According to Eqs 1, 5, the residual interference signal is y j t = w ¯ H ∑ k j k t a j k (7)

There are two ways to interpret principle of spatial interference suppression. One is to figure out w ¯ satisfies w ¯ ≠ 0 and the following equation set w ¯ H a j 1 j 1 t = 0 w ¯ H a j 2 j 2 t = 0 ⋮ w ¯ H a j K j K t = 0 (8)

The constrains make sure that the antenna gain of direction described by a j k is 0, thus suppress the interference coming from this direction. The other way is to solve w ¯ that makes the residual interference power zero: p j r e s = E ∑ k = 1 K w ¯ H a j k j k ( t ) ) ⋅ ∑ k = 1 K w ¯ H a j k j k t ) * = 0 (9) where p j r e s is the residual interference power.

Denote the interference steering vector matrix as A j = a j 1 a j 2 ⋯ a j K (10)

According to Eq. 6, the covariance matrix of the interference signal is non-negative. As a result, Eq. 8 is equivalent to the following form: w ¯ H A j = 0 r (11) where 0 r is a row vector, A j is an and matrix of dimension N×K. Denote the rank of A j as r a n k A j , the maximum value of r a n k A j is N. Analyze the solution set of Eq. 8. If r a n k A j ≤ N-1, w ¯ satisfies Eq. 8 and w ¯ ≠ 0 , thus the interferences can be completely suppressed. If r a n k A j = N, Eq. 8 is satisfied only if w ¯ = 0 , thus the navigation signal cannot be retained while suppressing interference, and anti-interference processing becomes invalid. It can be seen that for mutually independent interferences, the anti-jamming ability of the N-element array antenna depends not only on the number of interferences, but also on the spatial correlation of the interference signal steering vector. The precondition for the N-element array to suppress at most N-1 interference is that the steering vectors of the interference signal are linearly uncorrelated. The precondition for N-element array to suppress more than N-1 interferences is that the rank of the interference steering vector matrix is not greater than N-1.

Under the condition that the interferences are independent of each other, the definition of array sup-DOF interference is proposed according to spatial anti-jamming principle.

Definition 1: Assume interferences are independent of each other, and their steering vectors can be represented by a finite number of column vectors. When the number of interferences simultaneously received by the antenna array is greater than or equal to the number of array elements N, and at least N steering vectors are linearly uncorrelated, it is defined that the number of interferences surpass the array degree of freedom.

This definition can be described as follows: r a n k A j < N   i n t e r f e r e n c e s   d o n o t   s u r p a s s   t h e   a r r a y   d e g r e e   o f   f r e e d o m r a n k A j = N   i n t e r f e r e n c e s   s u r p a s s   t h e   a r r a y   d e g r e e   o f   f r e e d o m (12)

The above definition is only applicable to the case where interference signals are independent of each other. If the interference signal has correlation, or the steering vector of the signal cannot be represented by a limited number of column vectors, the interference suppression principle can be analyzed through the residual interference power of anti-jamming processing.

According to Eq. 7, the residual interference power is p j r e s = E y j t y j * t (13)

The above equation can be written as p j r e s = w ¯ H R j j w ¯ (14)

For GNSS array antenna receiver, the goal of anti-jamming process is to make the residual interference power zero, while retaining the navigation signal as much as possible. That is to say, the solution w ¯ satisfies w ¯ ≠ 0 while p j r e s = w ¯ H R j j w ¯ = 0 (15)

According to Rayleigh entropy theorem, the value range of interference residual power is λ min ≤ p j r e s ≤ λ max (16) where λ min and λ max is the minimum and maximum eigenvalues of R j j . The equal signs are taken when w ¯ equals to the corresponding eigenvector. In this case, the residual interference power of the anti-jamming process is optimized.

If the minimum eigenvalue of R j j is zero, the residual power is 0, and the interference signal is completely suppressed, which means that R j j is not a full-rank matrix; If the minimum eigenvalue of R j j is not zero, the interference signal cannot be completely suppressed, at this moment R j j is a full-rank matrix.

Thus, the sup-DOF interference can be defined by the rank of the interference covariance matrix:

Definition 2: Assume the number of interferences received by the array antenna simultaneously is K (K ≥ 1). If the rank of the covariance matrix of the received interference equals to the number of antenna elements N, the interferences surpass the array degree of freedom. The above interferences are collectively referred to as array sup-degree of freedom interference, or sup-DOF interference for short.

