For the transmitter side, let us first consider the case when the transmitting filters are constrained to be shifted versions of one prototype. From the expression in (6.19), we see that spectral leakage can be minimized by minimizing the stopband energy of the prototype filter F0′(z). Following a procedure similar to the design of receiver subfilters, we can write the stopband energy φf of the prototype F0′(z) as
φf = p†0Ap0, (8.25)
where
A = Z
ω∈Of
|F0(ejω)|2τ†
α(ω)τα(ω)dω. (8.26)
where Of denotes the stopband of the prototype filter. We can see that φf can be minimized by choosing p0 to be the eigenvector associated with the minimum eigenvalue of A.
Now consider the case when the subfilters are not constrained. The total spectral leakage is
Z
ω∈Ou
Sx(jω)dω, (8.27)
where Sx(jω) is the transmitted spectrum given in (6.19) and Ou denotes the band in which leakage is undesired. The total leakage can be minimized if we can minimize the individual contribution φk,f from each subchannel,
φk,f = Z
ω∈Ou
|Fk′(ejω)|2dω, (8.28) where Ou denotes the bands that are not used. We can write φk,f in a quadratic form like that in (8.25) and find the optimal subfilters. In this case the subfilters do not form a DFT bank, and neither do the new transmitting filters. An efficient implementation of the resulting transmitting bank can be found in [66].
8.9 Simulations
Example 1. Receiver Subfiltering–RFI reduction. In this example, we design the subfilters for RFI reduction at the receiver. The DFT size is M = 512 and cyclic prefix length is ν = 40. The order of the subfilters is β = 10. The channel used in this example is VDSL loop#1 (4500 ft) [49]. and the channel noise is AWGN of −140 dBm. Model 1 differential mode RFI interference is considered [49]. Four RFI sources are assumed in the simulations, at respectively 660, 710, 770 and 1050 KHz, of strength -60, -40, -70, and -55 dBm, respectively.
The sampling frequency is fs = 2.208 MHz.
We will consider two different subfilter designs. In the first design, the sub-filters Qk(z) are shifted versions of Q0(z) and only Q0(z) needs to be designed.
The subfilter Q0(z) is the solution to the minimization problem in (8.15). In this case the receiving filters form a DFT bank and the solution is the same as that in section 7.4. In the second design, the RFI source is known to the receiver and the subfilters Qk(z) are individually optimized by minimizing the objective function φk,h in (8.19). The SINRs (signal-to-noise-interference ratio) of the subchannels are as shown in Fig. 8.5. The first case is labelled ‘DFT bank (chapter 7 )’ while the second case ‘Subfilters (RFI known)’. For comparison, we have also shown the subchannel SINRs for the cases of rectangular, Hanning windows, and also the window from [71]. The receivers with subfilters enjoy higher SINRs for the tones that are close to the RFI frequencies, especially when the statistics of the RFI source is known and the subfilters are optimized individually. As a result, higher transmission rates can be achieved. The transmission rate of the first case is 7.44 Mbits/sec, and that of the second case is 8.54 Mbits/sec. The transmission rates for the cases of rectangular, Hanning windows, and [71] are 6.84, 7.16, and 7.27 Mbits/sec, respectively.
Example 2. Transmitter Subfiltering–spectral leakage suppression. The
0 50 100 150 200 250 0
10 20 30 40 50
tone index
SINR (dB)
Subfilters (RFI known) DFT bank (chapter 7) [68]
Hanning window Rectangular window
Figure 8.5: The subchannel SINRs.
block size M = 512 and prefix length ν = 40. The order α of the subfilters is 20. First we consider the case when the subfilters are shifted versions of the first subfilter P0(z) and thus the transmitting filters form a DFT bank. We form the positive definite matrix A and compute the eigenvector corresponding to the smallest eigenvalue to obtain p0. Second we design the subfilters by minimizing the individual φk,f in (8.28) for each subchannel. The first case is labelled ‘Subfil-ters (DFT bank)’ while the second case ‘Subfil‘Subfil-ters’. Fig. 8.6 shows the spectrum of the transmitter output. The subcarriers used are 38 to 90 and 111 to 255.
