Báo cáo hóa học: " Research Article Fast Signal Recovery in the Presence of Mutual Coupling Based on New 2-D Direct Data Domain Approach"

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Hindawi Publishing Corporation EURASIP Journal on Wireless Communications and Networking Volume 2011, Article ID 607679, 9 pages doi:10.1155/2011/607679 Research Article Fast Signal Recovery in the Presence of Mutual Coupling Based on New 2-D Direct Data Domain Approach Ali Azarbar,1 G. R. Dadashzadeh,2 and H. R. Bakhshi2 1 Department of Computer and Information Technology Engineering, Islamic Azad University, Parand Branch, Tehran 37613 96361, Iran 2 Faculty of Engineering, Shahed University, Tehran 33191 18651, Iran Correspondence should be addressed to Ali Azarbar, aliazarbar@piau.ac.ir Received 17 August 2010; Revised 9 December 2010; Accepted 18 January 2011 Academic Editor: Richard Kozick Copyright © 2011 Ali Azarbar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The performance of adaptive algorithms, including direct data domain least square, can be signiﬁcantly degraded in the presence of mutual coupling among array elements. In this paper, a new adaptive algorithm was proposed for the fast recovery of the signal with one snapshot of receiving signals in the presence of mutual coupling, based on the two-dimensional direct data domain least squares (2-D D3 LS) for uniform rectangular array (URA). In this method, inverse mutual coupling matrix was not computed. Thus, the computation was reduced and the signal recovery was very fast. Taking mutual coupling into account, a method was derived for estimation of the coupling coeﬃcient which can accurately estimate the coupling coeﬃcient without any auxiliary sensors. Numerical simulations show that recovery of the desired signal is accurate in the presence of mutual coupling. 1. Introduction On the other hand, many algorithms of the 1-D DOA estimation have been extended to solve the 2-D cases Adaptive antenna arrays are strongly aﬀected by the existence [10, 11]; however, a few have considered the eﬀect of of mutual coupling (MC) eﬀect between antenna elements; mutual coupling or any other array errors [12]. Besides, thus, if the eﬀects of MC are ignored, the system performance most of these proposed adaptive algorithms are based on will not be accurate [1, 2]. Research into compensation for the covariance matrix of the interference. However, these statistical algorithms suﬀer from two major drawbacks. First, the MC has been mainly based on the idea of using open circuit voltages, ﬁrstly proposed by Gupta and Ksienski [2]. they require independent identically-distributed secondary While this method has calculated the mutual impedance, data in order to estimate the covariance matrix of the the presence of other antenna elements has been ignored interference. Unfortunately, the statistics of the interference and a very simpliﬁed current distribution has been assumed may ﬂuctuate rapidly over a short distance, limiting the for each antenna elements. Many eﬀorts have been made availability of homogeneous secondary data. The resulting to compensate for the MC eﬀect for uniform linear array errors in the covariance matrix reduce the ability to sup- (ULA) and uniform circular array (UCA) [2–9]. In [3], an press the interference. The second drawback is that the adaptive algorithm was used to compensate for the MC eﬀect estimation of the covariance matrix requires the storage and in a ULA. In [7], the authors introduced a minimum norm processing of the secondary data. This is computationally technique MC compensation method, which is based on the intensive, requiring many calculations in real-time. Recently, technique in [2] for general arrays with arbitrary elements direct data domain algorithms have been proposed to and more accurate. In [9], a new method was proposed to overcome these drawbacks of statistical techniques [13–16]. compensate for the MC eﬀect which relied on the calculation The approach is to adaptively minimize the interference of a new deﬁnition of mutual impedance. however, the power while maintaining the array gain in the direction of authors did not deal with 2-D DOA estimation problem. the signal. The sample support problem is eliminated by
