EURASIP Journal on Applied Signal Processing 2004:17, 2675–2683 c(cid:1) 2004 Hindawi Publishing Corporation
An Approximate Algorithm for Robust Adaptive Beamforming
Tomoaki Yoshida NTT Access Network Service Systems Laboratories, Chiba 261-0023, Japan Email: tomoaki@ansl.ntt.co.jp
Youji Iiguni Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan Email: iiguni@sys.es.osaka-u.ac.jp
Received 11 February 2004; Revised 7 July 2004; Recommended for Publication by Mos Kaveh
This paper presents an adaptive weight computation algorithm for a robust array antenna based on the sample matrix inversion technique. The adaptive array minimizes the mean output power under the constraint that the mean square deviation between the desired and actual responses satisfies a certain magnitude bound. The Lagrange multiplier method is used to solve the con- strained minimization problem. An efficient and accurate approximation is then used to derive the fast and recursive computation algorithm. Several simulation results are presented to support the effectiveness of the proposed adaptive computation algorithm.
Keywords and phrases: robust array antenna, Lagrange multiplier method, Taylor series approximation, direction of arrival.
1. INTRODUCTION therefore exhibit slower convergence than the sample matrix inversion (SMI) technique [17, 18].
The directionally constrained minimization of power (DCMP) adaptive array adjusts the array weights to mini- mize the mean output power while keeping the antenna re- sponse to the direction of arrival (DOA) of the desired signal [1, 2]. When the true DOA is known a priori, the DCMP ar- ray achieves a good performance. More precisely, the array provides spatial filtering that maximizes the radar’s sensitiv- ity in the desired direction while suppressing interference sig- nals coming from other directions and measurement noises. However, if there is a mismatch between the prescribed and actual DOAs, the desired signal is viewed as an interference and then suppressed [3]. Even a small mismatch may cause a significant performance degradation.
We here consider the robust array antenna with the in- equality directional constraints [10, 11, 12, 13]. The robust array antenna is designed so that the mean output power is minimized under the constraint that the mean square devia- tion between the desired and actual responses satisfies a cer- tain magnitude bound. The constrained minimization prob- lem can be solved by using the Lagrange multiplier method. However, when the interference environment changes with time, we have to find a root of a nonlinear equation at each time step, which is computationally expensive. We thus apply second-order Taylor series approximations to the nonlinear equation to obtain the closed-form solution, and then derive an adaptive weight computation algorithm based on the SMI technique. The derived adaptive algorithm recursively com- pute the weight vector in O(N 2) computation time at each time step, where N is the number of array elements. Several simulation results are performed to show the effectiveness of the proposed adaptive computation algorithm.
For the solution, a number of robust array antennas that impose the directional derivative constraints [4, 5, 6, 7, 8, 9], the inequality directional constraints [10, 11, 12, 13], and the mean-square deviation constraints [14, 15, 16] have been de- veloped. These methods succeed in achieving flat main beam magnitude responses and decreasing the array sensitivity to look-direction errors. However, the adaptive weight compu- tation algorithm to solve the constrained minimization prob- lem at each time step is not provided, which is required to follow changing interference environment. Although some adaptive algorithms were presented in [6, 7, 10], they were derived based on the steepest descent technique and 2. DCMP ARRAY ANTENNA Consider a narrowband adaptive array antenna of N sensors. We define the kth array input at a discrete time t as xk,t and the kth weight as wk. We further define the array input vec- tor and the weight vector as xt = (x1,t, x2,t, . . . , xN,t)T and
