Class of blind maximal ratio combining methods for digital communication systems

S.K.Oh

Abstract: Simple iterative methods for blind maximal ratio combining based on a maximum likelihood principle and finite alphabet properties inherent in digital communication systems are presented. These methods can provide accurate estimates of channel parameters even with a small subset of data. The channel parameters and the data sequence are estimated simultaneously. Two methods, the joint combining and data sequence estimation method and the pre-combining and blind phase estimation method, are presented. Efficient initialisation schemes that can assure the convergence to the global optimum are presented. Simulation results demonstrate the performance of the two methods in terms of the symbol error rate and the estimated accuracy of the channel parameters.

1 Introduction

For coherent communication systems with diversity branches, maximal ratio combining (MRC) is the optimal linear combining technique, in which perfect knowledge of the channel parameters of respective branches is required [l, 21. Accurate estimates of the channel parameters can be easily obtained by invoking adaptive techniques based on decision-directed methods and pilot-symbol-assisted meth- ods [3-5]. The use of pilot signals or training sequences requires an additional channel andor bandwidth, thus reducing the throughput efficiency of the network. In addi- tion, for systems requiring the dynamic selection of combining branches [6-91, these adaptive techniques can no longer be used directly since they must be reinitialised after every selection of combining branches. The number of combining branches may even be varied.

In this context of dynamic selection and combining, MRC is applied to the selected combining branches only. The problem of dynamic selection and combining can be solved indirectly by combining the signals from all possible branches [3] . However, this requires huge computations proportional to the number of branches, and its perform- ance may be even degraded at lower SNR values due to destructive contribution of noisy branches. Therefore, a class of blind methods that can provide reliable estimates of the channel parameters, irrespective of dynamic selection of combining branches, is desirable.

In this paper, we propose two blind MRC methods based on a maximum likelihood (ML) principle and finite alphabet properties (FAPs) [lo], which can provide accu- rate estimates of the channel parameters even with a small subset of data. The joint combining and data sequence esti- mation (JC-DSE) method estimates iteratively the channel parameters and the data sequence from a finite observation

0 IEE, 2001 IEE Proceedings online no. 20010176 DOL 10.1049/ipm:20010176 Paper fmt received 2nd September 1999 and in revised form 17th July 2000 The author is with the School of Electronics Engineering, Ajou University, San 5, Wonchon-Dong, Paldal-Gu, Suwon, 442-949, Korea

by an alternating projection (AP) technique. The pre- combining and blind phase estimation (PC-BPE) method combines the diversity branches by using a good estimate of the combining vector up to a phase rotation and then iteratively estimates the unknown phase and the data sequence from the combined signal [l 11. Initialisation schemes for these two methods are presented to assure the convergence to the global optimum.

2 Problem formulation

Consider a baseband digital communication system with M diversity branches. A matched-filtered output of the ith branch can be expressed as

(1) where v,{n), a,, and 0, are an additive white Gaussian noise (AWGN) with zero mean and variance of c?, a gain, and a phase at the ith branch, respectively; s(n) is a transmitted data sequence. In the sequel, independent and identically distributed noises at all the branches are assumed. In com- pact form

where x(n) = [xl(n), ..., xM(n)], v(n) = [vl(n), ..., vdn)], and c = [q, ..., cM] with c, = a,eJ@. Collecting data over N symbol periods, we can rewrite eqn. 2 in matrix form

where X = [x(l), .,., x(N>IT, V = [v(l), ..., v(N>IT, and s = [s(l), ..., s(N)IT with the superscript T denoting transpose.

The blind MRC problem requires the joint estimation of the channel parameters c and the data sequence s from the finite observation X . In this paper, we consider only sym- metric QAM schemes such as 4-QAM and 16-QAM, although the proposed methods can be extended directly to nonsymmetric modulations. Hence, all the phase estimates 8 = [e,, ..., OM], obtained from any blind estimation meth- ods may suffer the phase ambiguity of d2.

