Second-order cyclic statistics based blind channel identification and equalization.

IDENTIFICATION AND EQUALIZATION

Luc Deneire Dirk T.M. Slock

Institut EURECOM, 2229 route des Cr^etes, B.P. 193, 06904 Sophia Antipolis Cedex, FRANCE

ABSTRACT

Blind channel identication and equalization based on sec- ond-order statistics by subspace tting and linear prediction have received a lot of attention lately. On the other hand, the use of cyclic statistics in fractionally sampled channels has also raised considerable interest. We propose to use these statistics in subspace tting and linear prediction for (possibly multiuser) channel identication. The main ben- et expected is to get rid of the dependence on the color of the additive noise, due to the properties of the cyclocorre- lations. We also present some simulations to illustrate the eectiveness of the method.

1. PROBLEM POSITION

We consider a communication system withpemitters and a receiver constituted of an array ofMantennas. The signals received are oversampled by a factor mw.r.t. the symbol rate. The channel is FIR of durationNT=mwhereT is the symbol duration. The received signal can be written as :

x

(n) = ^X1

k^=;1

h

(k)

u

(n^;k)+

v

(n) = ^X1

k^=;1

h

(n^;km)

a

k+

v

(n) where

u

(n) = ^X1

k^=;1

a

k(n^;km)

The received signal

x

(n) and noise

v

(n) are a M 1 vectors.

x

(n) is cycloctationary with period m whereas

v

(n) is assumed not to be cyclostationary with period m.

h

(k) has dimension Mp,

a

(k) and

u

(k) have dimensions p1.

2. CYCLIC STATISTICS

x

⁽n) is cycloctationary with period mwhereas

v

⁽n) is as- sumed not to be cyclostationary with periodm. Hence, the correlations :

R

xx(n;) = E

x

(n)

x

^H(n^;)

are cyclic innwith periodm(^Hdenotes complex conjugate tranpose). One can easily express them as:

* The work of Luc Deneire is supported by the EC by a Marie-Curie Fellowship (TMR program) under contract No ERBFMBICT950155

R

xx(n;) =

1

X

^=;1

1

X

^=;1

h

⁽n^;m)

R

aa()

h

^H(n^;m+m^;) +

R

vv()

We then express thek^thcyclocorrelation as :

R

^fxx^kg()⁴= 1m

m^X;1

l⁼⁰

R

xx(l;)e^;|2lkm = E^k

x

(l)

x

^H(l^;) whose value is :

R

^fxx^kg() = _m^{1 X1}

^=;1

1

X

^=;1

h

()

R

aa()

h

^H(+m^;)e^;|2km +

R

vv()(k) We can introduce a cyclic correlation matrix as :

R

^fxx^kg=⁴

2

6

6

6

4

R

^fxx^kg(0)

R

^fxx^kg(1)

R

^fxx^kg(K^;1)

R

^fxx^kg(^;1)

R

^fxx^kg(0)

R

^fxx^kg(K^;2)

... ... ... ...

R

^fxx^kg(1^;K)

R

^fxx^kg(2^;K)

R

^fxx^kg(0)

3

7

7

7

5

=^TK(

H

N

D

^f_DFT^kg )

R

^fuu^kgTKH(

H

N) +(k)

R

vv

where

R

^fuu^kg=

R

aaIm, whereis a block kronecker product, where the rst matrix is a block matrix and the second matrix is a elementwise matrix.

ruu(n;) = E^fu(n)^u(n^;)^g

=

1

X

i^=;1

1

X

j^=;1(n^;im)(n^;jm^;)^a(i)^aH(j) r_uu^fkg()=

Pml^=0;1Pi^Pj(l^;im)

| {z }

)l⁼i:m⁼⁰

(n^;jm^;)^a(i)^aH(j)raa(j^;i)w^kl wherew = e^;{2m

r_uu^fkg()=

1

X

j^=;1(jm + )raa(j)

TK(

H

N) is the convolution matrix of

H

N = [

h

(0)

h

(1)

h

(N^;1)] and

D

^f_DFT^kg = blockdiag[

I

pp^je^;|2mk

I

pp^jje^;|2(Nm;1)k

I

pp] This shows that the cyclic correlations at cycle frequency k ⁶= 0 are not aected by the additive non k=T cyclo- stationary noise, whatever its color.