Denote the rank of the interference covariance matrix as r a n k R j j . This definition can be described as r a n k R j j < N   i n t e r f e r e n c e s   d o n o t   s u r p a s s   t h e   a r r a y   d e g r e e   o f   f r e e d o m r a n k R j j = N   i n t e r f e r e n c e s   s u r p a s s   t h e   a r r a y   d e g r e e   o f   f r e e d o m (17)

According to Formula. 17, r a n k R j j is only related to the second-order statistical characteristics of the received array interference. The scope of application of this definition has no limit on the signal correlation and spatial correlation of the interference signals.

Based on the above analysis, the following conclusion can be drawn:

Conclusion 1: If the number of interferences is greater than or equal to the number of array elements (K ≥ N), the interference may not surpass the array degree of freedom. The existence of a specific direction allows the antenna array to completely suppress K interferences.

The specific incident directions in conclusion 1 can be divided into two categories. The first category is that the incident directions of interference are different, while their steering vectors are equal or conjugate; The second category is that the K steering vectors corresponding to different incident angles are correlated with each other and can be represented by N-1 vectors. Among them, it is difficult to find the incident direction of the second category by enumerating. Finding the incident direction of the second category has become an open problem in the field of differential geometry, which will not be further researched in this paper.

Taking the central UCA (Uniform Circular Array) of four elements as an example, the typical interference incidence direction is taken to further explain conclusion 1. The coordinates of array element are given in Eq. 18. Assume that the power of interferences in Figure 2 is 1, and the interference signals are independent of each other. In Figure 2A, the incident direction of interference 1 is θ j 1 , φ j 1 . Interference 1 and interference 2 are symmetrical about the xOy plane, interference 1 and interference 3 are opposite in the incident direction, interference 1 and interference 4 have the same incident elevation angle, and the azimuth difference is π. The incident directions of interference 2, 3, and 4 are − θ j 1 , φ j 1 , − θ j 1 , − φ j 1 and θ j 1 , φ j 1 + π respectively.

In the Oxyz coordinate system, the array element coordinate matrix is as follows. The first, second and third columns of the matrix are respectively the x, y, and z coordinates of the array element, and λ c is the signal carrier wavelength. P e l e = 0 0 0 1 2 λ c 0 0 − 1 2 λ c 3 2 λ c 0 − 1 2 λ c − 3 2 λ c 0 (18)

The interference steering vector is a j k = exp j 2 π f c P e l e r j k c (19) where c is the speed of light, r j k is the unit direction vector, and f c is the carrier frequency. r j k = cos ⁡ θ j k ⁡ cos ⁡ φ j k cos ⁡ θ j k ⁡ sin ⁡ φ j k sin ⁡ θ j k (20)

Simplify the steering vector expression, it can be written as a j k = exp j π ⁡ cos ⁡ θ j k 0 cos ⁡ φ j k sin φ j k − π 6 − sin φ j k + π 6 (21)

The steering vectors of interference 1–4 is a j 1 = a j 2 = 1 exp j π ⁡ cos ⁡ θ j 1 ⁡ cos ⁡ φ j 1 exp j π ⁡ cos ⁡ θ j 1 ⁡ sin φ j 1 − π 6 exp − j π ⁡ cos ⁡ θ j 1 ⁡ sin φ j 1 + π 6 ,   a j 3 = a j 4 = 1 exp − j π ⁡ cos ⁡ θ j 1 ⁡ cos ⁡ φ j 1 exp − j π ⁡ cos ⁡ θ j 1 ⁡ sin φ j 1 − π 6 exp j π ⁡ cos ⁡ θ j 1 ⁡ sin φ j 1 + π 6 (22)

It can be seen that the steering vectors of interference 1 and 2 are conjugate with interference 3 and 4. According to Eq. 6, the covariance matrix of each interference can be obtained as follows R j 1 = R j 2 = R j 3 = R j 4 (23)

Denote the steering vector matrix and covariance matrix of the above four interferences as A j _ 1 and R j j _ 1 respectively. Considering the sup-DOF interference definition 1 and definition 2, it can be obtained that r a n k A j _ 1 = 1 (24) r a n k R j j _ 1 = 1 (25)

Therefore, the above four interferences are equivalent to one interference. Interference 1–4 belong to the incident direction of first category interference mentioned at the end of Section 3.1.