3 ML estimation

xz(n) = azeJe ts (n) + v,(n), i = I,. . . , M

.(n) = s(n)c + v(n) (2)

x = s . c + v (3)

From eqn. 3 and the AWGN assumption, we can obtain the log likelihood function as

I19 IEE Proc.-Cornr?iuri.. Vol. 148. No. 3, June 2001

N 1 Is2

lnL=const-MNloga2-- C11z(n) -s(n) .cl12

(4) 71=1

Neglecting constant terms, the problem of maximising L with respect to c and s(n), y1 = 1, ..., N reduces to the fol- lowing least-square (LS) problem:

( 5 )

where the subscript F denotes the Frobenious norm of a matrix. Since elements of s are constrained to finite alpha- bets, this problem is a nonlinear separable optimisation problem with mixed discrete and continuous variables. This problem can be solved in two steps as follows. Minimising eqn. 5 with respect to c with s fixed, since c is uncon- strained, we obtain

where the asterisk denotes the complex conjugate trans- pose. We can then obtain the ML estimate of s by substi- tuting eqn. 6 back into eqn. 5 as

c = (s*s)-%*X ( 6 )

arg min 8 \ [ ( I - S S * ( S * S ) - ~ X ~ ~ $ (7) Then, c is obtained by substituting s obtained from eqn. 7 back into eqn. 6.

The global minimisation of eqn. 7 can be obtained by an exhaustive two-dimensional search. However, the number of possible s vectors grows exponentially with both N and the number of alphabets. This exhaustive search cannot be used even for modest size problems. Hence, a class of simpler iterative methods is desirable.

4 JC-DSE method

Assuming an initial estimate 2 of the channel parameters, an unconstrained LS estimate of data sequence, is obtained by minimising eqn. 5 with respect to s with 2 fixed as j U c = X2*(22*)-l. The constrained estimate s^ is obtained by projecting d,, onto finite alphabet space as

0 = p o j ( i U c ) = p ~ o j ( X C * ( i ? C * ) - ~ ) (8) where proj() denotes the projection of the unconstrained estimate onto finite alphabet space associated with modula- tion schemes. Then, a better estimate of c is obtained by substituting j of eqn. 8 into eqn. 6. We continue this proc- ess until E and/or s^ cannot be updated any more. This method may not converge to the global optimum with an arbitrary initial estimate Eo.

We now develop an efficient initialisation scheme that can assure the convergence to the global optimum. We begin by constructing the covariance matrix as

where E{.} is the statistical expectation of { . } and p is a real constant proportional to the signal power. From eqn. 9, we note that R , has one dominant and (M - 1) equal eigenvalues, that is

Hence, c can be completely determined from the principal eigenvector of Rxx as

where a, c i and $ are a normalising constant, a phase- normalised principal eigenvector of Rxx, and an unknown phase after phase normalisation, respectively. We normalise c i so that its modulus is equal to unity, llcII*112 = 1 (ampli- tude normalisation) and the strongest element defined as

R x x = E { X * X } = p c * c + 2 1 (9)

A1 = A,,, = pa2+02 , A 2 = . . . = AM = Is2 (10)

(11) c* = 01. cg * e34

140

C , ~ ( ~ ~ ~ ~ J with inzm = arg max,[lc,(z], i = 1, ..., M] has zero- phase (phase normalisation). In practice, a good estimate of c up to a phase rotation can be obtained from the principal eigenvector of the sample covariance matrix as

1 1 N

EXx = y * x*x = - ( s*s . c*c + V*V) (12) From eqn. 12, it can be seen that (UN) . (s*s) stands for an averaged signal power over N symbol periods.

For an initial selection of a, we can assume, without loss of generality, that the gain of the strongest branch is equal to unity. In this case, an appropriate value of a is chosen as

Bo = 1 / C p ( & I " (13) Alternatively, exploiting the property of eqn. 10, a can be estimated from the eigenvalues of RxxS (i.e. AIs = AI,," > @ 2 ... 2 A,$) as

where 62 = 1/(M - 1) ' CiM2 A! and PF denotes the power normalising factor associated with modulation schemes.

Now, assuming good estimates E; and k0, we choose a set of initial estimates, = ?,,*ej@,O, i = 1, ..., ndiv}, corre- sponding to ndiv different phases, {ko, i = 1, ..., ndiv}. This set of initial phases must be sufficient to assure the conver- gence to the global optimum. After respective estimations of pairs of (e, s) with all the ndiv initial estimates, the pair that minimises the LS criterion of eqn. 5, (2, s^) is decided as the final estimates of (e, s). We summarise the JC-DSE method as follows. JC-DSE method with multiple initial estimates

1. Compute tl>* = pe - vec(Rxyy) = pe - vec(X*x> 2 Determine ho from eqn. 13 or eqn. 14 3 For i = 1, ndiv

3u $i,o = {(i - OS)/ndiv} . (n/2), k = 0

3 c k = k + l 36 c. * & 2 * . eJ%O r,O 0 ' p

Si,/< = ~roAX?i,k (ci,,cci,L)-') e;,/' = (Si,k*ii,,')-' Si,k*X

3d Repeat 3c until (2i,k, s^i,k) = (2i,k-l, QCk-J

End 3e (E, s^i) = (c;,/c S i , d

4. (2, s^) = arg min(2j,Bi) IIX- ii . til12, i = 1, ..., ndiv In the above description, pe-vec(.) denotes the principal eigenvector of (.).