3. CHANNEL ESTIMATION BY SUBSPACE FITTING.

3.1. The classical way

Let's come back to the p = 1 case and write

H

⁰N =

H

N

D

^fDFT^kg , the indexkbeing deduced from the context.

One can write the (compact form of the) SVD of the cyclocorrelation matrix

R

^fxx^kg=

UDV

^H with the relations:

range^f

U

^{g= rangefT}K(

H

⁰N)^g and range^f

V

^g= range^fTK(

H

N)^g

We have assumed that ^TK(

H

⁰N)^g is full rank, which is the usual condition (indeed, one can easily verify than rank^fTK(

H

N)^g= rank^fTK(

H

⁰N)^g). We can then solve the classical subspace tting problem :

H

min⁰N;T^jjTK(

H

⁰N)^;

U

T^jj2_F

If we introduce

U

^?such that [

UU

^?] is a unitary matrix, this leads to

min

H

⁰N

H

⁰tN

2

4

D^?

X

i⁼¹

TN(

U

^?i^Ht)^T_N^H(

U

^?i^Ht)

3

5

H

⁰N^Ht

where

U

^?i is a K1 block vector with M 1 blocks, D^?=N+K^;1 and superscript^tdenotes the transposition of the blocks of a block matrix.

H

⁰N^Ht is then the eigenvec- tor corresponding to the minimum eigenvalue of the matrix between brackets.

The case p > 1 can be (partially) solved in a manner similar to [Slo94] and [Lou96].

3.2. A low complexity algorithm

The above derivation relies on the orthogonality between the signal and noise subspaces. Whenk⁶= 0, we have :

R

^fxx^kg=^TK(

H

⁰N)

R

^fuu^kgTKH(

H

N)

where the noise contribution to the covariance expression disappears. From this, we observe that the left column space of^TK(

H

⁰N) and of

R

^fxx^kgare the same. This leads to the following subspace tting criterion:

H

minN;

Q

^jjTK(

H

⁰N)^;

R

^^fxx^kg

BQ

^jj2_F (1) Hence

R fk^g

uu;Lm=^R_aa;L^Im

The matrix

B

has the same dimensions as^TK(

H

N) and is xed. Its choice inuences the quality of the channel esti- mate. The criterion is separable in

H

Nand

Q

. Minimizing w.r.t.

Q

gives :

Q

= (

F

^H

F

)^;1

F

^HTK(

H

⁰N);

F

= ^

R

^fxx^kg

B

Substitution in (1) leads to : min

H

N^jj

P

^?

F

^TK(

H

⁰N)^jj2_F

where

P

^?

F

⁼

I

^;

PF

^and

PF

⁼

F

(

F

^H

F

)^;1

F

^H. With the constraint^jj

H

N^jj=^jj

H

⁰N^jj= 1, we get:

H

^⁰N = arg max^jj

H

N^jj=1TK(

H

⁰N)^H

PF

^TK(

H

⁰N)

= arg max^jj

H

N^jj=1

H

⁰tN^F

H

⁰N^Ht

where^F can easily be constructed from

PF

^{andt is the}

usual block transpose operator. The solution is thus the maximum eigenvector of^F. As in [KOS96], the choice of

B

is set to

B

=^TK(

H

N), leading to a two step algorithm, the rst step where

B

is an arbitrary selection matrix, yielding a consistent estimate of the channel, the second step where

B

=^TK( ^

H

N).

The main benet of this procedure is the low complexity of the algorithm, compared to the rst one, as we don't need to make an eigendecomposition of the complete cyclic covariance matrix.

As noted in [KOS96], the use of the conjugate correla- tions (or conjugate cyclic correlations) for BPSK and MSK modulations can lead to the same conclusions for thek= 0 case.

Let's further note that, fork = 0, the use of this algo- rithm is also possible if we consider ~

R

^f0gxx =

R

^f0gxx ^;

R

vv, in the case of a known or estimated noise covariance matrix.

4. LINEAR PREDICTION

In this section, we assume uncorrelated symbol sequences.