2676 EURASIP Journal on Applied Signal Processing
w = (w1, w2, . . . , wN )T, respectively, where “T” denotes the transpose operator. The array output is then given by
yt = wHxt, (1)
(cid:5)
(cid:6)
(cid:3) θd+∆
The inequality constraint must be an active equality con- straint. If the constraint is not active, the solution to the op- timization problem becomes w = 0, which does not make sense. Hence we replace (5) by the equality constraint so that the Lagrange multiplier method is immediately applied. The Lagrangian function is then given by
(cid:4) (cid:4) (cid:4)2dθ − ε2 (cid:4)cH(θ)w − h
θd −∆
where “H” denotes the complex conjugate transpose. Con- sider a desired sinusoidal signal with a DOA θd. Putting the phase shift at the kth input as Φk(θd), the constraint of the DCMP array is formulated as H(w) = wHRw + λ , 1 2∆
cHw = h, (2)
= 0,
(6) where λ is the Lagrange multiplier. The solution to the con- strained minimization problem must satisfy the following re- lations:
2 , . . . , cT
1 , cT
(cid:3) θd+∆
(cid:4) (cid:4) (cid:4)2dθ = ε2. (cid:4)cH(θ)w − h
θd −∆
(7) ∂H(w) ∂w the constraint vector defined by cH = where c is (e− jΦ1(θd), e− jΦ2(θd), . . . , e− jΦN (θd)) and h is the desired re- sponse. Although we here treat a single constraint, the ex- tension to multiple (L) direction constraints is possible by replacing c by the L × N matrix (cT L )T, where L is the number of constraints. (8) 1 2∆
(cid:3) θd+∆
We put
θd −∆ (cid:3) θd+∆
When the DOA θd is given, the DCMP array determines the weight vector w so that the mean output power E[(yt)2] is minimized subject to the constraint (2), where E[·] de- notes the expectation operator. Using the Lagrange multi- plier method, the solution to the linearly constrained min- imization problem is obtained by [1, 2] S = R + c(θ)cH(θ)dθ, λ ∆ (9)
(cid:1) cHR−1c w = R−1c
(cid:2)−1h,
θd −∆
(3) u = c(θ)dθ λ ∆
(cid:2)
to have
(cid:1) |h|2 − ε2
h∗ h where R is the covariance matrix of xt, defined by R = E[xtxH t ]. Adaptive weight estimation algorithms to follow changing interference environment have been derived based on the SD and SMI techniques [1, 17]. wHSw − uHw + λ 2 2
3. ADAPTIVE ALGORITHM FOR ROBUST wHu − (cid:2)HS
(cid:1) w − hS−1u
(cid:2) (cid:1) w − hS−1u
|h|2
−
|h|2 − ε2
(cid:2) .
(10) H(w) = 1 2 = 1 2 ARRAY ANTENNA
(cid:1) uHS−1u + λ
3.1. Constrained minimization problem 2
(cid:7)
(cid:8)−1
(cid:3) θd+∆
(cid:3) θd+∆
Since S is positive definite and Hermitian, H(w) is mini- mized by putting
θd −∆
θd −∆
w = R + c(θ)cH(θ)dθ c(θ)dθ. λh ∆ λ ∆ The use of the equality constraint (2) causes performance degradation in the presence of look-direction errors. For the solution, a robust array antenna, which minimizes the mean output power under the constraint that the mean square de- viation between the desired and actual responses satisfies a certain magnitude bound, has been proposed [14, 15, 16]. This is formulated as (11)
(cid:6)
(cid:5) (cid:3) θd+∆
(cid:5) (cid:3) θd+∆
(cid:4) (cid:4)cT(θ)w − h
(cid:4) (cid:4)2dθ ≤ ε2,
θd −∆
θd −∆
(cid:6)
θd −∆ (cid:5) (cid:3) θd+∆
(cid:2)
−
(cid:1) |h|2 − ε2
wHRw (4) The constraint (8) is rewritten as (cid:6) min w (cid:3) θd+∆ h 0 = wH c(θ)c(θ)Hdθ w − wH c(θ)dθ subject to (5) 1 2∆
θd −∆
c(θ)Hdθ wh∗ + 2∆ ,
(12)
where ε and ∆ are small positive constants representing the severity of the constraint and the angle width considered in the constraint, respectively. While the equality constraint (2) restricts the output response to h only at the angle θd, the inequality constraint (5) makes the response close (in a least squares sense) to h in the angle range [θd −∆, θd +∆]. The re- sulting array therefore has robustness against look-direction errors. where “∗” denotes the complex conjugate. The Lagrange multiplier λ can be determined by substituting (11) into (12) and then solving it for λ. However, the closed-form solution is difficult to obtain due to its nonlinearity.