Denote the incident directions of four interference in Figure 2B are θ j 1 , φ j 1 , θ j 5 , φ j 5 , θ j 6 , φ j 6 and θ j 7 , φ j 7 respectively. The four incident directions have the following relationship: θ j 1 = θ j 5 = θ j 6 = θ j 7 (26) φ j 1 = − φ j 7 ,   φ j 5 = − φ j 6 (27) Denote the steering vector matrix and covariance matrix of the above four interferences as A j _ 2 and R j j _ 2 respectively, A j _ 2 = a j 1 a j 5 a j 6 a j 7 (28) R j j _ 2 = ∑ k = 1,5,6,7 a j k a j k H (29)

It can be simplified that R j j _ 2 = 1 0 0 r 14 0 1 0 r 24 0 0 1 − 1 0 0 0 0 (30) Among which r 14 = e j π ⁡ cos ⁡ θ j 1 3 ⁡ cos ⁡ φ j 5 + 3 sin ⁡ φ j 5 × − 1 + e j π ⁡ cos ⁡ θ j 1 3 ⁡ cos ⁡ φ j 1 − 3 ⁡ cos ⁡ φ j 5 + 3 sin ⁡ φ j 1 − 3 sin ⁡ φ j 5 − e − 2 3 j π ⁡ cos ⁡ θ j 1 ⁡ sin ⁡ φ j 5 + e j π ⁡ cos ⁡ θ j 1 3 ⁡ cos ⁡ φ j 1 − 3 ⁡ cos ⁡ φ j 5 − 3 sin ⁡ φ j 1 − 3 sin ⁡ φ j 5 e 2 j π ⁡ cos ⁡ φ j 1 ⁡ cos ⁡ θ j 1 − e 2 j π ⁡ cos ⁡ φ j 5 ⁡ cos ⁡ θ j 1 (31) r 24 = e 2 j π ⁡ cos ⁡ θ j 1 cos ⁡ φ j 1 + cos ⁡ φ j 5 × e − 2 j π ⁡ cos ⁡ θ j 1 ⁡ sin φ j 1 − π 6 − e − 2 j π ⁡ cos ⁡ θ j 1 ⁡ sin φ j 5 − π 6 + e 2 j π ⁡ cos ⁡ θ j 1 ⁡ sin φ j 1 + π 6 − e 2 j π ⁡ cos ⁡ θ j 1 ⁡ sin φ j 5 + π 6 − e 2 j π ⁡ cos ⁡ φ j 1 ⁡ cos ⁡ θ j 1 + e 2 j π ⁡ cos ⁡ φ j 5 ⁡ cos ⁡ θ j 1 (32)

It can be seen that the rank of the covariance matrix is 3, and the rank of the guidance vector matrix is 3. r a n k A j _ 2 = 3 (33) r a n k R j j _ 2 = 3 (34)

Interference 1, 5, 6, and interference 7 exist in the subspace of dimension-3, so they are equivalent to three interferences. They belong to the incident direction of the second category interference mentioned at the end of Section 3.1.

According to the typical interference deployment scenarios in navigation countermeasure, the sup-DOF interferences are usually gathered in certain space angles while their exact locations are uncertain [23]. To study the rules of anti-sup-DOF-jamming performance, it is simpler to depict a cluster of jammers by their DOA boundaries rather than concentrate on specific jammer configurations. As a result, the interference deployment is described in the following parameters. Suppose the interference number is K, the incident angle in elevation is θ j 1 , θ j 2 , ⋯ θ j K , the azimuth angle is φ j 1 , φ j 2 , ⋯ φ j k , and the power is p j 1 = p j 2 = ⋯ = p j K . In order to describe the positions of the K interferences, the central incident direction of the interferences is defined as θ j 0 , φ j 0 , and the interference distribution is simplified as Δ Θ j , Δ ϑ j . Their expressions are as follows. θ j 0 = ∑ k = 1 K θ j k K φ j 0 = ∑ k = 1 K φ j k K (35) Δ Θ = max θ j i − θ j l     i , l ∈ 1 , K , i ≠ j Δ ϑ = max φ j i − φ j l     i , l ∈ 1 , K , i ≠ j (36)

See Figure 3 for the schematic diagram of the central incident direction and distribution.

