Prior-Guided Deep Interference Mitigation
for FMCW Radars

A complex-valued convolutional neural network is generally composed of various complex-valued modules, including complex convolution, complex-valued activation functions and complex batch normalization [22].

To take advantage of the existing deep learning platform developed for real-valued NN (e.g., TensorFlow [25]), a complex convolution can be implemented by explicitly performing real-valued convolutions among the real and imaginary parts of the related terms. Specifically, the complex convolution between a complex filter $\mathbf{W}=\mathbf{A}+j\mathbf{B}$ and a complex vector $\mathbf{h}=\mathbf{x}+j\mathbf{y}$ can be expressed as:

where $j$ is the imaginary unit. $\mathbf{A}$ and $\mathbf{B}$ are real matrices, and $\mathbf{x}$ and $\mathbf{y}$ are real vectors, respectively.

Similar to the activation functions for real-valued CNNs, complex-valued activation functions introduce nonlinearity to complex-valued CNNs to increase their representation capabilities [22, 26, 27]. Complex Rectified Linear Unit (CReLU) is one of the most popular activation functions used in complex-valued CNNs [22], which applies traditional real-valued ReLU on both real and imaginary parts of a complex-valued input and is expressed as:

where $\Re(x)$ and $\Im(x)$ extract the real and imaginary parts of a complex number $x$. Compared to other complex-valued activation functions (e.g., zReLU [27] and modReLU [26]), CReLU generally achieves the best performance in inverse problems. Therefore, it is utilized in this paper as well.

The interference mitigation problem is tackled as a regression problem. As targets’ beat signals and interferences show distinct distributions in the $t$-$f$ domain, the $t$-$f$ domain spectral diagram obtained with the STFT algorithm is naturally used for IM. Considering the complex-valued nature of FMCW radar signals in the $t$-$f$ domain, a complex-valued fully convolutional network architecture is designed for interference mitigation with the basic complex-valued modules (see Fig. 3). The proposed CV-FCN is composed entirely of complex convolution layers, each of which except the last convolution layer is followed by the complex-valued activation function CReLU. The number of filters is fixed to one in the last convolution layer, which is used only to produce the output.

The $t$-$f$ spectrum of the interfered radar signal is set as the input of the network, and its counterpart of the associated reference (i.e., the clean signal) is used as the label. Since the existing deep learning tools do not support the complex-valued input data, the real and imaginary parts of input samples are separated into two channels. Meanwhile, the square kernels with size 3$\times$3 are used to deal with the two-dimensional input samples, and the zero-padding is used in the complex convolutional layer to ensure the output $t$-$f$ spectrum have the same shape as the input.

MSE is generally used as a loss function in the DL-based interference mitigation approaches [28], which calculates the difference between the output of the network and the related label. However, the performance of the network trained with the MSE as the loss function is limited by the size of the training dataset. With the increase of the network’s total parameters and training iterations, the overfitting problem cannot be avoided, making it challenging to extract critical features. Moreover, due to the lack of real interfered radar signals and related reference data in practice, synthetic data based on the analytical signal models are commonly generated to build the training dataset. However, the analytical model used for data synthesis is generally derived based on certain assumptions (e.g., perfect radar system and frequency-independent scattering property of targets) for simplification, which may make the synthetic dataset impractical to contain all the features of the data acquired in various realistic scenarios. Consequently, the performance of the networks trained using the MSE with only simulated radar signals would degrade when utilized to real data. To avoid the overfitting problem and improve the generalization of the trained network, explicitly incorporating the prior information could be of benefit for real data.

As shown in Section II, the interferences lead to time-varying beat frequencies after dechirping while the frequencies of targets’ beat signals are constant. The projection of the interference on the frequency axis is a line, while the projections of targets’ beat signals on the frequency axis are some points. Thus, interferences and targets’ beat signals show different sparsities along the frequency axis (i.e., different sparsities in the range profiles). To exploit this prior feature of interferences and beat signals in the $t$-$f$ domain, we introduce the $L_{2,1}$ norm of the recovered $t$-$f$ domain spectrum (i.e., the output of the proposed neural network) as a regularization term for the NN training in addition to the traditional loss function MSE. The complete loss function is expressed as

where $\|\mathbf{X}\|_{F}$ and $\|\mathbf{X}\|_{2,1}$ represent Frobenius norm and $L_{2,1}$ norm of a matrix $\mathbf{X}$, respectively, $\tilde{\mathbf{S}}\in\mathbb{C}^{M\times N}$ is the matrix of the recovered spectrum in the $t$-$f$ domain with the row and column related to the frequency and time dimensions, respectively, and $\mathbf{S}\in\mathbb{C}^{M\times N}$ is the label. $\|\mathbf{S}-\tilde{\mathbf{S}}\|_{F}^{2}$ is the MSE loss function, and $\|\tilde{\mathbf{S}}\|_{2,1}$ is the $L_{2,1}$ norm of $\tilde{\mathbf{S}}$, as shown in (7). $\lambda$ is a hyper-parameter used to make a trade-off between the MSE (i.e., data consistency) and the prior feature.

Due to the introduced regularization term $\|\tilde{\mathbf{S}}\|_{2,1}$, the overfitting problem can be avoided as much as possible. Moreover, as the regularization term $\|\tilde{\mathbf{S}}\|_{2,1}$ provides solid expert knowledge, it boosts the convergence rate of the network training (i.e., the network can be trained with fewer iterations) and the size of the dataset needed for training can be significantly reduced. Besides, the feature used for IM is the fusion of the prior information and the features extracted from the CNNs, which can motivate better generalization capability.

