Fm (z) / αm will have correspondingly larger energy
which means an increase in the power actually fed into the
channel. One correct approach would be to impose a power
constraint. Mathematically this is trickier than constraining
the power Pk in the symbols xk(n) .

18. Orthonormal DMT System
hk (m ) = f k* ( −n )
DFT filter bank :
18
f k ( n ) = f k (n )e jωk n
ω k = 2π k / M

19. 19

20. 


20
The DFT filter bank is used in DMT systems for certain
types of DSL services.
DFT can be implemented very efficiently using FFT
algorithm.
DFT based DMT system can take advange of the shape
of the effective noise noise spectrum and obtain a
performance close to the water-filling ideal.

21. Optimal Orthonormal DMT System



Orthonormality implies that the average variance of the
composite x(n) is the average of the variances of the
symbols xk(n) .
The actual power entering the channel is proportional to
the sum of powers Pk in the symbols xk(n) .
e jω) is fixed. Assume further that
For a given channel, Sqq(
M is fixed. For a given set of error probabilities and bit
2
rates, the required transmitted power depends only on
σ qk
the noise variances
. We have to find an
orthonormal filters bank such that this power is
minimized.
21

22. KLT Based DMT Systems
22

23. We replace the DFT matrix with another unitary matrix T such
that qk(n) and qm(n) are uncorrelated for all n when k≠m. Such
a matrix depends only on the power spectrum of the effective
noise q(n). It is called the M×M KLT matrix for q(n). Essentially
it is a unitary matrix which diagonalizes the correlation matrix
of q(n).
It can be shown that if T is chosen as the KLT matrix (and its
inverse used in the transmitter) then the required power P is
minimized.
23

24. Optimal Orthonormal DMT System
Using Unconstrained Filters
“Unconstrained”: noncausal and IIR are allowed ;
nonoverlapping brickwall filters are allowed.
“Optimal” : The best choice of {Hk(z) } such that transmitting
power is minimized.
jω
e
The answer again depends only on the effective noise spectrum
jω
Sqq( ) . In fact, it is the so-called principalecomponent filter
bank or PCFB for the power spectrum Sqq(
24
).

25. Partial sums of variances σ q0 , σ q0 + σ q,
1
2
25
2
2
2
2
2
σ q0 + σ q1 + σ q2 are
,…
larger than the corresponding partial sums for any other filter
bank in the class of ideal orthonormal filter banks.

26. Unlike the brickwall filter bank of Fig.7(c), each filter can have
multiple passbands. Thus, the PCFB partitions the frequency
domain in a different way according to the input spectrum.
Assume that the error probabilities and total allowed power are
fixed. It can be shown that the bit rate, which is proportional to
∑ bk
, is maximized by the PCFB.
k
Similarly, with appropriate theoretical modelling the information
capacity is also maximized by the PCFB.
26

27. Example
27

28. Assume that the desired probability of error is 10-9 in each band
and b0=6 bits/symbol and b1=2 bits/symbol. If the sampling rate
is 2MHZ, this implies a bit rate of 8 Mbits/sec.
It turns out that the power required by the PCFB is nearly 10
times smaller than the power required by the brickwall system.
For DMT systems with larger number of bands, the difference is
less dramatic.
28

29. 29

30. Originally intended for transmission of baseband speech (about
4 KHz bandwidth) more than 100 years ago, the twisted pair
copper wire has therefore come a long way in terms of
bandwidth ultization and commercial application. This has given
rise to the popular saying that the DSL technology turns copper
into gold.
30

31. 31

32. Filter Banks with Redundancy
32

33. 33
N/M is called the bandwidth expansion factor.

34. In a DMT system with redundancy, it is customary to use a simple FIR
or IIR equalizer D(z) such that D(z)C(z) is a good approximation of an
FIR filter of small length, say L.
This is called the channel shortening step.
Now, if the integer N is chosen as N=M+L-1, we have L-1 extra rows in
the matrix R(z). It is possible to choose these appropriately in such a
way that a simple set of M multipliers at the output of E(z) can equalize
the channel practically completely.
34

35. Filter Banks Precoders
35

36. 36

37. 37

38. The channel C(z) can usually approximated well by an FIR or IIR filter.
The zero forcing equalizer 1/C(z) is in general IIR and could even be
unstable.
Xia showed that for almost any channel (FIR or IIR) there exist FIR
filters Ak(z) and Bk(z) such that the channel is completely equalized.
In fact the well known class of fractionally spaced equalizers (FSE) is
a special case of the filter bank precoder with M=1 and uses N-fold
redundancy.
38

39. For filter bank precoder even if M=N-1 it is still possible to have such
FIR equalizers.
Giannakis showed that the redunduncy introduced by filter bank
precoders can be exploited to perform blind equalization when C(z) is
unknown.
39