Blind Single-input Multi-output (simo) Channel Identification With Application To Time Delay Estimation

BLIND SINGLE-INPUT MULTI-OUTPUT (SIMO) CHANNEL IDENTIFICATION WITH APPLICATION TO TIME DELAY ESTIMATION Chong- Yung Chi, Xianwen Chang and Chii-Horng Chen Department of Electrical Engineering & Institute of Communications Engineering National Tsing Hua University, Hsinchu, Taiwan, R.O.C. Tel: +886-3-5731156, Fax: +886-3-5751787, E-mail: [email protected]

ABSTRACT

2. BCE FOR SIMO LTI SYSTEMS

In this paper, with a given set of non-Gaussian measurements, a cumulant based single-input multi-output (SIMO) blind channel estimation (BCE) algorithm is proposed that uses multi-input multi-output (MIMO) inverse filter criteria (blind deconvolution criteria using higher-order cumulants) proposed by Tugnait, and Chi and Chen. Then a time delay estimation (TDE) algorithm is proposed that estimates P - 1 time delays from the phase information of the estimated single-input P-output ( P 2 2) system obtained by the proposed SIMO BCE algorithm. Some simulation results are presented to support the efficacy of the proposed SIMO BCE and TDE algorithms.

Let cum{yl, y ~..., , yp} denote the pth-order joint cumulant [3] of random variables y1 yz, ..., yp and

C p , 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{.} and F-'{o} denote the discrete-time Fourier transform and inverse Fourier transform operators, respectively. Assume that we are given a set of non-Gaussian measurements x [ n ] , n = 0, 1, ..., N - 1, modeled by (1) with the following assumptions:

(Al) u[n]is zero-mean, independent identically distributed 1. INTRODUCTION

Blind channel estimation (BCE) for single-input multi-output (SIMO) systems is a problem of estimating a P x 1linear time-invariant (LTI) system, denoted h[n] = (hl [n],hz[n], ...) h ~ [ n ]with ) ~ ,only a set of non-Gaussian vector output measurements ~ [ n=] ( z l [ n ]22[72], , ...,~ p [ n ]as) follows ~ Di)

~ [ n=]

h[k]u[n- k ]

+ w[n]

(1)

k=-m

where u[n]is the non-Gaussian driving input signal and w[n] = (wl[n],wz[n],...,~ p [ n ]is) additive ~ noise. The SIMO LTI system arises in science and engineering areas where multiple sensors are needed such as time delay estimation [l]and seismic signal processing, etc. In communications, multiple antennas receiving signals and fractionallyspaced signal processing at receiver can also be modeled as SIMO LTI systems [ 2 ] .

(i.i.d.), non-Gaussian and C p , p { u [ n ]#} 0 for a chosen ( p , q ) , where p and q are nonnegative integers and p+q23. (A2) The SIMO system h[n]is stable. (A3) The noise ~ [ nis] zero-mean Gaussian (which can be spatially correlated and temporally colored) and statistically independent of u[n].

wz[n],...,t ~ p [ n be ])~ a P x 1 FIR inLet v [ n ] = (2)l[n], verse filter (deconvolution filter) for which ~ [ n=] 0 for n < LI and n > Lz,and let e[.] be the inverse filter output, i.e., L2

e[n] =

V'[Z]

. x[n - 11

l=L1

=

2

s[k]. u[n- k ]

k=-m

+

c L2

vT[Z]w[n- Z ] (3)

l=L1

where s[n]is the overall system given by L2

0-7803-7011-2/01/$10.0002001E E E

v'[l]h[n - 11.

s[n]=

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

l=L1

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(4)

Chi and Chen [4]design the inverse filter ~ [ nby] maximizing the following multi-input multi-output inverse filter criteria (MIMO-IFC)

(S5) If

e-1

11 # ) ( w k )

where p and q are nonnegative integers and p+q 2 3. They also proposed a fast iterative MIMO-IFC based algorithm [5]for obtaining the optimum inverse filter ~ [ nfor] p+q 2 3 as x[n]is real and p = q 2 2 as x[n]is complex. Based on the relation between the optimum v[n]and the MIMO linear minimum mean square error equalizer reported in [5], one can show the following fact on which the BCE algorithm for SIMO systems below is based:

Two worthy remarks regarding the proposed SIMO BCE algorithm are as follows.

( R l ) The region of support associated with the estimate g[n]can be arbitrary as long as the FFT size C is chosen sufficiently large so that aliasing effects on the resultant G[n]are negligible.

(R2) The obtained estimate f i ( w ) is robust against Gaus-

Fact 1. Assume that V ( w ) = F { v [ n ] } is the optimum inverse filter associated with J p , p ( v [ n ]with ) L1 + -m and L2 + m. Let

GP@) = m P [ n l > R ( w ) = F { R [ k ] }= F { E [ x [ n ] x H[n- k ] ] } .

ck

then go to (s2),otherwise gib&) = H ( i ) ( w k )(except for a scale factor) and its C-point inverse FFT 6[n]are obtained.