An Approximate Algorithm for Robust Adaptive Beamforming 2677
(cid:9)
(cid:8)
(cid:10)(cid:7)
(cid:8)−1
Substituting (17) into (11) yields
(cid:7) 2∆ppH +
(cid:8)
= λh
∆3 ∆3 r R + G 2∆p + w (cid:4) λh ∆ λ ∆ 3 3 (cid:8)−1(cid:7) ∆2λ ∆2 G 2p + r (18)
(cid:7) R + 2λppH + (cid:7)
(cid:7)
= λh
= λhQ3V−1
3 (cid:8) 3 (cid:8)−1 ∆2λ ∆2 I + V−1G 2p + r 3 3 (cid:7) ∆2 V−1 (cid:8) . 2p + r 3
Putting the N-dimensional vectors vr, vq, and vp as
∆2λ ∆2λ vp = V−1p, V−1q, vr = 3 vq = 2∆2λ 3 3 V−1r, (19)
(cid:1)
(cid:2)−1.
the matrix Q3 in (18) is rewritten as
1 + · · · + cLcH
I + vrpH + vqqH + vprH (20) Q3 =
Therefore, we can compute Q3 in O(N 2) computation time by recursive use of the matrix inversion lemma: When the generalized singular value decomposition of R is obtained, the value of λ can be determined by finding a root of a nonlinear equation, referred to as “secular equa- tion” [19, 20]. A standard root-finding technique such as Newton’s method is applicable to the solution of the non- linear equation. Both root-finding algorithms and singular value decomposition algorithms use iterative methods, in which an iterative scheme is continued until convergence is obtained, that is, until the new value is very close to the previous value. When R changes with time as often hap- pens, root-finding and singular value decomposition need to be performed at each time step. The iterative methods re- quire O(N 2) computation time per iteration. The compu- tational complexity increases with an increase in the num- ber of iterations. Moreover, the use of the iterative meth- ods at each time step is not suited for adaptive array pro- cessing where the maximum processing time is crucial. We thus derive the adaptive computation algorithm by applying second-order Taylor series approximations to the nonlinear equation. We here consider a single constraint to derive the adaptive algorithm, as shown in (5). When there are multi- ple (L) direction constraints, we can use a similar technique to derive the adaptive algorithm by replacing c and ccH by c1 + · · · + cL and c1cH L , respectively, in (9), (10), (11), and (12).
, Q1 = I − vrpH 1 + pHvr
3.2. Computation of weight vector We define the N-dimensional vectors p, q, and r as , Q2 = Q1 − (21)
(cid:4) (cid:4) (cid:4) (cid:4)
(cid:4) (cid:4) (cid:4) (cid:4)
(cid:2) ,
θd q = , r = ,
(cid:1) p = c
θ=θd
θ=θd
. Q3 = Q2 − dc(θ) dθ d2c(θ) dθ2 Q1vqqHQ1 1 + qHQ1vq Q2vprHQ2 1 + rHQ2vp (13) 3.3. Computation of Lagrange multiplier and the (N × N) matrices G, V−1, and Q3 as We define several real values as
(cid:1)
G = rpH + 2qqH + prH, (14) β = pHR−1q, γ = pHR−1r, α = pHR−1p, (cid:2) ξ = α ϕ = γ + γ∗ + 2|β|2, v = 1 + 2λα. V−1 =
(cid:1) R + 2λppH
|h| ε ,
(cid:2)−1 = R−1 − 2λR−1ppHR−1 1 + 2λpHR−1p
, (15)
(cid:7)
(cid:8)−1
(22) ∆2λ . I + V−1G (16) Q3 = Then we have 3
pHV−1p = Using the second-order Taylor series expansion, we approxi- mately have
(cid:3) θd+∆
θd −∆
(cid:2)
(cid:1) θd
(cid:4) (cid:4) (cid:4) (cid:4)
(cid:1) θd = 2∆c
pHV−1q = (23) c(θ)cH(θ)dθ pHV−1r = α v , β v , γ v , ∆3
(cid:2) cH
θ=θd
≈ 2∆ppH +
(cid:3) θd+∆
(cid:7)
(cid:8)−1
θd −∆
(cid:4) I −
+ + · · · . phV−1GV−1p = d2 dθ2 c(θ)cH(θ) 3 ξ v2 ∆3 G, 3 Neglecting small quantities of order ∆4 in (16), we approxi- mately have ∆3 c(θ)dθ (cid:4) 2∆p + r. ∆2λ ∆2λ 3 I + V−1G V−1G. (24) Q3 = (17) 3 3
2678 EURASIP Journal on Applied Signal Processing
(cid:8)
(cid:7)
(cid:7)
(cid:8)
Substituting (24) into (18) yields
µR−1
−
∆2λ ∆2
t = 1, 2, . . . t = 1 R−1
(cid:7) R−1 t−1
1 − µ
t−1xtxH (1 − µ) + µxH
t R−1 t−1 t R−1 t−1xt
(cid:4) λhV−1
w (cid:4) λh V−1 2p + I − (cid:7) (25) 3 ∆2 r (cid:8) . r GV−1p +
(cid:8) V−1G 3 2p − 2λ∆2 3
(cid:2)
3
α = pHR−1 t p β = pHR−1 t q γ = pHR−1 t r (cid:1) γ + γ∗ ξ = α
+ 2|β|2
(cid:12)
(cid:11)
ϕ − 1
λ =
ϕ(ϕ + 1)|β|2 − ξ
We now obtain two different ways of computing w, that is, (18) and (25). The weight vector computed by (18) is more accurate than the one by (25), because (18) is derived using only approximations (17). We thus use (18) in the computa- tion of w and (25) in the computation of λ.
