In this section, we describe the GLE search method for DCAR and the multi-frequency coherent DOA ambiguous-solving method. We first provide a method for using GLE in receiving search, which can address the DCAR inefficiency issue. Moreover, it is feasible to transmit and receive an electromagnetic wave signal using the same DCAR antenna. In
Section 3.2, we provide a strategy for ambiguity-solving by employing several frequencies because there is DOA ambiguity at a single frequency. Using transceiver multi-frequency operating and the sliding window averaging approach, we resolve the ambiguous DOA described in
Section 3.2. In
Section 3.3, we investigate the beam migration issue of the grating lobe searching target and achieve coherent multi-frequency integration in the GLE search.
3.1. DCAR Transceiver Search Method with GLE
Three main issues with GLE search need to be resolved: how to employ the grating lobes in the DCAR search; how to distinguish whether the target is searched by the main lobe or the grating lobe; and how to accurately estimate the target’s angle.
Firstly, we provide a brief overview of how to utilize grating lobes in transmitting. Due to the uniformly sparse inter-array DCAR’s regular distribution of the grating lobes, the effectiveness of the transmitting search will improve if the grating lobes are employed for search.
Figure 4 demonstrates how to simultaneously search with grating lobes. As can be observed, a GLE search only needs three beamforming operations at three beam points to finish the target area’s search, but a main lobe search needs fifteen scans.
Since the beams are periodically repeated in the inter-array pattern, the DCAR search can be decomposed into two steps, namely inter-array and intra-array search for simplicity. The intra-array search provides a rough coverage of the search area with the wide beams of the subarray. In this step, we calculate the inter-array search weight vectors corresponding to
. Different subarrays share the same set of weight vectors. Then, we search with the narrow inter-array beam in the range of the wide beam. The inter-array weights are solved for
as shown in
Figure 5. The same inter-array search weights can be used for different subarray wide beam pointings. Importantly, if we need to point out that what we call step-by-step here actually refers to a fast beamforming weight calculation method, rather than a real scanning process.
In GLE search, what is modified is the beam adjusting process in the second step for the inter-array. The search range of the main lobe in GLE is no longer the whole wide main lobe of the subarray, instead, we only need to search for the range
. The rest area will be covered by the grating lobes.
Denote
as the transmit beam scheduling result from the GLE search
where
represents the main lobe scanning beam center pointing,
.
When the main lobe is pointing at the n-th arranged beam position
, the grating lobes will point at
as illustrated in
Figure 6.
The transmitting GLE search process can be described as calculating the vector of weights corresponding to the pointing of the beam center pointing
separately, and carrying out the transmitting DBF [
43] weight to complete the search process. The specific execution process can be found in the procedure given in Algorithm 1.
Receiving must take into account the resolution and identification of the target that the grating is searching for, where, unlike in transmitting, power coverage is simply taken into account. Targets searched by the grating lobes would ordinarily be suppressed as interference. This issue arises because, historically, we have assumed that a single search equates to a single target. It is vital to recognize this situation and obtain the real angle estimate of the target since the GLE search may search for several targets that are not in the main lobe.
Let us assume that a signal processing method exists to identify the case of the grating search target and that the angular information can be correctly estimated, which is the focus of the next section. Then, the GLE search at receiving is performed in almost the same way as at transmitting. A smaller beam hop
can also be selected during receiving process to avoid the power gain loss at the non-beam center area as described in
Section 2.3.1.
Algorithm 1: Transmitting Grating Lobes Searching Method |
INPUT: OUTPUT: Transmitting GLE search DBF weight Step 1. Subarray wide beam search scheduling Step 2. Subarray weight calculating For weight solving of the pointing (X) End Step 3. Inter-array GLE search scheduling Step 4. Inter-array weight calculating For weight solving of the pointing (X) End Step 5. DCAR search DBF weight (Kronecker product of ) For For (DBF point to ) End End
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In this scenario, the weight vector upon receiving DBF needs to be recalculated.
The receiving GLE search process at a single frequency can be described using the following algorithmic flow. The process of target detection and parameter estimation is described in the later sections.
