Optimal MSE DFE for multicarrier communication systems

Third JEEE Signal Processing Workshop on Signal Processing Advances in Wireless Communications,Taoyuan. Taiwan. March 20-23, 2001

Optimal MSE DFE for Multicarrier Communication Systems

^~

Vinod M. Prabhakaran Dept. of Electrical Engineering

Indian Institute of Science Bangalore 560012, India Email: mpQee.iisc.ernet.in

Abstract - Block transmission s y s t e m s which consist of a precoder transmit-filterbank a n d a block receiver have been used to c o m b a t frequency selective fading in wireless channels. Several a u t h o r s have addressed t h e problem of decision feedback equalizer (DFE) receivers for s u c h systems. We a t t e m p t t h i s p r o b l e m in a gen- eral framework and o b t a i n t h e o p t i m a l solution for the m i n i m u m mean-square error (MMSE) DFEs of finite complexity. We show that several previously derived results are special cases of our general solution.

1. INTRODUCTION

Block transmission systems, which consist of a precoder t r a n s mit filterbank and a block receiver, are a t the heart of many modern communication techniques for digital audio broadcast and digital subscriber line applications. The most popular among such systems is the OFDM/DMT used in Digital Au- dio Broadcasting.

In systems using an FIR transmit filter, decision feedback equalizers (DFEs) are known to offer low bit-error rates [l].

Previous results with DFEs in block transmission schemes were obtained with some assumptions on the transmitter or receiver structures. An infinite-length DFE receiver was assumed in [2]

and only an iterative solution was obtained. In 131, the author considers a specific form for the precoder where redundancy in the form of a known block of symbols is introduced between data blocks. DFE for a special class of precoder filterbanks was derived in [4]. The block length was chosen to be at least equal to the sum of the impulse response lengths of the channel and the precoder filter. In this paper, we attempt a general frame- work for minimum mean-square error (MMSE) DFEs of finite complexity for block transmission systems. No assumptions are made on the precoder filterbank, the channel (except that it is FIR) and the block size.

Our work is closely related to 15) in which FIR MMSE DFEs are derived for systems with an FIR transmit filter. We show that our solution reduces to that in [5] when the filterbank is made of just one filter. We also show that our solution reduces to the previously known results in [4] under the conditions imposed there on the precoder filterbank.

11.

PROBLEM

STATEMENT

Consider Fig. l(a) which shows the discrete-time model of a baseband block transmission scheme with a linear equal- izer. It was shown in [6] that most of the currenly used block transmission schemes can be put in this form. It is straightforward to show [6] that Fig. l ( b ) is an equivalent model. Here s(n) = [so(n) . . . S M - l ( n ) l T j {F(n)}ij = fj(nP

+

i); ⁰ <_ i

5

P

-

1; 0

5

j

5 M -

1, H(n) =

V. Umapathi Reddy Dept. of Electrical Communication

Engineering Indian Institute of Science

Bangalore 560012, India Email: vurQece.iisc.ernet.in

(a) Block Transmission System

.in)+)

t-(

^e(n)

M X 1 I ' x M P x P M x P M x I

P X 1

(b) Equivalent Model

Figure 1: Linear Equalizer

M x M

Figure 2: Decision Feedback Equalizer

1.

L

^h(nP

-

^P

+

^{1) h(nP - P}

+

^{2) . . '} h ( n P ) h(nP) h(nP

+

¹⁾ . .

.

h(nP

+

P

-

1) h(nP - 1) h ( n P ) ' .

.

h(nP

+

P

-

2)

v ( n ) = [v(nP)

.

. . v(nP

+

P - 1)IT, and { G ( n ) } , 3 = g,(%P

-

j); 0

5

^a

5

M - 1; 0

5

^j

5 P -

^1. It may be noted that the receiver section in Fig. l(a) is in a form different from that given in 161, but the expression for G ( n ) can similarly be found.