Note that the main body of this method, excluding the initialisation and post selection can be derived easily from the iterative least-squares with projection (ILSP) and itera- tive least-squares with enumeration (ILSE) methods [IO]. Assuming only a single signal instead of multiple synchro- nous signals and rearranging the notations, the main repeti- tion processes of both the ILSP and ILSE methods are reduced to step 3c of the JC-DSE method. Major differ- ences are the use of the multiple systematic initial estimates and the post-selection based on the LS criterion.

The JC-DSE method requires about U((2M + 1)N) com- plex multiplications for an essential iteration, i.e. step 3c, thus about ndiv*nicnv*U((2M + 1)N) complex multiplica- tions for all the iteration processes, assuming nicnv itera- tions to the convergence. This also requires additional complex multiplications of (M2N + EM) for 8; using the power method [I21 with E iterations and ndiv*(l.SMN) for the post-selection. In the international mobile telecommuni- cations-2000 (IMT-2000) systems, only three RAKE

IEE Proc.-Co?nmun.. Vol. 148, No. 3, June 2001

branches may be suffcient to collect most of signal powers (i.e. M = 3). In that case, using ndiv = 2, nicnv = 3 (Fig. 7) and E = 4, only O(60N) complex multiplications for 4-QAM are required. These computations are not so signif- icant for most communication systems.

5 PC-BPE method

Assuming the exact e,>* and noting llcI,*112 = 1, we obtain the combined signal X , as

where a, = a . ed4 and V, [= Vel>*] is the combined AWGN vector with zero mean and covariance of dZ. Then, the blind MRC problem reduces to the pre-combined LS prob- lem as

x, = xc; = s ' a , + v, (15)

This again is a nonlinear separable optimisation problem with mixed discrete and continuous variables. Minimising eqn. 16 with respect to a, with s fixed, we obtain

&, = ( s * s ) - b * x c (17) Substituting eqn. 17 into eqn. 16, we can obtain the ML estimate of s as

arg inin I [ (I - ( S ' S ) ~ ~ . S S * ) X , ~ ~ ~ (18)

For the PC-BPE method, assuming an initial estimate &, = &e-j4, the unconstrained estimate 9, is obtained by mini- mising eqn. 16 with respect to s with kc fixed as .fu, = X,&;-'. The constrained estimate 9 is then obtained by pro- jecting 9, onto finite alphabet space as

9

Then, a better estimate of a, is obtained by substituting eqn. 19 back into eqn. 17. We continue this process until 15, and/or 9 cannot be updated any more.

This method also uses ndiv initial estimates {&c,o,i = Bo . ed'bi, i = 1, ..., ndiv}. After respective estimations of a pair of (ac, s) with all the ndiv initial estimates, the pair that minimises the pre-combined LS criterion of eqn. 16, (bC, 0), is decided as the final estimates of (ac, s). We summarise the PC-BPE method as follows: PC-BPE method with multiple initial estimates:

1. Compute ?*>* = pe - vec(Rxys) = pe - vec(X * x) 2. Determine bo from eqn. 13 or eqn. 14

4. For i = 1, ndiv 3. x, = x. 2;

4a @i,o = {(i - O.S)/ndiv) ' (rd2), k = 0 4b a . = q, . e-j@i,O C A 0

4 c k = k + l %,k = prdxc ' %$-I>

%,i,k = (Si,k*Si,k)-l

4d Repeat 4c until (ac,i,k, si,k) = (ac,i,~c-l, si,k+I)

,End 4e (&c,i, S i ) = (ac,k, i , s/J

5. (&,, O) = arg min(bc,i,ii) IlX, - &c,i . Oil2, i = 1, ..., ndiv This method requires about O(2.5IV) complex multiplica- tions for an essential iteration, i.e. step 4c, thus about ndiv*nicnv* O(2.5N) complex multiplications for all the iter- ation processes. Therefore, this method can reduce the computational complexity by a factor of about A4 as compared to that of the JC-DSE method for an essential

iteration. This also requires additional complex multiplica- tions of (MN + EM*) for i!;, MN for X,, and ndiv*N for the post-selection. As before, using M = 3, ndiv = 2, nicnv = 3 and E = 4, only O(29N) complex multiplications for 4-QAM are required.

