The real-valued SVR can be obtained by solving the following optimization problem (Ramón and Christodoulou, 2006): (1) min w 1 2 ‖ w ‖ 2 + P ∑ i = 1 l ( ξ i + ξ i ′ ) . Subject to the constraints (2) y i − ( w T x i + b ) ⩽ ε + ξ i , ( w T x i + b ) − y i ⩽ ε + ξ i ′ , ξ i , ξ i ′ ⩾ 0 for i = 1 , 2 , … , l, where { ( x 1 , y 1 ) , … , ( x l , y l ) } is the training set and x i ∈ R n , w ∈ R n , y i ∈ R. P is a penalty factor. ε is the insensitive loss factor. ξ i and ξ i ′ are the slack variables of the output. P, ε, ξ i and ξ i ′ are all real numbers. In the formula (1), the optimal value can be obtained by finding the optimal value of its dual problem. According to the principle of real-valued SVR, an SVR model in complex field is built and the corresponding solution method is proposed. In order to simplify the expression, ‘Subject to the constraints’ will be abbreviated to ‘ s.t.’. The corresponding complex SVR task is formulated and expressed as (3) min 1 2 ‖ w ‖ 2 + C 1 ∑ i = 1 l ( ξ i + ξ i ′ ) + C 2 ∑ i = 1 l ( η i + η i ′ ) , Re ( y i − ( w T x i + b ) ) ⩽ ε + ξ i , Re ( ( w T x i + b ) − y i ) ⩽ ε + ξ i ′ , s.t. Im ( y i − ( w T x i + b ) ) ⩽ ε + η i , i = 1 , 2 , … , l , Im ( ( w T x i + b ) − y i ) ⩽ ε + η i ′ , ξ i , ξ i ′ , η i , η i ′ ⩾ 0 where x i ∈ C n , w ∈ C n , y i ∈ C. C 1 and C 2 are the penalty factor of the real and imaginary parts. ε is the insensitive loss factor. ξ i and ξ i ′ are the slack variables in the real part of the output, and the corresponding η i and η i ′ are in the imaginary part. C 1 , C 2 , ε, ξ i , ξ i ′ , η i and η i ′ are all real numbers. The formula (3) is a complex-valued quadratic programming problem, which also can be solved by solving its dual problem.

The Lagrange function of the equation (3) can be expressed as (4) L p = 1 2 ‖ w ‖ 2 + C 1 ∑ i = 1 l ( ξ i + ξ i ′ ) + C 2 ∑ i = 1 l ( η i + η i ′ ) − ∑ i = 1 l ( γ i ξ i + γ i ′ ξ i ′ ) − ∑ i = 1 l ( κ i η i + κ i ′ η i ′ ) + ∑ i = 1 l α i [ Re ( y i − w T x i − b ) − ε − ξ i ] + ∑ i = 1 l α i ∗ [ Re ( − y i + w T x i + b ) − ε − ξ i ′ ] + ∑ i = 1 l β i [ Im ( y i − w T x i − b ) − ε − η i ] + ∑ i = 1 l β i ∗ [ Im ( − y i + w T x i + b ) − ε − η i ′ ] , where α i , α i ∗ , β i , β i ∗ , γ i , γ i ′ , κ i and κ i ′ are the Lagrange multipliers. According to the dual definition of Wolfe (Brandwood, 1983; Huang and Zhang, 2007), the partial derivatives are: ∇ w ( L p ) = w − ∑ i = 1 l ϕ i x , ∇ b ( L p ) = − ∑ i = 1 l α i + ∑ i = 1 l α i ∗ − j ∑ i = 1 l β i + j ∑ i = 1 l β i ∗ . For real variables, the gradients can be obtained in the traditional way: ∇ ξ i ( L p ) = C 1 − γ i − α i , ∇ ξ i ′ ( L p ) = C 1 − γ i ′ − α i ∗ , ∇ η i ( L p ) = C 2 − κ i − β i , ∇ η i ′ ( L p ) = C 2 − κ i ′ − β i ∗ , where ϕ i = ( α i − α i ∗ ) + j ( β i − β i ∗ ), for all i = 1 , 2 , … , l. As all partial derivatives are equal to zero for saddle point conditions, the following parameters can be obtained: (5) w = ∑ i = 1 l ( α i − α i ∗ ) + j ( β i − β i ∗ ) x i , (6) ∑ i = 1 l ( α i − α i ∗ ) = ∑ i = 1 l ( β i − β i ∗ ) = 0 , (7) C 1 − γ i − α i = 0 , C 1 − γ i ′ − α i ∗ = 0 , (8) C 2 − κ i − β i = 0 , C 2 − κ i ′ − β i ∗ = 0 , for i = 1 , 2 , … , l. Let λ i = α i − α i ∗ , φ i = β i − β i ∗ , then (9) ϕ i = λ i + j φ i . Accordingly to the equations (5)–(9), the dual problem of the equation (4) can be solved. The solution can be presented as: (10) min 1 2 ∑ i = 1 l ∑ j = 1 l ϕ i H ϕ j ( x i , x j ) + ε ∑ i = 1 l ( | λ i | + | φ i | ) − Re [ ∑ i = 1 l ϕ i H y i ] , s.t. ∑ i = 1 l ϕ i = 0 , α i , α i ∗ , β i , β i ∗ ⩾ 0 , i = 1 , 2 , … , l . By introducing a kernel function K ( x i , x j ): C × C → R, the dual problem can be expressed as: (11) min 1 2 ∑ i = 1 l ∑ j = 1 l ϕ i H ϕ j K ( x i , x j ) + ε ∑ i = 1 l ( | λ i | + | φ i | ) − Re [ ∑ i = 1 l ϕ i H y i ] , s.t. ∑ i = 1 l ϕ i = 0 α i , α i ∗ , β i , β i ∗ ⩾ 0 , i = 1 , 2 , … , l . The objective function can be expressed by λ i and φ i , and the formula as follows: (12) 1 2 ∑ i = 1 l ∑ j = 1 l ϕ i H ϕ j K ( x i , x j ) + ε ∑ i = 1 l ( | λ i | + | φ i | ) − Re [ ∑ i = 1 l ϕ i H y i ] = 1 2 ∑ i = 1 l ∑ j = 1 l ( λ i λ j + φ i φ j ) K ( x i , x j ) + ε ∑ i = 1 l ( | λ i | + | φ i | ) − Re [ ∑ i = 1 l λ i y i ] − Im [ ∑ i = 1 l φ i y i ] .