The subcarriers with indices smaller than 38 are reserved for voice band and up-stream transmission, and those with indices between 91 and 110 are for egress (interference of DMT signals to wireless radio frequency transmission) control.
Also shown in the figure are the output spectrums when the rectangular window and transmitter window of [64] is used. The transmitter window in [64] requires no extra cyclic prefix but additional post-processing is needed at the receiver.
We see that the spectrum with the subfilters has a much smaller spectral leakage in unused bands.
0 0.2 0.4 0.6 0.8
−50
−40
−30
−20
−10 0
Frequency normalized by π.
Normalized trasmitted signal spectrum (dB)
Rectangular window [61]
Subfilters (DFT bank) Subfilters
Figure 8.6: The power spectrum of the transmitted signal.
8.10 Summary
In this chapter, we have presented a filterbank approach to the design of trans-mitter/receiver by introducing subfilters. The frequency separation among the subchannels can be considerably improved. Better separation among the trans-mitting filters translates to less spectral leakage in the transmitted spectrum while better separation among the receiving filters leads to improved RFI suppression.
As these are frequency based characteristics, the filterbank transceiver represen-tation provides a natural and useful framework for formulating the problem. The transmitter/receiver designs are converted to simple eigen-problems and closed form solutions have be obtained.
Chapter 9 Conclusion
In the first half of the thesis, we considered the problem of designing the transceiver with bit allocation. In the earlier works, the transceiver were designed for a given constellation or designed with real-valued bit allocation. In chapter 3, we designed the zero-forcing transceiver with bit allocation for maximizing bit rate under the high bit rate assumption. The optimal transceiver and bit allocation can be obtained in a closed form using simple Hadamard inequality and the Poincar´e separation theorem. In chapter 4, we designed the MMSE transceiver with bit allocation for maximizing bit rate. In this approach, we did not use the high bit rate assumption. We have shown that the optimal solution diagonalizes the channel matrix and optimal solution can be obtained by the water-filling solution.
For the rate maximizing problem. In chapter 5, we derived the dualities between the power-minimizing problem and the rate-maximizing problem with bit alloca-tion. We considered both the case without the integer bit constraint and the case when the integer bit constraint is imposed. We have shown that whether the bit allocation is integer-constrained or not, if a transceiver is optimal for the power-minimizing problem, it is also optimal for the rate maximizing problem and the converse is true. We also presented an algorithm to find the optimal solution for the power-minimizing problem and the rate-maximizing problem with integer bit constraint.
In the second half of this thesis, we considered the problems of designing the transmitting and receiving windows for the multicarrier systems. For the multi-carrier system, better separation among the transmitting filters translates to less spectral leakage in the transmitted spectrum while better separation among the receiving filters leads to improved RFI suppression. In chapter 7, we designed the receiving wondows for RFI suppression in the multicarrier system. We consider both the case when the receiver knows the statistics of the interference and the case when the statistics are not available to the receiver. In both cases the win-dows are channel independent and can be obtained in a closed form. In chapter 8, we proposed a filterbank approach to the design of transmitter and receiver by in-troducing subfilters. The filterbank transceiver representation provides a natural and useful framework. At the receiver side, we design the subfilters to mitigate RFI interference. The proposed filterbank approach here is more general. The design in chapter 7 is the special case of the filterbank approach when the subfil-ters are constrained to be the frequency shifted version of the first subfilter. At the transmitter side, we design the subfilters to minimize the spectral leakage.
The designs of the transmitting and receiving subfilters are converted to simple eigen-problems and closed form solutions can be obtained.
Future work:
• For the transceiver design with integer bit constraint, an exhaustive search is used to find the optimal solutions. In the future work, it is interesting to solve the problems with integer bit constraint in a closed form.
• The MIMO channel considered in this thesis is memoryless, in the future we will consider the case when the MIMO channel has a memory.
• The duality between the power minimization problem and rate maximiza-tion problem has been proposed in this thesis. In the future we will try to find the connection between other design criteria, for example, BER minimization or capacity maximization.