Algorithm 2: Transceiving GLE Search Signal Processing Method (Without DOA ambiguity Solving) |
Pre-processing: Transmitting DBF Searching, perform DBF process with weight in Algorithm 1 to accomplish transmitting search. Step 1. Beam schedule choosing: same beam scheduling with transmission? Yes to 2.b; No to 2.c; Step 2. Perform DBF process to the received signal with weight in Algorithm 1 to accomplish the receiving DBE search. Step 3. Receiving beam scheduling (more dense beam): , Step 4. Weight calculation: solve beam pointing to weights (Algorithm 1) Step 5. Perform the DBF process to the received signal with the weight in 2.d to accomplish the receiving search. Step 6. Parameter estimation, recognition of main lobe, or grating search and parameter
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3.2. Unambiguous DOA Estimation Method for Objectives with GLE Search
The principle of multi-frequency DOA ambiguity-solving is similar to multi-frequency Doppler ambiguity-solving or communication delay ambiguity solving [
44]. Assuming that the frequencies are
and the frequency varies with pulses
periodically repeated. The received data are noted as
, where
is a multi-dimensional matrix corresponding to the pulse frequency, array element, and range cell, respectively. The data matrix format is shown in
Figure 7, where the data at each frequency are noted as
, and
.
The ambiguous DOA estimations are obtained by performing DBF scanning on the received data. The specific process is performing beamforming at multiple receiving angles, as shown in (17) of the multi-channel data for each scanning beam, each frequency, and each range cell of the received data. Multi-dimensional search detection of the beamforming results in the range estimate and ambiguous DOA estimate of the target at each frequency being obtained.
If the target range between pulses has no migration within a coherent processing interval (CPI), then the range estimates at each frequency are equal. The transmitting scanning beam position can be obtained from DOA estimates. If the scheduling of the scanning beam upon receiving is the same as at transmission, then the DOA estimates will be equal to the corresponding beam center angle in . It is also possible to receive with a narrower angular interval for a more accurate DOA estimate. DOA ambiguity-solving must obtain the most accurate ambiguous DOA values possible. This will directly affect the accuracy of the final DOA estimation.
Extending the ambiguous DOA
to the range
according to the ambiguity period
to obtain all possible DOA value sequences,
represents the current subarray wide beam pointing, and
represents the subarray wide beam width
where
,
is equal to the angular difference between two neighboring lobes of the sparse inter-array pattern at frequency.
Set all the possible DOA sequences
into a vector
and rearrange them in ascending or descending order. Perform a sliding window process with a length of
I to vector
. Denote the DOA set of sliding window detection as
and calculate the mean of
. Take the sum of the absolute value of the difference between each element in
and the DOA set mean
as the judgment factor,
Figure 8 illustrates the process of multi-frequency sliding window averaging ambiguity-solving. After performing sliding window averaging on the vector
, we can obtain the actual DOA set of the different frequencies by finding the smallest judgment factor
. Take the mean value of this DOA set as the ambiguity-solved DOA estimation,
.
The minimum judgment factor of the real DOA set should be equal to 0 in the ideal situation. However, the ambiguous DOA obtained from parameter estimation is the beam pointing angle with the smallest angular spectrum value. The angular spectrum calculation is only performed at a finite number of scanning beam positions. Therefore, the DOA is usually with errors. In addition, the ambiguity period is an irrational value, and the ambiguity period is also with errors in practical operation. Thus, the judgement factor is not equal to 0, which is why the method mentioned earlier is used instead of directly using the mathematical solution when performing DOA ambiguity-solving.
The sliding window averaging ambiguity-solving process can be described using the following algorithmic procedure.