V−1 = R−1
2α + ∆2 (ϕ − 1) 12α3 t − 2λR−1 t ppHR−1 t 1 + 2λpHR−1 t p
Using (17), (23), and (25), we can approximately have
∆2λ
(cid:6)
vp =
V−1p
(cid:5) (cid:3) θd+∆
θd −∆
wH c(θ)c(θ)Hdθ
V−1q
(cid:2)
(cid:7)
w (cid:1) 8
3 vq = 2∆2λ 3 ∆2λ
= ∆λ2|h|2
(cid:8) ,
(cid:6)
(cid:5) (cid:3) θd+∆
∆2 8α2 v2 + ξ − v|β|2 3v3 (26)
c(θ)dθ w
V−1r vr = 3 Q1 = I − vrpH 1 + pHvr
θd −∆
(cid:2)
(cid:7)
Q2 = Q1 −
= ∆λh
(cid:8) .
(cid:1) ξ − 2v|β|2 2 3αv2
∆2 4α v +
Q3 = Q2 −
Q1vqqHQ1 1 + qHQ1vq Q2vprHQ2 1 + rHQ2vp
(cid:7)
(cid:8)
(cid:2)
(cid:7)
(cid:8)
Substituting (26) into (12) yields
2p +
wt = λhQ3V−1
∆2r 3
(cid:1) ξ − v|β|2 4 3v3
(cid:2)
(cid:7)
(cid:8)
λ2|h|2 ∆2 4α2 v2 +
Algorithm 1: Proposed adaptive algorithm.
− λ|h|2
(cid:1) ξ − 2v|β|2 2 3αv2
= ε2.
(cid:7)
(cid:8)
(cid:2)
(cid:1) |β|2(v + 1)v − ξ
(27) ∆2 + |h|2 4α v +
= 0.
1 − (28) v2 ϕ2 After some manipulation, (27) is reduced to + ∆2 (v − 1) 3α2v
(cid:12)
(cid:11)
(cid:11)
(cid:12)
Solving (28) for v yields where Rt is the estimates of R at time t and µ is a forget- ting factor such that µ (cid:5) 1. The computational complexity per sample is of order N 2. The direct computation of (31) causes the problem of numerical stability when using a short word-length processor. The use of the numerically stable up- dating scheme based on the UD or square-root decomposi- tion may be helpful. But we avoided the problem by using floating-point double precision arithmetics in the following simulation. ∆2. v = ϕ + ϕ(ϕ + 1)|β|2 − ξ (29) ϕ − 1 6α2
=
ϕ − 1 λ = . ϕ(ϕ + 1)|β|2 − ξ (30) Thus we have v − 1 2α 2α + ∆2 (ϕ − 1) 12α3
Algorithm 1 summarizes the proposed algorithm that re- cursively computes the weight vector wt from the array in- put xt in O(N 2) computation time. It is here noted that p, q, r, and ϕ can be computed a priori. We can consider that the true and approximated solutions are very close to each other because (18) and (30) are derived using second-order Taylor series approximations. This will be verified through computer simulations below.
4. COMPUTER SIMULATION We see that the Lagrange multiplier is expressed indepen- dently of the weight vector w. We can now obtain the closed- form solution to the constrained minimization problem (4), (5).