Appendix
Appendix A: Proof of Lemma 5.1
Let us consider the system (Fr, Λr), where Fr is the N × Mr matrix obtained by deleting the columns of F that correspond to the subchannels assigned with zero power and Λr is the Mr× Mr diagonal matrix obtained by deleting the columns and rows of Λs with zero power. The transmit power and bit rate of (Fr, Λr) is the same as (F, Λs). Define the new transmitter as
F = αF˜ r, (A-1)
where α > 0 is a scalar. Let us consider the new system ( ˜F, Λr) and fix the target error rate to be ǫ. The new transmit power is given by
Tr( ˜FΛrF˜′†) = α2Tr(FrΛrF†r). (A-2) So the new transmit power is a continuous and strictly increasing function of α.
Next we will show the new achievable bit rate is also a continuous and strictly increasing function in terms of α. The MSE matrix of the new system becomes
E = [α˜ 2N0−1F†rH†HFr+ Λ−1r ]−1. (A-3)
The derivative of ˜B with respect to α is
∂ ˜B
We have
∂ ˜σ2e
k
∂α = ∂ ˜E
∂α
kk
= −2αN0−1[A]kk, (A-8)
where A = ˜EF†rH†HFrE is positive semi-definite and thus [A]˜ kk ≥ 0. We now show that the diagonal elements of A can not all be zeros. Suppose [A]kk = 0 for k = 0, · · · , Mr− 1. This means all the norms of the columns of HFrE are zeros,˜ i.e., HFrE = 0. As ˜˜ E is invertible, HFrE = 0 implies HF˜ r = 0. In this case, no signal is transmitted and only the noise q is received by the receiver. Therefore, the diagonal elements of A cannot be all zeros. Substituting (A-8) into (A-5), we have
∂ ˜B
∂α > 0. (A-9)
Hence the bit rate ˜B of the new system is a continuous and strictly increasing
function of α. △△△
Appendix B: Proof of Lemma 5.2
Define the set
U = {i : σs2i > 0, 0 ≤ i ≤ M − 1}. (B-1) From (2.14), we have [E]lk = 0 for l /∈ U or k /∈ U. So (5.3) holds when l /∈ U or k /∈ U. We only need to consider (5.3) for l, k ∈ U. Let Fr and Λr be the reduced transmit matrix and reduced symbol autocorrelation matrix as we defined in Section2.2. Suppose the k-th symbol sk corresponds to the mk-th symbol in the system with transmitter Fr and autocorrelation matrix Λr. For l ∈ U, the noise variance σe2l is given by
where emk is the mk-th column of Er. Using (B-3) and (B-6), for l ∈ U we have
∂σe2l
∂σs2k =σs−4k emke†mk
ml,ml (B-7)
= σs−4k |[Er]ml,mk|2, (B-8)
= σs−4k |[E]l,k|2. (B-9) As σs−4k |[E]lk|2≥ 0, we conclude that σe2i is an increasing and continuous function of σ2sk. Using (B-9), we have
∂σe2k
∂σs2k = (σs2k/σe2k)−2 > 0, (B-10) which means σ2ek is strictly increasing on σs2k. The second order derivative of σe2k with respect to σs2k is
∂2σe2k
∂σ4sk = −2σs−6k σ4ek+ σ−4sk · 2σe2k ·∂σe2k
∂σs2k (B-11)
= 2σs−6k σ4ek(σe2k
σ2sk − 1). (B-12)
As σ2ek < σs2k for the MMSE receiver, we have
∂2σe2k
∂σs4k < 0, (B-13)
which implies σ2ek is a strictly concave function of σs2k. △△△
Appendix C: Proof of Lemma 5.3
Equalities hold in the power-minimizing problem.
Suppose (F∗, Λs∗, {b∗k}) is optimal for Apow and the minimized power is P∗. Let {ǫ∗k} and B∗ be the symbol error rates and bit rate achieved by the optimal solution. Then we have ǫ∗k ≤ ǫ and B∗ ≥ B0. First we show that ǫ∗k= ǫ for all k.
