Equalisation of Broadband Mimo Channels by Subband Adaptive Identification and Analytic Inversion

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 associated 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 identification 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

)| Ã 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 Ã more than the fullband

signals, we may downsample each of the signals by a fac-

tor Æ . Fig. 2 shows a simple subband system, which Fig. 3. Filter bank characteristic for K = 16 and N = 14

based on a prototype ﬁlter with Ô coefﬁcients.

Ò℄

ﬁlters input signal Ù through an analysis ﬁlter bank com-

Ã a Ò℄; k ; ; ;Ã

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, spectral aliasing limits the performance of any processing in the subband domain and in practice we oversample the signals

2.2. Modulated Filter Banks Ã

whereby we choose Æ slightly less than [6]. An exam-

The ﬁlters banks are often created using a generalised dis- ple of an analysis ﬁlter bank where Ã and

j !

A e

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 Ä Ã Ã Ã =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 channel. 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

Ò℄ 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

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

Å Ò℄ 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

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

Ô;k

h Ò ℄ h Ò℄

¾ ¿ Ô;k

Ô;k (9)

Ì Ì Ì

Ü Ò℄Ü Ò℄

h h h

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.

À 7

6 . . . . (5) 5

4 . . .. .

Ì Ì Ì

h h h

ÅÈ ¾È È

½ 3.3. Identiﬁcation Complexity

h h ℄ h ℄ h Ä ℄℄

ÑÔ ÑÔ ÑÔ h

where ÑÔ is The computational cost of the fullband NLMS algorithm is Øh

the channel impulse response between the Ñ transmitter

Øh

C È ÅÄ Å :

Ô h and receiver. Notice that for the purpose of channel FB (10) identiﬁcation we assume that the channel is static. One of the main advantages of the subband approach is a reduction in computational cost. Not only are the adaptive 3.2. Optimum Subband Identiﬁcation

ﬁlters generally shorter but the adaptive update can executed

Øh

=Æ

Next we derivethe subbandadaptive system which will adapt at of the rate of the fullband version due to the deci-

À Ý Ã Ò℄; Ô

to the unknown , using the received samples Ô mation. Of course the algorithm must be performed times È

as the desired signals and a known training sequence in parallel, once for each subband. The cost of the subband

Ü Ò℄; Ñ Å as the input signals Ñ . Fig. 4 shows one NLMS algorithm hence accrues to

of È multiple-input single-output (MISO) adaptive blocks

Å È ÈÃ

ÅÄ Å C

that run in parallel and are required to identify the MIMO C

h;× baÒk ËB

Æ Æ channel. Each MISO system adapts in subbands to the Å (11) sub-channels between the transmitters and a particular receiver. per fullband sampling period. To be able perform a di- The outputs of the subband adaptive units within the rect comparison between the fullband and subband cost, we

MISO system for receiver Ô are given by

Ä Ä h;×

must decide on a relationship between h and . An

Ä Ä Ä =Æ

h Ô

often quoted relationship is h;× , which

Ý Ò℄ h Ò℄Ü Ò℄; k Ã; Ô È; k Ô;k (6)

Ô;k would result in

Ì Ì Ì Ì

h Ò℄ h Ò℄ h Ò℄ h Ò℄℄

where Ô;k

½Ô;k ¾Ô;k Å Ô;k

ÈÃ

C Å Ä Ä =Æ Å

ËB h Ô

h Ò℄ h ;Ò℄ h ;Ò℄ :::

ÑÔ;k ÑÔ;k

and ÑÔ;k

;Ò℄℄ h Ä

Å È h;×

ÑÔ;k is the impulse response of the adaptive

C : Øh

baÒk (12)

ﬁlter which models the sub-channel between the Ñ trans-

Øh Øh k mitter and Ô receiver in the subband and depends on

per fullband sampling period. However the simulations pre-

Ò Ä

time due to the fact that the system is adaptive,and h;× is

Ä Ä =Æ h the length of the subband adaptive ﬁlter. There is some free- sented later use a value for h;× closer to . The anal-

ysis ﬁltering operation causes a ﬁxed computational over- Ä

dom in choosing in h;× and the value depends on factors

Ä Ä Ô

head, and for small h we may ﬁnd that the subband

Ä Ä Ô such as the size of h relative to , buta value in the range

adaptation is actually more costly than the fullband version.

Ä Ä =Æ; Ä Ä =Æ ℄

h h Ô h;× is usually chosen. Further

Ì Hence the subband method is generally more suitable for

Ü Ò℄ Ü Ò℄ Ü Ò℄ Ü Ò℄℄

½;k ¾;k Å ;k in (6) k , where

high bandwidth channels with a long impulse response.

