(2) Digital beamforming technique. After the data are filtered by the analog beamforming technique, they are sampled into the baseband through the RF channel and ADC. The original digital beamforming-based anti-jamming processing technique, that is, Equations (6) and (8), needs to be calibrated and then adopted to suppress interference in the digital domain. In the end, the final suppression of interference is completed.
3.1. Analog Beamforming Technique
Firstly, the analog beamforming technique is introduced. As shown in
Figure 2, the GNSS hybrid array receiver includes a total of
M SNFs, represented by
. Each SNF cannot be generated independently. The MNF not only needs to achieve partial suppression of interference in the analog domain but also strives to retain the original signal’s DOA information as much as possible. After GNSS and interference signals are filtered by the MNF, the signal can continue to retain the original DOA distribution in the baseband. Therefore, effective implementation of subsequent baseband anti-jamming algorithms can be ensured. The bampattern synthesis method is used to generate the MNF [
20].
In the synthesis process of the MNF, different steering vector forms are used as follows:
Here,
represents the steering vector corresponding to the
m-th SNF; that is, the corresponding antenna is used as the reference antenna for each
.
The SNF and MNF constraints are simultaneously applied to the synthesized pattern. The first SNF is generated by the SNF constraint. Then, using as a reference mask, the other notch filters are generated by the MNF constraints.
- (1)
SNF Constraints
The SNF constraints include constraints on both non-interference and interference areas. The non-interference area and interference area are constrained separately. The 3 dB main lobe width of the interference spatial spectrum is used as a basis to distinguish between the non-interference area and the interference area. The area outside the 3 dB main lobe width of the interference spatial spectrum is the non-interference area, and the other area is the interference area.
The constraint for the non-interference area is as follows:
where the lower and upper magnitude bounds are applied [
20]. In Equation (10), the non-interference area is divided into
P uniformly spaced angles,
represents the set of
P angles in the non-interference area, i.e.,
, and
is the ripple term.
The constraint for interference area is as
In Equation (11),
Q represents the number of interferences,
represents the set of
Q angles for
Q interference areas, i.e.,
, and
is the nulling term.
As shown in Equations (10) and (11),
P angles are used to constrain the non-interference area, but only one single angle is used to constrain each interference. This situation will lead to the inability of the objective function to converge and generate a deep enough null that meets the requirements. To solve this problem, the method of virtual interference is adopted, and the
K interferences at the same interference direction are assumed. In practice, the value of
K should be greater than or equal to the number of non-interference area constraint angles
P, that is,
K ≥
P. And then, Equation (11) is written as follows:
where
- (2)
MNF Constraints
After the first SNF
is generated based on the SNF constraints, the MNF constraints are used to generate
. The MNF constraints are used to partially suppress the interference and retain the original DOA distribution of the signal. Omitting the noise term
in Equation (1), the array input signal
can be expressed as follows:
Here,
represents the GNSS signal and interference signal. Substitute Equation (9) into Equation (14), the array input signal for each SNF in MNF can be represented as follows:
The output of the signal
after being filtered by each SNF
is
where
.
represents the beampattern of the
m-th SNF. Therefore, the vector form of the output signal, after being filtered by the MNF, is
In the MNF beampattern synthesis, the below constraints are applied.
Then, Equation (17) can be written as follows:
Through Equation (19), it can be seen that the output data of the MNF can retain the original DOA distribution.
The Equation (18) for the MNF constraints can be written as follows:
Here, Equations (20) and (21) represent the consistency constraints for non-interference and interference areas between the
m-th SNF and the first SNF, respectively.
and
represent the response error terms.
The
norm of
is adopted as the objective function, which is used to receive useful signals with high gain. The objective function is shown below.
Due to Equation (10), the objective function (22) is a nonconvex optimization problem, and the alternating directions method of multipliers (ADMM) optimization method is used to solve the objective function. It decomposes the problem into multiple sub-problems, iteratively solves the sub-problems in sequence, and obtains the final optimization result. The specific derivation process can be referred to in the literature [
20,
21]. The digital MNF
is obtained through Equation (22), and it requires less than 100 iterations. After quantization, the analog beamforming coefficient
is solved.
3.2. Digital Beamforming Technique
Secondly, a digital beamforming technique is introduced. The MVDR technique and PI technique in Equations (6) and (8) are still used in the digital beamforming part. However, through Equations (20) and (21), it can be seen that although we have retained the DOA distribution of the original GNSS signal as much as possible, a small amount of amplitude/phase error is inevitably introduced. Therefore, in order to maximize the interference suppression ability of the digital domain anti-jamming technique, two processes should be added: amplitude/phase error calculation and calibration.
The beampattern
corresponding to each analog SNF
should be firstly calculated as
Then, the calculation process of amplitude/phase error is as follows:
Here,
represents the amplitude/phase error between the
m-th SNF
and the first SNF
at angle
.
Finally, the updated MVDR algorithm is obtained as follows:
where
,
, and
represents the operation of transforming a vector into a diagonal matrix.
Due to the fact that a small amount of amplitude/phase error has no impact on the PI algorithm, Equation (8) is not updated in this section.
3.3. Implementation of Proposed Technique
Figure 2 illustrates the data processing flowchart of the robust wideband interference suppression method for the GNSS array antenna receiver based on a hybrid beamforming technique. The steps of the method are as follows:
Step 1: Confirm whether the interference’s DOA information has been obtained. If the interference’s DOA information is obtained, move to Step 5. If not, move to Step 2.
Step 2: Confirm whether ADC saturation occurs. If ADC saturation occurs, move to Step 3. If not, move to Step 4.
Step 3: The RF gain is adjusted from 30 dB to −10 dB to ensure that ADC does not experience saturation. The interference DOA information of the first epoch can be obtained.
Step 4: Set the weight of the 2D spatial notch filter to , and then move to Step 7.
Step 5: Confirm whether the RF gain is set to 30 dB and set the RF gain to 30 dB. The prior information on interference direction is first obtained from DOA estimation, and then the interference angle set and the non-interference angle set are determined based on the interference direction.
Step 6: Based on sets
and
, analog beamforming technique in
Section 3. 1 is run and
M 2D spatial notch filters
are obtained through multiple iterations.
Step 7: Quantify the weight ; the analog beamforming coefficient is obtained.
Step 8: Calculate the beampattern for each SNF based on Equation (23). And then calculate the amplitude/phase error matrix between each notch filter and the first SNF based on Equation (24).
Step 9: The analog MNF is written down to the corresponding phase shifters and attenuators in the hybrid array anti-jamming receiver.
Step 10: After filtering through the analog MNF, the signal is amplified, filtered, and sampled to the digital baseband signal .
Step 11: DOA estimation results can be obtained using a multiple signal classification (MUSIC) algorithm [
22] on baseband data, and then DOA estimation results are sent to Step 5 in real-time.
Step 12: Using Equation (25), a digital beamforming technique is performed on the baseband data .
The robust wideband interference suppression process based on the hybrid beamforming technique is completed.
This section presents a hybrid beamforming-based GNSS anti-jamming method. Compared with the conventional digital beamforming-based anti-jamming method, there will be an obvious increase in hardware structure and implementation complexity. In terms of hardware structure,
phase shifters and
attenuations need to be added to the AFE, as shown in
Figure 2. In terms of implementation, DOA estimation, analog MNF calculation, amplitude/phase error calculation, and calibration need to be added to the implementation, as shown in
Figure 4.