Blind Single-Input Multi-output (SIMO) Channel Identification With Application to Time Delay Estimation

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BLIND SINGLE-INPUT MULTI-OUTPUT (SIMO) CHANNELIDENTIFICATION WITH APPLICATION TO TIME DELAY ESTIMATION

Chong- Yung Chi, Xianwen Chang and Chii-Horng ChenDepar tment of Electrical Engineering &

Ins t i tu te of Comm unications EngineeringNational T sing H ua University, Hsinchu, Taiwan, R.O.C.

Tel: +886-3-5731156, Fax: +886-3-5751787, E-mail: [email protected]

ABSTRACTIn this paper, with a given set of non-Gaussian measure-ments, a cumulant based single-input multi-output (SIMO)blind channel estimation (BCE) algorithm is proposed thatuses multi-input mu lti-outpu t (MIMO) inverse filter criteria(blind deconvolution criteria using higher-order cumulants)proposed by Tugnait, and Chi and Chen. Then a time de-lay estimation (TD E) algorithm is proposed tha t estimatesP - 1 time delays from the phase information of the esti-mated single-input P-output ( P 2 2) system obtained bythe proposed SIMO BCE algorithm. Some simulation re-sults are presented t o support the efficacy of t he proposedSIMO BCE and TDE algorithms.

1. INTRODUCTIONBlind channel estimation (BCE) for single-input multi-out-put (SIMO) systems is a problem of estimating a P x 1 ineartime-invariant (LTI) system, denoted h[n]= (h l n], z[n],...)h ~ [ n ] ) ~ ,ith only a set of non-Gaussian vector outputmeasurements ~ [ n ] ( z l [ n ] ,2[72], ..,~ p [ n ] ) ~s follows

Di)

~ [ n ] h[k]u[n- ]+w[n] (1 )k=-m

where u[n] s the non-Gaussian driving input signal andw[n] = (wl[n],wz[n],..,~ p [ n ] ) ~s additive noise. TheSIMO LTI system arises in science and engineering areaswhere multiple sensors are needed such as time delay esti-mation [l] and seismic signal processing, etc. In communi-cations, multiple antennas receiving signals and fractionally-spaced signal processing at receiver can also be modeled asSIMO LTI systems [ 2 ] .

This work was supported by the National Science Councilunder Grant NSC 89-2213-E007-132.

0-7803-7011-2/01/$10.0002001EEE 293

2. BCE FOR SIMO LTI SYSTEMSLet cum{yl, y ~ ,..,yp } denote the pth-order joint cumulant[3]of random variables y1 yz, ...,yp andCp,q{y} = cum{yl = y2 = . ..= yp= y, yp+ l = yp +2 =

. . . = Yp+q = Y* } (2 )where y* is the complex conjugate of y. Let F{.} andF-'{o} denote the discrete-time Fourier transform and in-verse Fourier transform operators, respectively. Assumethat we are given a set of non-Gaussian measurements x[n] ,n = 0, 1, ..., N - 1, modeled by (1) with the following as-sumptions:( A l ) u[n]s zero-mean, independent identically distributed

(i.i.d.), non-Gaussian and Cp,p{u[n]} 0 for a chosen( p , q ) ,where p and q are nonnegative integers andp + q 2 3 .

(A2) The SIMO system h[n] s stable.(A3) The noise ~ [ n ]s zero-mean Gaussian (which can be

spatially correlated and temporally colored) and sta-tistically independent of u[n].