2 EURASIP Journal on Wireless Communications and Networking where x, A, s, and n denote the received signal vector, Desired Jammer 1 Jammer 2 Jammer M signal steering matrix, signal plus jammers vector and additive ··· white Gaussian noise vector, respectively, deﬁned as: z y x = [x11 (t ), x12 (t ), . . . , x1N (t ), x21 (t ), . . . , P1 ··· x2N (t ), . . . , xP1 (t ), . . . , xPN (t )]T , PN ··· s = [s(t ), J1 (t ), J2 (t ), . . . , JM (t )]T , . . .. . (2) . .. .. .. θ0 n = [n11 (t ), n12 (t ), . . . , n1N (t ), n21 (t ), . . . , ··· 21 ··· 2N n2N (t ), . . . , nP1 (t ), . . . , nPN (t )]T , ϕ0 ··· x 1(N − 1) 1N A = a θ0 , ϕ0 , a θ1 , ϕ1 , . . . , a θM , ϕM , 11 12 Figure 1: URA with N × P elements. where a θm , ϕm = a y θm , ϕm ⊗ ax θm , ϕm , m = 0, 1, 2, . . . , M , avoiding the estimation of a covariance matrix which leads T ax θm , ϕm = 1, β θm , ϕm , . . . , βN −1 θm , ϕm , to enormous savings in the required real-time computations. The performance of this algorithm is aﬀected by the MC T eﬀect, too [17] and must be compensated. a y θm , ϕm = 1, α θm , ϕm , . . . , αP−1 θm , ϕm . Unfortunately, the MC matrix tends to change with (3) time due to environmental factors, so full elimination of We deﬁne β(θm , ϕm ) = exp( j 2π (dx /λ) sin θm cos ϕm ) and its eﬀect and prediction of its variability are impossible. α(θm , ϕm ) = exp( j 2π (d y /λ) sin θm sin ϕm ) which represent Therefore, calibration procedures based upon signal pro- the phase progression of the signal between one element and cessing algorithms are needed to estimate and compensate for the eﬀect of the MC. The most likely way is to the next in the row and column, respectively. The a(θm , ϕm ) is mth signal’s direction manifold vector, superscript (·)T is the carry out some measurements for calibration. However, this procedure has the drawbacks of being time-consuming and transpose operation and the symbol ⊗ denotes the Kronecker very expensive [18]. Some other researches suggested self- tensor. Therefore, by suppression of time dependence in the calibration adaptive algorithms for damping the MC eﬀect phasor notation, complex vector of phasor voltage is: ⎛ ⎞ [19–21]. M In this paper, a new adaptive algorithm was proposed for x = s0 a θ 0 , ϕ 0 + ⎝ J m a θ m , ϕ m ⎠ + n, (4) the fast recovery of the signal with one snapshot of receiving m=1 signals in the presence of mutual coupling, based on 2- D D3 LS algorithm for URA. Then, utilizing the 2-D D3 LS where s0 and Jm are the complex amplitude of the desired signal and mth interferers, respectively. Next, the ﬁrst row algorithm properties, a novel technique for the coupling coeﬃcients estimation, without using any auxiliary sensors from each column is multiplied by β and subtracted from the second row; then the result of each column is multiplied by α is presented. and subtracted from the next column. This cancels out all the This paper is organized as follows. Section 2, conven- tional 2-D D3 LS algorithm is reviewed. In Section 3, a signals and only noise and interferers are left. The ﬁrst row of the matrix in (5) is the constraint to the desired signal which fast adaptive algorithm of direct data domain including mutual coupling eﬀect is presented. In Section 4, a new produces a gain factor of Q. For a conventional adaptive technique is presented for compensation of the MC eﬀect. array system, the K weights wk are used and the relationship between K with P and N can be chosen as K = K1 · K2 , In Section 5, numerical simulations illustrate these proposed K1 = (N + 1)/ 2, K2 = (P + 1)/ 2 [16]. Matrix equation can be techniques which can accurately recover the desired signal in constructed as: the presence of MC. ⎡ ⎤ b2 · · · bK2 −1 b1 b K2 ⎡ ⎤ ⎡⎤ ⎢ ⎥ w1 Q 2. 