In Fig. 2, the linear equalizer is replaced by a DFE. It is easy to show that practical implementation is possible only if we assume that B(n) = 0 , n

<

0, and B(0) is of the form B(0) = PTLP, where

L

is lower-triangular with ones alortg the diagonal and P is a permutation matrix. The order in which the decisions are made for a block of data is determined by the permutation matrix

P.

We will also make the following finite complexity assumption: the feed-forward filter, W ( n ) = 0 , n e { - ( N f - l ) , . . . , O } andthefeedbackfilter,B(n) =1O,n@

Let A(n) = H(n)

*

F(n) = xT'-,H(n - k)F(k). We will assume that A(n) = 0 , n @ (0,. . .

,

^{v } . This} amounts to (0,. .

',

^{N b } .}

A

0-7803-6720-0/0 1/$10.000200 1

lEEE

138

assuming that the transmit filters and the channel are causal

The parameters A (0

5

A

5

Nr

+

^U

-

^{1) and 6 (0}

5

6

5

mmSisM-1 {length ^Of fi(n))) then = (Lf -k&

-

l)jP

-

1.

M

-

1) together determine the delay of the feed-forward filter.

We define

1))

. .

. s H ( n

-

u)lH, y(n)

2

[ y H ( n

+

( N r

-

1)) From (1) and the above definitions, we can write

.

.

yH(n)lH, FIR. If the channel impulse response length is Lh, and Lf = and

s

= A ( N ~

-

1 +

-

N ~ ) M - AM

-

^6.2

Y(n) =

A+) +

³⁽ⁿ⁾ ( 5 ) where

s6(n) = [Sa(.) s6+l(n) ^{" '} s M - l ( n ) sO(n-1) ". s6-l(n-1)]T

A(0) A(l)

...

A(v) 0

...

We also assume that s(n) and v(n) are zero-mean and uncorre- 0 A(0) A ( l )

...

A(v)

...

lated vector random Drocesses.

A2

Referring to Fig. 2, we have

(') and y(n) = A(n)

*

s ( n )

+

^v(n)

Our goal here is to obtain the optimal solution for the DFE. +(n)

e

[ v H ( n

+ (Nf -

¹⁾⁾

. . .

vH(n)lH Though the criterion of optimization should be the minimization

'of bit-error rate, for reasons of tractability, we minimise the

arithmetic mean-square error (MSE) defined as TJ = E(eH(n)e(n)) = tr(E(e(n)eH(n)) Substituting (4) and ( 5 ) in (2), we get

q

2

E(eH(n)e(n)) where A

A where ^ai = E(i(n )GH( n)), R?j,

2

E(y(n)Y*(n)) =

E ( 3 ( n ) C H ( n ) ) . We will as- Taking the gradient of q with respect

to

^W and equating it e(n) = s6(n -k

Nf - -

A)

-

sa(n -k N f

-

- A) (3)

A & , , A H +

&+, fi9 e

E ( & ( ~ ) ~ H ( ~ ) ) = % * A H ,

b,,

= A

Thus, the problem is to choose W(n), B(n), A, 6, and P so as to minimize q

E(f(n)sH(n)) = RZ, and Rcc

sume that these correlation matrices are non-singular.

qmin = A min E(eH(n)e(n)) to the zero matrix, we get the optimum W

W(n),B(n),A,&P

Wept

= R$R?iBopt (7)

w.1 ubstituting the optimal feedforward filter of (7) in (6) and

B f H ( n ) - I6(n)

4 I 2

using the matrix inversion lemma, we can show that the'MMSE for a given B is

Figure 3: Decision Feedback Equalizer ^- A n equivalent form

It is easy to show that Fig. 2 is equivalent to Fig. 3 if B'(n) = P B ( n ) P T . P is a permutation matrix and B'(0) is lower-triangular with qnes along the diagonal. We can interpret this figure as follows. The permutation matrix P decides the order in which symbol decisions are made in each block by per- muting the sub-channels. The feedback loop makes decisions in order - i.e., the higher indexed decisions are made use of in making decisions of the lower indexed channels. PT restores the order. P is also a parameter t o be optimized because, in general, the choice of the order in which decisions are made will affect the MSE. When P = I and 6 = 0, the problem reduces to one of the problems solved in [9] in the context of multi-user communication'.

