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# Ch9

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### Ch9

1. 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. 2. The Noisy Channel 2
3. 3. 3
4. 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. 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. 6. Water filling rule: λ − Sqq ( e jω ) when ≥ 0 S xx ( e ) =  0 otherwise  jω 6
7. 7. M-fold Decimator, Expander and Multiplexer 7
8. 8. 8
9. 9. The Digital Transmultiplexer 9
10. 10. 10
11. 11. Discrete Multitone Modulation (DMT) 11
12. 12. Biorthogonality and Perfect DMT Systems 12
13. 13. 13 H k ( z ) Fm ( z ) |↓M = δ (k − m )
14. 14. Gkm ( z ) ≡ H k ( z ) Fm ( z ) g km ( M n ) = 0 14
15. 15. In a biorthogonal DMT system with zero-forcing equalizer, yk ( n ) = xk ( n ) + qk (n ) 15
16. 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. 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. 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. 19
20. 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. 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. 22. KLT Based DMT Systems 22
23. 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. 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. 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. 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. 27. Example 27
28. 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. 29
30. 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. 31
32. 32. Filter Banks with Redundancy 32
33. 33. 33 N/M is called the bandwidth expansion factor.
34. 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. 35. Filter Banks Precoders 35
36. 36. 36
37. 37. 37
38. 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. 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