(5)

gp[n] = sP[n](s*[n])P-l

- H(i-1)(wk)

k=O

sian noise because (9) is true regardless of the value of signal-to-noise ratio (SNR), although the inverse filter v[n]and the power spectrum R ( w ) depend on SNR.

(6) (7)

3. TIME DELAY ESTIMATION (TDE)

(8)

In time delay estimation, a single source signal, denoted qn],is received by P (2 2) spatially separate sensors. The received signal vector Z [ n ] can be modeled as

Then

Z [ n ]= E[n] where a is a non-zero constant.

= (qn], a1qn - d l ] ,...,a p - l q n

SIMO BCE Algorithm: With finite data ~ [ nobtain ], the inverse filter v[n]associated with J p , p ( v [ n ]using ) Chi and Chen's fast MIMOIFC algorithm [5], and its C-point FFT V ( w k ) , where wk = 27rk/C, k = 0 , 1 , ...,13 - 1. Obtain R ( w k ) using multichannel Levinson recursion algorithm [6].

h[k]u[n - k]

(13)

k=-m

in which h[n]is a stable LTI system and u[n]is zero-mean, i.i.d. non-Gaussian, and W [ n ]is a P x 1 additive Gaussian noise vector which can be spatially correlated and temporally colored. From (12) and (13), one can easily see that Z [ n ]can also be expressed as an SIMO model as follows

(Sl)Set i = 0. Set initial values H ( O ) ( w k )and conver(S2) Update i by i

+ W [ n ] (12)

03

qn] =

Step 2. Channel Estimation.

Eh

- dp-l])T

where a , and d,, i = 1,2, ..., P - 1 are amplitudes and time delays, respectively, qn] is a wide-sense stationary, colored non-Gaussian signal modeled by

Step 1. Blind Deconvolution.

gence tolerance

+W[n]

> 0.

+ 1. Compute

00

L [ k ] u [ n- k ] + G[n]

Z [ n ]= by (4)and its C-point inverse FFT ~ ( ~ - ' ) [ n ] .

( S 3 ) Compute gp[n]using ( 6 ) with s[n]= di-l)[n] and its C-point FFT GP(wk).

(14)

k=--m

where

-

h[n]= (h[n], aih[n - d i ] ,..., a p - i h [ n - d p - 1 1 ) ~ . (15)

(S4) Compute

Let

by (9) which is then normalized by I I H ( i ) ( W k ) 1 1 2 = 1.

x;z;

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c.

delay between L(')[n] and the true h[n]was artificially removed. The normalized mean-square error (NMSE) for the ith channel estimate x;[n]is defined as

It can be easily shown that

(bi[n],bz[n], ...,b ~ [ n ]=) ~F ' { B ( w ) }

b[n] =

(S[n],S[n - dl], ..., S[n- dp-l])?

=

(19)

TDE Algorithm:

-

Process Z[n]using the proposed SIMO BCE algorithm to estimate H ( w k ) , k = 0, 1, ..., L - 1, and then obtain its phase $ ( w k ) . 2btain g ( w k ) using (18) %ndits inverse L-point FFT b[n].Then the estimate di is obtained as

n

Then the overall NMSE (ONMSE) [7] can be obtained by averaging NMSEi over P channels as follows: l P ONMSE = NMSEi.

(25)

i=l

4. SIMULATION RESULTS

A . Simulation Results for the Proposed SIMO BCE Algorithm Consider a 2-channel MA(6) system taken from [7]whose transfer function was

H ( z )=

I

0.6140

+0.3684~~' +

- 0 . 2 5 7 9 ~ ~' 0.6140~-~0.8842~-~ +0.4421~-~ 0 . 2 5 7 9 ~ ~ ~

+

1

(21)

The driving input 241. was a real zero-mean, exponentially distributed i.i.d. random sequence with unit variance. The noise vector ~ [ n=]('w~[n], w2[n])' was assumed to be spatially independent and temporally white Gaussian. The synthetic data ~ [ nwere ] processed by the proposed SIMO BCE algorithm with p = 2, F F T length L = 64, L1 = 0 and LZ = 7 for the inverse filter v[n]and the initial condition H ( ' ) ( w k ) = 1 for all k . Thirty independent realizations were performed for N = 1024, 2048 and 4096, and SNR = 10 dB, 5 dB, 0 dB and -5 dB, respectively, where SNR is defined as P

Table 1 shows the ONMSEs for different values of data length N and SNR associated with the proposed SIMO BCE algorithm and Tugnait's method, respectively. One can see from Table 1 that the proposed SIMO BCE algorithm performs much better than Tugnait's method (smallerONMSE).