111. OPTIMAL

SOLUTION

Assuming that the past decisions are correct, we can express (3) as

e(n) = BHi(n)

-

W H y ( n ) ₍₄₎ where B [OMxAM+6 BH(0) B H ( l ) . . . BH(Nb) O M X S ] ~ , W

e

[ W H ( - ( N f

-

¹⁾⁾

...

WH(0)lH, S(n)

2

[ s H ( n + ( N f - '[9] appeared after this paper was submitted. Revisions were made to include connections with this important reference.

.

@ denotes the Kronecker tensor product. Note that R ~ , a , p is Hermitian.

Our objective is to find B such that (10) is minimized. B'(0) is restricted t o be lower-triangular with ones along the diagonal.

The solution is given by the following theorem.

Theorem 1 (Modified Linear Vector P r e d i c t o r ) If z(n) is a non-deterministic

[a]

zero-mean M x 1 vector random PTO-

cess, then the optimal predictor A ( n ) , such that A(n) = 0 , n

g'

(0,.

.

. , N} and A(0) is lower-triangular with ones along the di- agonal, which minimizes

2 A negative value for S indicates that the feedback filter is longer than necessary. In that case, we redefine Nb to remove the redundant terms in B(n) thereby making S = 0.

0-7803-6720-0/01/$10.0002001 IEEE 139

where ~ ( n ) = A H ( n )

*

z(n), i s where A = [AH(0)

. .

. A H ( N ) I H and making the following iden- tifications:

N = Nb

[ A H ( 0 ) . . . A H ( N b ) l H =

B

Here

The final equation is possible since R A , ~ , P is Hermitian. w e make the assumption that R A , ~ , P and R ~ : N ~ , ~ : N ~ are positive- definite. Then, the optimum B for a fixed A, 6, and P is3 RNN

: l

R ~ : N , ~ : N = :

(

^{R i l}

:::

R ~ : N , o = [RE .

.

. and Rij = E(z(n - i ) z H ( n - j)). L

)

^{Laf6,P ( 1 5)}

is given by the Cholesky decomposition IM

fi0 - R F ~ , ~ R ~ : & , ~ + , R ~ : N . O = L ~ D L (12)

Also

Smin = tr(D) Proof: We have

cRkjAopt(j) = 0, 1

5

k

5

N

where we have used the fact that ^R,j = Rg, 0

5

i

5

N , 0 5 j

5

N . We can put this in the form

R I : N , ~ : N ^{) : (}

(

^{: :}

)

⁼ -Ri:N,oAopt(0)

Since ~ ( n ) is non-deterministic, R ~ : N , ~ : N is positive-definite [8]

and hence non-singular. So we have

j=O

( )

^{= -R-'} l:N,l:NR1:N,OAoPt(0)

Using the optimum setting for A ( l ) ,

.

.

.

, A ( N ) in (13),

C

= tr(AH(0)(Roo - R ~ N , ~ R ~ : ~ I ~ : N R ~ : N , o ) A ( O ) ) (14)

where LA,6,Fb is given by the Cholesky decomposition b o - R~Nb,OR,~,,i:~,R1:N,,O = L E , ~ , P D A , ~ , P E A , ~ , P 1116) Also

7]A,6,P = t r ( D ~ , 6 , p ) (17)

min tr(Da,a,p) (18)

(13) Now, we have to solve for (Aopt, boptr Po,t) which minimizes

7]A,6,P

OgAM+dI(Nf-l+v)M P Vmin =

It appears that there is no closed-form expression possible for the optimum settings of these parameters. For a given P, (18) suggests an exhaustive search for the optimum value of A and 6.

