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
V. Umapathi Reddy
Dept. of Electrical Communication
Engineering
Indian Institute of Science
Bangalore 560012, India
Email: vurQece.iisc.ernet.in
Abstract - Block transmission systems which consist
of a precoder transmit-filterbank a n d a block receiver
have been used to combat 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 such systems. We a t t e m p t this problem in a general framework and o b t a i n t h e optimal solution for the
minimum mean-square error (MMSE) DFEs of finite
complexity. We show that several previously derived
results are special cases of our general solution.
(a) Block Transmission System
t-(
.in)+)
1. INTRODUCTION
M X 1
Block transmission systems, which consist of a precoder t r a n s
mit filterbank and a block receiver, are at 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 Audio 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 framework 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 equalizer. 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) =
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~
I'xM
e(n)
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
L
h(nP)
h(nP - 1)
+
+
. . . h(nP P
' . . h(nP + P
h(nP 1)
h(nP)
- 1)
- 2)
1.
h(nP - P + 1) h(nP - P + 2) . . '
h(nP)
v ( n ) = [v(nP) . . . v(nP + P - 1)IT,and {G(n)},3= g,(%Pj); 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 {-(Nf-l),...,O} andthefeedbackfilter,B(n) =1O,n@
-
(0,. . ', N b } .
A
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
assuming that the transmit filters and the channel are causal
FIR. If the channel impulse response length is Lh, and Lf =
mmSisM-1 {length Of fi(n))) then = (Lf -k& - l)jP- 1.
The parameters A (0 5 A 5 Nr
U - 1) and 6 (0 5 6 5
M - 1) together determine the delay of the feed-forward filter.
We define
+
s6(n)= [Sa(.)
s6+l(n) " '
sM-l(n)
sO(n-1) ". s6-l(n-1)]T
We also assume that s(n) and v(n)are zero-mean and uncorrelated vector random Drocesses.
Referring to Fig. 2, we have
+
y(n)= A(n)* s(n) v(n)
(')
Our goal here is to obtain the optimal solution for the DFE.
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
1)) . . . s H ( n- u)lH, y(n) 2 [yH(n+ ( N r - 1)) . . yH(n)lH,
and s =
A ( N~1 + -N ~ ) M
- AM - 6.2
From (1) and the above definitions, we can write
Y(n)= A+)
+ 3(n)
(5)
where
A(0) A(l)
... A(v) 0
0
A(0) A ( l ) ... A(v)
A2
and
+(n)
...
...
e [ v H ( n+ (Nf- 1)) ... vH(n)lH
Substituting (4) and ( 5 ) in (2), we get
TJ
= E(eH(n)e(n)) = tr(E(e(n)eH(n))
q 2 E(eH(n)e(n))
where
A
e(n) = s6(n -k Nf
-
- A) - sa(n
-k Nf
-
where
- A)
(3)
Thus, the problem is to choose W(n), B(n), A, 6, and P so as
to minimize q
A
qmin =
min
E(eH(n)e(n))
a
i
A
= E(i(n)GH(n)),R?j,
e
2
E(y(n)Y*(n)) =
b,,=
A
E ( 3 ( n ) C H ( n ) )We
. will asE(f(n)sH(n)) = RZ, and Rcc
sume that these correlation matrices are non-singular.
Taking the gradient of q with respect
W and equating it
to the zero matrix, we get the optimum W
A&,,AH
+
&+, fi9
E ( & ( ~ ) ~ H ( ~=) )% * A H ,
to
W(n),B(n),A,&P
Wept = R$R?iBopt
w.1
4I2
- I6(n)
BfH(n)
Figure 3: Decision Feedback Equalizer
form
-
An equivalent
It is easy to show that Fig. 2 is equivalent to Fig. 3 if
B'(n) = PB(n)PT. 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 permuting 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 )
.
(7)
ubstituting the optimal feedforward filter of (7) in (6) and
using the matrix inversion lemma, we can show that the'MMSE
for a given B is
(4)
where B [OMxAM+6 BH(0) B H ( l ) . . . BH(Nb) O M X S ] ~ ,
W [ W H ( - ( N f - 1)) ... WH(0)lH,S(n) 2 [ s H ( n +( N f -
e
'[9] appeared after this paper was submitted. Revisions were made
to include connections with this important reference.
