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Taylor polynomial expansion based waveform correlation cancellation for bistatic MIMO radar localization
Signal Processing 92 (2012) 1404–1410
Contents lists available at SciVerse ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Taylor polynomial expansion based waveform correlation cancellation for bistatic MIMO radar localization Bo Dang, Jun Li, Guisheng Liao n National Laboratory of Radar Signal Processing, Xidian University, Xi’an, Shaanxi 710071, China
a r t i c l e in f o
abstract
Article history: Received 4 June 2011 Received in revised form 23 November 2011 Accepted 24 November 2011 Available online 3 December 2011
A Taylor polynomial expansion (TPE) based waveform decorrelation algorithm with low complexity is proposed to cancel the effects of auto-correlation and cross-correlation of transmit waveform of multiple-input multiple-output (MIMO) radar in the receiver. Cayley–Hamilton theorem is first used to expand the direct matrix inversion to the form of polynomial matrix. And then, Taylor series is used to simplify the calculation of the polynomial matrix. Furthermore, a simplified method is proposed to select the normalized factor by exploiting the special properties of signal correlation matrix, where the convergence performance for the approximation error of the TPE decorrelator is also analyzed. Finally, the performance of waveform correlation cancellation is verified by applying it to bistatic MIMO radar multi-target localization. Simulation results demonstrate the effectiveness of the proposed algorithm. & 2011 Elsevier B.V. All rights reserved.
Keywords: Taylor polynomial expansion Waveform decorrelation Multiple-input multiple-output radar Low complexity
1. Introduction Multiple-input multiple-output (MIMO) radars [1–3] are attracting the attentions for its potential advantages, such as increased spatial diversity, extended virtual array aperture, super-resolution and obtaining the transmit angle information at the receiver. Unlike conventional phased-array radars, MIMO radar systems transmit different waveforms from different transmit elements. In turn, the auto-correlation and cross-correlation properties of transmit waveform sets will have great effect on the performance of MIMO radar systems [4]. There have been two typical methods for the reduction of MIMO radar waveform correlations. One is numerical suppression [5–7], which utilizes optimization approach to reduce the waveform auto-correlation and cross-correlation. However, it is difficult to design waveform sets with ideal peak sidelobe level of autocorrelation property as well as cross-correlation one for an arbitrary time delay. Another approach is the space-time coherence
n
Corresponding author: Tel./fax: þ 86 29 88201030. E-mail address: [email protected] (G. Liao).
0165-1684/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2011.11.023
cancellation, which uses specially designed codes to modulate traditional radar waveform and accumulates the pulse compression results to cancel the auto-correlation [8] and cross-correlation [9,10]. Space-time coding (STC) can help to mitigate waveform auto-correlation and crosscorrelation effects in MIMO radar effectively. However, the residual auto-correlation and cross-correlation of transmit waveform still exist, which influence the performance of multi-target localization. Unlike the methods above, a waveform decorrelation algorithm is proposed to cancel the interferences from nearby range bins effectively in the receiver [4]. However, due to the high complexity of direct matrix inverse (DMI) computation operation, it is difficult to invert the signal autocorrelation matrix. In this paper, Taylor polynomial expansion (TPE) based waveform correlation cancellation algorithm with low complexity is proposed to cancel the effects of auto-correlation and cross-correlation of transmit waveform in the receiver. It is shown that the complexity of the proposed method can be reduced considerably using Cayley–Hamilton theorem [11] and matrix form of Taylor series. Moreover, the effects of auto-correlation and cross-correlation of the transmit waveforms can be canceled effectively. The performance of
B. Dang et al. / Signal Processing 92 (2012) 1404–1410
the TPE method is verified by applying it to bistatic MIMO radar multi-target localization [4]. The rest of this paper is organized as follows: the bistatic MIMO radar signal model is presented in Section 2. In Section 3, a TPE based waveform correlation cancellation algorithm and multi-target localization is presented. The convergence performance of the TPE decorrelator is discussed in Section 4, where a novel method for the selection of the normalization factor to ensure convergence speed of matrix form of Taylor series is presented. The simulation results and performance evaluation are provided in Section 5. Finally, we conclude the paper in Section 6. 2. Bistatic MIMO radar signal model Consider a bistatic MIMO radar equipped with M transmit elements and N receive elements, where the spacing between adjacent elements at the transmitter and receiver is dt and dr, respectively. The array structure of bistatic MIMO radar used in this paper is illustrated in Fig. 1. We assume the range of the target is much larger than the aperture of transmit array and receive array. We assume that there are Pk targets located at (hkt,hkr) in the kth range bin, where hkt ¼ ½yk1t , ykpt T denote the angles of the target with respect to the transmit array (i.e. DODs) and hkr ¼ ½yk1r , ykpr T denote the angles with respect to the receive array (i.e. DOAs). Assuming K range bins have the targets. The receive steer vector and transmit steer vector can be expressed as akp ¼ ½1,ej2pdr sinykpr =l , ,ej2pðN1Þdr sinykpr =l T j2pdt sinykpt =l
bkp ¼ ½1,e
ð1Þ
j2pðM1Þdt sinykpt =l T
, ,e
ð2Þ
The transmit waveform can be represented as Sk ¼ ½s1k ,s2k , ,sMk T , where 2 3 0,. . .,0 , si ðlÞ, 0,. . .,0 |fflfflffl{zfflfflffl} |fflfflffl{zfflfflffl} 5 sik ¼ 4 k
Kk
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and l is the slow-time index. si ðlÞ ¼ ½si ð1Þ, si ðLÞT , i ¼ 1, ,M denote the transmitted coded pulse of the ith transmitter, where L represents the number of codes in one pulse period. The received target echoes can be described as follows: D¼
Pk K X X
bqkp akp bTkp Sk ej2pf qkp tl þN, l ¼ 1,2 L
ð3Þ
k¼1p¼1
where bqkp and fqkp represents the RCS of the pth target in the kth range bin in the qth pulse and Doppler frequency, respectively. NACN (L þ K) denotes the noise matrix and the columns of N are independent and identically distributed (i.i.d) circularly symmetric complex Gaussian random vectors with zero mean and an unknown covariance matrix. Considering the matrix form, (3) can be written as follows: D¼
K X
Hk Sk þ N
ð4Þ
k¼1
where Hk ¼
Pk P p¼1
bk akp bTkp ej2pf k tl . Matched by the trans-
mitted waveform Si expressed as Yi ¼ ð
K X
ði ¼ 1,2 ,KÞ, the output can be
H Hk Sk ÞSH i þ NSi
ð5Þ
k¼1
The matched results can be stacked as follows: Y ¼ ½Y1 ,. . .,YK ¼ HR þ NSH
ð6Þ
where 2 3 S1 6 7 S ¼ 4 ^ 5 2 C MKðL þ KÞ SK YACN MK and H¼[H1,y, HK]. The matrix R¼ SSHACMK MK can be obtained by constructing the transmit waveforms given in advance. Obviously, the correlation
θ
The pth target
ray Ar e v cei Re
θ
Transmit Array
The k range bin The k+1 range bin Fig. 1. Bistatic MIMO radar scenario.