Assume that the optimal solution of the dual problem (11) is α ¯ ( ∗ ) = ( α ¯ 1 , α ¯ 1 ∗ , … , α ¯ l , α ¯ l ∗ ) T and β ¯ ( ∗ ) = ( β ¯ 1 , β ¯ 1 ∗ , … , β ¯ l , β ¯ l ∗ ) T . The optimal solution of the primal problem can be obtained: (13) w = ∑ i = 1 l ( α ¯ i − α ¯ i ∗ ) + j ( β ¯ i − β ¯ i ∗ ) x i . If the training point is support vector on the boundary, then the threshold b ¯ can be computed: (14) b ¯ = y j − ∑ i = 1 l ϕ i K ( x i , x j ) + ( ε + j ε ) . Finally, a support vector regression function in complex data field can be obtained: (15) f ( x ) = ∑ i = 1 l ϕ i K ( x i , x ) + b ¯ .

Assume that the solution of the formula (11) is α ¯ ( ∗ ) and β ¯ ( ∗ ) , then the KKT condition of (3) can be given in formula (16). (16) Re ( y i − ( ∑ j = 1 l ϕ j K ( x j , x i ) + b ) ) ⩽ ε + ξ i , Re ( ( ∑ j = 1 l ϕ j K ( x j , x i ) + b ) − y i ) ⩽ ε + ξ i ′ , Im ( y i − ( ∑ j = 1 l ϕ j K ( x j , x i ) + b ) ) ⩽ ε + η i , Im ( ( ∑ j = 1 l ϕ j K ( x j , x i ) + b ) − y i ) ⩽ ε + η i ′ , α i { ε + ξ i − Re [ y i − ∑ j = 1 l ϕ j K ( x j , x i ) − b ] } = 0 , α i ∗ { ε + ξ i ′ − Re [ − y i + ∑ j = 1 l ϕ j K ( x j , x i ) + b ] } = 0 , β i { ε + η i − Im [ y i − ∑ j = 1 l ϕ j K ( x j , x i ) − b ] } = 0 , β i ∗ { ε + η i ′ − Im [ − y i + ∑ j = 1 l ϕ j K ( x j , x i ) + b ] } = 0 , ( C 1 − α i ) ξ i = 0 , ( C 1 − α i ∗ ) ξ i ′ = 0 , ( C 2 − β i ) η i = 0 , ( C 2 − β i ∗ ) η i ′ = 0 , − ξ i , − ξ i ′ , − η i , − η i ′ ⩽ 0 , − α i , − α i ∗ , − β i , − β i ∗ ⩽ 0 . for all i = 1 , 2 , … , l. Here, the inherent relationship among α i , α i ∗ , β i and β i ∗ will be considered firstly.

