Throughput Maximizing FIR Filterbank For MIMO LTI Wireline Channels

Throughput Maximizing FIR Filterbank For MIMOLTI Wireline ChannelsAhmed F.ShalashAnalog DevicesNew Jersey Design CenterSomerset,NJ08873 Email:ahmed.shalash@Mohammed NafieCairo UniversityDepartment of Electronics and CommunicationsGeeza,EgyptEmail:mnafie@.egAbstract—There is an ever growing need for higher speed communications over existing wireline plant.To achieve even higher throughput,bonding and joint op-timization of multiple-input-multiple-output cable bundles attracted attention recently.In this paper wefirst explain a technique which allows several users to use the same communication medium while at the same time allowing their transmission to be separated at the receiver.This technique would handle single-channel-multiple-users case. The technique is then extended to handle multiple input multiple output(MIMO)channel by jointly optimizing the transmitfilters to maximize the throughput over the MIMO channel.This optimization is suitable for multiple-channels-single-user and multiple-channels-multiple-users cases.A2-D MIMO channel example is given and sim-ulation results show that the joint optimization achieves higher throughput than independent optimization for each channel.I.I NTRODUCTIONWith the full deployment of wide band digital commu-nications over wireline channels,many new interesting technical as well as marketing problems arise.The problem of optimizing transmitfilters for increasing the throughput of the communication channel has attracted attention lately[1],[2],[3].The research in that area was primarily concerned with single user transmitting over additive white Gaussian noise channels.However,the need for multi-user or multi-task links can be envisioned for homes or small business users.One of these applica-tions is exhibited by the so-called channelized voice over digital subscriber loops(DSL).For these applications, the users can utilize one of the classical multiple access approaches but none are without drawbacks.A transmitfilter optimization technique for the multiple-access environment was shown in[4].However, with the growth of the DSL,the need for even higher data rates keeps mounting.This,coupled with the fact that the cost offiber deployment still makes it attractive to utilize the loop plant to achieve maximum possible throughput,led to the emergence of such project as the G.BOND.The G.BOND project,adopted by the International Telecommunication Union(ITU),supports the topology that one user would combine multiple DSL pairs to increase the data rate.This opens the door for jointly optimizing the transmitfilters to maximize the overall throughput over multiple links.The work done in the dynamic spectrum management project[5] clearly shows the advantage of jointly optimizing the MIMO channel to achieve better data rates.In this paper, numerical optimization is performed on FIRfilter bank to maximize the information rate.In this paper,we introduce an optimization technique that can be utilized to allow either multiple-access over a single wireline channel(single loop)or single/multiple user(s)operating on multiple loops.We model the prob-lem of the multiple loops as a MIMO channel problem. The authors cannot emphasize enough that even though the problem formulation as afilter bank,[6],[7],[8], looks similar to the known multi-dimensional modula-tion used in wireless communications,there are many fundamental differences between wireless and wireline problems.First the wireline channels are linear-time-invariant(LTI).Second,unlike the wireless channels, complete synchronization can be guaranteed for the wireline case.Third,unlike the wireless channel,the wireline channel can be measured and known a priori. Previous work show efficient ways to estimate the chan-nel characteristics[9][10].The receiver is assumed to be a multi-dimensional DFE[11].The remainder of the paper is divided as follows. Section II has the problem formulation.Section III has the solution methodology for fractionally-sampled multi-dimensionalfilter banks over a single channel.SectionIV extends the problem to multiple channels(loops) by formulating it as a MIMO problem.A2-D MIMO problem example is given to highlight the advantage of joint optimization.Section V has concluding remarks.II.P ROBLEM F ORMULATIONSeveral studies have been conducted to optimize the transmitterfilter to successfully mitigate the ISI and noise at the receiver with reasonable complexity[1][3]. From a theoretical stand point,one can achieve the maximum throughput over a given channel by either using infinite length transmitfilters or by sending the data in blocks afterfiltering it by non-stationaryfilters. Attempts had been made tofind optimal practicalfilters. For example,in the discrete multitone(DMT)scheme redundancy is added at the