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B. Dang et al. / Signal Processing 92 (2012) 1404–1410
matrix of waveforms R will contaminate the matrix H, in which the parameters to be estimated are included. 3. TPE based waveform correlation cancellation and multi-target location
Ns 1 1 X _ Y TPE ¼ YðI þXÞ1 ¼ Y ðXÞi
From (6), it is obvious that the effects of auto-correlation and cross-correlation of the transmit waveform, which is included in the correlation matrix R, can be canceled by multiplying the inverse of R [4] _ Y DMI ¼ YLDMI ¼ YR1 ¼ H þNSH R1 ð7Þ However, the DMI decorrelator is difficult to implement as the inversion of R has very high computation load. The computation complexity for DMI decorrelator via Cholesky-decomposition [14] is Oð4=3ðMKÞ3 þ 8 ðMKÞ2 10=3MKÞ. In the following section, a low computation load method is proposed to cancel the effect of the correlation waveforms. 3.1. Taylor series based polynomial expansion A class of multistage linear receivers can be obtained by applying a polynomial expansion in R to the matched filter outputs. Here the soft estimator L is produced by considering the Cayley–Hamilton theorem [11] L¼
Ns X
wi Ri
ð8Þ
i¼0
where NS denotes the number of stages of the decorrelator and hence the accuracy of the approximation, and wi represents the weight of the decorrelator. A polynomial expansion approaching in R to the matched filter bank output Y can be applied to approximate the linear mapping matrix L in DMI case. It can be proved that when NS is great enough [11] Ns X
wi Ri R1
ð9Þ
i¼0
Therefore, the linear decorrelator matrix of TPE can be written as ! Ns X _ Y TPE ¼ YLTPE ¼ ðHR þ NSH Þ wi Ri ð10Þ i¼0
From (10), the matrix multiplication to calculate Ri still has a complexity of O((MK)3). In turn, we take the Taylor series into account to operate (10) [14] ð1 þxÞ1 ¼
1 X
ðxÞi
9x9 o 1
ð11Þ
i¼0
For matrix form, (11) can be expressed as ðI þXÞ1 ¼
1 X
ðXÞi
9lj ðXÞ9o 1
8j
ð12Þ
i¼0
where lj(X) represents the jth eigenvalue of X. If we assume (IþX) 1 ¼ aR 1 in DMI case, XACMK MK can be written as X ¼ aRI
where the selection of factor a is set to satisfy 9lj ðXÞ9o 1 8j. By restricting the sum in (12) up to the NS order, the modified linear decorrelator matrix of TPE can be written as
ð13Þ
a
a
ð14Þ
i¼0
However, the calculation of ( X)i for iZ2 comprises matrix multiplication, which has the complexity of O((MK)3). Here, we rewrite (14) as 1 _ Y TPE ¼ ðYYX þ YX2 YX2i1 þYX2i Þ
a
ð15Þ
We can conclude (15) as 1 _ _ Y TPE ðiþ 1Þ ¼ YY TPE ðiÞX
a
ð16Þ
where YACN MK. The computation complexity of the TPE decorrelator is O(Ns N (MK)2). Obviously, the complexity of TPE method is directly proportional to the number of stages NS, which is determined by a. We will discuss the selection of a in detail in the next section. 3.2. ESPRIT based multi-target location To locate the targets, an ESPRIT based multi-target location method [4] is used to estimate the DOD and DOA of the targets. For simple discussion but without loss of generality, the target in the first range bin is considered. From (10) we can obtain the data of the first range bin Z1 ¼ H1 þ V1 ¼
P1 X
b1p a1p bT1p ej2pf 1p t þ V1
ð17Þ
p¼1
_ where Z1 and V1 are the first M columns of Y TPE and H NS LTPE, respectively. We extract the first N 1 rows of Z1 and define them as a new matrix Z11 and define the last N 1 rows of Z1 as a new matrix Z12. Z11 and Z12 can be reshaped as g11 ¼row(Z11) and g12 ¼row(Z12), where row (ci,j)¼[c1,1,y,c1j,y,ci,1,y,ci,j]T denotes the operator that stacks the row of a matrix into a column vector. When the signals of Q pulses period are transmitted, we have X1g ¼ ½g11 , , g1Q and X2g ¼ ½g21 , , g2Q . The covariance matrix of X1g and X2g can be written as follows: RX11 ¼ EðX1g XH 1g Þ ¼ CX11 þ N11
ð18Þ
RX12 ¼ EðX2g XH 2g Þ ¼ CX12 þ N12
ð19Þ
After eliminating the effect of the noise [4], we can obtain CX11 and CX12. To obtain the close form solution of DOA and DOD of targets, we construct a matrix C ¼ CX12 C#X11 , where C#X11 is the pseudo inverse of CX11. Let fu1 ,u2 , uP g and fv1 ,v2 , vP g be the left and right singular vectors of CX11, respectively. Just as the method proposed in [12], we can write CK1 ¼K1u. Thereby, estimates of the DOAs and DODs can be obtained by eigen-decomposition C¼UKUH, where K ¼ diagðl1 , , lP1 Þ represent the P1 nonzero eigenvalues of C and the corresponding eigenvectors. The DOAs of the
B. Dang et al. / Signal Processing 92 (2012) 1404–1410
targets are
Im
y1ri ¼ arcsin +ðli Þl=2pdr , i ¼ 1, ,P 1
X X 1 N1 1 M1 arcsin N1 j ¼ 1 M1 k ¼ 1
+
! uðj1ÞM þ k þ 1,i l , uðj1ÞM þ k,i 2pdr
M
| z − vii |≤
ð20Þ
where +(li) denotes the phase of li. l denotes the wavelength. The DODs can de estimated by all the elements of U: y1ti ¼
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vii ≥
M
| vij | i =1,i ≠ j
| vij |
All eigenvalues are on the positive axis
∀j
i =1,i ≠ j
Re
i ¼ 1, ,P 1
vii
ð21Þ
λ max( R )