Let v[n]= (2)l[n],z[n],.., ~ p [ n ] ) ~e a P x 1 FIR in-verse filter (deconvolution filter) for which ~ [ n ] 0 forn < L I and n > Lz, nd let e[] be th e inverse filter out-put, i.e.,

L 2

l=L1e[n] = V'[Z] .x[n- 1

L2= 2 s[k] u[n- ]+ c T[Z]w[n- ] (3)k = - m l=L1

where s[n] s the overall system given byL2

l = L1s[n] v'[l]h[n- 1. (4)

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Chi and Chen [4 ]design the inverse filter ~ [ n ]y maxi-mizing the following multi-input multi-output inverse filtercriteria (MIMO-IFC)

( 5 )where p and q are nonnegative integers and p+ q 2 3. Theyalso proposed a fast iterative MIMO-IFC based algorithm[5 ] or obtaining t he optimum inverse filter ~ [ n ]or p+q 2 3as x[n] s real and p = q 2 2 as x[n] s complex. Basedon the relation between the optimum v[n] nd the MIMOlinear minimum mean square error equalizer reported in [ 5 ] ,one can show the following fact on which the BCE algorithmfor SIMO systems below is based:Fac t 1. Assume that V ( w )= F { v [ n ] } s the optimuminverse filter associated with J p , p ( v [ n ] ) ith L1 + -m andL2 +m. Let

g p [ n ] = sP[n](s*[n])P-l (6 )GP@) = m P [ n l > ( 7 )R ( w ) = F { R [ k ] }= F { E [ x [ n ] x Hn- ] ] } . ( 8 )Then

where a is a non-zero constan t.SIMO BCE Algorithm:Step 1.Blind Deconvolution.

With finite da ta ~ [ n ] ,btain th e inverse filter v[n] sso-ciated with J p , p ( v [ n ] )sing Chi and Chen's fast MIMO-IFC algorithm [5], and its C-point FFT V ( w k ) ,wherewk = 27rk/C, k = 0 , 1 , ..., 3 - 1. Obtain R ( w k ) usingmultichannel Levinson recursion algorithm [6].

Step 2. Channel Estimation.(Sl) et i = 0. Set initial values H ( O ) ( w k ) nd conver-

gence tolerance Eh > 0.(S2) Update i by i + 1. Compute

by (4) nd its C-point inverse FFT ~ ( ~ - ' ) [ n ] .( S 3 ) Compute gp[n] sing ( 6 )with s[n]= di-l)[n] and

its C-point FFT G P ( w k ) .(S4) Compute

by (9) which is then normalized byx;z; I IH(i ) (Wk)112 = 1.

(S5) If e-1 11 # ) ( w k ) -H(i-1)wk) c kk= O

then go to (s2), therwise gib&) H ( i ) ( w k ) ex-cept for a scale factor) an d its C-point inverse FFT6[n] re obtained.

Two worthy remarks regarding th e proposed SIMO BCEalgorithm are as follows.( R l ) The region of support associated with the estimateg[n] an be arbitrary as long as the FFT size C is

chosen sufficiently large so that aliasing effects onthe resultant G[n] re negligible.

(R2) The obtained estimate f i ( w ) is robust against Gaus-sian noise because (9) is true regardless of the valueof signal-to-noise ratio (SNR) , although the inversefilter v[n] nd the power spectrum R ( w ) depend onSNR.

3. TIME DELAY ESTIMATION (TD E)In time delay estimation, a single source signal, denotedqn] , s received by P (2 2) spatially separate sensors. Thereceived signal vector Z [ n ]can be modeled as

Z [ n ]= E[n]+W [ n ]= (qn],1qn - l ] , ..,a p - l q n - p - l ] ) T +W [ n ] (12)

where a , an d d ,, i = 1,2, ...,P - are amplitudes and timedelays, respectively, qn] s a wide-sense stationary, colorednon-Gaussian signal modeled by

03qn] = h[k]u[n k ] (13)k = -m

in which h[n] s a stable LTI system and u[n]s zero-mean,i.i.d. non-Gaussian, and W [ n ] s a P x 1 additive Gaussiannoise vector which can be spatially correlated and tempo-rally colored.

