Diversity of Linear MIMO Receivers

Multimedia Communications Laboratory
University of Texas at Dallas


The gains afforded by multi-antenna (MIMO) systems in the presence of rich fading environments has now been well known. MIMO systems can thus be designed that allow much higher rates than single-antenna systems, examples of which are known in the form of BLAST and its variants [1,2].

The main idea behind BLAST is successive decoding and interference cancellation. While BLAST has been successful in giving higher rates, it also has high computational complexity at the receiver. This complexity arises from the difficulty of separating the multiple signal streams that are sent simultaneously across the MIMO channel and canceling their cross-interference.

The problem of interference cancellation existed long before the MIMO channel became a subject of intense interest. A convenient method for addressing interference is linear equalization, e.g. zero-forcing and minimum mean-square error (MMSE) equalization. This work addresses linear receivers in the context of MIMO systems (Figure 1). The basic idea is that the interference suppression is confined to the linear block, while it is possible to do other operations, e.g. decoding of error control codes, on the separated streams after the linear receiver. The idea of linear MIMO receivers has been broached in [3,4], among others. The main contribution of our work [5] is the analysis of the performance of this system, which has been an open problem.

Linear MIMO system

Figure 1. Linear MIMO Receiver


This investigation has given rise to interesting and partially unexpected results.

For zero-forcing receivers, the analysis as well as simulations show that the diversity is always M-N+1, where M and N are respectively the number of transmit and receive antennas. This is the best achievable diversity regardless of the encoding and decoding strategy.

For the MMSE receiver, however a more interesting scenario arises. If the input streams are separately encoded, then the diversity of the MMSE receiver is also M-N+1, exactly the same as the zero-forcing receiver.

However, if we allow the input streams to be encoded jointly, then the diversity of the system has interesting and somewhat unexpected behavior. In this case, the diversity is dependent on the spectral efficiency that we wish to push through the system. If we desire high spectral efficiencies, then the diversity of the MMSE receiver is still M-N+1. However, at low enough spectral efficiencies, higher diversities are obtained, and in fact we show that at low-enough spectral efficiencies, a maximum diversity of MN is available.

While this work is somewhat reminiscent of the diversity-multiplexing tradeoff of Zhang and Tse [6], we should note the important difference that here we speak of spectral efficiency, i.e., rate, not the multiplexing gain (prelog factor of capacity). Therefore the picture is very different from [6].

An example of the simulations that confirm this work is provided below in Figure 2. In this figure, M=N=2 and the curves show rates R=1,2,4,10 bits/sec/Hz. The system model involves combined spatial encoding of the data streams on the transmit side and linear receivers. It is clearly seen that zero-forcing has constant diversity slope regardless of the rate, while the MMSE receiver has a rate-dependent diversity. For comparison purposes, we also show the outage curves for the maximum likelihood receiver, which acts as a lower bound to both linear receivers and is known to achieve full diversity at the fixed rates considered in this work. It is observed that at low rates MMSE performance approximates the maximum likelihood decoder, while at high rates it diverges from the ML performance and is closer to the zero-forcing performance.

Relay with private messages

Figure 2. Simulation results for a 2x2 system with rates R=1,2,4,10 b/s/Hz

For more details and the full analysis, the interested reader is referred to [5].

Open Problem

This work [5] addressed the diversity of MMSE receivers in the asymptote of low and high rate, but the complete characterization of the diversity of MMSE receivers as a function of rate (and possibly block length) remains an open problem at the time of the writing of this web page.


  1. G. Foschini, G. Golden, R. Valenzuela, and P. Wolniansky, ``Simplified processing for high spectral efficiency wireless communication employing multi-element arrays,'' Journal on Selected Areas in Communications, vol. 17, pp. 1841--1852, Nov. 1999.

  2. J. H. Winters, J. Salz, and R. D. Gitlin, ``The impact of antenna diversity on the capacity of wireless communication systems,'' IEEE Transactions on Communications, vol. 43, pp. 1740--1750, February/March/April 1994.

  3. M. Rupp, C. Mecklenbrauker, and G. Gritsch, ``High diversity with simple space time block-codes and linear receivers,'' in Proc. IEEE GLOBECOM, San Francisco, CA, November 2003, pp. 302--306.

  4. E. K. Onggosanusi, A. G. Dabak, T. Schmidl, and T. Muharemovic, ``Capacity analysis of frequency-selective MIMO channels with sub-optimal detectors,'' in Proc. IEEE ICASSP, 2002, pp. 2369--2372.

  5. A. Hedayat and A. Nosratinia, ``Outage and diversity of linear receivers in flat fading MIMO channels,'' IEEE Trans. Signal Processing, vol. 55, no. 12, Dec. 2007, pp. 5868-5873.

  6. L. Zheng and D. Tse, ``Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels,'' IEEE Transactions on Information Theory, vol. 49, no. 5, pp. 1073--1096, May 2003.

Last modified 2008

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