where u(j 1)M þ k,i is the (j 1)Mþ kth row and ith column element of U. 4. Convergence analyses of the TPE decorrelator 4.1. Selection of a In (16), the factor a has to satisfy 9lj ðXÞ9o 1 8j , which means 9lmax ðXÞ9o 1, where lmax ðXÞ represents the largest eigenvalue of R. In turn, we have 9almax ðRÞ19 o 1. Since the signal correlation matrix R is always positive definite and Hermitian, all eigenvalues are real and positive. Hence, all eigenvalues are on the positive axis of the complex plane. Thereby, the factor a can be chosen on 0oao
2
ð22Þ
lmax ðRÞ
To achieve faster convergence speed, we need more accurate estimation of lmax ðRÞ. However, the computation of lmax ðRÞ is complicated and is therefore often substituted with an upper bound 0 r lmax ðRÞ rtraceðRÞ
M X
9vij 9
ð24Þ
i ¼ 1,iaj
where z ¼vii is the center of each Gershgorin circle. The N Gershgorin circles are concentric, as the diagonal elements of correlation matrix R are the same. Furthermore, for MIMO radar systems, the signal autocorrelation matrices with weak cross-correlation are normally diagonal-dominant vii Z
M X
9vij 9
8j
diagonal-dominant. From Fig. 2, the largest eigenvalue can be restricted to
lmax ðRÞ r2vii ¼
2traceðRÞ MK
ð26Þ
Hence, we can set the scaling factor a to
a¼
MK traceðRÞ
ð27Þ
The effectiveness of the proposed method will be verified by simulations in Section 5. 4.2. Determination of the convergence speed
ð23Þ
where trace(R) represents the trace of R. In [13], the factor a was set as 2=traceðRÞ which can be easily calculated but it is very coarse, while in [14] the optimal factor a was taken as 2=ðlmax ðRÞ þ lmin ðRÞÞ, where lmin ðRÞ is the smallest eigenvalue of R. These results are well-known and have previously been presented in [15]. However, the estimation of the largest eigenvalue needs a complexity of O((MK)2). Both these methods set the factor a to satisfy (22). In this paper, we make a simple approach to 2=lmax ðRÞ. According to the Gershgorin theorem [16], all eigenvalues of an arbitrary N-dimensional square matrix VN N ¼(vi,j) lie in the N Gershgorin circles which can be expressed as 9zvii 9 r
Fig. 2. Example for eigenvalue distribution of matrix R.
ð25Þ
i ¼ 1,iaj
Note that the TPE decorrelator is valid when the signal auto-correlation matrices are full rank and normally
In [14], it has been proved that the radius of the circle in Fig. 2, which includes all eigenvalues of R, determines the convergence speed of polynomial expansion (PE) detector. So the different waveform sets will lead to different convergence speed of PE detector. Moreover, the convergence speed of the TPE decorrelator can be influenced by the normalization factor a. In the following, the determination of the convergence speed of the TPE decorrelator is discussed. Considering the upper bound for the approximation error of the polynomial expansion decorrelator given in [14], the estimation error of the ith _ stage of Y TPE can also be bounded as
oi _ oi _ _ _ :Y TPE ð1ÞY TPE ð0Þ:r :XY=a: :Y TPE Y DMI : r 1o 1o ð28Þ where : : denotes the norm of a vector or a matrix. o is an arbitrary norm satisfy :X:r o o 1. Since, it can be observed from (13) that the value of :X: varies with a. This implies that the approximation error of the TPE can be reduced by optimizing a. Furthermore, the factor o in (28) causes an exponential decrease in the approximation error of the TPE decorrelator with the number of stages. 5. Simulations It is assumed that three transmit antennas and three receive antennas are configured with half-wavelength
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spacing between adjacent elements. Furthermore, two optimized polyphase code sets (Deng’s coeds [6] and Khan’s codes [7]) and Hadamard codes with L¼128 and M ¼3 are utilized (marked as D-, K- and H-, respectively). All target are assumed to have unit RCS. The signal-tonoise ratio (SNR) is defined as the ratio of one transmitter signal power and the noise power. We also assume that
the targets occupy the first eight range bins (K¼ 8). Each range bin has five targets. The root mean square error (RMSE) of the DOD and DOA of the targets in the third range bin is shown in the following figures. In Figs. 3 and 4, we compare the target location performances of six cases. The stage Ns is set to 10. In the first three cases (marked as D-TPE, K-TPE, D-DMI,
101
D-TPE DOD D-TPE DOA K-TPE DOD K-TPE DOA D-DMI DOD D-DMI DOA D-NWD DOD D-NWD DOA D-STC DOD D-STC DOA H-Orth DOD H-Orth DOA
RMSE (degree)
100
10-1
10-2 -10
-5
0
5
10
15
20
SNR (dB) Fig. 3. Average RMSE with the influence of SNR (the number of pulse¼ 200, Ns ¼10).
101 D-TPE DOD D-TPE DOA K-TPE DOD K-TPE DOA D-DMI DOD D-DMI DOA D-NWD DOD D-NWD DOA D-STC DOD D-STC DOA H-Orth DOD H-Orth DOA
RMSE (degree)
100
10-1
10-2 50
100
150
200 250 300 the number of pulse
350
400
Fig. 4. Average RMSE with the influence of the number of pulse (SNR¼ 10 dB, Ns ¼10).