2-D Direct Data Domain Algorithm ⎢ D1 DK2 ⎥ ⎢ ⎥ ⎢⎥ D2 · · · D(K2 −1) ⎢ ⎥ ⎥ ⎢ w2 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎥ ⎢D D(K2 +1) ⎥ × ⎢ ⎥ = ⎢ ⎥, Consider a URA consisting of N × P equally spaced elements D3 · · · DK2 ⎢ ⎥ ⎢ . ⎥ ⎢.⎥ 2 ⎢ ⎥ ⎢ . ⎥ ⎢.⎥ ⎢. with the spacing of dx in rows (in the x direction) and d y in ⎥ ⎢ . ⎥ ⎢.⎥ ⎢. ⎥ ⎣ ⎦ ⎣⎦ ⎢. columns (in the y direction). The array receives a signal from ⎣ ⎦ wK 0 a known direction (θ0 , ϕ0 ) and M interferers from unknown D(K2 −1) DK2 · · · D(P−2) D(P−1) directions (θm , ϕm ), m = 1, 2, . . . , M as shown in Figure 1. (5) The output of the array voltage can be expressed as where b1 = 1 β · · · βK1 −1 , bi = αi−1 b1 , x = As + n, (6) (1)
EURASIP Journal on Wireless Communications and Networking 3 ⎡ ⎤ xi1 − β−1 xi2 − α−1 x(i+1)1 − β−1 x(i+1)2 xiK1 − β−1 xi(K1 +1) − α−1 x(i+1)K1 − β−1 x(i+1)(K1 +1) ··· ⎢ ⎥ ⎢ ⎥ . . Di = ⎢ ⎥. (7) . . ⎢ ⎥ . . ⎣ ⎦ xi(K1 −1) − β−1 xiK1 − α−1 x(i+1)(K1−1) − β−1 x(i+1)K1 xi(N −1) − β−1 xiN − α−1 x(i+1)(N −1) − β−1 x(i+1)N ··· For simplicity β(θ0 , ϕ0 ) = β and α(θ0 , ϕ0 ) = α. Because the matrix in (5) is not square, the conjugate gradient method (CGM) is used to solve the matrix equation and to obtain the cx y cy cx y weighting solution. It has a good convergence characteristic and converges to the minimum norm solution, even for the singular problem [13]. Now, the amplitude of the recovered cx cx signal is as [16]: K1 K2 1 cx y cy cx y s0 = wi xi+[(i−1)/K1 ](K1 −1) , (8) Q i=1 where w = [w1 , w2 , . . . , wK ]T is the weights vector in the absence of coupling and subscript, [·], denotes rounding down to the integer: Figure 2: Map of mutual coupling. K1 K2 α[(i−1)/K1 ] βi−1−[(i−1)/K1 ]K1 wi . Q= (9) i=1 where C1 and C2 are N × N submatrices of C and can be given 3. 2-D Fast Signal Recovery Algorithm by in the Presence of Mutual Coupling C1 = Toeplitz([1, cx , 0, . . . , 0]), If one assumes that C denotes the mutual coupling matrix (13) (MCM) of the array, the output will be as: C2 = Toeplitz c y , cxy , 0, . . . , 0 . x = CAs + n, (10) ⎛ ⎞ Then, the following equation is derived to recover the desired M x = s0 Ca θ0 , ϕ0 + ⎝ Jm Ca θm , ϕm ⎠ + n. signal in the presence of mutual coupling (Proof in the (11) appendix), notwithstanding to compute the inverse matrix of m=1 MC. Hence, this equation could be reduced the computation Svantesson [6] showed that the coupling between the of the algorithm neighboring elements with the same interspace is almost the same and the magnitude of the mutual coupling coeﬃcient K1 K2 1 between two far apart elements is so small that can be s0 = wci · xi+[(i−1)/K1](K1 −1) , (14) approximated to zero. Thus, a banded symmetric Toeplitz Qc i=1 matrix can be used as a model for the mutual coupling of ULA and URA. In this paper, each sensor is assumed to be where wc = [wc1 , wc2 , . . . , wcK ]T is the weights vector when aﬀected by the coupling of the 8 sensors around it, which is coupling is known and shown in Figure 2. We deﬁne MCM as [12]: K1 K2 ⎡ ⎤ C1 C2 0 · · · 0 α[(i−1)/K1 ] βi−1−[(i−1)/K1 ]K1 wci 0 0 Qc = 1 + βcx + αc y + αβcxy ⎢ ⎥ ⎢C2 C1 C2 · · · 0 0 0⎥ i=1 ⎢ ⎥ ⎢ ⎥ ⎢. . ⎥ (K1 −1)K2 . C = ⎢ . .. ⎥ .. . , (12) ⎢. ⎥ . α[(i−1)/(K1 −1)] βi−1−[(i−1)/(K1 −1)](K1 −1) . + cx + αcxy ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 · · · C2 C1 C2 ⎥ i=1 ⎣ ⎦ × wci+1+[(i−1)/ (K1−1)] 0 0 0 ··· 0 C2 C1 PN ×PN
