We now consider the case when the statistics of RFI interference are not available to the receiver (uninformed receiver). In this case, the frequency and amplitude
of RFI are not known. We can minimize the total interference by minimizing the
Then the stopband energy φh can be rewritten as φh =
The elements of Q are given by
[Q]mn =
(−sin(m−n)ωπ(m−n) s, m 6= n,
1 −ωπs, m = n. (7.26)
The window vector g can be written as
g = d + Eb, (7.27)
where dT = [ 0 1TM ], and ET= [ Iβ 0 −Iβ ].
As a result, the objection function can be given in terms of b,
φh = (d + Eb)TQ(d + Eb), (7.28)
Similarly to the informed window, using the method of optimization in [77], we can obtain the following optimal uninformed solution b that minimizes the stopband energy
b = −(ETQE)−T(ETQTd). (7.29) In this case, neither the channel nor the RFI information is needed for obtaining the window.
7.5 Simulations
In this section, we will evaluate the proposed window design technique. The channels used for our evaluations are seven VDSL loops [49]. The DFT size M = 1024, cyclic prefix ν = 80, and window length β = 10. The channel noise consists of AWGN of -140 dBm, FEXT and NEXT crosstalk as described in [49].
The time domain equalizer of length 20 is used to shorten the channel to length less than 70 [79]. The RFI interference is of differential mode with strength -55dBm [49]. Three RFI sources with frequencies at 1.44, 1.9, and 2.0MHz are considered. In this simulation, the RFI signal is generated as in [49]. We will first use VDSL loop1 of length 4500ft as an example to examine the frequency response of the proposed window and demonstrate the effect on subchannel interference and SINR.
Frequency response: Suppose the statistics of RFI is available to the re-ceiver. We compute w using (7.20) and obtain the informed window form (7.1).
Fig. 7.4 shows the frequency response of the informed window g. For comparison, we have also shown the frequency responses of the Hanning window, Blackman window, and Kaiser window with shape parameter β = 5 [75]. We can see that the informed window has a faster roll-off in low frequency while the other three windows have much smaller sidelobes in high frequency. However, the roll-off in high frequency will not be important when the sidelobes are so small that RFI
is not the dominating noise. As the proposed window has the characteristics of fast roll-off in low frequency, fewer tones will be dominated by RFI as we will see next.
0 0.2 0.4 0.6 0.8
−80
−60
−40
−20 0 20 40 60 80
Frequency normalized by π
Magnitude response (dB)
Hanning window Informed window Kaiser window Blackman window
Figure 7.4: Frequency response of receiving windows.
Subchannel Interference: We compute the interference power at the re-ceiver outputs for the receiving windows. Fig. 7.5 shows the RFI interference power of individual tones for the informed window, uninformed window, win-dow in [71], Hanning winwin-dow, Blackman winwin-dow, and Kaiser winwin-dow with shape parameter β = 5. In Fig. 7.5(a), we compare with the window in [71] and Han-ning window. In Fig. 7.5(b), we compare with Blackman window and Kaiser window. We can see that the informed window and uninformed windows have lower RFI power than the other four windows near the RFI source frequencies.
Also shown in Fig. 7.5(a) and Fig. 7.5(b) are the combined effects of channel noise (AWGN, FEXT, and NEXT) and the residual ISI for the informed window, uninformed window, window in [71], Hanning window, Blackman window, and Kaiser window, which are labeled as “other noise (informed)”, “other noise (un-informed)”, “other noise (Window [71])”, “other noise (Hanning)”, “other noise
(Blackman)”, and “other noise (Kaiser)”. In both Fig. 7.5(a) and Fig. 7.5(b), the curves of “other noise” overlap with each other and are indistinguishable in the figure. From Fig. 7.5, we can see that RFI is dominating in the tones around the RFI frequencies. For the tones away from the interference sources, other noise is dominating. As a result, higher attenuation of the window in high frequency is of little significance. In this case, the commonly used Hanning window and Black-man window are over designed in high frequency region. The proposed windows, due to their faster roll-off in low frequency, has fewer RFI dominating tones.
Subchannel SINRs: Fig. 7.6 shows the SINRs of the individual tones for both informed and uninformed windows. For comparison, in Fig. 7.6(a)(b), we have also shown the SINRs of the window in [71], Hanning window, Blackman window and Kaiser window with shape parameter β = 5. From Fig. 7.6(a)(b) we see that the SINRs of the informed and uninformed window are higher than those of the other windows near the RFI source frequency, i.e., in the tones where RFI interference is dominating. This is due to the fact that the proposed windows achieve a better trade-off in low frequency and high frequency. Therefore, we can transmit more bits in the neighboring tones by using the proposed windows.
The two curves corresponding to the two proposed windows almost overlap with each other. This shows that the use of uninformed window leads to only a minor performance degradation.