450
B. Dang et al. / Signal Processing 92 (2012) 1404–1410
respectively), the waveform decorrelation methods of TPE and DMI have been used to cancel the interference of nearby range bins with Deng’s codes and Khan’s codes, respectively. In the forth case, using Deng’s codes (marked as D-NWD), there is no waveform decorrelation operation and the DOD and DOA are estimated directly after range compression. In the fifth case (marked as D-STC), the STC based correlation reduction method in [8] is used for Deng’s codes. In the last case (marked as H-Orth), we use
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the Hadamard codes in the transmitters and the targets exist only in the second range bin. In turn, there are no nearby range bin interference and the effects of autocorrelation and cross-correlation of the waveforms. It can be observed from Figs. 3 and 4 that the ESPRIT method does not work well using Deng’s codes and Khan’s codes without waveform decorrelation, even if the transmit code sets had been optimized carefully. With the help of STC to mitigate waveform cross-correlation [8], the location
RMSE (degree)
100
N =1 D-TPE DOD N =2 D-TPE DOD
10-1
N =3 D-TPE DOD N =4 D-TPE DOD N =5 D-TPE DOD H-Orth DOD N =1 D-TPE DOA N =2 D-TPE DOA N =3 D-TPE DOA N =4 D-TPE DOA N =5 D-TPE DOA
10-2 -10
H-Orth DOA
-5
0
5 SNR (dB)
10
15
20
Fig. 5. Convergence speed of average RMSE with the influence of SNR (the number of pulse ¼200).
N =1 D-TPE DOD N =2 D-TPE DOD
100
N =3 D-TPE DOD N =4 D-TPE DOD N =5 D-TPE DOD
RMSE (degree)
H-Orth DOD N =1 D-TPE DOA N =2 D-TPE DOA N =3 D-TPE DOA N =4 D-TPE DOA N =5 D-TPE DOA H-Orth DOA
10-1
50
100
150
200 250 300 the number of pulse
350
400
Fig. 6. Convergence speed of average RMSE with the influence of the number of pulse (SNR¼ 10 dB).
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B. Dang et al. / Signal Processing 92 (2012) 1404–1410
performance can be improved greatly. However, as shown in Figs. 3 and 4, the STC based waveform correlation reduction still lead to performance loss compared to DMI and H-Orth cases due to the residual auto-correlation and cross-correlation. On the other hand, the performance of DTPE and K-TPE can approach the DMI and H-Orth cases very well using the TPE based waveform decorrelation. Furthermore, Figs. 5 and 6 show the convergence performance of the TPE based waveform correlation cancellation with variable of SNR and the number of pulse, respectively, where Ns is set from 1 to 5. We can see that the proposed method converges rapidly to the optimal condition which has no nearby range bin interference. Although the computation complexity is much lower when Ns is set from 1 to 4, the performance loss for target location is too high. When Ns is set to 5, the proposed method can have almost the same performance as the H-Orth case. It is also shown that the TPE decorrelator with a computational complexity of O(5N (MK)2) can achieve almost the same performance as the DMI decorrelator with a computational complexity of Oð4=3ðMKÞ3 þ 8 ðMKÞ2 10=3MKÞ. Under the simulation condition in this section, the computation complexity of the DMI method is about three times of the TPE method. Especially, for large matrices, the computational complexity of the DMI decorrelator is large and the proposed TPE decorrelator is a good alternative. 6. Conclusion A TPE based low complexity waveform correlation cancellation algorithm is proposed for MIMO radar receivers. It is shown that the computation load of the proposed method is much lower than that of DMI method using of Cayley–Hamilton theorem and Taylor series in matrix form. The impact of interferences from nearby range bins of interest which are caused by the autocorrelation and cross-correlation of transmit waveforms can be canceled effectively by the proposed method.
Acknowledgments This work is supported by a grant from the National Science Fund for Distinguished Young Scholars (No. 60825104), the Major Science Basic Research Development
Program of China (973 Program) (No. 2011CB707001) and the Fundamental Research Funds for the Central Universities. The authors would like to thank the anonymous reviewers for their thoughtful and to-the-point comments and suggestions, which greatly improved the manuscript. References [1] A. Haimovich, R. Blum, L. Cimini, MIMO radar with widely separated antennas, IEEE Signal Processing Magazine 25 (1) (2008) 116–129. [2] Jian Li, P. Stoica, MIMO radar with colocated antennas, IEEE Signal Processing Magazine 24 (5) (2007) 106–114. [3] Haidong Yan, Jun Li, Guisheng Liao, Multitarget identification and localization using bistatic MIMO radar systems, EURASIP Journal on Advances in Signal Processing (2008) (Article ID 283483, 8 pp). [4] Jun Li, Guisheng Liao, Kejiang Ma, Cao Zeng, Waveform decorrelation for multitarget localization in bistatic MIMO radar systems, in: Proceedings of the 2010 IEEE Radar Conference, Washington, May 2010, pp. 21–24. [5] J. Li, P. Stoica, X. Zheng, Signal synthesis and receiver design for MIMO radar imaging, IEEE Transactions on Signal Processing 56 (1) (Aug. 2008) 3959–3968. [6] Deng Hai, Polyphase code design for orthogonal netted radar systems, IEEE Transactions on Signal Processing 52 (2004) 3126–3135. [7] H.A. Khan, Y. Zhang, C. Ji, C.J. Stevens, D.J. Edwards, D.O. Brien, Optimizing polyphase sequences for orthogonal netted radar, IEEE Signal Processing Letters 13 (10) (Oct. 2006) 589–592. [8] X. Song, S. Zhou, P. Willett, Multipath interference mitigation for a shared spectrum MIMO radar system, in: Proceedings of the 2010 IEEE Radar Conference, Washington, May 2010. [9] X. Song, S. Zhou, P. Willett, Enhanced multistatic radar resolution via STC, in: Proceedings of the 2010 IEEE Radar Conference, CA, Pasadena, May 2009. [10] X. Song, S. Zhou, P. Willett, Reducing the waveform cross correlation of MIMO radar with space-time coding, IEEE Transactions on Signal Processing 58 (8) (Aug. 2010) 4213–4224. [11] Wei Xing, Generalization of Cayley–Hamilton theorem for multivariate rational matrices, IEEE Transactions on Automatic Control 54 (3) (March 2009) 631–634. [12] Ming Jin, Guisheng Liao, Jun Li, Joint DOD and DOA estimation for bistatic MIMO radar, Signal Processing 89 (2) (Feb. 2009) 244–251. [13] Z.D. Lei, T.J. Lim, Simplified polynomial-expansion linear detectors for DS-CDMA systems, Electronics Letters 34 (16) (Aug 1998) 1561–1563. [14] G.M.A. Sessler, F.K. Jondral, Low complexity polynomial expansion multiuser detector for CDMA systems, IEEE Transactions on Vehicular Technology 54 (4) (2005) 1379–1391. [15] V.K. Varma, Gottumukkala Hlaing, Minn, Naofal Al-Dhahir, Reduced complexity polynomial expansion approximation to MMSE-DFE, in: Proceedings of the 2009 IEEE Vehicular Technology Conference Fall (VTC 2009-Fall), Anchorage, AK, pp. 1–5. [16] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, UK, 1985.