Consider vectors of signals

X

Lm(n) = [

x

⁽n)^H

x

⁽n^;1)^H

x

⁽n^;Lm+ 1)^H]^H;

U

Lm⁺N^;1(n) = [

u

(n)^H

u

(n^;Lm^;N+ 1)^H]^H, then we have the equation

X

Lm(n) = ^TLm(

H

N)

U

Lm⁺N^;1(n).

The linear prediction equations are then :

x

^(n)^j

X

Lm⁽n^;1)=

p

¹

x

(n^;1) ++

p

Lm

x

(n^;Lm)

x

~(n)^j

X

Lm⁽n^;1)=

x

(n)^;

x

^(n)^j

X

Lm⁽n^;1)

x

~(n)^j

X

Lm⁽n^;1)= [IM ^;

P

Lm]

| {z }

=

P

Lm

X

Lm⁺¹(n)

To nd the prediction lter, we express orthogonality of the prediction error to the past using the cyclocorrelation. This gives the normal equations :

E^k

x

^~(n)

X

HLm⁺¹⁽n⁾ = [IM^;

P

Lm]R^fxx;Lm^{kg +1}= [^2fx;Lm^{~ kg}00]

Where

^2fx;Lm~^kg = E^{4 k}

x

~⁽n)~

x

^H(n)

= ^[I_M^;

P

Lm^]E^k

X

Lm⁺¹⁽n⁾

X

HLm⁺¹⁽n^{) [}I_M^;

P

Lm^]H

=

P

Lm

R

^f_xx;Lm^{kg +1}

P

HLm

The prediction error can be expressed as : ~

x

⁽n) =

h

(0)

u

(n) + (

H

N ^;

P

Lm^T(

H

N))

U

Lm⁺N^;1(n^;1), where

H

N= [

h

(1)

h

(N)00].

The cyclovariance of the error becomes : E^k~

x

(n)~

x

^H(n) =²a

m

h

(0)

h

^H(0) +²a

m

H

⁰N

H

^HN

where

H

N=

H

N+

P

Lm^T(

H

N). Minimizing this cyclo- variance with respect to

P

Lm leads to

H

N = 0.

Hence, from the prediction quantities, we can determine a zero forcing (ZF) equalizer and the channel similarly to [SP94].

5. SIMULATIONS 5.1. Subspace tting

In this section, we restrict ourselves to thep= 1 case, using a randomly generated real channel of length 5T, following a raised cosine with 90 % excess bandwith.

The receiver hasM = 3 antennas and we oversample by m= 3. We draw the NRMSE of the channel, dened as

NRMSE =

v

u

u

t1 50

50

X

l⁼¹

jj

h

^^(l);

h

^jj2_F=^jj

h

^jj2_F

where ^

h

^(l) is the estimated channel in the lth trial. In the gures below, the NRMSE in dB has been calculated as 10log 10(NRMSE). The estimation is based on the classic subspace tting of the k = 1 cyclocorrelation ma- trix. This matrix is calculated from a burst of 100 QAM-4 symbols (note that if we used real sources, we would have used the conjugate cyclocorrelation, wich is another means of getting rid of the noise, provided it is circular). For these simulations, we used 50 Monte-Carlo runs and a channel length of 18 (neglecting the 5 rst and 6 last near zero val- ues of the channel). The estimations for an SNR of 20 dB are reproduced hereunder.

0 5 10 15 20 25 30

−0.2 0 0.2

0 5 10 15 20 25 30

−0.5 0 0.5

0 5 10 15 20 25 30

0

This channel estimate is then followed by a linear MMSE- ZF multichannel equalizer to give us the MSE at the output in the gure below and an estimation of the BER (for SNR=

0,5 and 10 dB), this latter is obtained by simulation on 100000 QAM-4 symbols for each channel estimate : i.e. 10⁷ bits for each SNR.

0 5 10 15 20 25

−30

−25

−20

−15

−10

−5 0 5 10

SNR in dB

SNR in dB

Output MSE, estimated channel vs true channel

BER=0.0034 BER=0.103

BER=0.3

Furthermore, we simulated two colored noise scenarios, the rst one in a spatially correlated noise with correlation matrix 1 :7 :49

:7 1 :7 :49 :7 1

!

the other with a spatio-temporally ltered noise by hn= :346 ^;:180 :057 ^;:057 ^;:365 ^;:358

:298 :068 ^;:231 :053 :090 ^;:229 :387 ^;:100 :413 :172 :032 :079

!

