Abstract
A compressive sensing based array processing method is proposed to lower the complexity, and computation load of array system and to maintain the robust antijam performance in global navigation satellite system (GNSS) receiver. Firstly, the spatial and temporal compressed matrices are multiplied with array signal, which results in a small size array system. Secondly, the 2-dimensional (2D) minimum variance distortionless response (MVDR) beamformer is employed in proposed system to mitigate the narrowband and wideband interference simultaneously. The iterative process is performed to find optimal spatial and temporal gain vector by MVDR approach, which enhances the steering gain of direction of arrival (DOA) of interest. Meanwhile, the null gain is set at DOA of interference. Finally, the simulated navigation signal is generated offline by the graphic user interface tool and employed in the proposed algorithm. The theoretical analysis results using the proposed algorithm are verified based on simulated results.
1. Introduction
The global navigation satellite system (GNSS) has been widely used in military, satellite attitude control mission, and civilian applications such as vehicle positioning and mobile navigation. The user can utilize a GNSS receiver to know the current location, including longitude, latitude, and time. At present, two bands, L1 and L2, are employed in global positioning system (GPS) to meet the high-precision positioning demands such as during military operations. However, most of low cost GNSS receivers are equipped with only single frequency (L1) band, which results in low antijam capability. The accuracy and precision of user location can be affected by unintentional or intentional interferences. Thus, the antijam module needs to be embedded in the receiver to maintain positioning performance. The antenna array and pre- or postprocessing techniques are common methods to mitigate or cancel interferences. Among them, only antenna array technique can obtain the best antijam performance (about 30 to 50 dB). However, the pre- or postprocessing techniques are widely used in current interior GNSS receiver design because of low cost and complexity. The spatial-temporal adaptive processing (STAP) techniques have been proposed to perform 2D filtering on radar or navigation signals to mitigate narrowband/wideband (NB/WB) interferences, multipath, and other uncertainties [1–7]. Nevertheless, it is necessary to increase the number of antennas and time delay elements to obtain better antijam performance for antenna array processing [8]. Thus, the major challenges for a perfect GNSS receiver design include reducing the weight, complexity, and computation time in antenna array processing to maintain good acquisition performance and interference resistance.
In 2008, Candes and Wakin proposed a compressive sensing approach for acquiring the sparse signal and reconstruction from the compressed measurements [9]. This technique includes three components: the sparse transform of the signal, including ordered discrete Hadamard transform (DHT) and discrete cosine transform (DCT), the sparse signal with linear/nonlinear measurement, and signal reconstruction, such as robust uncertainty principles [10], orthogonal matching pursuit method [11], and iterative thresholding [12]. This technique has been widely used in biomedicine, radar, wireless communication, and other field applications [13–15]. The compressive sensing (CS) based approach has been applied to array processing to estimate Doppler-direction-of-arrival (DDOA) for beamforming by using less number of array antennas and less number of samples/snapshots for each antenna [16, 17]. In this paper, the CS based spatial-temporal array processing with minimum variance distortionless response (MVDR) is proposed to mitigate the narrowband/wideband interferences. The time delay elements, different from the previous research [18], are added into each antenna, allowing us to effectively mitigate interference through STAP compressed measurement using a MVDR beamformer.
2. Methodology
2.1. Signal Model
Assume that th navigation satellite signal is received from the antenna frontend at th () time instant, which can be represented as follows: where depicts the number of light of sight (LOS) navigation satellite signal in the sky, is signal power, and depict the navigation message bit and pseudo random noise (PRN) code of th satellite with time delay , represents analog intermediate frequency (IF), and is interference term which consists of WB/NB interferences and Gaussian noise. denotes the Nyquist sampling rate (Hz) which is equal to , where and denote a short observation and the data length of signal reception, respectively.
As for array processing, the aggregate signal received at th input of array system of antenna elements is given by where denotes the incoming signals on th antenna element and is an additional interference term of -dimension. denotes the array response vector with respect to th navigation signal source at elevation and azimuth . Assume that the basic matrix is determined in the angle domain of size () as where and represents the number of azimuth and elevation search grid and illustrates the steering vector from the direction of arrival at azimuth and elevation . Meanwhile, the basic matrix in the frequency domain of size () is defined as where is frequency search grid and denotes the Fourier basis vector for . Combining (3) and (4), (1) can be rewritten as matrix with a size of and as its coefficient on the th row and the th column: where is a sparse matrix including nonzeros coefficients. Assume that the DOA of th signal at and Doppler frequency are aligned on the search grids (if and are large enough), indexed by and , respectively, and .