4 EURASIP Journal on Wireless Communications and Networking K1 (K2 −1) Table 1: Parameters for the desired signal and interferer. α[(i−1)/K1 ] βi−1−[(i−1)/K1 ]K1 wci+K1 + c y + βcxy Magnitude Phase θs ϕs i=1 75◦ 45◦ Signal 1–10 V/m 0 (K1 −1)(K2 −1) 43◦ −77◦ α[(i−1)/(K1 −1)] βi−1−[(i−1)/(K1 −1)](K1 −1) Jammer1 1000 V/m 0 + cxy i=1 × wci+K1 +1+[(i−1)/ (K1−1)] . 10 (15) 9 The conventional recovering of the signal is as the following: 8 1 T −1 Recovered of the signal s0 = wCx , (16) 7 K Q 6 where [·]K denotes, K rows from the vector. C−1 is computationally intensive and requires many calculations 5 in the real-time because evaluation of the inverse requires an Θ([PN ]3 )process (here Θ(·) denotes “on the order of ”). 4 Therefore, (14) can be replaced with (16) and the number of 3 processes would be an Θ(K1 K2 ). 2 4. Mutual Coupling Compensation 1 1 2 3 4 5 6 7 8 9 10 In this section, a new method is presented to estimate the Intensities of the signal coupling coeﬃcients from the properties of the 2-D D3 LS 2D-D3LS without MC algorithm. If the mutual coupling eﬀect is ignored, the 2D-D3LS with MC term (xi j − β−1 xi( j +1) ) − α−1 (x(i+1) j − β−1 x(i+1)( j +1) ), for i = 1, 2, . . . , P − 1 and j = 1, 2, . . . , N − 1 will have no signal Figure 3: Recovered strength of the desired signal in the absence and presence of mutual coupling. components. However, in the presence of MC, for the edge elements in the URA, the above term can be written as the following: x11 − β−1 x12 − α−1 x21 − β−1 x22 Consider a URA with 7 × 7 elements in which the spacing between each two elements in rows and columns is λ/ 2. The −1 −1 = s0 α β cxy + Interferers, array receives the desired signal with one jammer. The signal to noise ratio is 20 dB and other parameters are listed in x11 − β−1 x12 = − β−1 cx + αβ−1 cxy s0 + Interferers, Table 1. The number of adaptive weights chosen for our simu- x11 − α−1 x21 = − α−1 c y + α−1 βcxy s0 + Interferers, lation will be 16 [16]. Jammer is 60 dB stronger than the (17) intensity of the desired signal. The magnitude of incident signal varies from 1 V/m to 10 V/m; but jammer intensities As is seen in (17), when there are no interferers, the equations are constant as given in Table 1. Figure 3 shows the accuracy can be solved. In this paper, it is assumed that dx = d y = d; of the recovered signal in the presence of MC using new so cx = c y . The above equations can be solved in order formulation (18) with comparison to the ideal recovering. to estimate cx , c y , and cxy . Once the system estimates the Figure 4 shows the result of the recovered signal in the coupling coeﬃcient, it needs only one snapshot of the data in presence of MC, using a new proposed algorithm with order to obtain an acceptable solution. So, when the coupling comparison to the ideal recovering. The expected linear is unknown, ﬁrst we can estimate mutual coupling from (17) relationship is clearly seen and the jammer has been nulled and then, the fast recovering of the signal is as the following: and signal recovered correctly. K1 K2 1 Later on, the performance of the proposed method is s0 = wci · xi+[(i−1)/K1 ](K1 −1) , (18) illustrated by the various simulations. The amplitude of Qc i=1 the desired signal accuracy is measured by the root mean- squared error (RMSE), and L = 100 is the number of Monte where s0 is the estimation of s0 and Qc is Qc with replacement of cx , c y , cxy with cx , c y , cxy . Carlo runs. Figure 5 shows the RMSE of the estimated coupling coeﬃcients versus signal-to-noise ratio (SNR). Figure 6 5. Numerical Examples shows the RMSE of the estimated amplitude of the desired In this section, the capability of MC compensation for signal, versus SNR. For high SNR, error is very low and in the proposed algorithm will be tested with two examples. case there is no noise, new formulation is equal to the ideal.