According to the Corollary, we can analyse the distribution of the real and imaginary parts of the sample points on the boundary. At the same time, in order to facilitate the expression of the formula, a simple notation is introduced: (22) E i = E ( x i ) = f ( x i ) − y i . Therefore the complex domain KKT conditions can be expressed as: (23) | Re ( E i ) | ⩽ ε ( | λ i | = 0 ) , | Re ( E i ) | = ε ( 0 < | λ i | < C 1 ) , | Re ( E i ) | ⩾ ε ( | λ i | = C 1 ) , | Im ( E i ) | ⩽ ε ( | φ i | = 0 ) , | Im ( E i ) | = ε ( 0 < | φ i | < C 2 ) , | Im ( E i ) | ⩾ ε ( | φ i | = C 2 ) .

The formula (11) can be solved by using SMO (Cristianini and Shawo-Taylor, 2005; Yang and Tian, 2004) method.

SMO algorithm is a special case of decomposing algorithm. The core of the algorithm is to update two Lagrangian multipliers corresponding to two sample points at a time, while others remain unchanged. In the complex domain, the Lagrangian multiplier variables corresponding to sample points ( x i , y i ) and ( x j , y j ) are ( λ i , ϕ i ) and ( λ j , ϕ j ). It is important for SMO algorithm to choose the two appropriate training points and solve the corresponding optimal problem with two Lagrangian multiplier variables. In the following, these two problems are solved.

Firstly, suppose that there are already two training points ( x 1 , y 1 ) and ( x 2 , y 2 ). In order to simplify the formula, let K i j = K ( x i , x j ). Therefore the objective function (12) can be expressed as (24) W ( λ 1 , λ 2 , ϕ 1 , ϕ 2 ) = 1 2 λ 1 2 K 11 + 1 2 λ 2 2 K 22 + λ 1 λ 2 K 12 + ε [ | λ 1 | + | λ 2 | ] − [ λ 1 Re ( y 1 ) + λ 2 Re ( y 2 ) ] + 1 2 φ 1 2 K 11 + 1 2 φ 2 2 K 22 + φ 1 φ 2 K 12 + ε [ | φ 1 | + | φ 2 | ] − [ φ 1 Im ( y 1 ) + φ 2 Im ( y 2 ) ] + λ 1 ∑ i = 3 l λ i K 1 i + λ 2 ∑ i = 3 l λ i K 2 i + φ 1 ∑ i = 3 l φ i K 1 i + φ 2 ∑ i = 3 l φ i K 2 i + W ( const ) , where W ( const ) has no relationship with ( x 1 , y 1 ) and ( x 2 , y 2 ). Let the initial feasible solution of this problem is ( λ 1 old , ϕ 1 old ) and ( λ 2 old , ϕ 2 old ), and the optimal solution of the problem is the new values ( λ 1 , ϕ 1 ) and ( λ 2 , ϕ 2 ) of the two variables. In order to satisfy the constraint, the new values must satisfy (25) λ 1 old + λ 2 old = s 1 old = λ 1 + λ 2 , ϕ 1 old + ϕ 2 old = s 2 old = ϕ 1 + ϕ 2 , where s 1 old and s 2 old are constant. The formula (25) is substituted into the objective function (24) and takes partial derivation of W with respect to λ 2 . By simplifying, we finally get (26) λ 2 = λ 2 old + 1 κ ( Re ( E 1 ) − Re ( E 2 ) − ε [ sgn ( λ 2 ) − sgn ( λ 1 ) ] ) . Similarly, we can update ϕ 2 : (27) ϕ 2 = ϕ 2 old + 1 κ ( Im ( E 1 ) − Im ( E 2 ) − ε [ sgn ( ϕ 2 ) − sgn ( ϕ 1 ) ] ) , where κ = k 11 + k 22 − 2 k 12 and sgn ( . ) is signum function.

Second, we should solve the problem of selecting the two training points in the iteration. According to the definition of gradient, we choose first training point which not only violates KKT conditions but also can maximize the module of gradient. (28) ∂ W ∂ λ i = ∑ j = 1 l λ j K i j + 0 + ε sgn ( λ i ) − Re ( y i ) = Re ( E i ) − Re ( b ) + ε sgn ( λ i ) , (29) ∂ W ∂ φ i = ∑ j = 1 l φ j K i j + 0 + ε sgn ( φ i ) − Im ( y i ) = I m ( E i ) − I m ( b ) + ε sgn ( φ i ) , for i = 1 , 2. Since the ultimate aim of training is to find the training points that make objective function minimize, we choose the second training point which can make greatest change of formula (24).