6 Simulation results and discussions

We present several simulation results to demonstrate the performance of the proposed methods on the SER and the estimated accuracy of the channel parameters. For compar- ison, we also present the symbol error performance with the exact channel parameters and Cramer-Rao bounds (CRBs) which were derived in our work (excluded due to space constraints).

In our simulations, we consider 4-QAM and 16-QAM. Through extensive simulations, we chose 2 and 4 as appro- priate (i.e. minimum) ndiv values to assure the global convergence for 4-QAM and I6-QAM, respectively. In every experiment, a total of 1 x 106 symbols are used to obtain the SER as a function of SNRb (i.e. SNR per bit at the strongest branch). Also, the means and variances of the channel parameters are computed using the same set of symbols. We consider N = 4, 8, 16, 32, and 64 for 4-QAM, and N = 10, 16, 32 and 64 for 16-QAM. We use three diversity branches (M = 3) which have (1.0, O O ) , (0.5, 140") and (0.866, -90') pairs of the amplitudes and phases, respectively. The above phase angles give the largest phase deviation from the selected initial phases for the modula- tions considered.

For the initial selection of a, we use the scheme of eqn. 13 due to its simplicity. From the simulation results using eqn. 14 under all the above scenarios, we have found that two selection schemes do not display any recognisable differences in the SER and the accuracy of the channel parameters.

Fig. 1 shows the SERs of the JC-DSE method as a func- tion of SNRb for 4-QAM and 16-QAM. From this, we see that this method has almost the same performance as the ideal performance (using the exact parameters) with all N

0 1 2 3 L 5 6 7 8 9 1 0 1 1 1 2 SNR,dB

Symbol error rate of ihe JC-DSEmethodiis afunctwn of SNRh with

N = 8 for CQAM; N = IO for 16-QAM

Fig. 1 N as apcnmeter for 4-QAM and 16-QA .-O....

.... .... N = 32 -0-- N = 64 -X- ideal

..__ 0 .... N z 4

....A .... Nr16

141 IEE Pioc.-Commun., Vol. 148. No. 3, June 2001

considered, irrespective of modulation types. The degrada- tion with a quite small N (e.g. N = 4 for 4-QAM and N = 10 for 16-QAM) is still tolerable. The performance improves as N increases. Figs. 2 and 3 show the root- mean-squared (RMS) errors of the channel parameters for 4-QAM. From this, we see that the RMS errors for both the amplitude and phase angle are very close to the corre- sponding CRBs, thus concluding that the estimates obtained by the JC-DSE method are very efficient. However, the RMS errors of the phase estimates diverge

01 I I I 8 I I I

0 1 2 3 4 5 6 7 SNR,dB

RMS errors of the WnplinrCie es th ted by the JC-DSE method as a Fig. 2 function of SNR,, with N as a pwmeter f i r 4-QAM ..-M-- estimated ( N = 4) 4- CRB(N=4) - -0- - estimated ( N = 8) -0- CRB(N=X) -.A-- estimated ( N = 16) -A- CRB (N = 16) .-V.-. estimated ( N = 32) -V- CRB (N = 32) -+- estimated ( N = 64) -0- CRB (N = 64) ::

30

- .

01 I I I I I I I

0 1 2 3 4 5 6 7 SNR,dB

Fig.3 tion of SNRb with Nas aparmter f i r 4-QAM -M-- estimated (N = 4) -0- CRB(N-4) - -0- ~- estimated ( N = 8) - -0-- CRB ( N = 8) ..-A-.. estimated (N = 16) -A- CRB ( N = 16) -.V- estimated (N = 32) --V- CRB ( N = 32) -*-- estimated (N = 64) -0- CRB (N = 64)

142

RMS errors of phuse angle estimated by JC-DSE method as a func-

rapidly from the corresponding CRBs as SNR decreases. With smaller N, the degree of divergence gets severer.

Fig. 4 shows the SERs of the PC-BPE method as a func- tion of SNRb for 4-QAM and 16-QAM. From this, we see that this method also has almost the same performance as the ideal performance with all N considered, irrespective of modulation types. From Figs. 1 and 4 do not show any noticeable difference in the SER performance between the two methods. From simulation results, however, we have found that the JC-DSE method is slightly better than the PC-BPE method. Fig. 5 shows the RMS errors of the channel parameters obtained by the PC-BPE method.