In order to simplify the analysis of the central incident direction and distribution, the interference signals are assumed to be independent and the intersection angle of adjacent interferences are equal. A two-element antenna is taken as an example for analysis.

The positions of the two elements are shown in Figure 4. Its coordinates are as follows, where λ c is the signal carrier wavelength. P 2 e l e = x 1 y 1 z 1 x 2 y 2 z 2 = 0 0 0 0 0 1 2 λ c (37)

The steering vector of the kth interference is a j k = exp j π 0 sin ⁡ θ j k (38)

The characteristic polynomial of the interference covariance matrix is a one-variable quadratic equation about the eigenvalue λ : f λ = λ I − R j j = λ 2 − M 11 + M 22 ⋅ λ + R j j (39) Wherein, M 11 and M 22 are the algebraic cofactors of two diagonal elements respectively, ⋅ representing the determinant of the matrix.

Take two independent interferences as an example. Set the interference power as p j 1 and p j 2 respectively. The interference central direction and distribution are set as follows. θ j 0 = θ j 1 + θ j 2 2 (40) Δ Θ j = θ j 2 − θ j 1 (41)

Combine Eq. 6 and Eq. 39, let f λ = 0 , then the optimal residual interference power is obtained: p j r e s θ j 0 , Δ Θ j = p j 1 + p j 2 − p j 1 + p j 2 ) 2 − 2 p j 1 p j 2 ( 1 − cos ⁡ ( π ⁡ sin ⁡ ( θ j 0 − Δ Θ j 2 ) − π ⁡ sin ⁡ ( θ j 0 + Δ Θ j 2 ) (42)

The partial derivative of the above equation is obtained from θ j 0 and Δ Θ j : ∂ p j r e s θ j 0 , Δ Θ j ∂ Δ Θ j = − p j 1 p j 2 π ⁡ sin π ⁡ sin θ j 0 − Δ Θ j 2 − π ⁡ sin θ j 0 + Δ Θ j 2 ⋅ cos θ j 0 − Δ Θ j 2 + cos θ j 0 + Δ Θ j 2 2 p j 1 + p j 2 ) 2 − 2 p j 1 p j 2 ( 1 − cos ⁡ ( π ⁡ sin ⁡ ( θ j 0 − Δ Θ j 2 ) − π ⁡ sin ⁡ ( θ j 0 + Δ Θ j 2 ) ∂ p j r e s θ j 0 , Δ Θ j ∂ θ j 0 = p j 1 p j 2 π ⁡ sin π ⁡ sin θ j 0 − Δ Θ j 2 − π ⁡ sin θ j 0 + Δ Θ j 2 ⋅ cos θ j 0 − Δ Θ j 2 − cos θ j 0 + Δ Θ j 2 p j 1 + p j 2 ) 2 − 2 p j 1 p j 2 ( 1 − cos ⁡ ( π ⁡ sin ⁡ ( θ j 0 − Δ Θ j 2 ) − π ⁡ sin ⁡ ( θ j 0 + Δ Θ j 2 ) (43)

It can be seen from the observation that it is difficult to simplify p j r e s θ j 0 , Δ Θ j to the multiplication of two one-variable functions. The central incident direction is closely coupled with the interference distribution.

If Δ Θ j ≠ 0 , let ∂ p j r e s θ j 0 , Δ Θ j ∂ θ j 0 = 0 (44)

The central interference incident direction that minimizes the optimal residual interference power can be solved θ j 0 = π 2 + π l ,   l = 0 , ± 1 , ± 2 , ⋯ (45)

Combine Eq. 45 with Eq. 42, it can be obtained that p j r e s ( θ j 0 , Δ Θ j ) = 0 (46)

Extending to K interferences (K ≥ 2), the optimal residual interference power is p j r e s = ∑ k = 1 K p j k − ∑ k = 1 K p j k 2 − 2 ∑ u = 1 , v = 1 u ≠ v K p j u p j v 1 − cos π ⁡ sin ⁡ θ j u − π ⁡ sin ⁡ θ j v (47)