2.2. Compressive Spatial-Temporal Array Processing
Figure 1 depicts the proposed CS based array system architecture. Firstly, the spatial compressed matrix with a size of () is determined to reduce the original dimension of spatial array matrix to a lower dimension of array matrix. Then, channel analog data is converted to digital samples instead of original channel data. Subsequently, the analog to digital converter (ADC) with a Nyquist sampling rate is changed into an analog to information (A2I) converter with a sub-Nyquist sampling rate by a switch. The original signal samples of temporal domain are divided into block due to A2I converter to carry out the block-wise compression. The temporal compression matrix of size () is defined and indicated as a sampling process in A2I converter with ratio of Nyquist sampling rate. The A2I converter performs subsampling process that transforms the samples per block to samples; it results in total samples for output. The independent and identically distributed (i.i.d.) coefficients in spatial-temporal compression matrices ( and ) are assumed to be randomly distributed (e.g., Bernoulli, Gaussian, etc.), which satisfies restricted isometry property (RIP) with high probability [19]. During the hardware implementation process, the on/off selection switch or a phase shifter/attenuator can be used to implement the multiplication with coefficients from spatial compressive matrix . The mixer and integrators can be used to be implemented in multiplication with coefficients from due to the fact that A2I converter operates on the analog signal after ADC. The A2I converter then outputs the digital signal. The smaller and are, the lower the implementation complexity of A2I converter is. To simplify the notation, the digital baseband version is employed in all equations. Assume that the array signal at th index in (6) is rewritten as where denotes the block index, and the length of each block is , which is equal to coarse acquisition (C/A) code period. is the uncompressed signal of size :After the compressed spatial-temporal process, (7) and (8) become where indicate the th compressive spatial-temporal matrix with a size of . Then (9) can be rewritten as where and denote compressed angle and frequency basis matrix, respectively.
Consider Assume frequency basis matrix is divided into submatrices, each with rows. Then, the matrix depicts th submatrix of . Equation (10) can be rewritten as
2.3. 2-Dimensional MVDR Beamformer
Assume that the DOA of a desired signal and Doppler frequency are known a priori. The optimum MVDR beamformers and can be found by employing iterative procedures. Thus, spatial gain vector can be found by solving the constrained minimization problem: where depicts the null steering vector with the coefficient , , and with a size of . The Lagrange multiplier method [20] is utilized to solve minimization problem in (14) and the solution of MVDR beamformer is as follows: In addition, the temporal gain vector can also be found via constraint minimization problem: where is angular frequency of special interest, denotes transposition without conjugation, and , , and with a size of . Meanwhile, the method of Lagrange multiplier is employed to solve minimization problem in (16) and the solution is explicitly given by the MVDR beamformer: The gain vector is a beamformer for and is a certain DDOA for beamforming.
The spatial correlation matrix is calculated to obtain the new gain vector . Therefore, the is utilized as a bandpass filter centered at frequency to acquire the filtered signal . The temporal correlation matrix is calculated to find the new gain vector . Consequently, to compute the temporal gain vector, is used as spatial filter steered to to obtain the beamformed signal . The MVDR method is utilized to solve and which looks for a spatial-temporal filter to reject the great amount of out-of-band power while passing component at angle or frequency with no distortion. In the following, assume that the compressed spatial-temporal gain vector is equal to , the total output power of the system is , where , and the corresponding optimal signal to interference-plus-noise ratio (SINR) is given by where denotes -dimensional compressed spatial-temporal steering vector of the special interest and is signal power of interest. The minimum sum of error square is as follows:
2.4. Performance Comparison
This section explores the difference between proposed method and other traditional STAP algorithms in terms of system complexity and computation efficiency. The advantage of proposed compression method is that it can reduce the computation time during matrix multiplication of STAP algorithms and also the logic gate count. The performance difference of reduced-rank power minimization (PM) [21], MSNWF [22], MVDR [23], and compressive MVDR (C-MVDR) of beamforming algorithm with regard to convergence rate, steady state error, numerical stability, and system complexity is demonstrated and compared in Table 1. It can also be concluded that array signal after the compression process passing through MVDR beamformer can yield better beam pattern. The error convergence rate of proposed method is superior to that without compression. Also, C-MVDR and MSNWF both are lower in steady state error in contrast to MVDR and reduced-rank PM without compression. The system complexity of proposed method is lower than that of other methods. Table 2 illustrates the relation between system complexities of different STAP algorithms in different update procedures [24]. The implementation difference of each algorithm is assessed using the number of multiplier (MUX), adder (ADD), and data memory (MEM). Table 2 shows that, with the reduction of and , the total computation time of C-MVDR will decrease by cube root as opposed to that of MVDR. In addition, the total hardware implementation complexity will reduce a lot.