transmitter to map the trans-mitted symbols on orthogonal tones.The DMT can be viewed in light offilter bank theory as in[2].In[12],an optimization technique was introduced to maximize the throughput of noisy ISI channels by passing the white input sequence through an FIR linear transmitfilter.The MMSE-DFE is the assumed receiver as it is a widely used receiver topology.Adopting the standard discrete time representation,the y output of an additive-noise dispersive channel at time step k is given byy k=νm=0h m x k−m+n k,(1)where h m=[h l−1,m···h0,m]t is the m th vector of channel coefficients,assuming an oversampling factor of l,andνis the channel memory.The channel input x at time step k is given byx k=νtn=0p n u k−n,(2)where{p i}v t i=1are the transmitfilter coefficients and {u k}is an assumed white unit-energy transmitted se-quence.Under the assumption of prior knowledge of the channel and noise characteristics,the channel throughput can be maximized.As shown in[1]the channel through-put over a block of N output symbols is given byI=1(N+ν)log2(|I N+ν+H h R−1nn HR xx|),(3)where I N+νis the identity matrix of size N+ν,()h denotes the hermitian transpose,H is the channel matrix whosefirst block-row is equal to the(ν+1)length chan-nel impulse response appended by zeros,and R nn and R xx denote the Toeplitz auto-correlation matrices for the noise and the input sequences,respectively.The channel throughput can be maximized by numerically solving the constrained and nonlinear optimization problem using the sequential quadratic programmingalgorithm[12].stationarynoisex(m) x(m+1) x(m+2).....x(m+N−1)Fig.1.Operation of1-D fractionally sampled FIR transmitfilterIII.M ULTI-USER O PTIMIZATIONA.Fractional OptimizationMore optimization can be achieved when the FIR filter is fractionally sampled instead of being symbol rate sampled.Logically,increasing the sampling rate of the FIR will increase the optimization degrees of freedom, hence allowing increase in the channel throughput.The throughput function can be written as in(3),except that nowR xx=P∗R uu∗P h(4) where P is a matrix corresponding to the fractionally spaced transmitfilter to be optimized,and u is the oversampled input bit sequence.Therefore R uu will be a diagonal matrix,with zeros everywhere except in the positions corresponding to symbols.Figure1shows the operation of the1-D fractionally sampled FIRfilter. As an example,the optimization was performed on the linear channel given byh(z)=0.1+0.3z−1+0.5z−2.(5) The resulting information rates for different SNR are plotted in Figure2.As shown in thefigure,the fractional optimization gives a consistent throughput improvement compared to the popular raised-cosine signal[4].B.Multi-Dimensional SignalingUp until now,most of the research in multiuser communications has focused on wireless channels.Many of the ideas developed for wireless mediums cannot be carried over to wireline communications.An obvious way to see this is that,in wireless communications, each user has their own channel characteristics while in wireline they all share the same wire,and hence the same channel.Second,the wireline channel,prac-tically,is not time-varying and synchronization can be easily achieved between different users.The optimizationFig.2.Throughput for fractionally sampled transmitfiltersFig.3.The advantage of the optimized signaling compared to the time division multiple accesstechnique explained above can be used to generate1-dimensional FIR signal.To be able to accommodate multiple users over the same channel,an increase in the system dimensionality has to be made.We propose two ways for obtaining the multi-dimensional FIRfilters. First,direct optimization with a modification to equa-tion(3)can be performed.Specifically,and since we assume independent usersR xx=i P i R ui u iP h i(6)where P i and R ui u i are,respectively,the transmitfiltermatrix and the autocorrelation function of user i.This technique will outperform the time division multiplexing from information rate view point.Figure3shows the advantage of using the optimization multi-dimensional FIR transmitfilters over the time multiplexed single transmitfilter.The other technique offersflexibility in the signals generation and allows incremental addition of users and easier receiver equalization.We use the frequency response of the FIRfilter as a template and employ the signal generation technique explained in[13] to incrementally generate any required number of sig-nals.The optimization problem is stated mathematically asfindingmin{f i}{max(|F i−R|)}s.t.f i is orthogonal to f j,j<i,where F i is the frequency characteristics of thefilter f i and R is the template being used.Notice that when users join the system incrementally,the original users on the system need only to change their rate such that the total rate is equal to the Nyquist