Table 1 shows the bit rates for seven VDSL loops [1] with window length β = 10, where VDSL loop 1 to 4 are of length 4500 ft. The sampling frequency is fs = 4.416 MHz. For comparison purpose, we have also included the bit rates of the rectangular window, Hanning window, Blackman window, Kaiser window, and the window in [71]. In addition, the bit rates for the case when there is no RFI interference are also shown in the table. From the table, we can see that the proposed windows have better performance for all the test loops.
0 100 200 300 400 500
Figure 7.5: Subchannel interference power of the DMT system with windowing.
(a) Informed window, uninformed window, window in [71], and Hanning win-dow. (b) Informed window, Blackman window, and Kaiser window with shape parameter β = 5.
7.6 Summary
We have proposed a window design method for RFI suppression in DMT sys-tems. The proposed windows strike a balance between low frequency and high frequency response. Thus, fewer tones are dominated by RFI and better bit rates
0 100 200 300 400 500
Figure 7.6: Subchannel SINRs of the DMT system with windowing. (a) Informed window, uninformed window, window in [71], and Hanning window. (b) Informed window, Blackman window, and Kaiser window with shape parameter β = 5.
is achieved. We consider both the case when the receiver knows the statistics of the interference (informed receiver) and the case when the statistics are not available to the receiver (uninformed receiver). In both cases the windows are channel independent and can be obtained in a closed form. Windows designed for uninformed receiver (interference-independent window) has the advantage that
Loop 1 2 3 4 5 6 7 informed 20.74 20.42 18.94 11.25 26.60 22.75 17.97 uninformed 20.60 20.40 18.90 11.22 26.58 22.70 17.94 rectangular 19.72 19.49 17.95 10.40 26.38 21.92 16.94 Hanning 20.23 19.96 18.59 10.90 26.48 22.33 17.42 Blackman 20.14 19.86 18.48 10.80 26.39 22.32 17.46 Kaiser β = 5 20.38 20.02 18.78 10.97 26.46 22.33 17.68 window [71] 20.24 20.23 18.82 11.06 26.52 22.60 17.79 No RFI 23.34 22.78 21.49 13.45 27.59 22.39 20.57
Table 7.1: Bit rate (Mbits/sec) on VDSL loops.
the window coefficients need not be updated when the statistics of the RFI in-terference changes. We also shows not knowing the statistics of the RFI source leads to only a minor performance degradation.
Chapter 8
A Filterbank Approach to
Window Designs for Multicarrier Systems
In chapter 7, we have designed the receiving windows for RFI suppression at the receiver. At the transmitting side, spectral leakage is also an important issue in the multicarrier system, and transmitting windows have been used to mitigate the out of band spectral leakage. Better frequency separation among the transmitting filters leads to a smaller out-of-band spectral leakage and also less interference to radio frequency transmission. In this chapter, we will propose a unified filterbank approach to the design of transmitting/receiving windows for multicarrier systems. The approach used here will be more general. We will introduce the so-called subfilters. The use of subfilters will enhance the frequency selectivity of the transmitting and receiving filters. It can be shown that the receiving windows in chapter 7 are special cases of this filterbank approach. The filterbank viewpoint provides an additional insight into the transmitter design for spectral leakage reduction as well as to the receiver design for interference suppression.
8.1 System Model
From section 6.3 and section 6.4, we know that the spectral leakage at the trans-mitter and the number of subchannels affected by RFI at the receiver depend on the sidelobes of the transmitting and receiving filters. To have a better frequency selectivity, we will design the transmitter and receivers using the filterbank repre-sentation in Fig. 6.3. Employing the polyphase identity [76], we observe that the transfer function Tk,i(z) from the i-th transmitter input si(n) to the k-th signal yk(n) at the receiver is given by
Tki(z) = [Hk(z)C(z)Fi(z)]↓N, (8.1) where the notation [A(z)]↓N denotes the N-fold decimated version of A(z). Note that the DMT system has zero inter-block and inter-subchannel ISI, and the transmitter inputs are the same as the receiver outputs sk(n) = ˆsk(n) when there is no channel noise. As yk(n) = λksˆk(n), we have
Tki(z) = λkδ(k − i). (8.2)
Summarizing, we can obtain the following lemma.
Lemma 8.1 Consider the system in Fig. 6.3. The transfer function Tk,i(z) from thei-th transmitter input si(n) to the k-th signal yk(n) at the receiver is given by Tki(z) = λkδ(k − i), 0 ≤ k, i ≤ M − 1. (8.3) The result holds for any FIR filter C(z) of order L ≤ ν. The constant λk are the M-point DFT of c(n).
So long as the order of C(z) is not larger than ν, the system is free from inter-block interference and inter-subchannel interference. This means that, if we cascade another filter before or after the channel, as long as the product of this extra filter and C(z) has order no larger than ν the overall system remains ISI free. We will use this observation later to design transmitters and receivers in the following