According to the above formula, if cos π ⁡ sin ⁡ θ j u − π ⁡ sin ⁡ θ j v = 1 , the optimal residual interference power reaches the minimum value p j r e s , min = 0 ; If cos π ⁡ sin ⁡ θ j u − π ⁡ sin ⁡ θ j v = − 1 , the optimal residual interference power reaches the maximum value p j r e s , max . The maximum value p j r e s , max ≤ ∑ k = 1 K p j k , and the condition for the equality is K = 2 (see Appendix A for detailed proof). Therefore, if the total power of interference is fixed, two interferences can achieve better jamming effect than multiple interferences for two-elements array.

Interference cancellation ratio (ICR) is defined as the ratio of input interference power to residual interference power: I C R = ∑ k = 1 K p j k p j r e s (48)

Setting two typical interference configurations in Table 2, the above analysis results are numerically illustrated.

Assume that the interference power is equal, and the intersection angle of adjacent interferences are equal. The variation of optimal residual interference power and ICR against the interference distribution and central incident direction is shown in Figure 5. Figures 5(A−D) show the numerical calculation results of configuration ① and configuration ② respectively.

As shown in Figure 5A, for a two-element array, if the interference number K = 2, the maximum optimal residual interference power is 1 and the minimum value is 0. If the incident direction of interference is symmetrical about ±90°, sin ⁡ θ j k = sin π − θ j k = sin ⁡ θ j k , thus two interferences are equivalent to one interference. Note that in Figure 5B, ICR should be positive infinity at extreme points, where Δ Θ = 0 and θ j 0 = ± 90 ∘ . It can be seen from Figures 5C, D that when the interference number K ≥ 2, the maximum value of optimal residual power is less than 1, which shows that it is less effective than two interferences, the analysis at Eq. 47 is verified. For the case of multi element array, if the interference number K ≥ N and the total interference power is 1, it can be either concluded that the maximum optimal residual interference power is 1 (see Appendix B for detailed proof), and the minimum is 0 (according to Section 3.2).

Conclusion 2: If the number of interferences is greater than or equal to the number of array elements, the interference residual power obtained by optimal spatial filter is closely related to the central incident direction and the interference distribution. The maximum value of the interference residual power is the sum of the interference power, and the minimum value is 0. In particular, when the element number N = 2, the maximum residual power equals to the sum of interference power only if the number of interference K = 2.

The above conclusion show that for the evaluation of sup-DOF anti-jamming capability, the interference incidence direction will cause huge differences in the evaluation results, and the anti-interference capability needs to be evaluated separately for different interference deployment scenarios; For the deployment of jammer, if the number and power of several jammer are determined, the interference efficiency can be improved by reasonably setting the incident direction of interference.

It can be seen from Section 4.1 that in order to study the rule of sup-DOF interference suppression performance, it is necessary to test multiple groups of interference incidence directions, and the workload of simulation calculation is huge. Eq. 45 shows that the interference suppression performance of the linear array takes π as the cycle. If the anti-jamming performance of the plane array also has periodicity, it can reduce the repetitive calculation and improve the simulation efficiency. In order to solve this problem, this section analyzes the periodicity of the interference suppression performance of the navigation receiver array antenna, assuming that the interference signals are independent of each other.

First, take the central UCA of four elements as an example. The element positions are shown in the orange circle in Figure 6, and the coordinates are shown in the Eq. 18.

Suppose the incident direction in elevation remains constant, if the azimuth angle φ j k deviates 2 π 3 + 2 π 3 m   m = 0 , ± 1 , ± 2 , ⋯ from the initial direction, then the deviated steering vector is a j k ′ = exp − j π λ c cos ⁡ θ k 0 − sin φ j k + π 6 cos φ j k sin φ j k − π 6 (49) a j k ′ equals to line exchange of a j k , which can be written as follows: a j k ′ = T s 1 ⋅ a j k (50) where T s 1 is the matrix representing row exchange. T s 1 = 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 (51) T s 1 is a unitary matrix, T s 1 H = T s 1 − 1 .

