2004 IEEE Sensor Array and Multichannel Signal Processing Workshop
E'ILTERBANK PRECODING FOR MIMO FREQUENCY SELECTIVE CHANNELS:
MINIMUM REDUNDANCY AND EQUALIZER DESIGN
Hjaya krishna. A, K. 1.! S.Hari
vkrishna@protocol.ece.iisc.ernet.in, hari@ece.iisc.ernet.in,
Dept. of ECE, Indian Institute of Science, Bangalore - 560012.
ABSTRACT
This paper deals with the problem of designing zero forcing
finite impulse response (ZF-FIR) equalizers for multiple input multiple output (MIMO) FIR channels, in a polynomial
matrix framework. When channel knowledge is available
at the transmitter, a precoding operation, which introduces
redundancy, can he performed to enable FIR equalization
at the receiver. We derive the expression for the minimum
rcdundancy required to render an arbitrary MlMO channel
matrix polynomially invertible. The non-uniqueness of the
FIR inverse can be uscd to design equalizers based on different criteria such as noise minimization, low delay and low
complexity. We provide ;L solution for the equalizcr which
minimizes the output noise powcr. Simulation results are
provided to demonstrate the effectiveness of the proposed
design.
On the other hand, methods which perform FIR equalization at the receiver [3][5]in general require n/r > N and the
channel matrix to be irreducible. 'A polynomial matrix H ( z )
is polynomially invertible if and only if it is irreducible, i.e,
it has full rank for all z # 0, including z = 00.
We present a filterbank precoding framework which enables FIR equalization for an arbitrary channel (which may
even be singular) of any dimension. At the transmitter, the
sourccc signal vector is processed by an FIR prccoder before being transmitted. The precoder is chosen in such a
way that the precoded channel is FIR invertible. The minimum redundancy required to accomplish FIR invertibility
is derived. It is shown that the design freedom available for
the precodcr can be used to aid in the design of the noiseminimizing equalizer.
2. THE PRECODING FRAMEWORK
1. INTRODUCTION
Multiple antenna communication systems have attracted a
lot of attention due to the substantial increase in data rates
that they facilitate. But in the presence of intersymbol interference (ISI), equalization becomes a challenging task.
Space-time processing methods [ 1][2] tackle IS1 by converting the frequcncy sclcctivc MIMO channel into a set olparallel flat tiding channels. On the other hand, FIR equalizer
designs [ 3 ] view the FIR channel as a polynomial matrix in
the delay operator 2 - l .
Availability of channel knowledge at the transmitter can
also be exploited to combat IS1 and to improve system performance. Space-time precoding methods [ 11 use the channel information to allocate power along the singular modcs
of the block-Sylvester matrix corresponding to the channel.
But because of the implicit IJR nature of the decoding operation, there is a trade-off between the achievable rate and
the performance of thc system. In [4],the knowledge of the
11.1x N channel H ( z ) is utilized to design a pre-equalizer,
i.e, a polynomial matrix T ( z )such that H ( z ) T ( z )= I , at
the transmitter. But this method requires hd 5 N , and assumes that the channel coefficient matrices are orthogonal.
0-7803-8545-4/04/$20.00
02004 IEEE
Consider the MIMO signal model given by
L1/-1
y(*n) =
H(7L
- k j s ( k ) +.(en)
(1)
k=O
whcre x ( k > i s the N-length input vector, y ( n ) is the Mlength received signal vector, N ( k ) , k = O , l , ..., LH - 1
is the time domain representation of the M x N frequency
selcctive MIMO channel, and v ( n ) is the M-length noise
vector. In the absence of noise, we can write
Y ( z )= H(zjX(z)
1
(2)
in the z-domain. Now, an FIR equalizer, (left inverse) for
H ( z ) exists if and only if il.12 N and W ( z ) is irreducible.
Thereforc, given an arbitrary H ( z ) ,we wish to find an N x
I - ( K 5 N ) precoder matrix G ( z ) , so that the 11.1 x Ir'
composite channel matrix C(z) = W ( ; ) G ( z ) has an FIR
left inverse. N - K is the redundancy introduced by the
precoder. If the channel matrix N(z ) is already FIR invertible, then we can chose G ( z ) = I (N = K ) , so that no
redundancy is introduced. The question we address here is:
what is the maximum K (minimum redundancy N - IT)
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for which G(z) is irreducible?. Before we tackle this question; we will look at Some properties of the Smith form of a
polynomial matrix.
We can always choose
2.1, Smith form
so that g i ( z ) I c , ( t ) , 1 5 i 5 K .