rate of the usable bandwidth of the channel.Thefilters are always the same no matter how many users are using the channel.IV.M ULTI-C HANNEL O PTIMIZATIONThe problem defined in the previous section can be extended to cover operating over MIMO channel.An ex-ample of2-D MIMO channel is shown in Figure4.The channel clearly resembles the known definition of direct connections and far end crosstalk(FEXT)connections associated with DSL environment operating in simplex mode.Only FEXT is considered in this example,how-ever the problem can easily be extended to cover full-duplex communications with near-end crosstalk(NEXT) as well.The throughput as defined in equation(3)can be re-defined to account for the crosstalk portion asI1=1(N+ν1)log2(|I N+ν1+H h1R−1n1n1H1R x1x1|)(7) I2=1(N+ν2)log2(|I N+ν2+H h2R−1n2n2H2R x2x2|),(8)Where R n1n1and R n2n2will each include the external noise term and the cross talk term.Under the assumption of independence among the signal sources and between the signal source and the noise,we can writeR n1n1=R nn+((P2H21)h R u2u2P2H21)(9)R n2n2=R nn+((P1H12)h R u1u1P1H12)(10) where R nn is the external noise auto-correlation,assum-ing same noise is added to both paths,H21and H12are the cross talk channel matrices for both paths,and P1 and P2are the transmitfilter matrices.Fig.4.2-input/2-output MIMO channelA.Optimization FlavorsThere are many possibilities for optimization depend-ing on the particular setup.For multiple users case,we can assume same information rate for all users,i.e.fair-ness among users.This will be similar to the constraint put on the optimization performed for the single-channel-multiple-users case.If the computational complexity is of a concern,we can assume same transmit filter for all users,making it less computationally intensive to perform the optimization.Another setup involves,similar to the G.BOND project,that the user owns all the channels and thus we need to optimize the aggregate information rate.There are other possibilities that can be entertained and are beyond the scope of this paper.The authors intend to discuss these cases in an upcoming paper.Fig.5.The forward channel and the far end crosstalk channelB.Numerical ExampleAssume for simplicity that the direct channel is the same as in equation (5)and the crosstalk channel is given byh (z )=0.1−0.1z −1−.2z −2.(11)The direct and crosstalk channels are shown in Fig-ure 5.The optimization was performed by assuming a symmetric MIMO system (H 1=H 2and H 12=H 21)and by forcing the two filters P 1and P 2to be identical.This guarantees that I 1=I 2and hence can be used to guarantee fairness among different users.Even with such limitations,the jointly optimized filter outperforms the individually optimized filter as shown in Figure 6.The received signal and noise profiles are shown in Figure 7.Fig.6.The joint optimization versus individual optimization for I 1(=I 2in this case)When the optimization is done for I 1+I 2,without con-straining I 1=I 2,even better performance can be achieved as shown in Figure 8.This case can be used,for example,when a single user is using multiple pairs and the only concern is to maximize the total throughput.It is worth noting that for high signal-to-noise ratios,the solution converges to the case where P 1=P 2and I 1=I 2.Note that the information rate in Figure 8is I 1+I 2rather than just I 1as in Figure 6.In Figure 6I 1=I 2by definition.C.Higher DimensionsIt is straight forward to increase the dimensionality of the optimization problem.For practical DSL environ-ment,e.g.a single user owning 2-pair,or 4-pair cable,the MIMO optimization can be performed off line using a central processing unit.Notice that the optimization suitable for each case would depend on the particularFig.7.The PSD profile at the y1point in Figure 4scenario.To give a simple example of that,assume that we want to maximize the throughput of user 1who owns a 2-pair while at the same time keeping fairness between him and user 2with a 4-pair.Then you would maximize the throughput of the user 1while constraining the user 2throughput to be half per channel (or perhaps a little bit less than half)compared to user 1.As the channel is assumed to be practically LTI,then this operation needs to be performed once.For practical systems,temperature changes might contribute to channel variations.However,temperature changes are usually very slow and can be neglected for all practical purposes.Fig.8.The joint optimization of I 1+I 2when P 1and P 2are differentV.C ONCLUSIONIn this paper,the problem of optimizing the FIR trans-mit filter bank was extended from a single channel envi-ronment to MIMO environment.The problem was firstdefined for the single channel,fractionally sampled case.The throughput optimization was performed for multi-dimensional 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