From (12) and (13), one can easily see tha t Z [ n ] an alsobe expressed as an SIMO model as follows

00Z [ n ]= L [ k ] u [ n- ]+G[n] (14)

k=--m

where-h[n]= (h[n],ih[n - i ] , ..,a p - i h [ n - p - 1 1 ) ~ . (15)Let

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It can be easily shown thatb[n] = (bi[n],bz[n],..,b ~ [ n ] ) ~F ' { B ( w ) }

= (S[n],S[n dl], ...,S[n- dp-l])? (19)TDE Algorithm:

Process Z[n] sing the proposed SIMO BCE algo-rithm to estimate H ( w k ) , k = 0, 1, ..., L - 1, andthen obtain its phase $ (w k ) .2btain g ( w k )using (18) %nd its inverse L-point FF Tb[n]. hen th e estimate di is obtained as

-

4. SIMULATION RESULTSA . Simulation Results for the Proposed SIMO BCEAlgorithm

Consider a 2-channel MA(6) system taken from [7] hosetransfer function was

0.6140 + 0 . 3 6 8 4 ~ ~ '- 0 . 2 5 7 9 ~ ~ ' 0 . 6 1 4 0 ~ - ~ 0 . 8 8 4 2 ~ - ~

+ 0. 44 21 ~- ~ 0 . 2 5 7 9 ~ ~ ~ 121)

I( z )=The driving input 241 was a real zero-mean, exponentiallydistr ibuted i.i.d. random sequence with uni t variance. Th enoise vector ~ [ n ]( 'w~[n],2[n])' was assumed to be spa-tially independent and temporally white Gaussian. Thesynthetic data ~ [ n ]ere processed by the proposed SIMOBCE algorithm with p = 2, FFT length L = 64, L1 = 0and LZ = 7 for the inverse filter v[n] nd the initial con-dition H ( ' ) ( w k )= 1 for all k . Thirty independent realiza-tions were performed for N = 1024, 2048 and 4096, andSNR = 10 dB, 5 dB, 0 dB and -5 dB, respectively, whereSNR is defined as

P

i = lFor comparison, h[n] s also estimated by Tugnait's BCEmethod [7]as follows:

E [zi[k]e*k - n]]E [le[kll21

hi[%?]=where e is the optimum inverse filter output associatedwith J z , .

Let c")[n] enote the estimate of h[n] t the Zth re-alization normalized by a constant energy, and the time

delay between L(')[n]nd the true h[n] as artificially re-moved. The normalized mean-square error (N M S E ) for theith channel estimate x;[n]s defined as

n

Then the overall NMSE (ONMSE) [7] can be obtained byaveraging NMSEi over P channels as follows:

(25)l PONMSE = - NMSEi.i= l

Table 1 shows the ONMSEs for different values of datalength N and SNR associated with the proposed SIMOBCE algorithm and Tugnait's method, respectively. Onecan see from Table 1that the proposed SIMO BCE algo-rithm performs much better t han Tugnait's method (smaller-ONMSE).B. Simulation Results for the Proposed TD E Algo-rithm

Assume that there were 2 sensor elements ( P = 2),the amplitude a1 = 1, the true time delay d l = 5 andthe driving input 4 was a real zero-mean, exponentiallydistributed i.i.d. r andom sequence with unit variance. Thesystem h[n] (see (13)) was a non-minimum phase ARMA(3,2)system taken from [I]

(26)1- 2.952-1 + 1.9~-*H ( z )= 1- 1.32-' + 1 . 0 5 ~ - ~0 . 3 2 ~ - ~and noise G[n]was coherent (i.e., Gl[n]= G2[n]) nd Gl[n]was generated as the output of a first-order MA model [l ]

H,(z) 1+ 0.82-1 (27)driven by white Gaussian noise. The synthetic data Z[n]were processed by the proposed TDE algorithm with p = 2,FFT length L: = 32, L1 = 0 and L2 = 9 for the inversefilter v[n] nd th e initial condition H(O)(wk )= 1 for all k .Thirt y independent runs were performed for N = 2048 and4096, and SNR= 0 dB and -5 dB. For comparison, d l isalso estimated by Tugnait's TD E methods [l] s follows:

hdl = arg{mdax{Ti[d]}},= 1 or 2 (28)where

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Table1. ONMSE associated with the proposed SIMO BCE algorithm an d Tugnaits method, respectively.Proposed algorithm Tugnaits method