3.4. Summary of the proposed adaptive algorithm
To follow changing interference environment, we recursively estimate R−1 by
−
µR−1
t = 1 R−1
(cid:8) ,
(31)
(cid:7) R−1 t−1
t−1xtxH (1 − µ) + µxH
t R−1 t−1 t R−1 t−1xt
We consider a desired signal with a frequency 100 MHz, a power 1, and a DOA θd = 90◦, and an interference with a frequency 100 MHz, a power 10, and a DOA θi = 150◦. We set h = 1, N = 4, ∆ = 0.5◦, ε = 0.02, T = 2 nanoseconds. We chose the element spacing equal to one-half wavelength, and added a white noise with mean 0 and variance 0.01(= σ 2 n) to the array input. 1 − µ
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An Approximate Algorithm for Robust Adaptive Beamforming 2679
Figure 1: Array pattern.
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Conventional Robust
90 89 θr (degree)
Figure 2: Comparison of SINRs.
P(0.01, 0.5) P(0.02, 0.5) P(0.05, 0.5)
(b)
(cid:14)
(cid:4) (cid:4)2
= E
(cid:15)(cid:4) (cid:4)st
When the desired signal st is coming from a direction θ, the covariance matrix of the array input is represented by
R(θ) = E
(cid:13) xtxH t
(cid:16) c(θ)c(θ)H.
(32)
Figure 3: SINR for various values of ε. (a) True solution. (b) Ap- proximated solution.
(cid:16)
(cid:4) (cid:4)2
(cid:4) (cid:4)2
(cid:4) (cid:4)2.
(cid:15)(cid:4) (cid:4)yt
(cid:15)(cid:4) (cid:4)st
= wH
o R(θ)wo = E
(cid:16)(cid:4) (cid:4)wH
o c(θ)
Let the optimal weight vector computed off-line be wo. The array pattern with respect to θ is then represented by signal-to-interference-plus-noise ratio respectively. The (SINR) is then defined by G(θ) = E . SINR = (35) (33) Pd Pi + Pe
Figure 1 shows the array pattern of the robust array. We see that the array antenna places a null in the direction of the interference, 150◦, while keeping a large antenna response to the desired direction, 90◦.
(cid:13)
The array input xt is decomposed into the sum of the desired signal component dt, the interference component it, and the observation noise component et. The powers of dt, it, and et are expressed as (cid:14) Pd = wHE
(cid:13) itiT t
(cid:14) w,
(34) dtdT w, t Pe = wHE Pi = wHE (cid:14) w,
(cid:13) eteT t
Let the actual and prescribed DOAs of the desired signal be θr and θd, respectively. We put θd = 90◦ to design the con- straint vector c, and computed the weight vector w for vari- ous values of θr. Figure 2 plots the SINR as the function of θr. The result for the conventional array computed by (3) is also shown for comparison purposes. It is found that the robust array offers a flat SINR in the look direction, although there is a tradeoff in the noise rejection capability of the processor in look directions which are far away from the desired signal. Figure 3 shows the SINRs for ε = 0.01, 0.02, and 0.05 with ∆ = 0.5◦, where Figures 3a and 3b are the results of the
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(b)
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2680 EURASIP Journal on Applied Signal Processing
Figure 5: SINR for various values of SNR. (a) True solution. (b) Approximated solution.
Figure 4: SINR for various values of ∆. (a) True solution. (b) Ap- proximated solution.
errors is increased as ∆ is larger, while resolution capability is decreased. Figure 5 shows the SINRs for σ 2 n = 0.01, 0.1, and 1 with ε = 0.02 and ∆ = 0.5◦, where Q(c) denotes the result n = c. Figure 6 shows the SINRs for N = 4, 6, and 8 with for σ 2 n = 0.01, where R(d) denotes the result ε = 0.02, ∆ = 0.5◦, σ 2 for N = d. We see that robustness is decreased as σ 2 n is larger or N is larger. We also see that the exact and approximated solutions are very close to each other except for the case of N = 8. exact and approximated solutions, respectively, and P(a, b) denotes the result for ε = a and ∆ = b. The exact solution was obtained by (11) and (12), and the approximated solu- tion was obtained by (18) and (30). We see that robustness against look-direction errors is increased as ε is smaller, while resolution capability of the desired and interference signals is decreased. Therefore, we have to make a tradeoff between ro- bustness and resolution capability in determining the value of ε. We also see that the exact and approximated solutions are very close to each other.
We quantitatively evaluated the approximation errors of the Lagrange multiplier and the weight vector computed by the proposed algorithm. Table 1 summarizes the true and Figure 4 shows the SINRs for ∆ = 0.3◦, 0.5◦, and 1.0◦ with ε = 0.02. We see that robustness against look-direction
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An Approximate Algorithm for Robust Adaptive Beamforming 2681
Figure 6: SINR for various numbers of array elements. (a) True solution. (b) Approximated solution.