EURASIP Journal on Wireless Communications and Networking 5 30 10 9 RMSE (%) of coupling coeﬃcient 25 8 Recovered of the signal 20 7 6 15 5 10 4 3 5 2 0 10 15 20 25 30 35 40 1 1 2 3 4 5 6 7 8 9 10 S/N (dB) Intensities of the signal cx , c y cxy Figure 4: Recovered strength of the desired signal with the proposed algorithm in the presence of mutual coupling. Figure 5: RMSE of the coupling coeﬃcients versus the SNR. 20 RMSE (%) of amplitude of recovered signal 6. Conclusion 18 16 In this paper, the problems of 2-D D3 LS algorithms were 14 studied for recovering of the signal in the presence of mutual 12 coupling and driving a new formulation to recover the signal in the presence of MC. Without using the moment of method 10 and impedance matrix calculation, coupling coeﬃcients 8 can be automatically estimated and without computing the 6 inverse matrix, the desired signal can be recovered. Because we did not use the inverse MC matrix, the amount of 4 computation would be reduced. Moreover, simulation results 2 were conﬁrmed when SNR was high and the RMSE of the 0 method was very close to the ideal D3 LS in the absence of 10 15 20 25 30 35 40 MC. S/N (dB) Ideal D3LS Proposed algorithm Appendix Figure 6: RMSE of the recovered amplitude versus the SNR. In this appendix, (8) and (14) are proved. Consider a URA consisting of 5 × 5 elements. The array receives one signal (s) from a known direction (θ0 , ϕ0 ) and one interferer ( j ) (this proof can be extended similarly). From (1), let the received By taking mutual coupling into account, from (11) for each signal at the array in the presence of mutual coupling for each column element be ﬁrst column: xnp = snp + jnp , for n = 1, . . . , 5, p = 1, . . . , 5 , s11 = 1 + βcx + αc y + αβcxy s, (A.1) s12 = βs11 + cx + αcxy s, where snp , jnp are the received signal and jammer at the npth s1 p = βs1( p−1) , for p = 3, 4, 5, element, expressed as 2nd column: s11 = s = gs e jwt , sn( p+1) = βsnp , s(n+1) p = αsnp . s21 = αs11 + c y + βcxy s, (A.2)
6 EURASIP Journal on Wireless Communications and Networking s22 = βs21 + cxy + αcx + α2 cxy s, subtracted from the next column, in the absence of mutual coupling, this will cancel out all the signals and only noise s2 p = βs2( p−1) , for p = 3, 4, 5, and interferer will be left 3 rd column: xnp − β−1 xn( p+1) − α−1 x(n+1) p − β−1 x(n+1)( p+1) , (A.4) s31 = αs21 , for