Ü Ò℄ Ò

Ñ;k is the subband ﬁlter state vector at time corre-

Ü Ã

sponding to Ñ;k . We may now deﬁne subband error Øh

signals for the Ô receiver 4. ANALYTIC INVERSION

e Ò℄ Ý Ò℄ Ý Ò℄ k Ã; Ô È:

Ô;k Ô;k Ô;k The second partof the task is to analyticallyinvertthe subband- (7) identiﬁed MIMO channel. We are constraining the equaliser yp[n] analysis desired signal

subband 1

hˆ 1p,1[n] ˆ x1[n] h2p,1[n] yˆp,1[n] ep,1[n] analysis

hˆ Mp,1[n]

subband 2

hˆ 1p,2[n] ˆ x2[n] h2p,2[n] ep,2[n] analysis yˆp,2[n] e [n] hˆ [n] synthesis p Mp,2 error M signal transmitter signals subband K

hˆ 1p,K[n] xM[n] analysis hˆ 2p,K[n]

yˆp,K[n] ep,K[n] hˆ Mp,K[n]

Fig. 4. A MISO subband adaptive system arranged for system identiﬁcation.

Ô k

to be a linear FIR system, and there are two methods we use Ñ and for subband , yielding ¾

to invert the channel. These two methods have been demon- ¿

À À À

¾½;k Å ½;k ;k

strated for fullband channel inversion in [8], but we shall ½½

6 7

À À À

¾¾;k Å ¾;k ½¾;k

6 7

brieﬂy review them and apply the subband technique here.

À ;

6 7

k . . . . (14) 4

. . .. . 5

À À À

½È ;k ¾È ;k Å È ;k

Å Ä Ä À ÈÄ

;× h;× g ;× k where is of dimensions g

4.1. Time-Domain Inversion Ä

and g ;× is the subband MISO equaliser length for each sub-

channel. From this we may develop an expression for the

Øh

È k The time-domain inversion technique, so-called as it arranges optimum solution for the Å equaliser ﬁlters for the

the time-dispersiveness of the broadband MIMO channel subband ¾

a two dimensional matrix, calls for the creation of convo-

Ý À

g Á À À ; À d

Ñ;k k Ñ

k (15) k

lutional matrices. Each sub-channel has its own convolu- ¾

tional matrix and these are augmented and stacked to create Ü

d a larger “parent” MIMO convolutional matrix [8,9]. where Ñ is a channel selection and delay vector and

refers to the pseudo-inverse [10] which is required for valid

Øh Ñ

The convolutional channel matrix between trans- À

À À Å > È

cases where but k may be rank-deﬁcient.

Øh Øh k

mitter and Ô receiver for the subband is given by

This calculation must be performed for all Ñ and

Ã Ã

k to create a MIMO equalisers, one for each

¾ ¿

h subband. These can then be placed between analysis and

ÑÔ;k 7

6 . synthesis ﬁlter banks to create the full equaliser system.

À 7

6 .

ÑÔ;k

6 7

The computationalcomplexity of the subband time-domain

À :

ÑÔ;k (13)

6 7

. . . ¿

Ä 5 4 method is . Since the subband ﬁlters are potentially

. . . g

. . . Æ

À up to times shorter than the fullband equalisers it is ev-

ÑÔ;k

Æ ident that the cost of inverting a single subband is

lower, but this must be performed Ã times, hence the com-

Æ =Ã We may construct a parent convolutional matrix À over all putational saving afforded by the subband techniqueis . Although this method will result in an optimum subband so- 0.4 2 lution the cost of the inversions, even though much less than 0.3 the fullband method, can still grow rapidly with increasing

0.2 Ä

g ;× to unacceptable levels. Since we are dealing with high bandwidth channels this is likely to be a problem. 0.1 power profile |h[n]|

0 0 50 100 150 200 250 300 4.2. Frequency-Domain Inversion time nT / [ns] s

A solution to the problem of the computational cost grow- 10 ing rapidly for the time-domain inversion is to use lower 0 complexity method, and frequency-domain inversion is a −10 )| / [dB] s

good candidate to this end. We may transform the impulse f/f π −20 response of each of sub-channel for each subband into its j2

−30

|H(e Ä bandlimited spectral representation using a length g ;× FFT, −40 and hence formulate the problem in the frequency-domain. 0 10 20 30 40 50 60 70 80 90 100

frequency f / [MHz]