Since even for small values of M , the number of possible P's is large ( M ! ) , an exhaustive search for the optimal P may not be possible. A heuristic approach can be used to get a reasonable P. For instance, in multicarrier systems, if the inter-block in- terference in not excessive, estimates of the sub-channel SNRs, if available, can be used as a criterion for choosing the order in which decisions are made within a block. We may choose to make decisions on the strongest sub-channel first, followed by the next strongest, and so on.

If an exhaustive search is to be carried out for P, we need only calculate the R o o - R~Nb,OR1:~b,l:NbR1:Nb,O matrix for P = I.

We can show that for other values of P it can be obtained by pre-multiplying the above matrix by P and post-multiplying by PT.

If Nb = v , the search for optimal A and 6 for a given P involves performing only one Cholesky decomposition (that of

IN^+"

^{8 P)(RG1}

+

AHRGiA)-l(I~f+v €9 P')). This can be shown along the lines of a similar result in [5].

The optimal feedback filter Bopt is given by (15) when the Since z(n) is non-deterministic, the (N+l) M (N+l)M ^matrix optimal settings are subtituted. horn (7), the optimum feed-

& _R'Nio

)

is positive definite [8]. Hence, the Schur forward filter is

(

R ~ : N . o R ~ : N , ~ : N

complement of this matrix with respect to R ~ : N , ~ : N , namely

wept

= ( A R G i A H

+

Fks)-'A&iBopt (19)

i-

R:N,OR;h,l:NR1:~,~

is also positivedefinite. So the Cholesky factorization in (12) _exists. It is now easy to see that Aopt(0) is L-'. Substituting in (14),

where

B o p t = [OMxAOptM+6 ( I N b + 1 8 P o p t ) B ~ t ( I N b + 1 ~ p T p t ) O l d ~ S . , t ] ~

and Sopt = ( N f

-

1

+

^U

-

Nb

-

Aopt)M

-

6,t.

Cmin = tr( D)

The MMSE is given by (17) with the optimal settings s u b Fast algorithms for performing the Cholesky decompositions, stituted.

The optimal B can now be written by rewriting (13) as which exploit the structure in the matrices, appear in [9].

0-7803-6720-O/0

1/$1 O.OOQZO0 1 IEEE

3The subscript indicates the dependence of this solution ^OIL A, 6, and P.

1 40

Iv. DISCUSSION OF SOME SPECIAL CASES

MMSE Linear Equalizer

We can derive the MMSE linear equalizer by setting B ( n ) = Ib(n). We can choose P = I without any loss of generality.

B(n) =II5(n)

=+

B = E A (20) where EA = [0 . . . 0 I 0

.

. . OIT, with the M x M identity ma- trix being positioned after A zero matrices, each of size M x M . The optimum linear equalizer for a given A can then be written from (7) as

W,H =

( q i ~ ~ ~ ~ ~ ) ~

(21)

= E ~ % ~ A ~ ( A R ~ ~ A ~

+

R++)-’ (22) and the corresponding MSE is

VA =

~ T ( E , H & ~ E ~ -

W H ~ j , j , W ) (23) When s ( n ) is white, this reduces to the expressions derived in 171 .4

A Single FIR Filter Precoder (A4 = 1)

When A4 = 1, the transmitter is simply an FIR filter. Now the feed-forward filter w(n) and feedback filter b(n) are vectors.

We will indicate this by using lower-case letters. Clearly, I5 = 0 and P = [l].

Let the Nb x Nb matrix R and the Nb x 1 vector r be sub- matrices of the (Nb

+

¹⁾ x (Nb

+

1) matrix RA defined in ( l l ) ,

From ( E ) , the optimum feedback. filter for a given A is (24) The optimum feed-forward filter, for a fixed A, is then given by (7)

+iff = b:&nAH(kRgnAH

+

R+c)-’

BA = Roo - rhR-’r

( 2 5 )

(26) and the corresponding MSE by

Using the identity R,’ =

( R o o

-

rHR-’r)-’ (rHR-’r - &o)-’rHR-’

( R - r&o-’rH)-’

(

(r&O-’rH - R)-’r&o-’

it can be shown that

This is precisely the solution obtained in [5].

