EQUALISATION OF BROADBAND MIMO CHANNELS BY SUBBAND ADAPTIVE IDENTIFICATION AND ANALYTIC INVERSION Viktor Bale and Stephan Weiss Communications Research Group, School of Electronics & Computer Science University of Southampton, Southampton SO17 1BJ, UK g fvb01r,sw1 @ecs.soton.ac.uk ABSTRACT they were white at the transmitter they would be colouredby the channel, and hence we would expect slow convergence. This paper introduces the subband method of performing In fact it may be so slow that for realistic channels even adaptive identification and analytic inversion of broadband after several tens or hundreds of thousands of algorithm it- MIMO channels. It shows that the techniques can poten- eration the adaptation mean squared error (MSE) still may tially lower the computational cost while improving the per- not reach an acceptably low value [3]. In addition to this formance for highly frequency-selectivechannel with a long the computational cost involved in performing such a long impulse response. It covers subband adaptive identification adaptation can quickly become unacceptably high for large and shows two methods to invert a broadband MIMO chan- MIMO channels with long impulse responses. A further po- nel, the time-domain and frequency-domain methods. Fi- tential problem is that for fading mobile channels even if the nally results are shown for adaptation MSE, channel-equaliser equaliser is calculated beforehandusing an analytic method, MSE and BER performance. the adaptive inversion may not be able to track the equaliser at a fast enoughrate and it may soon become useless. Using 1. INTRODUCTION a fast converging algorithm such as the RLS also has asso- ciated problems as the complexity is greater than the NLMS Potentially great capacity increases throughthe use of Multiple- and hence for large broadband MIMO system the cost may Input Multiple-Output (MIMO) systems have now become be unacceptable. Further the RLS may exhibit worse perfor- well-known. Much of the work in this area has assumed that mance than LMS-type algorithms when tracking dynamic the sub-channels which comprise the MIMO channels have system [4]. a flat frequency transfer characteristic. Since one the poten- A promising solution proposed in [3] was to employ tial applications of MIMO system is to increase the date rate subband adaptive techniques to invert the channel. The con- through the channel to amounts which were not previously vergence rate was shown to be greater and the computa- possible, it seems far for more realistic that we would use tional cost lower than the fullband method. Hence the sub- broadband sub-channels which are frequency-selective. To band inversionmay be better able to track a dynamicequaliser be able to realise the high capacities promised by broadband at a lower cost. In order to use this method for tracking MIMO channels, we need to develop a high performance though, we must first initialise the equaliser to the opti- low-complexity technique for finding a suitable broadband mum at a point in time. Although subband adaptive inver- MIMO equaliser. Further if the equaliser is to be used in a sion showed improved convergence over the fullband ap- mobile environment we must assume that the channel will proach, it is still too slow to use this method to initialise the temporally dynamic or fading. This further exacerbates the equaliser in the first instance. Hence we must use an alterna- problem of finding and tracking the optimum equaliser so tive method which is the subject of this paper. We propose that satisfactory performance is maintained. to use a subband adaptive identification of the broadband A common and simple approach is to use the adaptive MIMO channel, which can be performed using many fewer NLMS algorithm to adapt to the inverse of the broadband iterations than the inversion, followed by a computationally MIMO channel and track it as the channel fades. The prob- efficient analytic inversion. The whole process must also lem with this is that the convergencerate of the NLMS algo- be performed in subbands as the subband adaptive tracking rithm is related to the ratio between the minimum and max- system must be initialised with the subband equaliser coef- imum of the PSD of the input signal to the algorithm [1,2]. ficients. The system consider is shown in Figure 1. In an adaptive inversion set-up this would be the received Sec. 2 briefly introduces the technique of subband pro- signal at the output of the MIMO channel, and since even if cessing, while Sec. 3 covers subband adaptive identifica- tion and explains the