Ä Å È

For each subband we now have g ;× scalar valued matrices, one for each frequency-bin of the FFT, which can easily be inverted using a standard matrix inversion algo- Fig. 5. CIR sampled at 100 MHz (top) and magnitude re- rithm. Hence the zero-forcing inverse for each frequency sponse (bottom) of a sample channel generated by the SV bin in each subband is given by the pseudo-inverse of that model. channel, i.e.

effect and may degrade the performanceof the subband sys-

j ¾ À ½ À d

G ℄ e À f ℄À h℄ À f ℄: f k

k (16) k k tem more than we would have otherwise expected. By how Unfortunatelythe matter is complicated by a phenomenon much the performanceis worsenedwill be shown in the sim- that affects frequency-domain multiplications known as cir- ulations. cular convolution,or the wrap-aroundeffect [11]. In essence The main motivating factor for the use of the frequency- a convolution in the frequency-domain differs from the true domain inversion method is that is requires less computa- linear convolution when performed in time-domain at either tional power than the time-domain method. It can be shown

end of the resultantsignal or system, and the shorter the FFT that the FFTs are the computationally dominant part of the

Ä Ä g ;× inversion process and hence the complexity is g ;× . the worse the problem becomes. A solution is to use a regu- ¾ larisation factor in the inversion, giving a new deﬁnition for Evidently the savings offered by the subband approach are the inverse [12] now much smaller than for the time-domaininversionmethod.

In fact, we shall see in the simulations that it is possible that

j ¾ d À ½ À

Ä Ä

G f ℄ e À f ℄À h℄ ¬ Á À f ℄

g Ô k

k (17) for short equalisers where the savings may not

k k outweigh the analysis and synthesis ﬁltering cost overhead.

where ¬ is the regularisation factor. The method of regular-

isation is a powerful technique for ﬁnding the solutions of ¾

¾ 5. SIMULATION RESULTS

ill-posed problems. Also it can be shown that if ¬ ,

Ü i.e. the noise-to-signal ratio (NSR) then at low SNR values 5.1. MIMO Channel Model and Simulation Parameters the system MSE performance approaches that of the optimum time-domain method [13]. At some critical SNR, the We use the Saleh-Valenzuela statistical indoor model to gen- MSE performance will start to degrade as the NSR falls be- erate realistic channel impulse responses (CIRs) [14]. The low a value that results in the best performance possible us- SV model produces a clusters of rays according to an ex- ing the subband frequency-domain method in a noiseless ponential distribution. Following the suggestions in [14],

environment. At this critical SNR we would gain a bet- we have simulated two sets of CIRs at sampling rates of

ter performance result by switching to a ﬁxed regularisation 100MHz and 1GHz with a mean cluster arrival time

factor. ns, a mean ray arrival time ns, a cluster power

One pertinent difference between the subband and full- decay time constant ns and the ray power decay

band frequency-domaininversion is due to the relative lengths time constant ns [14]. The magnitude response of a of the equaliser. Since the subband adaptive ﬁlters are gen- typical SV channel, band-limited and sampled at 100 MHz,

erally shorter than the channel impulse response length, we is shown in Fig. 5. We generate four of these channels inde-

may often also use shorter subband equaliser ﬁlters than pendently to create a broadband MIMO channel. Sim-

would have been necessary for the fullband system. Un- ulations are then run on an ensemble of such MIMO fortunately a shorter equaliser will worsen the wrap-around channel realisations. 10 fullband adaptation, 100 MHz SV adaptation, channel cost (MACs) fullband adaptation, 1 GHz SV subband, 100 MHz SV 846 0 subband adaptation, 100 MHz SV

[dB] subband adaptation, 1 GHz SV fullband, 100 MHz SV 1,012

MSE subband, 1 GHz 2,034 ξ −10 fullband, 1 GHz SV 9,652

−20 Table 1. Computational cost (MACs) per fullband sampling

−30 period of fullband and subband adaptive identiﬁcation for

the 100 MHz and 1 GHz-sampled MIMO channels.

−40 ensemble adaptation MSE, inversion, channel cost (MACs)

−50 5

: 0 0.5 1 1.5 2 2.5 3 3.5 4 TD Sub, 100 MHz SV 4

fullband sampling periods, n x 10 ¿

FD Sub, 100 MHz SV

: TD Full, 100 MHz

Fig. 6. MIMO channel identiﬁcation adaptation MSE for ¿

: FD Full, 100 MHz SV

fullband and subband adaptations with 100 MHz and 1 6

: TD Sub, 1 GHz SV

GHz-sampled SV channel. ¿

FD Sub, 1 GHz SV

TD Full, 1 GHz SV

For the analytic inversions, we have selected a delay FD Full, 1 GHz SV

d Ä =

g ;× to address the non-minimum phase characteristic of the channel. For the NLMS adaptations we use Table 2. Evaluation of the computational cost of various