Suppose the k0-th subchannel is assigned with nonzero power σs∗2
k0 > 0 and the error rate is ǫ∗k0 < ǫ. Consider a new system with the same transmitter F∗ and bit allocation {b∗k}, but power allocation is changed to
˜ The bit rate of the new system is still B∗ as bit allocation is not changed. Next we will show that there always exists α < 1 such that the same error rate requirement will be satisfied, i.e., Using (C-1) and (C-4) we have
˜ (C-3), and the error rate constraint for the k0-th subchannel can be rewritten as
˜ σs2
k0 ≥ γ, (C-6)
where
k0 > γ, which implies that α < 1. Using (C-4) we have a smaller transmit power ˜P < P∗ and still satisfy all the constraints in Apow. This contradicts the assumption that P∗ is the minimal power when B0 is given.
Hence we have that ǫ∗k= ǫ for all k. Next we prove that the bit rate B∗ is equal to B0. Assume the bit rate is
B∗ > B0. (C-8)
Consider a new system with transmitter ˜F = αF∗ and ˜Λs = Λs∗, where α > 0 is a scalar. For the target error rate ǫ, we know from Lemma 5.1 that the bit rate of the new system is a strictly increasing and continuous function of α. So we can properly choose α < 1 such that the new bit rate ˜B = B0. In this case, the required power is smaller than P∗. This contradicts the assumption that P∗ is the minimal power when B0 is given. Hence the equality in the bit rate constraint will hold when the design is optimal.
Equality holds in the rate-maximizing problem.
Suppose (F∗, Λs∗
, {b∗k}) is optimal for Arate and the maximized bit rate is B∗. Let P∗ and {ǫ∗k} be the transmit power and error rates of the optimal solution.
Then we have P∗ ≤ P0and ǫ∗k≤ ǫ. First we show that ǫ∗k = ǫ for all k. Suppose for the k0-th subchannel, σs∗2
k0 > 0 and ǫ∗k0 < ǫ. From (2.17) we know that the error rate ǫk is a continuous and increasing function of the number of bits allocated when SNR βk is fixed. For the same F∗ and Λs∗, we can increase the number of
bits allocated to the k0-th subchannel such that the new error rate ˜ǫk0 satisfies ǫ∗k0 < ˜ǫk0 ≤ ǫ. (C-9) The error rates of other subchannels are not affected while a higher bit rate be achieved. This contradicts the assumption that B∗ is the maximal bit rate when the power constraint P0 is given. Hence we have that ǫ∗k = ǫ for all k. Now let us show P∗ = P0. Assume the transmit power is
P∗ < P0. (C-10)
Consider the new system with transmitter ˜F = αF∗ and ˜Λs = Λs∗, where α = pP0/P∗ > 1. The power of the new system is ˜P = P0. From Lemma 5.1, the bit rate of the new system is a strictly increasing function of α, and we have ˜B > B∗. This contradicts the assumption that B∗ is the maximal bit rate when the power constraint P0 is given. Hence the power constraint will hold when a solution is
optimal for Arate. △△△
Appendix D: Proof of Lemma 5.4
Suppose (F∗, {σs∗2k}, {b∗k}) is optimal for Apow,int. Let ǫ∗k be the error rate on the k-th subchannel of the optimal system. Then ǫ∗k is given by
ǫ∗k = 4 does not work for property B∗ = B0 and a different proof is needed. Suppose
B∗ > B0, (D-3)
Suppose σs∗2
k0 > 0 and b∗k0 > 0 for some k0-th subchannel. Consider a new system with the same transmitter F∗, but the bit allocation is changed to
˜bk = b∗k0 − 1, k = k0,
b∗k, otherwise, (D-4)
and power allocation is changed to
˜ Next, we will show that with appropriate choice of α, the error rate ˜ǫkof the new system still satisfies the error rate constraint in Apow,int. Using (2.17), ˜ǫk is can
be expressed as will be smaller than ǫ∗k if the quantity in the Q function of (D-8) is larger than or equal to that in the Q function of (D-1), i.e.,
1 that (D-9) is satisfied. For example, we can choose
α = 1 Rearranging (D-13), we can see that (D-9) is satisfied for k = k0. Therefore, we have ˜ǫk < ǫ∗k = ǫ for all k. This means (F∗, {˜σs2k}, {˜bk}) can achieve a smaller transmit power and still satisfies all the constraints in Apow,int. This contradicts the assumption that (F∗, {b∗k}, {σ∗2sk}) is optimal for Apow,int. Hence the total bit
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