41n [7], the expressions were derived for the optimum discrete-time fixed order FIR linear equalizer minimising the MSE in the presence of near- and far-end crosstalk.

.

0-7803-6720-0/0 1 /$10.000200 1

IEEE

141

Trailing-zeros Case

In [4], it is assumed that the precoder filters are such that P = Lh

+ Lf -

1 which gives v = 0. Then, the best choice of the feed-forward and feedback lengths is

Nf

= 1 and Nb = 0, and consequently the only choice of

A

and 6 are zero. For simplicity we choose P = I.

Now from ( 1 l) ,

RA =

(K;’ +

AH(0)&-~A(O))-’ ₍₂₉₎ where

as

= E ( s ( n ) s H ( n ) ) , and Rvv = E(v(n)vH(n)). So, RA is M x M which implies that &O = RA. Then (16) becomes

~0~ = (~:l+ AH(o)&-:A(o))-l = L ~ D ~ L ~ ₍₃₀₎

The optimum feedback and feed-forward filters from (15) and (7) respectively, are

Bopt = LO1 (31)

Wopt =

EY,‘R,.B,~

(32)

Vman = tr(D0) (33)

and the MMSE is

The expressions (31) to (33) match the results in (41.

V. CONCLUSIONS

We have considered the general case of decision feedback equal- izers of finite complexity for block transmission systems and o b tained the optimal solution. We have shown that several results derived elsewhere are special cases of our general solution.

ACKNOWLEDGMENTS

The authors would like to thank Prof. Thomas Kailath for a helpful discussion during the preparation of the paper.

REFERENCES

N. AI-Dhahir and J . M. Cioffi, “Block transmission over disper- sive channels: Transmit filter optimization and realization, and MMSEDFE receiver performance”, IEEE ^{h n s .} Inform. ^The- ory, vol. 42, pp. 137-160, Jan. 1996.

J . Yang and S. Roy, “Joint transmitter-receiver optimization for multi-input multi-output systems with decision feedback,” IEEE h n s . Inform. Theory vol. 40, pp. 1334-1347, Sep. 1994.

G. K. Kaleh, “Channel equalization for block transmission sys- tems,” IEEE J. Select. Areas Commun. vol 13. pp. 110-121, J a n 1995.

A. Stamoulis, G.B. Giannakis, and A. Scaglione, “Self-recovering transceivers for block transmissions: filterbank precoders and decision-feedback equalizers,” Proc. of the 2nd IEEE-SP Work- shop on Signal PTOC. Advances i n Wireless Communications, An- napolis, MD, May 1999.

N. AI-Dhahir and J . M. Cioffi, “MMSE decision-feedback equal- izers: Finite-length results,” IEEE Bans. Inform. Theory, vol.

A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Redundant filterbank precoders and equalizers Part I: Unification and optimal designs,” IEEE Trans. Sig. PTOC., vol. 47, pp. 1988-2006, Jul.

1999.

M. L. Honig, P. Crespo, and K. Steiglitz, “Suppression of near- and far-end crosstalk by linear pre- and post-filtering,” IEEE J.

Select. A r m Commun., vol. 10, pp. 614-629, Apr. 1992.

A. Gersho and R. M. Gray, Vector Quantization and Signal Com- pression, Kluwer Academic Publishers, Boston, 1992

N. AI-Dhahir and A. H. Sayed, “The Finite-Length Multi-Input Multi-Output MMSEDFE,” IEEE h n s . Sig. PTOC., vol. 48, pp. 2921-2936, Oct. 2000.

41, pp. 961-975, Jul. 1995.