Contents lists available at SciVerse ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Taylor polynomial expansion based waveform correlation cancellation for bistatic MIMO radar localization Bo Dang, Jun Li, Guisheng Liao n National Laboratory of Radar Signal Processing, Xidian University, Xi’an, Shaanxi 710071, China
a r t i c l e in f o
abstract
Article history: Received 4 June 2011 Received in revised form 23 November 2011 Accepted 24 November 2011 Available online 3 December 2011
A Taylor polynomial expansion (TPE) based waveform decorrelation algorithm with low complexity is proposed to cancel the effects of auto-correlation and cross-correlation of transmit waveform of multiple-input multiple-output (MIMO) radar in the receiver. Cayley–Hamilton theorem is first used to expand the direct matrix inversion to the form of polynomial matrix. And then, Taylor series is used to simplify the calculation of the polynomial matrix. Furthermore, a simplified method is proposed to select the normalized factor by exploiting the special properties of signal correlation matrix, where the convergence performance for the approximation error of the TPE decorrelator is also analyzed. Finally, the performance of waveform correlation cancellation is verified by applying it to bistatic MIMO radar multi-target localization. Simulation results demonstrate the effectiveness of the proposed algorithm. & 2011 Elsevier B.V. All rights reserved.
Keywords: Taylor polynomial expansion Waveform decorrelation Multiple-input multiple-output radar Low complexity
1. Introduction Multiple-input multiple-output (MIMO) radars [1–3] are attracting the attentions for its potential advantages, such as increased spatial diversity, extended virtual array aperture, super-resolution and obtaining the transmit angle information at the receiver. Unlike conventional phased-array radars, MIMO radar systems transmit different waveforms from different transmit elements. In turn, the auto-correlation and cross-correlation properties of transmit waveform sets will have great effect on the performance of MIMO radar systems [4]. There have been two typical methods for the reduction of MIMO radar waveform correlations. One is numerical suppression [5–7], which utilizes optimization approach to reduce the waveform auto-correlation and cross-correlation. However, it is difficult to design waveform sets with ideal peak sidelobe level of autocorrelation property as well as cross-correlation one for an arbitrary time delay. Another approach is the space-time coherence
n
Corresponding author: Tel./fax: þ 86 29 88201030. E-mail address: [email protected] (G. Liao).
0165-1684/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2011.11.023
cancellation, which uses specially designed codes to modulate traditional radar waveform and accumulates the pulse compression results to cancel the auto-correlation [8] and cross-correlation [9,10]. Space-time coding (STC) can help to mitigate waveform auto-correlation and crosscorrelation effects in MIMO radar effectively. However, the residual auto-correlation and cross-correlation of transmit waveform still exist, which influence the performance of multi-target localization. Unlike the methods above, a waveform decorrelation algorithm is proposed to cancel the interferences from nearby range bins effectively in the receiver [4]. However, due to the high complexity of direct matrix inverse (DMI) computation operation, it is difficult to invert the signal autocorrelation matrix. In this paper, Taylor polynomial expansion (TPE) based waveform correlation cancellation algorithm with low complexity is proposed to cancel the effects of auto-correlation and cross-correlation of transmit waveform in the receiver. It is shown that the complexity of the proposed method can be reduced considerably using Cayley–Hamilton theorem [11] and matrix form of Taylor series. Moreover, the effects of auto-correlation and cross-correlation of the transmit waveforms can be canceled effectively. The performance of
B. Dang et al. / Signal Processing 92 (2012) 1404–1410
the TPE method is verified by applying it to bistatic MIMO radar multi-target localization [4]. The rest of this paper is organized as follows: the bistatic MIMO radar signal model is presented in Section 2. In Section 3, a TPE based waveform correlation cancellation algorithm and multi-target localization is presented. The convergence performance of the TPE decorrelator is discussed in Section 4, where a novel method for the selection of the normalization factor to ensure convergence speed of matrix form of Taylor series is presented. The simulation results and performance evaluation are provided in Section 5. Finally, we conclude the paper in Section 6. 2. Bistatic MIMO radar signal model Consider a bistatic MIMO radar equipped with M transmit elements and N receive elements, where the spacing between adjacent elements at the transmitter and receiver is dt and dr, respectively. The array structure of bistatic MIMO radar used in this paper is illustrated in Fig. 1. We assume the range of the target is much larger than the aperture of transmit array and receive array. We assume that there are Pk targets located at (hkt,hkr) in the kth range bin, where hkt ¼ ½yk1t , ykpt T denote the angles of the target with respect to the transmit array (i.e. DODs) and hkr ¼ ½yk1r , ykpr T denote the angles with respect to the receive array (i.e. DOAs). Assuming K range bins have the targets. The receive steer vector and transmit steer vector can be expressed as akp ¼ ½1,ej2pdr sinykpr =l , ,ej2pðN1Þdr sinykpr =l T j2pdt sinykpt =l
bkp ¼ ½1,e
ð1Þ
j2pðM1Þdt sinykpt =l T
, ,e
ð2Þ
The transmit waveform can be represented as Sk ¼ ½s1k ,s2k , ,sMk T , where 2 3 0,. . .,0 , si ðlÞ, 0,. . .,0 |fflfflffl{zfflfflffl} |fflfflffl{zfflfflffl} 5 sik ¼ 4 k
Kk
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and l is the slow-time index. si ðlÞ ¼ ½si ð1Þ, si ðLÞT , i ¼ 1, ,M denote the transmitted coded pulse of the ith transmitter, where L represents the number of codes in one pulse period. The received target echoes can be described as follows: D¼
Pk K X X
bqkp akp bTkp Sk ej2pf qkp tl þN, l ¼ 1,2 L
ð3Þ
k¼1p¼1
where bqkp and fqkp represents the RCS of the pth target in the kth range bin in the qth pulse and Doppler frequency, respectively. NACN (L þ K) denotes the noise matrix and the columns of N are independent and identically distributed (i.i.d) circularly symmetric complex Gaussian random vectors with zero mean and an unknown covariance matrix. Considering the matrix form, (3) can be written as follows: D¼
K X
Hk Sk þ N
ð4Þ
k¼1
where Hk ¼
Pk P p¼1
bk akp bTkp ej2pf k tl . Matched by the trans-
mitted waveform Si expressed as Yi ¼ ð
K X
ði ¼ 1,2 ,KÞ, the output can be
H Hk Sk ÞSH i þ NSi
ð5Þ
k¼1
The matched results can be stacked as follows: Y ¼ ½Y1 ,. . .,YK ¼ HR þ NSH
ð6Þ
where 2 3 S1 6 7 S ¼ 4 ^ 5 2 C MKðL þ KÞ SK YACN MK and H¼[H1,y, HK]. The matrix R¼ SSHACMK MK can be obtained by constructing the transmit waveforms given in advance. Obviously, the correlation
θ
The pth target
ray Ar e v cei Re
θ
Transmit Array
The k range bin The k+1 range bin Fig. 1. Bistatic MIMO radar scenario.
