IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 5, MAY 2006 1645 Filter Bank Precoding for FIR Equalization in High-Rate MIMO Communications A. VijayaKrishna and K. V. S. Hari, Senior Member, IEEE Abstract—In this paper, the problem of designing finite-im- pulse-response (FIR) equalizers for multiple-input multiple-output (MIMO) FIR channels is considered. It is shown that an arbi- trary MIMO frequency-selective channel can be rendered FIR equalizable by a suitable filter bank (FB) precoding operation that introduces redundancy at the transmitter. The expression for the minimum redundancy required to ensure FIR invertibility is derived. The analysis is extended to the case of MIMO multi- carrier modulation. Optimum zero-forcing (ZF) and minimum mean-squared error (MMSE) solutions for the FIR equalizer are derived. Simulation results are provided to demonstrate that the proposed scheme achieves better performance than the block-pro- cessing methods while supporting a higher data rate. Index Terms—Filter bank, finite-impulse-response (FIR) equal- ization, multiple-input multiple-output (MIMO), polynomial ma- trix, pseudocirculant matrix, Smith form. I. INTRODUCTION W ITH the ever-increasing demand for higher data rates, multple-input multiple-output (MIMO) designs are per- haps the most viable options for future wireless communication systems [1]. Various channel impairments like multipath and dispersion, resulting in intersymbol interference (ISI), make signal processing for MIMO communications a challenging task. Techniques like MIMO-orthogonal frequency-division multiplexing (MIMO-OFDM) and space–time (ST) precoding have been developed to deal with MIMO frequency-selective channels [2], [3]. In this paper, we provide a filter bank (FB) framework for MIMO communications. In the single-input single-output (SISO) case, the FB pre- coding approach to multicarrier modulation is quite well de- veloped [4], [5]. FB transceivers can be designed to provide a much higher data rate (number of symbols per channel use) than the block processing methods like OFDM that require re- dundancy of the order of channel length [4], [6]. In addition, the FB framework contains the block processing methods as special cases, thus providing a larger context for studying the tradeoffs involved in system design. Among the FB approaches to MIMO communications, the knowledge of the channel is utilized in [7] to design a pre-equalizer, i.e., a polynomial matrix such that , at the transmitter. However, this method requires , and assumes that the channel coefficient Manuscript received December 1, 2004; revised May 20, 2005. The associate editor coordinating the review of this manuscript and approving it for publica- tion was Dr. Markus Rupp. The authors are with the Department of Electrical Communication En- gineering, Indian Institute of Science, Bangalore 560 012, India (e-mail: vkrishna@ece.iisc.ernet.in; hari@ece.iisc.ernet.in). Digital Object Identifier 10.1109/TSP.2006.871971 matrices are orthogonal. In [8], a blind finite-impulse-response (FIR) equalizer is designed using polynomial matrix tech- niques. Pohl et al.[9] uses the Kronecker form of matrix pencils to design a zero-forcing (ZF) equalizer. These methods, which perform FIR equalization at the receiver, in general require and the channel matrix to be irreducible. In [10], an iterative procedure for the joint design of precoder and equalizer is developed with the assumption that the channel is communicable. In [11], the concept of biorthogonal partners is used to design an FIR fractionally spaced equalizer. In this paper, we consider the problem of designing a FB pre- coding framework that achieves FIR equalization of an arbitrary MIMO FIR channel without imposing any constraints on the nature of the channel. The channel can be of any dimension, and it can even be singular. At the transmitter, the availability of channel knowledge is utilized to design an FIR precoder in such a way as to make the precoded channel FIR invertible. We derive the minimum redundancy required to accomplish FIR invertibility. In practical scenarios wherein the assumption of random channel coefficients holds, it will be seen that full rate can be achieved for rectangular channels in the FB precoding framework, i.e., no redundancy is required. However, when the channel is square, a redundancy of one symbol per channel use is required. In addition, it will be seen that the assumption of channel knowledge at the transmitter can be dispensed with in practical scenarios. A redundancy of one symbol per channel use for square chan- nels implies considerable rate loss when the channel dimensions are small. This motivates us to extend the FB precoding frame- work to the case of MIMO multicarrier modulation. Due to the blocking operation inherent in multicarrier modulation, the ef- fective channel becomes a block pseudocirculant matrix [12]. By investigating the properties of the Smith form of block pseu- docirculant matrices, we derive the expression for minimum re- dundancy required to enable FIR equalization at the receiver. It will be seen that the MIMO multicarrier approach retains the full rate advantage for rectangular channels, while increasing the achievable data rate in case of square channels. In addi- tion, the MIMO multicarrier approach provides a better frame- work than the basic FB approach for trading off rate for perfor- mance. Compared with the ST methods, which require redun- dancy of the order of channel length [2], the MIMO multicarrier framework provides comparable performance while supporting a much higher data rate. The equalizer for the precoded channel is in the form of a FIR left inverse, and hence is not unique. This non-uniqueness can be utilized to design equalizers based on different criteria. We show that the design freedom available at the precoder can be 1053-587X/$20.00 © 2006 IEEE