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@ 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 Predictor) If z(n)
is a non-deterministic [a] zero-mean M x 1 vector random PTOcess, 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 diagonal, 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.
where A = [AH(0). . . A H ( N ) I Hand making the following identifications:
N = Nb
where ~ ( n=)A H ( n *) z(n),is
[AH(0)...AH(Nb)l=
HB
Here
(
R~:N,~
=: N
:
Ril
:::
:l
RNN
and Rij = E(z(n - i ) z H ( n- j)).L
R ~ : N=
, o[RE . . .
is given by the Cholesky decomposition
fi0
- R F ~ , ~ R ~ : & , ~ + , R ~ :=
N .LO~ D L
The final equation is possible since R A , ~ ,isP Hermitian. w e
P R ~ : N ~ are
, ~ positive:N~
make the assumption that R A , ~ ,and
definite. Then, the optimum B for a fixed A, 6, and P is3
IM
)
Laf6,P
(15)
(12)
where LA,6,Fb is given by the Cholesky decomposition
Also
Smin
= tr(D)
b o - R~Nb,OR,~,,i:~,R1:N,,O
= L E , ~ , P D A , ~ , P E A , ~1116)
,P
Also
Proof: We have
7]A,6,P
= tr(D~,6,p)
(17)
Now, we have to solve for (Aopt,boptrPo,t)which minimizes
(13)
7]A,6,P
Vmin =
min
OgAM+dI(Nf-l+v)M
tr(Da,a,p)
(18)
P
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
cRkjAopt(j) = 0, 1 5 k 5 N
j=O
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 interference in not excessive, estimates of the sub-channel SNRs,
R I): N
: ,( ~:: N
:
= -Ri:N,oAopt(0)
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
Since ~ ( nis) non-deterministic, R ~ : N , ~is: positive-definite
N
[8] the next strongest, and so on.
and hence non-singular. So we have
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
= -R-'l:N,l:NR1:N,OAoPt(0)
pre-multiplyingthe above matrix by P and post-multiplying by
where we have used the fact that R,j = Rg, 0
j 5 N . We can put this in the form
5 i 5 N, 0 5
( )
( )
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
C = tr(AH(0)(Roo- R ~ N , ~ R ~ : ~ I ~ : N R ~ : N ,(14)
o ) A (shown
O ) ) along the lines of a similar result in [5].
The optimal feedback filter Boptis 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.oR ~ : N , ~ : N
complement of this matrix with respect to R ~ : N , ~
namely
:N,
wept = (ARGiAH 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 where
Aopt(0)is L-'. Substituting in (14),
B o p t = [OMxAOptM+6 ( I N b + 1 8 P o p t ) B ~ t ( I N b + 1 ~ p T p tO
) ld~S.,t]~
Using the optimum setting for A(l), . . . , A ( N ) in (13),
(
+
)
+
Cmin
+
and Sopt= ( N f - 1 U - Nb - Aopt)M- 6,t.
The MMSE is given by (17) with the optimal settings s u b
stituted.
Fast algorithms for performing the Cholesky decompositions,
which exploit the structure in the matrices, appear in [9].
= tr(D)
The optimal B can now be written by rewriting (13) as
3The subscript indicates the dependence of this solution OIL A, 6,
and P.
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Iv. DISCUSSION
OF
Trailing-zeros Case
SOME SPECIAL CASES
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)
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 ( 1l ) ,
+ -
MMSE Linear Equalizer
=+
B =EA
(20)
where EA = [0 . . . 0 I 0 . . . OIT, with the M x M identity matrix 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
RA = (K;’
(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 ~ (30)
L ~
( q i ~ ~ ~ ~ ~(21) )The~optimum feedback and feed-forward filters from (15) and
W,H =
+
E ~ % ~ A ~ ( A R
R++)-’
~ ~ A ~ (22)
=
(7) respectively, are
and the corresponding MSE is
VA =
+ AH(0)&-~A(O))-’
~ T ( E , H & ~ E ~- WH~j,j,W)
Bopt = LO1
Wopt = EY,‘R,.B,~
(23)
When s(n) is white, this reduces to the expressions derived in
171.4
A Single FIR Filter Precoder (A4= 1)
(31)
(32)
and the MMSE is
Vman = tr(D0)
The expressions (31) to (33) match the results in (41.
(33)
V. CONCLUSIONS
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 submatrices of the (Nb 1) x (Nb + 1) matrix RA defined in ( l l ) ,
+
We have considered the general case of decision feedback equalizers 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
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)-’
(25)
and the corresponding MSE by
BA
= Roo - rhR-’r
(26)
Using the identity
R,’ =
( (Roo -
rHR-’r)-’
(r&O-’rH - R)-’r&o-’
(rHR-’r - &o)-’rHR-’
( R - r&o-’rH)-’
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.
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