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matrix of waveforms R will contaminate the matrix H, in which the parameters to be estimated are included. 3. TPE based waveform correlation cancellation and multi-target location
Ns 1 1 X _ Y TPE ¼ YðI þXÞ1 ¼ Y ðXÞi
From (6), it is obvious that the effects of auto-correlation and cross-correlation of the transmit waveform, which is included in the correlation matrix R, can be canceled by multiplying the inverse of R [4] _ Y DMI ¼ YLDMI ¼ YR1 ¼ H þNSH R1 ð7Þ However, the DMI decorrelator is difficult to implement as the inversion of R has very high computation load. The computation complexity for DMI decorrelator via Cholesky-decomposition [14] is Oð4=3ðMKÞ3 þ 8 ðMKÞ2 10=3MKÞ. In the following section, a low computation load method is proposed to cancel the effect of the correlation waveforms. 3.1. Taylor series based polynomial expansion A class of multistage linear receivers can be obtained by applying a polynomial expansion in R to the matched filter outputs. Here the soft estimator L is produced by considering the Cayley–Hamilton theorem [11] L¼
Ns X
wi Ri
ð8Þ
i¼0
where NS denotes the number of stages of the decorrelator and hence the accuracy of the approximation, and wi represents the weight of the decorrelator. A polynomial expansion approaching in R to the matched filter bank output Y can be applied to approximate the linear mapping matrix L in DMI case. It can be proved that when NS is great enough [11] Ns X
wi Ri R1
ð9Þ
i¼0
Therefore, the linear decorrelator matrix of TPE can be written as ! Ns X _ Y TPE ¼ YLTPE ¼ ðHR þ NSH Þ wi Ri ð10Þ i¼0
From (10), the matrix multiplication to calculate Ri still has a complexity of O((MK)3). In turn, we take the Taylor series into account to operate (10) [14] ð1 þxÞ1 ¼
1 X
ðxÞi
9x9 o 1
ð11Þ
i¼0
For matrix form, (11) can be expressed as ðI þXÞ1 ¼
1 X
ðXÞi
9lj ðXÞ9o 1
8j
ð12Þ
i¼0
where lj(X) represents the jth eigenvalue of X. If we assume (IþX) 1 ¼ aR 1 in DMI case, XACMK MK can be written as X ¼ aRI
where the selection of factor a is set to satisfy 9lj ðXÞ9o 1 8j. By restricting the sum in (12) up to the NS order, the modified linear decorrelator matrix of TPE can be written as
ð13Þ
a
a
ð14Þ
i¼0
However, the calculation of ( X)i for iZ2 comprises matrix multiplication, which has the complexity of O((MK)3). Here, we rewrite (14) as 1 _ Y TPE ¼ ðYYX þ YX2 YX2i1 þYX2i Þ
a
ð15Þ
We can conclude (15) as 1 _ _ Y TPE ðiþ 1Þ ¼ YY TPE ðiÞX
a
ð16Þ
where YACN MK. The computation complexity of the TPE decorrelator is O(Ns N (MK)2). Obviously, the complexity of TPE method is directly proportional to the number of stages NS, which is determined by a. We will discuss the selection of a in detail in the next section. 3.2. ESPRIT based multi-target location To locate the targets, an ESPRIT based multi-target location method [4] is used to estimate the DOD and DOA of the targets. For simple discussion but without loss of generality, the target in the first range bin is considered. From (10) we can obtain the data of the first range bin Z1 ¼ H1 þ V1 ¼
P1 X
b1p a1p bT1p ej2pf 1p t þ V1
ð17Þ
p¼1
_ where Z1 and V1 are the first M columns of Y TPE and H NS LTPE, respectively. We extract the first N 1 rows of Z1 and define them as a new matrix Z11 and define the last N 1 rows of Z1 as a new matrix Z12. Z11 and Z12 can be reshaped as g11 ¼row(Z11) and g12 ¼row(Z12), where row (ci,j)¼[c1,1,y,c1j,y,ci,1,y,ci,j]T denotes the operator that stacks the row of a matrix into a column vector. When the signals of Q pulses period are transmitted, we have X1g ¼ ½g11 , , g1Q and X2g ¼ ½g21 , , g2Q . The covariance matrix of X1g and X2g can be written as follows: RX11 ¼ EðX1g XH 1g Þ ¼ CX11 þ N11
ð18Þ
RX12 ¼ EðX2g XH 2g Þ ¼ CX12 þ N12
ð19Þ
After eliminating the effect of the noise [4], we can obtain CX11 and CX12. To obtain the close form solution of DOA and DOD of targets, we construct a matrix C ¼ CX12 C#X11 , where C#X11 is the pseudo inverse of CX11. Let fu1 ,u2 , uP g and fv1 ,v2 , vP g be the left and right singular vectors of CX11, respectively. Just as the method proposed in [12], we can write CK1 ¼K1u. Thereby, estimates of the DOAs and DODs can be obtained by eigen-decomposition C¼UKUH, where K ¼ diagðl1 , , lP1 Þ represent the P1 nonzero eigenvalues of C and the corresponding eigenvectors. The DOAs of the
B. Dang et al. / Signal Processing 92 (2012) 1404–1410
targets are
Im
y1ri ¼ arcsin +ðli Þl=2pdr , i ¼ 1, ,P 1
X X 1 N1 1 M1 arcsin N1 j ¼ 1 M1 k ¼ 1
+
! uðj1ÞM þ k þ 1,i l , uðj1ÞM þ k,i 2pdr
M
| z − vii |≤
ð20Þ
where +(li) denotes the phase of li. l denotes the wavelength. The DODs can de estimated by all the elements of U: y1ti ¼
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vii ≥
M
| vij | i =1,i ≠ j
| vij |
All eigenvalues are on the positive axis
∀j
i =1,i ≠ j
Re
i ¼ 1, ,P 1
vii
ð21Þ
λ max( R )
where u(j 1)M þ k,i is the (j 1)Mþ kth row and ith column element of U. 4. Convergence analyses of the TPE decorrelator 4.1. Selection of a In (16), the factor a has to satisfy 9lj ðXÞ9o 1 8j , which means 9lmax ðXÞ9o 1, where lmax ðXÞ represents the largest eigenvalue of R. In turn, we have 9almax ðRÞ19 o 1. Since the signal correlation matrix R is always positive definite and Hermitian, all eigenvalues are real and positive. Hence, all eigenvalues are on the positive axis of the complex plane. Thereby, the factor a can be chosen on 0oao
2
ð22Þ
lmax ðRÞ
To achieve faster convergence speed, we need more accurate estimation of lmax ðRÞ. However, the computation of lmax ðRÞ is complicated and is therefore often substituted with an upper bound 0 r lmax ðRÞ rtraceðRÞ