potential computational cost advan- a [n] N N s 0[n] tage. Sec. 4 developsthe time-domainand frequency-domain 0 methods of inverting the subband representation of a MIMO u[n] u[n] a [n] N N s1[n] system, states the costs of the inversions and also explains 1 some associated problems and methods to overcome them. Finally, in Sec. 5 simulations results are presented before a [n] N N sK-1[n] discussing conclusions in Sec. 6. K-1 analysis filter bank synthesis filter bank 2. SUBBAND TECHNIQUE Fig. 2. Subband decomposition by mean of analysis and 2.1. Oversampled Subband Decomposition synthesis filtering banks. In essence the subband approach involves the partitioning 0 )| à ω of the input signals into a finite number, , of frequency j −20 (e bands or subbands. This is similar to transforming the prob- k |A −40 lem into the frequency domain and is performed in practice 10 −60 by a class of band-pass filters, but unlike this the signals re- 20log −80 main as time-domain sequences. Since the subband signals 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 normalized frequency ω/π are now bandlimited by a factor à more than the fullband signals, we may downsample each of the signals by a fac- à tor Æ . Fig. 2 shows a simple subband system, which Fig. 3. Filter bank characteristic for K = 16 and N = 14 Ä based on a prototype filter with Ô coefficients. Ò℄ filters input signal Ù through an analysis filter bank com- à a Ò℄; k ¼; ½; ¡ ¡¡ ;à ½ prising band-pass filters k , dec- Æ imates the signals by Æ , upsamples by and reconstructs 2.3. Complexity the original fullband signal by passing through synthesis fil- Ã Æ × Ò℄ Although would result in the greatest computa- ters k and summing. Any signal processing task can be performed on the decimated subband signals [2]. tional savings when performingan adaptivealgorithm, spec- tral aliasing limits the performance of any processing in the subband domain and in practice we oversample the signals 2.2. Modulated Filter Banks à whereby we choose Æ slightly less than [6]. An exam- ½ Æ ½ The filters banks are often created using a generalised dis- ple of an analysis filter bank where à and j ! A e µ ´ crete Fourier transform (GDFT) [5], which have the advan- is shown in Fig. 3, where magnitude responses k ¯ Æ a Ò℄ tage that all the band-pass filters can be created by modu- — k of only the first 8 filters are shown. The compu- lating a common prototype filter to the correct frequency. tational cost involved in the analysis and synthesis filtering Secondly they have the desirable property that the synthesis process is [7] filter bank is simply the parahermitian of the analysis banks C ´¾Ä · à ÐÓ Ã · à µ=Æ; Ô baÒk (1) when the system is expressed using a polyphase representa- ¾ Ä tion [2]. per fullband sample period, where Ô is the prototype filter length. MIMO MIMO 3. ADAPTIVE IDENTIFICATION channel equaliser 3.1. Multi-Channel Filtering adaptive The first step involved in finding the optimum equaliser in system analytic ¢ È identification inversion subbands is to adaptive identify a Å MIMO chan- nel. We use a multi-channel form of the NLMS algorithm, whereby the adaptive filter state vectors of each channel copy coefficients are stacked, which effectively transforms the problem into a single-channel form. The Å inputs transmitted through the MIMO channel excite È signals at the receivers, which can Fig. 1. System setup with adaptive MIMO system iden- be expressed as tification and analytic inversion to calculate the MIMO Ò℄ ÀÜÒ℄ · Ò℄; equaliser. Ý (2) Å Ä h Ò℄ ¾ C Å where Ü contains stacked input signal vec- Following the usual Weiner-Hopf type analysis [1] for each Ä tors of equal to the channel length h subband we arrive at the optimum adaptive filter solution Ì Ì Ì Ì ½ ÜÒ℄ Ü Ò℄ Ü Ò℄ ¡¡¡ Ü Ò℄℄ (3) ½ ¾ Å h Ê Ô ; Ô;k ;ÓÔØ Ô;k (8) ÜÜ;k Ü Ò℄ Ü Ò℄ Ü Ò ℄ ¡¡¡ Ü Ò Ä · Ñ Ñ Ñ h and Ñ Ê where ÜÜ;k is the auto-correlation matrix of the filter state È Ì ½´½µÅ Ò℄ ¾ C ; Ô ℄℄ . The noise vector contains Øh k Ü Ò℄ Ô Ô;k vector in the subband k , and is the cross-correlation È noise samples taken from a white Gaussian source vector between the subband filter state vector and desired Ì Ý Ò℄ Ò℄ Ò℄ Ò℄ ¡¡¡ Ò℄℄ : Ô;k ¾ È ½ (4) signal . Hence the update step of the subband multi- channel NLMS algorithm follows as Ý Ò℄ The received signal vector at time Ò, , is length P and £ Ò℄ Ò℄e defined analogously to (4). Finally, the channel is defined Ü k Ô;k h Ò · ℄ h Ò℄ · ¾ ¿ Ô;k Ô;k (9) À Ì Ì Ì Ü Ò℄Ü Ò℄ h h ¡¡¡ h k k ½½ ¾½ Å ½ Ì Ì Ì 6 7 h h ¡¡¡ h ½¾ ¾¾ Å ¾ 6 7 where is the normalised adaptation step-size coefficient.