1 0 - 6 6 - 0 'I 2 3 L 5 6 7 8 9 1 0 1 1 1 2

SNR,dB Symbol error rate of PC-BPE method as a function of SNRb with N Fig. 4

as a parameter for 4-QAM and 16-QAM .... 0 .... N = 4 .... .... ~ = 3 2 .-O- N = 8 for 4-QAM N = 10 for 16-QAM -0- N = 64 ....A .... N = 16 -X- ideal

0'30r

01 I I I I I I I 0 1 2 3 4 5 6 7

SNR,dB Fi - 5 P&PE as a f i t w n of SNRb with N ar a parameter for 4-QAM --B- estimated (N = 4) -0- CRB(N-4) - -0- - estimated (N = 8) -0- CRB(N=8) --A..- estimated (N = 16) --A- CRB ( N = 16) -V.- estimated (N = 32) -7- CRB ( N = 32) -+.- estimated (N = 64) -0- CRB (N = 64)

RMS errors of mplittude es th ted by PC-BPE method and rmd3ed

IEE Proc.-Commun., Vol. 148, No. 3, June 2001

From Fig. 5, we see that the RMS errors of the amplitude estimates are very close to the corresponding CRBs, and do not show any noticeable difference as compared to those by the JC-DSE method. From Fig. 6 (dotted lines), how- ever, the RMS errors of the phase estimates obtained by the PC-BPE method diverge largely from the correspond- ing CRBs over various values of SNR and N. This large deviation is due to the inaccuracy of the initial estimate t i . From Figs. 1-6, we note that the SER performance is not severely affected in spite of relatively large deviations in phase angles. This shows the relative importance of the

01 I I I I I I 1

0 1 2 3 4 5 6 7 SNR,dB

RMS errors of am litude esthted by PC-BPE method undmod@d Fi .6 P&PE as a fiuzctwn of Si& with N a uurmter for 4-OAM

modified PC-BPE - -m- - estimated (N = 4) 13- CRB(N-4) - -0- - estimated (N = 8) -0- CRB(N-8) --A- - estimated (N = 16) -A- CRB (N = 16) - -V- - estimated ( N = 32) -V- CRB (N = 32) - -+- - estimated (A‘ = 64) -0- CRB (N = 64)

> C

c .- > C

c .-

3

- -.- - 2

0 1 2 3 4 5 6 1

3

- -.- - 2

0 1 2 3 4 5 6 1 SNR,dB

Fig.7 Average number o lterutwm to convergence nicnv o proposed two metho& us ajhctwn of SI& with N a s aparumeter A r 4 d A L ....D.... JC-DSE (N = 4) -Z& PGBPE(N=4) ....O..- JC-DSE (N = 8) -0- PC-BPE (N = 8) -..A.... JC-DSE (N = 16) -A- PC-BPE (N = 16) ....V.... JC-DSE (N = 32) -0- PC-BPE (N = 32) ....+.-. JC-DSE (N = 64) -0- PC-BPE ( N = 64)

amplitude in the combining performance rather than the phase. Fig. 6 (dashed lines) shows the RMS errors of the phase estimates obtained by a modified PC-BPE method that performs additionally steps 3c and 3d of the JC-DSE method using the sequence estimated by the PC-BPE method. From this, we see that the RMS errors of the phase estimates by the modified method are better than those by the PC-BPE method and are almost the same as those by the JC-DSE method. Using the modified method, the SER and the RMS errors of the amplitude estimates are almost the same as those of the above two methods.

Fig. 7 shows the average nicnv using the initial estimates obtained by eqn. 13 for 4-QAM. From this, we see that nicnv increases as N increases and SNR decreases. The increased nicnv with increased N is due to the fact that the number of parameters to be updated is increased. We also see no noticeable difference between the two methods.

7 Conclusions

We have presented two iterative methods for blind MRC, based on the ML principle and the FAPs, that is, the JC- DSE and PC-BPE methods. In these, the channel parame- ters of selected diversity branches and the data sequence are estimated simultaneously. Efficient initialisation schemes that can assure the convergence to the global optimum were presented. From the simulation results, we see that the proposed methods can accurately estimate the channel parameters for MRC even with a quite small subset of data. The PC-BPE method reduces the computational complexity for an essential iteration by a factor of about M as compared to that of the JC-DSE method without notice- able performance degradation. Therefore, this class of blind MRC methods can be used effectively in a system that requires the dynamic selection of combining branches.

8 Acknowledgment

This work was supported by the Basic Research Program of the Korea Science and Engineering Foundation and by the Brain Korea 21 project.

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