Table 1: Approximation accuracies.
N
4
4
4
4
4
4
4
4
6
8
σ 2 n 0.01 0.01 0.01 0.01 0.01 0.01 0.1 1 0.01 0.01
ε 0.02 0.01 0.03 0.05 0.02 0.02 0.02 0.02 0.02 0.02
∆ 0.5 0.5 0.5 0.5 0.3 1 0.5 0.5 0.5 0.5
(cid:17)λ 24.6107 49.963 16.2252 9.52998 24.5805 24.7523 25.1836 30.9070 24.5654 24.5626
λ 24.5686 49.6534 16.2136 9.52965 24.5686 24.5986 25.1605 30.9052 24.5569 24.5561
|w − (cid:17)w|2 7.44582e-08 3.14965e-07 3.14146e-08 1.02032e-08 1.33783e-09 1.51836e-05 9.75074e-10 1.31363e-11 3.25641e-06 2.76087e-05
|w − (cid:17)w|2/|w|2 2.97194e-07 1.23124e-06 1.28040e-07 4.33819e-08 5.35673e-09 5.86991e-05 3.91061e-09 5.27997e-11 1.92091e-05 0.000189067
Figure 8b that the conventional method fails when there is a mismatch between the prescribed and actual DOAs, while the proposed method exhibits almost the same convergence performance due to its robustness against look-direction er- rors.
5. CONCLUSION
approximated Lagrange multipliers, the squared error be- tween the true and approximated weights, and the normal- ized error. The approximation is found to be very accurate. Figure 7 plots the normalized error between the true and ap- proximated weights as the function of the angle width ∆, where Figure 7a is the result for ε = 0.01, 0.02, 0.05, Figure 7b is the result for σ 2 n = 0.01, 0.1, 1, and Figure 7c is the result for N = 4, 6, 8. It is evident that the normalized error increases with an increase of ∆.
Finally, we compared the robust array trained by the pro- posed algorithm to the conventional array trained by the SMI algorithm in convergence performance. Figure 8 depicts the convergence trajectories of the SINR, where Figures 8a and 8b are the results for θr = 90◦ and θr = 91◦, respectively. We used the same parameters as in Figure 2. We see from Figure 8a that both methods show almost the same perfor- mance in the absence of look-direction errors. We see from We have derived the adaptive weight computation algorithm for the robust array antenna based on the SMI technique by using second-order Taylor series approximations. The adap- tive algorithm can recursively compute the weight vector in only O(N 2) computation time. Simulation results have shown that we have to tune parameters ∆ and ε so that a good tradeoff between robustness and resolution capability is achieved, and that robustness depends upon the array size and the SNR.
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2682 EURASIP Journal on Applied Signal Processing
Figure 8: Convergence comparisons. (a) θr = 90◦. (b) θr = 91◦.
) g o l (
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The inequality constraint for the case of broadband sources was considered in [14, 16]. Using the same approx- imation method, the result for a narrowband source will be extended to broadband sources.
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REFERENCES
[1] O. L. Frost III, “An algorithm for linearly constrained adaptive array processing,” Proceedings of the IEEE, vol. 60, no. 8, pp. 926–935, 1972.
∆ (degree)
[2] K. Takao, M. Fujita, and T. Nishi, “An adaptive antenna ar- ray under directional constraint,” IEEE Trans. Antennas and Propagation, vol. 24, no. 5, pp. 662–669, 1976.
N = 4 N = 6 N = 8
(c)
[3] H. Cox, “Resolving powers and sensitivity to mismatch of optimum array processors,” Journal of the Acoustical Society of America, vol. 54, no. 3, pp. 771–785, 1973.
I
(ε
= (a) Case n = 0.01, 0.1, 1). (c) Case III
Figure 7: Approximation accuracies: 0.01, 0.02, 0.05). (b) Case II (σ 2 (N = 4, 6, 8).
[4] S. P. Applebaum and D. J. Chapman, “Adaptive arrays with main beam constraints,” IEEE Trans. Antennas and Propaga- tion, vol. 24, no. 5, pp. 650–662, 1976.
An Approximate Algorithm for Robust Adaptive Beamforming 2683