0 5 10 15 20 25

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1 0

SNR in dB

NRMSE in dB

channel estimation NRMSE for k=1

* −− filtered noise + −− white noise o −− Spatial colored noise

For comparison purposes, we also made estimations with k = 0 and for the classical subspace tting based on the covariance matrix with Mmchannels. We illustrate the eect of the (raised cosine transmission) lter on the overall channel estimation performance. The tree bottom curves correspond to the estimation of the convolution of channel and lter. The two top curves corresond to the estimation of the (original) channel alone (no tranmission lter present).

0 5 10 15 20 25

−14

−12

−10

−8

−6

−4

−2 0

SNR in dB

NRMSE in dB

channel estimation NRMSE

* : k=0 o : k=1 + : classic R ... : original h, k=0 ._. : original h, k=1

5.2. Linear prediction

Simulations on this method proved to be rather poor (for k= 1), in the sense that we have to use rather large bursts of symbols to get a proper estimation. For illustration, we include the evolution of the NRMSE as the burst length grows for a well-behaved channel (of length 2T and ran- domly generated). This simulation is done with SNR=25, M=3, m=3.

10² 10³ 10⁴

−9

−8

−7

−6

−5

−4

−3

burst length

NRMSE in dB

channel estimation NRMSE vs burst length

6. CONCLUSION

We have proposed two new channel identication schemes based on cyclic statistics which are independent of the color of the additive noise. The subspace tting gives good per- formance, but the benet of the independence on the noise color is to be moderated by the loss due to the rather low cyclic spectral power atk⁶= 0.

On the other hand, our approach, withk= 0 gives better results than the classical scheme. This is mostly due to the fact that we can better rene the channel length estimate (and should be preceded by a good channel length estima- tion algorithm). Indeed, if we use a channel of the form (whereM = 2 andm= 2)

whereis a near zero value, the cyclocorrelation approach can aord to restrict to the central part of the channel, but the classical approach will try to nd theMmmultichan-

nel : ⁰

B

@

1

C

A or

0

B

@

1

C

A

with 2 more (near zero) parameters to estimate, which will globally give a worse estimation. The cyclic correlation ap- proach withk= 0 simply corresponds to the use of a mean correlation matrix estimated at _Tm separated lags.

Unfortunately, at least in mobile environments, where the burst lengths are short, the linear prediction method pro- posed gives poor results. When long data pieces are avail- able, it could work. A possible direction to improve this method is the use of the best linear predictor proposed by [Mia93], where an equivalent multivariate stationary pro- cess is built, which catches more prediction properties as the classical multivariate process obtained by vectorization of the oversampled signals.

REFERENCES

[Gar94] William A. Gardner, editor. Cyclostationarity in Communications and Signal Processing. IEEE Press, 1994.

[Gia95] Georgios B. Giannakis. Linear Cyclic Correlation Approaches for Blind Identication of FIR Chan- nels. In 29th Asilomar Conference on Signals, Systems and Computers, pages 420{424, 1995.

[KOS96] Martin Kristensson, Bjorn Ottersten, and Dirk Slock. Blind Subspace Identication of a BPSK Communication Channel. In Proc. Asilomar 96, Asilomar, USA, November 1996.

[Lou96] Ph. Loubaton. Quelques aspects methodologiques concernant la separation des melanges convolutifs.

Ecole des Techniques avancees Signal Image Pa- role, Grenoble, septembre 1996.

[Mia93] A.G. Miamee. Explicit Formula for the Best Linear Predictor of Periodically Correlated Se- quences. SIAM Journal of Mathematic Analysis, 24(3):703{711, May 1993.

[Slo94] Dirk T.M. Slock. Blind Joint Equalization of Mul- tiple Synchronous Mobile Users Using Oversam- pling and/or Multiple Antennas. InTwenty-Eight Asilomar Conference on Signal, Systems & Com- puters, 1994.

[SP94] D.T.M. Slock and C.B. Papadias. Blind fractionally-spaced equalization based on cyclo- stationarity. InProc. Vehicular Technology Conf., pages 1286{1290, Stockholm, Sweden, June 1994.