M X
9vij 9
ð24Þ
i ¼ 1,iaj
where z ¼vii is the center of each Gershgorin circle. The N Gershgorin circles are concentric, as the diagonal elements of correlation matrix R are the same. Furthermore, for MIMO radar systems, the signal autocorrelation matrices with weak cross-correlation are normally diagonal-dominant vii Z
M X
9vij 9
8j
diagonal-dominant. From Fig. 2, the largest eigenvalue can be restricted to
lmax ðRÞ r2vii ¼
2traceðRÞ MK
ð26Þ
Hence, we can set the scaling factor a to
a¼
MK traceðRÞ
ð27Þ
The effectiveness of the proposed method will be verified by simulations in Section 5. 4.2. Determination of the convergence speed
ð23Þ
where trace(R) represents the trace of R. In [13], the factor a was set as 2=traceðRÞ which can be easily calculated but it is very coarse, while in [14] the optimal factor a was taken as 2=ðlmax ðRÞ þ lmin ðRÞÞ, where lmin ðRÞ is the smallest eigenvalue of R. These results are well-known and have previously been presented in [15]. However, the estimation of the largest eigenvalue needs a complexity of O((MK)2). Both these methods set the factor a to satisfy (22). In this paper, we make a simple approach to 2=lmax ðRÞ. According to the Gershgorin theorem [16], all eigenvalues of an arbitrary N-dimensional square matrix VN N ¼(vi,j) lie in the N Gershgorin circles which can be expressed as 9zvii 9 r
Fig. 2. Example for eigenvalue distribution of matrix R.
ð25Þ
i ¼ 1,iaj
Note that the TPE decorrelator is valid when the signal auto-correlation matrices are full rank and normally
In [14], it has been proved that the radius of the circle in Fig. 2, which includes all eigenvalues of R, determines the convergence speed of polynomial expansion (PE) detector. So the different waveform sets will lead to different convergence speed of PE detector. Moreover, the convergence speed of the TPE decorrelator can be influenced by the normalization factor a. In the following, the determination of the convergence speed of the TPE decorrelator is discussed. Considering the upper bound for the approximation error of the polynomial expansion decorrelator given in [14], the estimation error of the ith _ stage of Y TPE can also be bounded as
oi _ oi _ _ _ :Y TPE ð1ÞY TPE ð0Þ:r :XY=a: :Y TPE Y DMI : r 1o 1o ð28Þ where : : denotes the norm of a vector or a matrix. o is an arbitrary norm satisfy :X:r o o 1. Since, it can be observed from (13) that the value of :X: varies with a. This implies that the approximation error of the TPE can be reduced by optimizing a. Furthermore, the factor o in (28) causes an exponential decrease in the approximation error of the TPE decorrelator with the number of stages. 5. Simulations It is assumed that three transmit antennas and three receive antennas are configured with half-wavelength
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spacing between adjacent elements. Furthermore, two optimized polyphase code sets (Deng’s coeds [6] and Khan’s codes [7]) and Hadamard codes with L¼128 and M ¼3 are utilized (marked as D-, K- and H-, respectively). All target are assumed to have unit RCS. The signal-tonoise ratio (SNR) is defined as the ratio of one transmitter signal power and the noise power. We also assume that
the targets occupy the first eight range bins (K¼ 8). Each range bin has five targets. The root mean square error (RMSE) of the DOD and DOA of the targets in the third range bin is shown in the following figures. In Figs. 3 and 4, we compare the target location performances of six cases. The stage Ns is set to 10. In the first three cases (marked as D-TPE, K-TPE, D-DMI,
101
D-TPE DOD D-TPE DOA K-TPE DOD K-TPE DOA D-DMI DOD D-DMI DOA D-NWD DOD D-NWD DOA D-STC DOD D-STC DOA H-Orth DOD H-Orth DOA
RMSE (degree)
100
10-1
10-2 -10
-5
0
5
10
15
20
SNR (dB) Fig. 3. Average RMSE with the influence of SNR (the number of pulse¼ 200, Ns ¼10).
101 D-TPE DOD D-TPE DOA K-TPE DOD K-TPE DOA D-DMI DOD D-DMI DOA D-NWD DOD D-NWD DOA D-STC DOD D-STC DOA H-Orth DOD H-Orth DOA
RMSE (degree)
100
10-1
10-2 50
100
150
200 250 300 the number of pulse
350
400
Fig. 4. Average RMSE with the influence of the number of pulse (SNR¼ 10 dB, Ns ¼10).
450
B. Dang et al. / Signal Processing 92 (2012) 1404–1410
respectively), the waveform decorrelation methods of TPE and DMI have been used to cancel the interference of nearby range bins with Deng’s codes and Khan’s codes, respectively. In the forth case, using Deng’s codes (marked as D-NWD), there is no waveform decorrelation operation and the DOD and DOA are estimated directly after range compression. In the fifth case (marked as D-STC), the STC based correlation reduction method in [8] is used for Deng’s codes. In the last case (marked as H-Orth), we use
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the Hadamard codes in the transmitters and the targets exist only in the second range bin. In turn, there are no nearby range bin interference and the effects of autocorrelation and cross-correlation of the waveforms. It can be observed from Figs. 3 and 4 that the ESPRIT method does not work well using Deng’s codes and Khan’s codes without waveform decorrelation, even if the transmit code sets had been optimized carefully. With the help of STC to mitigate waveform cross-correlation [8], the location
RMSE (degree)
100
N =1 D-TPE DOD N =2 D-TPE DOD
10-1
N =3 D-TPE DOD N =4 D-TPE DOD N =5 D-TPE DOD H-Orth DOD N =1 D-TPE DOA N =2 D-TPE DOA N =3 D-TPE DOA N =4 D-TPE DOA N =5 D-TPE DOA
10-2 -10
H-Orth DOA
-5
0
5 SNR (dB)
10
15
20
Fig. 5. Convergence speed of average RMSE with the influence of SNR (the number of pulse ¼200).