3. Simulation Results
The analysis and assessment of the proposed system are illustrated and verified in this section. Uniform rectangular array (URA) with half of wavelength spacing is utilized in the proposed system under no mutual coupling condition. The number of antenna elements is associated with the number of narrowband/wideband interferences that can be cancelled by the STAP algorithm. All antennas are assumed to be ideal and have an omnidirectional radiation pattern. In general, the number of broadband interferences that can be eliminated by the spatial-temporal filtering corresponds to [4]. Assume that IF frequency of GNSS signal is 1.25 MHz with a sampling rate of MHz and the reception data length is ms (equal to samples). The compression matrices and are generated using random Gaussian distribution. The Monte Carlo simulation is conducted for 500 runs to assess the performance of proposed system under jammed environment.
First, the GPS YUMA almanacs data [25] and MATLAB software are employed to simulate the distribution of LOS satellites (total thirteen) in the sky. All GPS satellite signals are received at different azimuth and elevation with −15~−10 dB SNR in average. Assume that incident direction of wideband interference is at azimuth 330 degree and 53 elevation degree and the direction of arrival of narrowband interference is at azimuth 110 degree and 38 elevation degree. All the INR of interference is set at 30 dB. The number of tap delay element is . Figures 2(a) and 2(b) show the 3D gain pattern as a function of azimuth and elevation angle with and without compression measurement under 4 antennas, respectively. The spatial and temporal compression ratio is set as and (, , and ), respectively. Figures 2(a) and 2(b) illustrate that both methods can effectively mitigate interference. In particular, Figure 2(b) shows that the proposed method yields better wideband and narrowband interference mitigation performance. When the antenna number is increased to 9 and the spatial and temporal compression ratio is set as and (, , and ), respectively, a null has been steered toward its DOA in a particular frequency (−47.3 dB for WBI and −45.1 dB for NBI) for each interference (see Figures 2(c) and 2(d)). Thus, the GNSS signals arriving from the same direction can still be distinguished in the frequency domain by a compression spatial-temporal process. For the wideband interference, a null should be placed in its entire frequency band (see Figure 3).
(a)
(b)
(c)
(d)
| (a) Original ( taps) |
| (b) Compressed ( taps) |
| (c) Compressed ( taps) |
| (d) Compressed ( taps) |
In Figure 3, at an elevation of 53 degree where the wideband interference is received, a null is steered toward that direction for all frequencies. Meanwhile, increasing the number of delay tap can obtain the deep null gain toward interference direction. However, the increasing number of time delay taps will result in more sidelobes within the Doppler frequency range (±0.2 KHz) of the desired signal. When the estimation of AOA and Doppler shift is inaccurate, the directivity outputs SINR, which can be observed in Figure 4. However, the increasing number of time delay taps will result in more sidelobes within the Doppler frequency range (±0.2 KHz) of the desired signal. When the estimation of AOA and Doppler shift is inaccurate, the directivity gain of the desired signal will reduce a lot, which will also reduce output SINR, as observed in Figure 4.
A comparison of the two realizations in terms of SINR improvement varying with the number of delay tap is depicted in Figure 4. The SINR improvement is defined as the difference between the output and input SINR [4, 26]. It is found that increasing the number of delay taps continuously does not improve the antijam performance, which simultaneously denotes output SINR. It is due to the increment of time delay taps that can lead to the desired signal component to be partially mitigated or cancelled, which results in poor SINR improvement performance. Selecting the appropriate number of delay taps is important to enhance system antijam performance.
The comparison of hardware implementation complexity of proposed system and traditional STAP under different space compression ratio and time compression ratio is illustrated in Figure 5. The definition of total complexity is the sum of the number of adders, subtractors, and multipliers. This value indicates that the hardware implementation cost will be lower if the space and time compression ratio is lower. Nevertheless, when the space and time compression ratio is below the lower limit, the proposed method fails to mitigate interference. The analysis is feasible if the number of antennas and time delay taps is more than four and five, respectively.
4. Conclusions
The combination of compressive spatial-temporal filter with an MVDR processing can mitigate narrowband and wideband interferences in GNSS receiver. As long as spatial-temporal compression matrices can satisfy RIP with high probability and adopt suitable tap number, the proposed method can obtain robust interference mitigation performance. The compression process consumes system computation time that can significantly reduce time after MVDR for a compressed sample. In terms of total system computation time, the use of compression sensors is still better than not using them. The reduced hardware implementation complexity and computation load are more apparent with the increasing number of antennas and time delay taps. In the proposed method, the interferences can be distinguished by a spatial-temporal process within the same DOA of signals and a null gain is placed in its entire frequency band. In addition, the compression method applied to spatial-temporal signal reception model can reduce the hardware complexity and computation time up to 7.35-fold with a temporal compressive ratio of 0.72 during interference mitigation process under four antenna elements. This method is conducive for future antijam module miniaturization.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to thank the Ministry of Science and Technology of Taiwan for financial support to this work (Grant no. NSC 102-2221-E-020-007).