SNR (dB)N 10 5 0 -5 10 5 0 -5

~ ~~

1024 0.0358 0.0460 0.1568 0.7055 0.0438 0.0606 0.1846 0.88882048 0.0183 0.0228 0.0650 0.5481 0.0210 0.0291 0.0767 0.79794096 0.0109 0.0139 0.0354 0.2812 0.0110 0.0163 0.0400 0.4647

Table 2. Mean, standard deviation a nd RMSE for 21 associated with Tugnaits methods an d the proposed TDE algorithm,respectively.

True Time Delay dl = 5N = 2048 N = 4096

SNR (dB) TDE Method Mean (T RMSE Mean (T RMSET1[dl 4.8333 1.5992 1.5811 4.8667 0.9732 0.9661

0 Tz[dl 5.0333 1.6078 1.5811 5.0000 0.0000 0.0000Proposed Algorithm 5.0000 0.0000 0.0000 5.0000 0.0000 0.0000TI [dl 6.5000 6.0272 6.1128 4.8667 5.5238 4.4497

-5 T2[dl 4.9667 5.8101 5.7126 3.3667 4.2221 4.4609Proposed Algorithm 4.1667 1.8952 2.0412 4.6667 1.2685 1.2910

Table 2 shows mean, st andard deviation (0 )and root-mean-square error (RMSE) for $1 associated with Tugnaitsmethods and the proposed TDE algorithm, respectively.One can see from Table 2 tha t th e proposed TD E algorithmperforms much bette r t han Tugnaits methods (smaller vari-ance and RMSE).

5. CONCLUSIONSWe have presented an SIMO BCE algorithm using cumulantbased MIMO-IFC (see (5)) which is robust against Gaus-sian noise, and a TDE algorithm that estimates P - 1 timedelays only using th e phase information of the single-inputP-output ( P2 2) system estimated by the proposed SIMOBCE algorithm. Simulation results show that t he proposedSIMO BCE algorithm and TDE algorithm outperform Tug-naits channel estimation method and TDE methods, re-spectively.

6. REFERENCES[l ] J. K . Tugnait, Time delay estimation with unknown

spatially correlated Gaussian noise, ZEEE Trans. Sig-nal Processing, vol. SP-41, pp. 549-558, Feb. 1993.

L. Tong and S. Perreau, Multichannel blind identifi-cation: From subspace to maximum likelihood meth-ods, Proc. ZEEE, vol. 86, no. 10, pp. 1951-1968, Oct.1998.C. L. Nikias and A. P. Petropulu, Higher Order Spec-tral Analysis: A Nonlinear Signal Processing Frame-work, Prentice-Hall, Englewood Cliffs, New Jersey,1993.C.-Y. Chi and C.-H. Chen, Blind MA1 and IS1 sup-pression for DS/CDMA systems using HOS based in-verse filter criteria, ZEEE Trans. Signal Processing (inrevision).C.-Y. Chi and C.-H. Chen, Cumulant based inversefilter criteria for blind deconvolution: properties, al-gorithms, and application to DS/CDMA systems, toappear in ZEEE Trans. Signal Processing, July 2001.S. M. Kay, Mo dem Spectral Estimation, Prentice-Hall,1988.J. K. Tugnait, Identification and deconvolutionof multichannel linear non-Gaussian processes usinghigher order statistics and inverse filter criteria, ZEEETrans. Signal Processing, vol. 45, No. 3, pp. 658-672,Mar. 1997.

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Blind Single-Input Multi-output (SIMO) Channel Identification With Application to Time Delay Estimation