Algorithm 3: Multi−Frequency Sliding Window Averaging Ambiguity-Solving |
Step 1. DOA estimation: Obtain the ambiguous DOA estimates at each frequency Step 2. Extend the ambiguous DOA: extend to the range according to the ambiguity period Step 3. Set all the possible DOA sequences and rearrange them in ascending or descending order Step 4. Sliding window DOA set, Step 5. Calculate the mean of Step 6. Calculate the judgment factor of Step 7. Find the smallest judgment factor Step 8. Obtain the ambiguity-solved DOA
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3.3. Multi-Frequency Coherent Integration with GLE Search
Multi-pulse integration is an important processing method in radar signal processing. The basic principle is summing multiple pulses to improve the SNR by exploiting the property that the target signal is correlated between pulses and that the SNR is not. Since the signal representation is complex and the system operating parameters between pulses may differ, the signal phase between pulses may differ. If the pulses are summed in the same phase, the integration is called coherent integration [
45] and can obtain the maximum SNR improvement. If only summing the pulse amplitude without considering the phase information, the integration is called non-coherent integration.
The signal phase in GLE search is more complex than the traditional multi-pulse coherent integration scenario. For the targets in the main lobe scanning area, the ambiguous DOAs are the same, which means that they are scanned at the same scanning beam in GLE search. However, the targets in the grating lobe scanning area are not scanned at the same scanning beam position in GLE search due to the different distribution of the grating lobes at different frequencies. In other words, the target in the grating lobe scanning area has a situation similar to the inter-pulse beam migration, which needs to be considered for coherent integration.
The array manifold of the DCAR at different frequencies is
where
and
represents the time delay of the reference element.
First, integrate the multiple channel signal within the single pulse. The receiving DBF is performed along the element dimension, and the weights use the transmitting DBF weights (for equal transmission and receiving a scanning beam). When the integration of multiple-frequency data between pulses is not considered, parameter estimation can be roughly described as searching along the range dimension to obtain the range estimation, and searching along the beam dimension to obtain the DOA estimation.
The target signal of each
i-
frequency after DBF can be expressed as
where
represents the complex envelope signal of the target echo,
. According to the array signal theory,
is valid when the detection signal satisfies the narrow-band condition.
represents the amplitude gain of the target signal after receiving and transmitting DBF, and we assume that the receiving scanning beam scheduling is the same as the transmission.
Writing all target signals of all frequencies after DBF in vector form, we have
More SNR gain can be obtained by summing multiple pulses, as shown in
Figure 9. Taking the target in the main lobe scanning area as an example, the SNR gain can be utilized when summing the data at three frequencies. The result of the coherent summation
of
is
is not easily available in practical signal processing, so the coherent weight vector shown in (24) is unlikely to be available in practical applications. However, if the frequency is uniformly stepped, then the phase difference between frequencies is also uniform. The multi-frequency coherent integration can be accomplished by performing FFT on the data .
Its beam position after DBF at different frequencies may be different for the grating lobe scanning area target. Moreover, it is impossible to directly integrate the data without knowing the beam position (i.e., the DOA estimates) where the target is located. In this scenario, we can iterate all possible beam positions to find the actual beam position combination of the target at each frequency, as shown in
Figure 10. The area covered by color in
Figure 10 represents the beam position of the target.
After determining the current beam position combination, the target’s corresponding receiving scanning beam position can also be determined. The FFT is performed on the I data identified by the current combination. Furthermore, the maximum accumulation amplitude can be obtained for the combination of the target true beam positions.
The processing of the receiving the signal with multi-frequency coherent integration in GLE search is described in the Algorithm 4.
Algorithm 4: Receiving Signal Processing with Multi-Frequency Coherent Integration |
Pre-processing: Transmitting and receiving GLE search using Algorithms 1 and 2 For search beam Step 1. Receiving DBF: Weighted summing of the received signal for each channel End For beam combination Step 2. Find the receiving scanning beam position corresponding to the current combination and perform FFT on i- frequency data for each range cell Step 3. Search along the range dimension of integrating data at multiple frequencies to obtain the range estimation Step 4. Ambiguous DOA estimates: the combination of the receiving beam positions corresponding to is the ambiguous DOA estimation result . Step 5. Unambiguous DOA estimates using Algorithm 3 End Step 6. Determine the actual beam combination by finding the maximum result
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The signal power gain of a single pulse (frequency) is
where
is the signal power of a single channel.