Since the minimum rank of H ( z ) is T,
Given a p x q polynomial matrix A(z) with normal rank T ,
T 5 min(p, q), we can write [6]
A(z) = E ( z ) B ( z ) F ( z )
(3)
where E ( z ) and F ( z ) are unimodular matrices (square irreducible matrices) of sizesp x p and g x q respectively. B ( z )
is a y x 4 matrix of the form
B ( z ) = diay(n~(zj,cr2(z),
. . . ,a,(z), 0,. . - 0 )
such that c y i ( z ) I ni+l(z)
(i.e, a i ( z ) divides cxi+l(z)).The
m o n k polynomials cyi(z) arc called thc kvnrilrnt polyrromiUES of A ( z ) . Wc say that A ( z ) is equivalent to B ( r ) and
write A ( z ) B ( z ) .
Since E ( z ) and F ( z ) are unimodular matrices, it follows from Q. 3 that rank[A(z)] = rank[B(z)] Vz. Thercfore, if the minimum rank of A(B) ovcr all values o f z is r,,
it can be shown that q ( z ) = Q ~ ( z =
) . I - = cyTnL(z)= 1.
If an 'ri x k (n > k ) matrix P ( z ) is irreducible, then its
Smith form is
-
P(z'
[
IkXk
O(n-k)xk
1
hl(Z) =
(4)
.... = h,,,*(z)= 1
and
hr_+1(z), "'.', hr(z)
where W ( z )is an arbitrary IC x Ii' unimodular matrix.
Thcn, we have
r
IF,, X T r r ~
O(n.1-
T,,
T,,,
) X r,,
1
If A' 5 T , , ~, we have
irrcducible.
be the miiiimunr rank of the given M x
H ( z ) . An N x K precoder G ( z ) which
retzders the coniposite channel nlntrix C(Z) = H ( z ) G ( z )
poiynomially irivertible exists fatidonly (f K 5 rTn.
Theorem 2 Let
N FIR charinel
-
Now, since r m is the minimum rank of H(zz),the submatrix
formed by the first T , columns
~
of D ( z ) is
Therefore, C ( z )
2.2. Minimum Redundancy
1
Now we will prove the if part.
Let H ( z ) = U ( z ) D ( z ) V ( z be
) the Smith form decomposition of the channel matrix. Choose
Proof follows directly from the Binet-Cauchy formula
[61.
#
where T is the normal rank of H ( z ) . Hence, hi(t) I ~ ( z )
for 1 5 i 5 K only if K 5 T,. Therefore, C ( z ) is irreducible only if K 5 T ~ .
The following result is stated in [ 6 ] .
Theorem 1 Let Pi(z),,i = I,2, ~ L I W invciriunt polytzornid s {ai~(~),
aia(t),....,~+~(.z)], i = 1,2, where ~i is the
normal rank o f e ( z ) . Let P ( z ) = P1(z)P~(z)
have nornial
rank T , with ivivarianfpol~~ronzials
{cq ( z ) , ., cyT(z)}.. Then,
for 1 5 k 5 T a i d i = 1,2, crik(z) I a k ( z ) .
we have
-
I K x hO(A1-K) x K
I
. Therefore, C ( z ) is
a
This result can be seen as a generalization to the MIMO
case of the minimum redundancy results obtained for the
multicarrier modulation case [7J.
2.3. Minimum rank and channel zeros
Proofi
We will first prove thc only i f part.
Let { c i ( z ) } {hi(.)}
,
and (.gi(z)) denote the invariant
polynomials ofC(z), H ( z )and G(a)respectivcly. Suppose
C ( z )is irreducible. Then
'"' - [
IKXK
O(M-K)xK
I
Given an M x N channel H ( z ) with normal rank T , let
A(z) denote the g.c.d of the minors of order T of H ( z ) (if
H ( z ) is square and has full normal rank, then A(z) is its
determinant). Expanding A(;) in term5 of its factors, we
have
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The rank of H ( z ) drops below the normal rank T only at
the channel zeros 6,. z = 1.2,. . . , s. We now relate the
minimum rank of H ( t ) to the maximum among the multiplicities of the channel zeros.
Corollary 1 Let I- and r, be the normal rank and the
mininziirn rank o f f H ( z ) respectively. Let A(z) be as in Eq.
5. Then we have r,,, 2 7’ - maxl{r,}
A random non-square polynomial matrix is irreducible
with probability 1 [S]. Therefore, given an arbitrary M x N
channel H ( z ) with ~$1# iV, filterbank precoding utilizes
all the degrees of freedom i.e, the throughput is min(M, N )
symbols per channel use. If AI < N , then the channel
can be completely pre-equalized i.e, H ( z ) G ( z ) = 1. If
h1 > N , FIR equalization can be accomplished without
precoding. Compared to space-time processing methods
[1][2], our scheme provides better throughput at a much
lesser complexity.
When M
N , we need a redundancy of 1 i.e, K =
N - 1. For large N , the rake of our scheme is in the same
range as that of the space-time precoding method [ I ] .