If all K interferences deviate 2 π 3 + 2 π 3 m   m = 0 , ± 1 , ± 2 , ⋯ relative to the original incident direction, denote R j j ′ as the deviated interference covariance matrix. The relationship between R j j ′ and the original covariance matrix R j j is as follows. R j j ′ = T s 1 R j j T s 1 − 1 (52)

Matrix R j j ′ is similar to R j j , so they have same (smallest) eigenvalues. As a result, this deviation of incident direction in azimuth does not change the optimal residual interference power and ICR.

In the same way, it can be proved that if the azimuth incidence angle deviates π 3 + 2 π 3 m   m = 0 , ± 1 , ± 2 , ⋯ , the deviated steering vector a j k ″ is equal to the line exchange of the conjugate of a j k . a j k ″ = T s 2 ⋅ a j k * (53) T s 2 = 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 (54)

Similarly, if all K interferences are deviated π 3 + 2 π 3 m   m = 0 , ± 1 , ± 2 , ⋯ relative to the original incident direction, denote R j j ″ as the deviated interference covariance matrix. The relationship between R j j ″ and the original covariance matrix R j j is as follows. R j j ″ = T s 2 R j j T T s 2 − 1 (55)

Matrix R j j ″ is similar to R j j T . What’s more, the transpose transformation does not change the eigenvalues of the matrix, so the minimum eigenvalue of R j j ″ is the same as that of R j j . This deviation of incident direction in azimuth does not change the optimal residual interference power or ICR.

To sum up, if the relative relationship between the interference incident directions is certain, the residual interference power of four-element central UCA is periodic in the azimuth direction. The period is π 3 .

Secondly, take the five-element central UCA as an example for analysis. The array element position is shown in the golden circle in Figure 6, where one element is located at the origin, and the other four elements are uniformly distributed on the circumference with half-wavelength radius. By the same derivation method, when the incident direction of the interference is deviated π 2 + 2 π 3 m   m = 0 , ± 1 , ± 2 , ⋯ from the original incident direction, the new interference covariance matrix is similar to the original covariance matrix, and the minimum eigenvalue is the same, so the period of the interference residual power is π 2 .

Take the interference number K = 10 as an example to illustrate the numerical calculation of the above analysis results. See Figure 6 for array antenna geometry. Interference parameters are given in Table 3.

The relative incident direction of the interference remains constant, and the central incident direction of the azimuth is changed. The changes of residual interference power and ICR against central incident direction in azimuth are shown in Figure 7. Wherein, Figures 7A, B show that the anti-jamming performance period of four-element central UCA is π 3 , while Figures 7C, D show that the anti-jamming performance period of five-element central UCA is π 2 , which verifies the above analysis.

In spite that the central UCA is taken as an example for theoretical derivation and numerical calculation, it can be seen from the derivation process that the azimuth period of the suppression performance is only related to the number of elements uniformly distributed on the circumference, and whether or not to deploy elements at the center of the circle has no effect on the period size. If the number of elements uniformly distributed on the circumference is M (M ≥ 3), it can be further generalized that the interference suppression performance period of UCA or central UCA is. π M M   i s   o d d 2 π M M   i s   e v e n (56)

Correspondingly, the anti-jamming performance repeats for N p cycles when the interference azimuth changes from 0 to 2 π towards the central incident direction. N p = 2 M M   i s   o d d M   M   i s   e v e n (57)

The following conclusion can be further generalized:

Conclusion 3: Assume the interference signals are independent of each other. For an UCA with half-wavelength radius circumference, if the number of elements uniformly distributed on the circumference is M (M ≥ 3), and the number of elements at the center of the circle is 0 or 1, then the interference suppression performance period in azimuth is 4 π M ⋅ 3 + ( − 1 M + 1 . Correspondingly, interference suppression performance repeats 3 + ( − 1 M + 1 ) M / 2 cycles when the central incident direction turns over the range of 2π in azimuth. If the number of elements at the center of the circle is 1, the above conclusion is also applicable to M = 1,2, and the array is degenerated into an ULA (Uniform Linear Array).

This conclusion can be applied to the evaluation of antenna array anti-jamming performance. In the study of the relationship between antenna array anti-jamming performance and azimuth incidence angle, it can reduce the repetitive test or simulation calculation, and improve the evaluation efficiency by 3 + ( − 1 M + 1 ) M / 2 times.