N =1 D-TPE DOD N =2 D-TPE DOD
100
N =3 D-TPE DOD N =4 D-TPE DOD N =5 D-TPE DOD
RMSE (degree)
H-Orth DOD N =1 D-TPE DOA N =2 D-TPE DOA N =3 D-TPE DOA N =4 D-TPE DOA N =5 D-TPE DOA H-Orth DOA
10-1
50
100
150
200 250 300 the number of pulse
350
400
Fig. 6. Convergence speed of average RMSE with the influence of the number of pulse (SNR¼ 10 dB).
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performance can be improved greatly. However, as shown in Figs. 3 and 4, the STC based waveform correlation reduction still lead to performance loss compared to DMI and H-Orth cases due to the residual auto-correlation and cross-correlation. On the other hand, the performance of DTPE and K-TPE can approach the DMI and H-Orth cases very well using the TPE based waveform decorrelation. Furthermore, Figs. 5 and 6 show the convergence performance of the TPE based waveform correlation cancellation with variable of SNR and the number of pulse, respectively, where Ns is set from 1 to 5. We can see that the proposed method converges rapidly to the optimal condition which has no nearby range bin interference. Although the computation complexity is much lower when Ns is set from 1 to 4, the performance loss for target location is too high. When Ns is set to 5, the proposed method can have almost the same performance as the H-Orth case. It is also shown that the TPE decorrelator with a computational complexity of O(5N (MK)2) can achieve almost the same performance as the DMI decorrelator with a computational complexity of Oð4=3ðMKÞ3 þ 8 ðMKÞ2 10=3MKÞ. Under the simulation condition in this section, the computation complexity of the DMI method is about three times of the TPE method. Especially, for large matrices, the computational complexity of the DMI decorrelator is large and the proposed TPE decorrelator is a good alternative. 6. Conclusion A TPE based low complexity waveform correlation cancellation algorithm is proposed for MIMO radar receivers. It is shown that the computation load of the proposed method is much lower than that of DMI method using of Cayley–Hamilton theorem and Taylor series in matrix form. The impact of interferences from nearby range bins of interest which are caused by the autocorrelation and cross-correlation of transmit waveforms can be canceled effectively by the proposed method.
Acknowledgments This work is supported by a grant from the National Science Fund for Distinguished Young Scholars (No. 60825104), the Major Science Basic Research Development
Program of China (973 Program) (No. 2011CB707001) and the Fundamental Research Funds for the Central Universities. The authors would like to thank the anonymous reviewers for their thoughtful and to-the-point comments and suggestions, which greatly improved the manuscript. References [1] A. Haimovich, R. Blum, L. Cimini, MIMO radar with widely separated antennas, IEEE Signal Processing Magazine 25 (1) (2008) 116–129. [2] Jian Li, P. Stoica, MIMO radar with colocated antennas, IEEE Signal Processing Magazine 24 (5) (2007) 106–114. [3] Haidong Yan, Jun Li, Guisheng Liao, Multitarget identification and localization using bistatic MIMO radar systems, EURASIP Journal on Advances in Signal Processing (2008) (Article ID 283483, 8 pp). [4] Jun Li, Guisheng Liao, Kejiang Ma, Cao Zeng, Waveform decorrelation for multitarget localization in bistatic MIMO radar systems, in: Proceedings of the 2010 IEEE Radar Conference, Washington, May 2010, pp. 21–24. [5] J. Li, P. Stoica, X. Zheng, Signal synthesis and receiver design for MIMO radar imaging, IEEE Transactions on Signal Processing 56 (1) (Aug. 2008) 3959–3968. [6] Deng Hai, Polyphase code design for orthogonal netted radar systems, IEEE Transactions on Signal Processing 52 (2004) 3126–3135. [7] H.A. Khan, Y. Zhang, C. Ji, C.J. Stevens, D.J. Edwards, D.O. Brien, Optimizing polyphase sequences for orthogonal netted radar, IEEE Signal Processing Letters 13 (10) (Oct. 2006) 589–592. [8] X. Song, S. Zhou, P. Willett, Multipath interference mitigation for a shared spectrum MIMO radar system, in: Proceedings of the 2010 IEEE Radar Conference, Washington, May 2010. [9] X. Song, S. Zhou, P. Willett, Enhanced multistatic radar resolution via STC, in: Proceedings of the 2010 IEEE Radar Conference, CA, Pasadena, May 2009. [10] X. Song, S. Zhou, P. Willett, Reducing the waveform cross correlation of MIMO radar with space-time coding, IEEE Transactions on Signal Processing 58 (8) (Aug. 2010) 4213–4224. [11] Wei Xing, Generalization of Cayley–Hamilton theorem for multivariate rational matrices, IEEE Transactions on Automatic Control 54 (3) (March 2009) 631–634. [12] Ming Jin, Guisheng Liao, Jun Li, Joint DOD and DOA estimation for bistatic MIMO radar, Signal Processing 89 (2) (Feb. 2009) 244–251. [13] Z.D. Lei, T.J. Lim, Simplified polynomial-expansion linear detectors for DS-CDMA systems, Electronics Letters 34 (16) (Aug 1998) 1561–1563. [14] G.M.A. Sessler, F.K. Jondral, Low complexity polynomial expansion multiuser detector for CDMA systems, IEEE Transactions on Vehicular Technology 54 (4) (2005) 1379–1391. [15] V.K. Varma, Gottumukkala Hlaing, Minn, Naofal Al-Dhahir, Reduced complexity polynomial expansion approximation to MMSE-DFE, in: Proceedings of the 2009 IEEE Vehicular Technology Conference Fall (VTC 2009-Fall), Anchorage, AK, pp. 1–5. [16] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, UK, 1985.