combinations of time-domain (TD) and frequency-domain

: . For the fullband processing and the 100 MHz-

(FD), and subband and fullband inversion, for the 100 MHz

Ä Ä g ;f sampled SV channel we use h and ,

and 1 GHz-sampled SV channel. Ä

where g ;f is the fullband equaliser length, and for the 1

Ä Ä g ;f

GHz-sampled SV channel we use h and

Ã . For the equivalent subband system we use ,

are a continuous process, but as we simply need to reach

Æ Ä

based on a prototype ﬁlter with Ô taps a speciﬁed MSE and the fullband method converges much resulting in the characteristic given in Fig. 3 and have set

quicker, we ﬁnd that the total operation count may in fact be

Ä > Ä =Æ Ä > Ä =Æ

h;f g ;× g ;f h;× and for the

greater for the subband technique. For example, for the 100

Ä > Ä =Æ h;f 100 MHz-sampled SV channel, and h;×

MHz channel, -20 dB MSE is reached in 14% of the time it

Ä Ä =Æ g ;f and g ;× for the 1 GHz-sampled SV chan- takes the fullband method, and hence the overall computa- nel. tion count for the fullband is 16.8% of that for the subband. For the 1 GHz-sampled channel though this ratio becomes 5.2. Adaptive Identiﬁcation about 245%, meaning that for this long channel the subband

The fullband and subband adaptive identiﬁcation MSE be- adaptation has lower cost in all respects.

haviour for a 100 MHz and 1 GHz-sampled channels

is shown in Figure 6. As expected the fullband techniques 5.3. Analytic Inversion

Ä > Ä =Æ h;× show the quickest convergence because g ;× and also as the input signals are white the subband method will The system MSE after using the various analytic inversion not demonstrate an convergence improvement. methods discussed in this paper is shown in Figure 7. The The real power of the subband system in this case arises system MSE refers to the error between the input to the from the reduction in computational cost. Table 1 shows the MIMO channel and the channel-equaliser response to this cost per fullband sampling period in multiply-accumulate input. The method labelled “Full” corresponds to fullband

computations (MACs) of the subband and fullband adaptive inversion of a known channel and the time-domain inver-