n = 1, 2, . . . , 4, p = 1, 2, . . . , 4. s32 = βs31 + α cxy + αcx + α2 cxy s, The weight vectors should be in a way that produces zero s3 p = βs3( p−1) , for p = 3, 4, 5, output; therefore, a reduced rank matrix is formed in which the weighted sum of all its elements would be zero. In order 4th column: to make the matrix not singular, the additional equation is introduced through the constraint that the same weights s41 = αs31 , when operating on the signal produced a gain factor Q, which is the ﬁrst equation. Therefore, (5) will be s42 = βs41 + α2 cxy + αcx + α2 cxy s, ⎡ ⎤ ⎡⎤ w1 Q ⎡ ⎤ s4 p = βs4( p−1) , for p = 3, 4, 5. ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ b1 b2 b3 ⎥ ⎢w2 ⎥ ⎢ 0 ⎥ ⎢ ⎢ D3 ⎥ × ⎢ . ⎥ = ⎢ . ⎥, (A.3) ⎢D1 D2 ⎥ ⎢ ⎥ ⎢⎥ (A.5) ⎦ ⎢ . ⎥ ⎢.⎥ ⎣ ⎢ . ⎥ ⎢.⎥ ⎣ ⎦ ⎣⎦ D2 D3 D4 (a) Absence of the Mutual Coupling. If the one row from each w9 0 column is multiplied by β and subtracted from the next row and then the result of each column is multiplied by α and ⎡ ··· β2 1 ⎢ ⎢ x11 − β−1 x12 − α−1 x21 − β−1 x22 x13 − β−1 x14 − α−1 x23 − β−1 x24 ··· ⎢ ⎢ ⎢ x12 − β−1 x13 − α−1 x22 − β−1 x23 x14 − β−1 x15 − α−1 x24 − β−1 x25 ··· ⎢ ⎢ ⎢ ⎢ x21 − β−1 x22 − α−1 x31 − β−1 x32 x23 − β−1 x24 − α−1 x33 − β−1 x34 ··· ⎢ ⎢ ⎢ x22 − β−1 x23 − α−1 x32 − β−1 x33 x24 − β−1 x25 − α−1 x34 − β−1 x35 ··· ⎣ ··· αβ2 α x21 − β−1 x22 − α−1 x31 − β−1 x32 x23 − β−1 x24 − α−1 x33 − β−1 x34 ··· x22 − β−1 x23 − α−1 x32 − β−1 x33 x24 − β−1 x25 − α−1 x34 − β−1 x35 ··· x31 − β−1 x32 − α−1 x41 − β−1 x42 x33 − β−1 x34 − α−1 x43 − β−1 x44 ··· x32 − β−1 x33 − α−1 x42 − β−1 x43 x34 − β−1 x35 − α−1 x44 − β−1 x45 ··· (A.6) ⎤ ··· α2 α2 β2 ⎥ ⎥ x31 − β−1 x32 − α−1 x41 − β−1 x42 x33 − β−1 x34 − α−1 x43 − β−1 x44 ··· ⎥ ⎥ ⎥ x32 − β−1 x33 − α−1 x42 − β−1 x43 x34 − β−1 x35 − α−1 x44 − β−1 x45 ··· ⎥ ⎥ ⎥ x41 − β−1 x42 − α−1 x51 − β−1 x52 x43 − β−1 x44 − α−1 x53 − β−1 x54 ⎥ ··· ⎦ x42 − β−1 x43 − α−1 x52 − β−1 x53 x44 − β−1 x45 − α−1 x54 − β−1 x55 ··· ⎡ ⎤ ⎡⎤ w1 Q ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢w2 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢⎥ ×⎢ . ⎥ = ⎢ . ⎥. ⎢ ⎥ ⎢⎥ ⎢ . ⎥ ⎢.⎥ ⎢ . ⎥ ⎢.⎥ ⎣ ⎦ ⎣⎦ w9 0