B . Simulation Results for the Proposed TDE Algorithm Assume that there were 2 sensor elements ( P = 2), the amplitude a1 = 1, the true time delay dl = 5 and the driving input 1 . 4 was a real zero-mean, exponentially distributed i.i.d. random sequence with unit variance. The system h[n] (see (13)) was a non-minimum phase ARMA(3,2) system taken from [I]

H(z)=

+

1 - 2.952-1 1 . 9 ~ - * 1 - 1.32-' 1 . 0 5 ~ --~0 . 3 2 ~ - ~

+

(26)

and noise G[n]was coherent (i.e., Gl[n]= G2[n])and 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 inverse filter v[n]and the initial condition H(O)(wk)= 1 for all k . Thirty independent runs were performed for N = 2048 and 4096, and SNR= 0 dB and -5 dB. For comparison, d l is also estimated by Tugnait's TDE methods [l]as follows:

i=l h

dl = arg{mdax{Ti[d]}}, i = 1 or 2

For comparison, h[n]is also estimated by Tugnait's BCE method [7] as follows:

hi[%?]=

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

where

E [le[kll21

where ].[e is the optimum inverse filter output associated with J z ,2. Let c")[n]denote the estimate of h[n]at the Zth realization normalized by a constant energy, and the time

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(28)

Table 1. ONMSE associated with the proposed SIMO BCE algorithm and Tugnait’s method, respectively. Proposed algorithm

Tugnait’s 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

0.8888

2048

0.0183

0.0228

0.0650

0.5481

0.0210

0.0291

0.0767

0.7979

4096

0.0109

0.0139

0.0354

0.2812

0.0110

0.0163

0.0400

0.4647

Table 2. Mean, standard deviation and RMSE for respectively.

21 associated with Tugnait’s methods and the proposed TDE algorithm,

True Time Delay dl = 5

N = 4096

N = 2048 SNR (dB)

0

-5

(T

RMSE

Mean

4.8333

1.5992

1.5811

Tz [dl

5.0333

1.6078

Proposed Algorithm

5.0000

TI [dl

(T

RMSE

4.8667

0.9732

0.9661

1.5811

5.0000

0.0000

0.0000

0.0000

0.0000

5.0000

0.0000

0.0000

6.5000

6.0272

6.1128

4.8667

5.5238

4.4497

T2 [dl

4.9667

5.8101

5.7126

3.3667

4.2221

4.4609

Proposed Algorithm

4.1667

1.8952

2.0412

4.6667

1.2685

1.2910

TDE Method

Mean

T1[dl

L. Tong and S. Perreau, “Multichannel blind identification: From subspace to maximum likelihood methods,” Proc. ZEEE, vol. 86, no. 10, pp. 1951-1968, Oct.

Table 2 shows mean, standard deviation ( 0 ) and rootmean-square error (RMSE) for $1 associated with Tugnait’s methods and the proposed TDE algorithm, respectively. One can see from Table 2 that the proposed TDE algorithm performs much better than Tugnait’s methods (smaller variance and RMSE).

1998.

C. L. Nikias and A. P. Petropulu, Higher Order Spectral Analysis: A Nonlinear Signal Processing Framework, Prentice-Hall, Englewood Cliffs, New Jersey, 1993. C.-Y. Chi and C.-H. Chen, “Blind MA1 and IS1 suppression for DS/CDMA systems using HOS based inverse filter criteria,” ZEEE Trans. Signal Processing (in revision).

5. CONCLUSIONS

We have presented an SIMO BCE algorithm using cumulant based MIMO-IFC (see (5)) which is robust against Gaussian noise, and a TDE algorithm that estimates P - 1 time delays only using the phase information of the single-input P-output ( P 2 2) system estimated by the proposed SIMO BCE algorithm. Simulation results show that the proposed SIMO BCE algorithm and TDE algorithm outperform Tugnait’s channel estimation method and TDE methods, respectively.

C.-Y. Chi and C.-H. Chen, “Cumulant based inverse filter criteria for blind deconvolution: properties, algorithms, and application to DS/CDMA systems,” to appear in ZEEE Trans. Signal Processing, July 2001.

S. M. Kay, Modem Spectral Estimation, Prentice-Hall, 1988. J. K. Tugnait, “Identification and deconvolution of multichannel linear non-Gaussian processes using higher order statistics and inverse filter criteria,” ZEEE Trans. Signal Processing, vol. 45, No. 3, pp. 658-672, Mar. 1997.

6. REFERENCES [l] J. K. Tugnait, “Time delay estimation with unknown spatially correlated Gaussian noise,” ZEEE Trans. Signal Processing, vol. SP-41, pp. 549-558, Feb. 1993.

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