3. EQUALIZER DESIGN
cz=o
.-I
F ( L F - ~ )]
Theorem3 r f C is tall and furl rattk arid the noise v ( n )
in Eq. I is white, the equalizer which minimizes the output
noise power is given by F = J(CHC)-’CH.
Proo$
The output of the equalizer can be written as
Y ( z )= F ( Z ) C ( Z ) X ( Z )f F ( z ) V ( z )
(8)
Using Eq. 6 and Eq. 7, we can rewrite Eq. 8 as
Y(.)
= FCX(n)-k
rn({2)
(9)
where X ( n ) and U(71)are the blocked versions of z ( n )and
v ( n ) , of dimensions K ( L c LF - 1) x 1 and M L p x 1
respectively. Thc covariance matrix of the noise component
of y ( a )is given by
+
The design of thc equalizer can be based on different criteria such as low delay, low complexity and noise minimization. The polynomial matrix framework enables one to utilize many of the results in systems theory literature regarding the construction of inverse systems [9]. In this section,
wc derive the noise-minimizing equalizer, using the design
freedom available at thc precoder.
Given the M x K irreduciblc precodcd channel C ( z ) ,
the equalizer design problem is to find a polynomial matrix
L ,., - 1
F(r) =
F ( i ) z p i such that F ( z ) C ( z ) = I . This
can also be written as FC = 3,where
F = \ F ( 0 ) F(1)
rank if C(2 ) is column-reduced. Any polynomial matrix can
be brought to colurnn-reduced form by multiplying it to the
right by a suitable unimodular matrix [6]. From the proof of
theorem 2. it follows that we have the freedom of choosing a
K x A” unimodular factor of G(z). We can choose that unimodular matrix such that C ( z ) becomes column-reduced,
resulting in a full rank C. In the following section, superscript H denotes Herrnitian transpose.
M y = E[FUV”FH]= F F H
(10)
in view of the white noise assumption. The ooise-minimizing
equalizer is obtained by minimizing t r [ F F Hsubjcct
J
to the
constraint .FC = J . Using the Lagrange multiplier method,
wc get
2 3 - CCH = 0
(1 1)
where C is the K x K ( L c + L F - 1) Lagrange multiplier
matrix. From Eq. 11, we have
(6)
23c
and 3 = [ I 0 . . . 0 1. 7 and J are K x M L F ,and
C is M L F x K ( L c L F - 1).C is tall if L F 2 -.K ( L r - 1 )
C is full rank if and only if [ lo]
= LCHC
(12)
From Eq. 11 and Eq. 13, we have
+
i
C ( z ) is irreducible and column-reduced.
L F 2 maxlSj5nr-K v$ - 1, where v$ are the Kronecker indices of the dual space of span(C(z)).
The degrees of all the columns of C ( z )are equal.
The third restriction can be overcome by deleting the null
columns of C as explained in [ 3 ] . Therefore, given an irreducible C ( z ) and with L F chosen large enough, C is full
4. SIMULATION RESULTS
Simulations were carried out with 3 x 2 channel coefficient
matrices Hi with independent Gaussian distributed complex
coefficients (Rayleigh fading) with an exponential power
delay profile. LH was chosen to be 4.Source signals were
generated from a BPSK constellation. The noise generated
1
I
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the minimum redundancy required to enable FIR equalization of an arbitrary channel. We have shown that our scheme
achieves better rates than block-based schemes Iike spacetime precoding, with lesser complexity.
6. REFERENCES
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I
25
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SNU ldB1
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we implemented the space-time (ST) prccoding and dccoding [ 1, Lemma 31.
Since Hi wcre randomly generatcd and H,-,,,-l was full
rank, the channel was both irreducible and column reduced.
Hence, noprecoding was pcrforrned and the minimum noise
FIR (MN-FIR) equalizer was used. ST precoding involved
transmission along the eigcnmodes of ‘H“7-l where ‘H is the
block-Sylvester channel matrix. The block size was chosen
to be 32.
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nels, our scheme is full rate, i.e, the throughput for
[he 3 x 2 channel is 2 symbols per channel use (2
bitslseclHz). In the S T precoding scheme, the rate
is limited by the conditioning o f X H R and by the
zero padding used to avoid IBI. Fig. 1 shows the BER
curves for the MN-FIR equalizer (with a rate of 2
b/s/Hz) and the ST dccoder (with rates of 1.70 b/s/Hz
and I .76 b/s/Hz).
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The ST decoding method involves thc eigen decomposition of R H X ,of dimension N ( L c P - 1) x
N ( L c + P - 1) with P >> Lc. Therefore its computational complexity and latency arc much higher than
that of the MN-FIR equalizer. But it has the advantage o f trading off rate for performance.
+
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5. CONCLUSION
In this paper, we have presented a filterbank precoding scheme
for MIMO frequency selective channels. We have derived
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