EURASIP Journal on Wireless Communications and Networking 7 Then, performing the matrix multiplication in (A.6) for the Similarly, the same can be done for the third row of the matrix (A.5), and so forth. In the absence of mutual coupling ﬁrst row of the matrix will give (cx = c y = cxy = 0). From (A.3) and (A.10) (x11 − s11 ) · w1 + x12 − βs11 · w2 + x13 − β2 s11 · w3 w1 + βw2 + β2 w3 + αw4 + αβw5 + αβ2 w6 + (x21 − αs11 ) · w4 + x22 − αβs11 · w5 (A.7) + α2 w7 + α2 βw8 + α2 β2 w9 = Q. + x23 − αβ2 s11 · w6 + x31 − α2 s11 · w7 + x32 − α2 βs11 · w8 + x33 − α2 β2 s11 · w9 = 0. With performing the matrix multiplication in (A.6) for the (A.11) second row of the matrix the following is obtained: Then, (A.11) will be as simple as (x11 w1 + x12 w2 + x13 w3 ) + (x21 w4 + x22 w5 + x23 w6 ) x11 − β−1 x12 − α−1 x21 − β−1 x22 w1 + (x31 w7 + x32 w8 + x33 w9 ) x12 − β−1 x13 − α−1 x22 − β−1 x23 w2 + = s w1 + βw2 + β2 w3 + αw4 + αβw5 + αβ2 w6 x13 − β−1 x14 − α−1 x23 − β−1 x24 w3 + + α2 w7 + α2 βw8 + α2 β2 w9 x21 − β−1 x22 − α−1 x31 − β−1 x32 w4 + 9 x22 − β−1 x23 − α−1 x32 − β−1 x33 w5 + =⇒ wi xi+2[(i−1)/3] = sQ . i=1 −1 −1 −1 x23 − β x24 − α x33 − β x34 w6 + (A.12) −1 −1 −1 Therefore, the desired signal can be recovered by x31 − β x32 − α x41 − β x42 w7 + K2 K1 x32 − β−1 x33 − α−1 x42 − β−1 x43 w8 1 + s= wi xi+[(i−1)/K1 ](K1 −1) . (A.13) Q i=1 −1 −1 −1 x33 − β x34 − α x43 − β x44 w9 = 0. + (b) Presence of the Mutual Coupling. When there is mutual (A.8) coupling, the matrix (A.5) can be formed and the (A.3) and (A.10) can be written in a similar way So (x11 − s11 ) · w1 + x12 − βs11 − cx + αcxy s · w2 + x13 − β2 s11 − β cx + αcxy s · w3 j11 w1 + j12 w2 + j13 w3 + j21 w4 + j22 w5 + j23 w6 + j31 w7 + j32 w8 + j33 w9 + x21 − αs11 − c y + βcxy s · w4 −1 −β j12 w1 + j13 w2 + j14 w3 + j22 w4 + j23 w5 + x22 − αβs11 − β c y + βcxy s + j24 w6 + j32 w7 + j33 w8 + j34 w9 − cxy + αcx + α2 cxy s · w5 − α−1 j21 w1 + j22 w2 + j23 w3 + j31 w4 + j32 w5 + x23 − αβ2 s11 − β2 c y + βcxy s + j33 w6 + j41 w7 + j42 w8 + j43 w9 + α−1 β−1 j22 w1 + j23 w2 + j24 w3 + j32 w4 + j33 w5 −β cxy + αcx + α2 cxy s · w6 + j34 w6 + j42 w7 + j43 w8 + j44 w9 = 0. + x21 − α2 s11 − α c y + βcxy s · w7 (A.9) + x22 − α2 βs11 − αβ c y + βcxy s Az α−1 = 0, β−1 = 0, and wi = 0, (A.9) will be true for all w / / / −α cxy + αcx + α2 cxy s · w8 if and only if each summation in the parenthesis is equal to zero. Therefore, the ﬁrst summation will be used + x23 − α2 β2 s11 − αβ2 c y + βcxy s −αβ cxy + αcx + α2 cxy s · w9 = 0. j11 w1 + j12 w2 + j13 w3 + j21 w4 + j22 w5 (A.10) + j23 w6 + j31 w7 + j32 w8 + j33 w9 = 0. (A.14)
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