Abstract: In the field of wireless communications, the multiple-input, multiple output (MIMO) channel is experiencing increased interest due to the drammatic increases in capacity that result from adding multiple transmit and receive antennas to wireless systems. Early work in the area centered on channels with flat fading characteristics. Here, it was found that channel capacity increases linearly with the number of antennas used [1]. However, the simple transmit and receiver architectures that result from the assumption of flat fading are not easily reproduced when it is relaxed. Thus, one topic of recent work has been appropriate receiver (and in some cases, transmitter) architectures for channels with intersymbol interference (ISI, e.g., frequency-selective fading).
The author compares the performance of the optimal finite-length MIMO minimum-mean-square-error decision feedback equalizer (MMSE DFE) with channel transmission using a Discrete Matrix Multitone (DMMT) system in the presence of small amounts of ISI.
In [1], it is shown that the optimum structure for MIMO transmission in a flat-fading environment uses a channel decomposition that involves signal processing both at the receiver and transmitter.� In the more general situation with frequency-selective fading, the optimum architecture is less obvious.� Current system implementations in which multi-user detection (a form of a MIMO system) is performed typically involve CDMA, which can eliminate ISI by spreading it in frequency, or else lessen its impact drastically.� In these cases, the typical architecture utilized is a decision-feedback equalizer.� However, alternatives have been discussed, the most promising of which seems to be the MIMO analogue of discrete multitone, or discrete matrix multitone.� Here, the ISI is essentially eliminated through spatial frequency division.
In [2], fast methods are presented for computing the optimal FIR filter coefficients for a MIMO MMSE-DFE.� Three architectures are discussed: the general multi-channel analogue of the SISO DFE; a situation where channels are decoded sequentially each symbol period, and thus current decisions are available for previously decoded channels (similar in spirit to the architecture in [3]), and the situation in which all channels� current decisions are simultaneously known (how this would occur in practice is unclear).� Three things must be optimized in the system for a given FIR feed-forward filter length: the filter delay, the feedback filter coefficients, and the feed-forward filter coefficients.� If one assumes perfect decisions, having error uncorrelated with channel output (orthagonality, or E[ekyk*] = 0) implies that the feed-forward coefficients can be found from the formula:
Wopt* = Bopt* Rxy Ryy-1
As previously mentioned, the MIMO DFE architecture is implemented only on the receiver.� Since it�s clear that an optimal system design will involve some transmitter-side processing, the question to answer is what the optimal system (transmitter and receiver) design is.� In [5], the MIMO analog of DMT is presented.� This system involves a small time overhead (circularizing the channel, or pre-transmitting the last few symbols in a block before the block), but allows for an elegant solution.� In this situation, the inverse fast fourier transform, and the fast fourier transform are basis vectors, and thus can diagonalize, the channel matrices.� Mathematically, if H is composed of Hnm, the sub-matrix representing the circular transmission of a block of N data points from transmitter n to receiver m, then:
F H F* = [Enm]
where Enm represents the block diagonal matrices of eigenvalues.� Thus, permuting the result with permutation matrices PT and PR results in a block diagonal matrix, G.
PR F H F* PT = G
Each of the MRxMT blocks can then be diagonalized using the singular value decomposition, and thus the diagonalization of the channel is completed.� If :
G = U S V*
then,
Y = U* PR F H F* PT V X = S X
After channel estimation, the system implementation only really requires the calculation of the svd of the channel coefficient matrices, which will be small (MRxMT).� Thus, the implementation of DMMT may actually be more feasible, in a changing channel environment, than the MIMO DFE, a somewhat surprising result.
Random 2x2 channel impulse responses were generated for 40 different channels.� The impulse response was assumed to be three taps, with normally distributed real coefficients.� After the coefficients were calculated, each impulse response was normalized to unit energy.� This channel model was used because of its simplicity (and ease of calculation).� Future work would involve more realistic models.� Transmission was assumed to be BPSK.� Because the channels are unit energy, a measure of receiver SNR is simply the inverse of the noise power.
The pseudo-optimum (feed-back fixed to one tap) MIMO DFE coefficients were calculated for a 6 tap feed-forward filter.� The channel diagonalization for a DMMT implementation with a block size of 16 symbols was also calculated.� Note that optimally, the input signals to each DMMT subchannel would be adapted in power and/or bits/symbol given the knowledge of the diagonal channel transmission functions.� However, this was not implemented due to time constraints.
20,000 random bits were generated, and the number of bit errors was calculated at the output of the DFE and DMMT.
The results are shown in the figure below.� Notice that the DMMT almost always performs slightly more poorly than the DFE.� (This is in addition to the data rate penalty that is incurred because of the block prefixes.)� However, in certain (perhaps pathological) cases, the DFE performs very poorly, whereas the DMMT is consistent.� This suggests the classic problem with decision-feedback, error propagation, is not only present in MIMO situations, but even worsens.
Implementation
Notes
The DMMT was implemented somewhat smoothly, without any large apparent errors.� Results are as expected, and channel decomposition computation time was not excessive (as might have been feared).
Conclusion
The results of this study somewhat surprisingly suggest that the performance of DMMT, a system in which there are optimal transmitter and receiver filter operations is not as good, or at least certainly not better, than the MIMO DFE, a receiver-only system.� However, as previously mentioned, a critical portion of DMMT system implementation, power and/or bit rate adaptation, was not implemented, and might have improved performance.� Furthermore, because the bit error rates tested where somewhat high (on the order of 1 in 1000), the problem of error propagation, though noticeable, was not highlighted as it might have been if the system was tested at very low bit error rates (in which one error cascades into many).� Finally, the channel that was used for comparison is not realistic (though it is unlikely that this would have affected the results).� Future work would have to address DMMT adaptation, and better channel models, and perhaps even channel and filter coefficient estimation times.
[1] EE359 course reader.
[2] N. Al-Dhahir and A. Sayed, "The Finite-Length Multi-Input Multi-Output MMSE-DFE", IEEE Trans. Signal Proc., vol. 48, pp. 2921-2936, Oct. 2000.
[3] A. Lozano and C. Papadias, "Space-Time Receiver for Wideband BLAST in Rich-Scattering Wireless Channels", VTC2000, vol. 1, pp. 186-190, June 2000.
[4] G. Foschini et al., "Simplified Processing for High Spectral Efficiency Wireless Communication Employing Multi-Element Arrays", IEEE J. Selected Areas in Comm., vol. 17, pp. 1841-1852, Nov. 1999.
[5] G. Raleigh and J. Cioffi, "Spatio-Temporal Coding for Wireless Communication", IEEE Trans. Comm., vol 46, pp. 357-366, March 1998.