It is assumed that the noise of different pulses and different antenna elements are independent and identically distributed zero-mean Gaussian white noise. The noise power of a single channel is
, then the noise after DBF is a zero-mean Gaussian with variance
. The noise power gain after DBF is
In general, the signal-to-noise ratio gain that can be utilized to obtain ambiguous DOA and range estimates based on single-pulse data is
When the target is in the direction of the center of the scanned beam, the maximum SNR is achieved as
The signal-to-noise gain that can be utilized to obtain the estimates of multi-frequency coherent integration is
3.4. Multiple Target Scenarios for GLE Search Implementation Expansion
When multiple targets are simultaneously in the detection area, there will be multiple ranges and DOA estimation results after target detection. In this section, we divide the multiple targets into two scenarios, depending on how they are processed. The different colors in the figure express the different estimates of DOA obtained at different frequencies.
3.4.1. Multiple Targets in the Different Range Cell
Firstly, we discuss the multiple targets located at the different ranges of cell scenarios shown in
Figure 11a. Without loss of generality, we assume the number of multi-frequencies to be 3. The detection results obtained at each frequency are
Each set of ambiguous DOA estimates in the detection result corresponds to a range estimate,
According to the matching of range estimation and DOAs, we can perform the multi-frequency coherent integration and the ambiguity-solving operation separately. The signal-to-noise ratio gain in this scenario is the same as in the single-target scenario.
3.4.2. Multiple Targets in the Same Range Cell
When multiple targets are in the same range cell, it is impossible to determine the combination of multiple DOA correspondences in this detection result, i.e., the scenario shown in
Figure 11b.
The detection results obtained at each frequency are
The DOA estimates that the number in the same range estimate is greater than the number of frequencies. We cannot determine the scenario of DOA combination based on the detection results of (32), which results in us being unable to perform the ambiguity-solving operation.
We can employ the multi-hypothesis [
46] approach of multiple-target tracking into the multiple targets scenario to solve the problem. The process of multi-hypothesis detection consists of two parts that are consistent with the single-target scenario: multi-frequency coherent integration and multi-frequency ambiguity-solving.
Multi-frequency coherent integration for scenario B is different from scenario A. The multiple targets of each frequency after DBF can be expressed as
where
M is the number of targets.
Since multiple targets exist in the same range cell, that means multiple target signals correspond to the same detection estimate
. Then we can have
The signal-to-noise ratio gain in this scenario after coherent integration is
Multi-frequency ambiguity-solving for scenario B is similar to multi-hypothesis tracking (MHT). In MHT, all possible hypotheses are identified and associated with the scenario of multiple targets within each data frame in the tracking data. Only the true target association hypothesis can obtain the maximum association likelihood probability. In our problem, we can make multiple hypotheses about the DOA estimates
where
is the set consisting of the DOA detected under
, and ‘×’ represents the Descartes product of sets.
Then, performing the ambiguity-solving operation for each hypothesis
, and obtaining the minimum set factors of the hypothesis
,
where
H represents the cardinal number of H.
The true DOA combination is the combination corresponding hypotheses to the smallest M value . The number of targets in the current range cell M is equal to the maximum of , , .
The specific execution process of multiple targets is described as follows.
Algorithm 5: Multiple Targets Multi-Frequency Coherent Integration with GLE Search |
Pre-processing: Transmitting and receiving GLE search using Algorithms 1 and 2 For search beam 1: (unambiguous DOA estimation) For range cell 1: (range cell number) Step 1. Coherent integration with FFT operation; Step 2. Ambiguous DOA estimates If max() → multiple-target scenario case is scenario B Step 3. Find the receiving scanning beam position corresponding to the current combination and perform FFT on i- frequency data for each range cell; Step 4. Range estimation: searching along the range dimension to obtain the range estimation ; Step 5. Constructing hypothesis sets H and obtaining the minimum set factors of the current hypothesis Step 6. Unambiguous DOA estimates: the ambiguity-solved DOA-corresponding hypotheses to the smallest M value . Otherwise, → a multiple-target scenario case is scenario A Step 7. Multi-frequency coherent integration and range estimation using Algorithm 4 Step 8. Unambiguous DOA estimates using Algorithm 3 End End
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