In this subsection, a regression test is performed to verify the ability of the support vector regression model to handle noise by using two function f 1 ( x ) = e j x / 20 + N, x ∈ R and f 2 ( x ) = sin ( x ) x + N, x ∈ C. For f 1 ( x ), x ∈ [ 0 , 60 ]. For f 2 ( x ), x = x r +j x i , x r and x i are real random variables, and x r ∈ [ − 4 , 4 ], x i ∈ [ − 1 , 1 ], j as the imaginary unit. N is the white Gaussian noise with zero mean and variances σ = 0.2. The regression curves are shown in Figs. 1 and 2, which indicate that with the proposed solution method, the regression data can agree well with the original data.

At the same time, a Monte Carlo experiment for 50 times are conducted to demonstrate the stability of the proposed solution, based on the root mean-square-error (RMSE) of complex-data defined as: (30) RMSE = 1 N ∑ t = 1 N | x t − x ˆ t | 2 , where x ˆ t is the estimated value of x t .

The resultant RMSE of the complex-data is depicted in Fig. 3, which shows that the square error is small and has a high stability.

In all of the above experiments, a real Gaussian Radial Basis Kernel (RBF) function is selected for SMO algorithm and the Gaussian RBF is expressed as (31) K ( x , y ) = exp ( − ‖ x − y ‖ 2 r 2 ) , where r is the Gaussian kernel parameter. For f 1 ( x ), C 1 = C 2 = 2, ε = 1 E − 5, r = 50. For f 2 ( x ), C 1 = C 2 = 10, ε = 0.1, r = 0.8.

Next, we compare the performance of the proposed method with CSVR (Bouboulis, 2015). The CSVR method exploits a Xcomplex kernel function and transforms the complex problem into two real SVR tasks. In order to compare the performance of the two algorithms under the same standard, the Gaussian Radial Basis kernel function is selected and C 1 = C 2 . The training samples are generated by f 2 ( x ), and x r ∈ [ − 1 , 1 ], x i ∈ [ − 1 , 1 ]. In this experiments, r = 0.6 and ε = 0.1 for both proposed solution and CSVR. Table 1 shows the difference in RMSE between the two algorithms under 50 Monte Carlo experiments. It can be concluded from the Table 1 that the precision of the proposed solution is roughly the same as that of CSVR.

In this section, the proposed complex-SVR will be applied to the field of anti-interference to verify its effect in practical application. The anti-interference technology is an important part of array signal processing. According to the literature (Ramón and Christodoulou, 2006), the output vector of an L-element array can be described in matrix notation as: (32) X = A s S + A j J + N , where S is desired signals, J is interfere signals, N is noise vector, and A s is the response vector of the desired signals, A j is the response vector of the interfere signals. The purpose of the anti-interference is to obtain desired output S and suppress interfere signal. Therefore, the output of the array processor is (33) y = W H X = d + ε , where W is the weight vector of the array, d is the estimated value of the desired signal S, ε is the estimation error. In practical applications, we only know the output vector of the array. Therefore, in order to achieve interference suppression, it is important to know how to use the output vector X of an L-element array to get the optimal weight vector W.

Firstly, it is important to generate the training data. For an L-elements Unit Circular Array (UCA), a dataset X ( k ) = [ x 1 ( k ) , x 2 ( k ) , … , x l ( k ) ] T can be obtained, where 1 < k < N and N is the snapshot number and x i ( k ) is the received value of the i − t h element at moment k. The autocorrelation matrix of all samples can be computed by (34) R = 1 N ∑ k = 1 N X ( k ) X ( k ) H .

According to the PI algorithm (Zahm, 1973; Compton, 1979), the optimal weight is (35) W = R − 1 Const Const T R − 1 Const , where Const = [ 1 , 0 , … , 0 ] T . And then a set of pairs { R , W } can be chosen as training samples.

Secondly, we will show how to use Complex-SVR to achieve anti-interference. The narrowband signal is chosen as the jammer, whose direction is θ = 120 0 , φ = 50 0 , where θ is the azimuth, φ is the elevation angle. In this experiment, a 4-element UCA is chosen and the data X is divided into 100 sections (where N = 12800), so as to have 128 snapshots in each section. Then, based on the equations (34) and (35), 100 sample data can be obtained. In the experiment, 70 sample data are chosen as training samples, and the remaining 30 points as the testing samples. In this experiment, C 1 = C 2 = 20, ε = 1 E − 4, r = 0.8. The arrays patterns of PI and the proposed complex-valued SVR are illustrated in Fig. 4. where Fig. 4(a) is a pattern of the azimuth angle when the elevation angle is fixed. and Fig. 4(b) is a pattern of the elevation angle when the azimuth angle is fixed. The antenna gain is defined as (36) G = 10 log ( ‖ W H A j ‖ 2 ) . It can be seen from Fig. 4 that complex-valued SVR can get deeper nulls in the direction of the interfering signals than PI.