identification of the 100 MHz and 1 GHz-sampled SV sion (TD Full) represents the optimum performance possi- channels. We see the for the 100 MHz-sampled channel the ble from any of the systems. The curves labelled “SuS” subband adaptation requires about 84% of the computations refers to subband identification followed by subband inver- needed for the fullband technique. For the 1 GHz-sampled sion, where as “SuF” represent subband identification af- channel, this becomes 21%. Hence we see that the relative ter which the equivalent fullband system is found by pass- savings offered by the subband method increase as the chan- ing a impulse through the analysis filter bank, the subband nel impulse response, and hence equaliser length increases. filters and the synthesis filter bank, and finally this is in- This is a fair comparison assuming that the adaptations verted in the fullband. Also shown are curves for frequency- domain inversion regularised by the NSR (FDN) and also also the BER performance is quite satisfactory. Finally, by a heuristically optimised fixed value which results in this method of finding a subband equaliser for broadband the best possible performance in a noiseless environment MIMO channels is useful for initialising an adaptive equaliser (FDO). We have no result for the fullband inversion of the for dynamicallyfading channels where we would use subband- known channel for the 1 GHz-sampled SV channel as the adaptive inversion to track the equaliser. Subband-adaptive channel is of such length that it would be computationally inversion has shown the ability to converge faster from coloured unfeasible, however we have included SuF results for this input signals and hence may be able to facilitate superior channel for interest. We see that generally at lower SNR tracking performancethan the equivalent fullband system [3]. value the FDN curves approach the optimum performance of the TD curves, and that as the SNR rises the FDN curves 7. REFERENCES begin to worsen due to circular convolution but the FDO [1] S. Haykin, Adaptive Filter Theory, Prentice Hall, Englewood curves continue to improve until the best achievable MSE is Cliffs, 3rd edition, 1996. reached. [2] S. Weiss and R. W. Stewart, On Adaptive Filtering in Over- The cost of performing these inversions is shown in Ta- sampled Subbands, Shaker Verlag, Aachen, Germany, 1998. ble 2, where the complexity orders are evaluated. For the [3] V. Bale and S. Weiss, “A Low-Complexity Subband Adap- time-domain inversion significant savings are offered by the tive MIMO Equaliser for Highly Frequency-Selective Chan- nels,” in Spectral Methods and Multi-rate Signal Processing, subband method as expected. 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Vetterli, “Adaptive Filtering in Subbands band identifications in Table 1 which must be performed for with Critical Sampling: Analysis, Experiments and Applica- many thousands of iterations it is evident that the identifications to Acoustic Echo Cancelation,” IEEE Transactions on tion will dominate the overall computational cost. Signal Processing, vol. SP-40, no. 8, pp. 1862–1875, August 1992. [7] S. Weiss, “Analysis and Fast Implementation of Oversam- 5.4. BER Performance pled Modulated Filter Banks,” in Mathematics in Signal Pro- cessing V, J. G. McWhirter and I. K. Proudler, Eds., chap- Figure 8 show the BPSK BER performance for the various ter 23, pp. 263–274. Oxford University Press, March 2002. systems developed in this paper. The systems are the same [8] V. Bale and S. Weiss, “Comparison of Analytic Inversion as those in Figure 7, but the simulations are now performed Techniques for Equalisation of Highly Frequency-Selective only over a “realistic” SNR range of 0 dB to 30 dB. We see MIMO Systems,” in Proceedings of Workshop on Sig- that for the 100 MHz-sampled SV channel the performances nal Processing for Wireless Communications, King College, are all quite similar. Only the FDO curves are worse as London, UK, June 2004, pp. 150–155. [9] O. Kirkeby, P. A. Nelson, F. Orduna-Bustamante, and the point where a fixed regularisation factor outperformsthe H. Hamada, “Local Sound Field Reproduction Using Dig- noise power value occurs at about 30 dB. The performance ital Signal Processing,” Journal of the Acoustical Society of the 1 GHz-sampled SV channel is the best due to the of America, vol. Vol.100, no. No.3, pp. 1584–1593, March much longer equaliser filters. 1996. [10] G. Strang, Linear Algebra and Its Applications, Academic Press, New York, 2nd edition, 1980. 6. DISCUSSION [11] J. J. Shynk, “Frequency-Domain and Multirate Adaptive Fil- tering,” IEEE Signal Processing Magazine, vol. 9, no. 1, pp. In this paper we have introduced a method of adaptively 14–37, January 1992. identifying a broadband MIMO channel and analytically in- [12] O. Kirkeby, P. A. Nelson, H. Hamada, and F. Orduna- verting to create an MIMO equaliser in subbands. The per- Bustamante, “Fast Deconvolution of Multichannel Systems Using Regularization,” IEEE Transactions on Speech and formance results and computational costs indicate that for Audio Processing, vol. 6, no. 2, pp. 189–194, March 1998. channels with a short impulse response, or for channels that [13] V. Bale, Computationally Efficient Equalisation of Broad- are not particularly frequency-selectivethe subband approach band Multiple-Input Multiple-Output Systems, Mini-Thesis, is of limited use. However, for much longer channels, such University of Southampton, UK, 2004. as the 300 coefficient 1 GHz-sampled SV channel, the ad- [14] A. A. M. Saleh and R. A. Valenzuela, “A Statistical Model vantages offered by technique are much more impressive. for Indoor Multipath Propagation,” IEEE Journal on Se- Not only does subband processing lower the computational lected Areas of Communications, vol. 5, no. 2, pp. 128–137, cost involved in identifying and inverting the channel, but February 1987. TD Full 100 MHz SV FDN Full 100 MHz SV 20 FDO Full 100 MHz SV TD SuS 100 MHz SV FDN SuS 100 MHz SV FDO SuS 100 MHz SV 10 FDN SuF 100 MHz SV FDO SuF 100 MHz SV TD SuS 1 GHz SV FDN SuS 1 GHz SV FDO SuS 1 GHz SV 0 FDN SuF 1 GHz SV FDO SuF 1 GHz SV

−10

−20 ensemble system MSE [dB]

−30

−40 0 10 20 30 40 50 60 SNR [dB]

Fig. 7. System MSE versus SNR behaviour for subband and fullband analytic inversion for a 100 MHz and 1 GHz-sampled SV channel.

0 10

−2 10

TD Full 100 MHz SV FDN Full 100 MHz SV −4 10 FDO Full 100 MHz SV

BER TD SuS 100 MHz SV FDN SuS 100 MHz SV FDO SuS 100 MHz SV FDN SuF 100 MHz SV −6 FDO SuF 100 MHz SV 10 TD SuS 1 GHz SV FDN SuS 1 GHz SV FDO SuS 1 GHz SV FDN SuF 1 GHz SV FDO SuF 1 GHz SV −8 10 0 5 10 15 20 25 30 SNR [dB]

Fig. 8. BER versus SNR performance for subband and fullband equalisers for a 100 MHz and 1 GHz-sampled SV channel.