Before being fed into the network, the complex-valued input samples are normalized. Specifically, the normalization method can be described as:

where $\mathbf{Y}$ is the matrix of the STFT spectrum of beat signals contaminated by interferences, and $m$ and $n$ are the row and column indices of an entry of the matrix.

After being processed by the network, the matrix of the recovered spectrum is multiplied by the denominator in equation (8) for further processing.

In the network training process, the complex weight initialization strategy[22] was used to initialize the parameters of complex convolution layers. The Adam algorithm with a fixed learning rate of 0.001 and 32 input samples per batch was used for training. We end the training at epoch 100 after good convergence was observed. Moreover, all the models were trained on a single NVIDIA 2080Ti graphics processing unit (GPU). The code was implemented using Keras and Tensorflow tools.

Due to the difficulties in acquiring both interfered radar echoes and their related references in practice, especially for dynamic scenarios, in this paper we decided to use synthetic FMCW radar signals for the proposed neural network training and then employ both synthetic and measured data for test.

For data generation, a victim FMCW radar with the parameters described in Table I was considered. To emulate the scenarios with various scatterers and different interferences, each parameter of targets and interfering signals was randomly chosen from a uniform distribution $\mathcal{U}[a,b]$ for continuous variables in a closed interval or $\mathcal{U}(a,b)$ for continuous variables in an open interval or $\mathcal{U}\{a,b\}$ for discrete variables, where $a$ and $b$ define the bounds of an interval. The detailed intervals of the values of the parameters of targets and interfering signals are shown in Table II, where $K$ and $T_{sw}$ refer to the chirp rate and sweep duration of the victim radar in Table I. Moreover, complex white Gaussian noise was added to synthetic signals to account for system noise and measurement errors. To characterize the interference-contaminated signals in the presence of complex white Gaussian noise, Signal to Noise Ratio (SNR) and Signal to Interference plus Noise Ratio (SINR) are used as metrics. The SNR ranges from -20 dB to 20 dB with step size of 5 dB while the SINR takes values randomly from a uniform distribution as shown in Table II.

After synthesizing the time-domain radar signals according to the setups described above, their time-frequency spectra are generated through the STFT algorithm. Specifically, the STFT was implemented by using a 256-point hamming window with a hop size of one for signal segmentation and 256-point fast Fourier transform (FFT) for spectrum calculation (see Table I). Since the beat signals of targets and interferences are synthesized according to controllable parameters, we can obtain both the interfered signals and the associated references (see the example in Fig 5).

The experimental data in this paper were collected with the full-polarimetric PARSAX radar in TU Delft, which has two orthogonally polarized transmitting channels and four receiving channels for full polarimetric signal acquisition. We simultaneously use a horizontally polarized (H-pol) channel to emit a fixed FMCW signal and the vertically polarized (V-pol) channel to transmit an arbitrary FMCW-type waveform with various chirp rates, time duration, bandwidth, and time delay relative to the beginning of the signal in the H-pol channel.

The full-polarimetric signals scattered from the illuminated scene arrive at the receiving antenna at the same time. After passing through an orthomode transducer, the H-pol (i.e., HH and VH) and V-pol (i.e., HV and VV) scattered signals can be separated. However, the HH and VH (correspondingly HV and VV) signals inevitably interfere with each other at the receiving channels. As the HH (correspondingly VV) signals are generally much stronger than the VH (correspondingly HV) signals, the interference impact of HH (VV) on the VH (HV) is generally much severer. So, the acquired HV signals are used to construct the experimental dataset used in this paper. The radar data were measured by illuminating three scenes: an industrical chimney, a rotating wind turbine, and a street with moving cars. In total, 500-sweep radar data were measured with various interference signals for each sweep. As the latter two scenes were dynamic, acquiring the related references with our radar system was impractical, which is generally the case in practice. As an example, Fig. 6 illustrates the street scenario at a time instant and the $t$-$f$ spectrum of the acquired signal. As the references are unavailable, the experimental data are only used to test the trained neural networks.

In principle, the shape of the STFT spectra of radar signals is determined by the number of sampling points in a sweep and parameters of the STFT algorithm, and most CNNs can only process input samples of specific shapes. In order to process radar signals of different shapes in the $t$-$f$ domain, the matrix of the STFT spectrum is split into a combination of smaller matrices before being fed into the network, which can be described as Algorithm 1. Each element in the matrix of the recovered spectrum is predicted according to both the past and future time-frame information, and the phase is guaranteed to be continuous. In our experiments, the $N_{p}$ is set to 4, and the shape of the input sample (i.e., $M\times M$) is 256$\times$256.

To quantitatively evaluate the performance of different interference mitigation methods, the SINR of a recovered radar signal relative to the clean reference is used as a performance metric. The SINR cannot only measure the remaining interferences and noise in the recovered signal, but also represent the signal distortion. It is defined as:

where $\tilde{\mathbf{s}}$ is the recovered signal in the time domain, and $\mathbf{s}$ is the corresponding reference. Note the SINR is inversely proportional to the error vector magnitude (EVM) [9].

To find an optimal network architecture based on the proposed CV-FCN for FMCW radar interference mitigation, a parameter search is performed using MSE as the loss function.

Firstly, the problem of how to design the number of filters in each convolution layer is investigated. Three different network architectures denoted as Type I, II, and III are implemented for comparison. Except for the last layer, the number of filters in each convolution layer is fixed to constant in Type I, and the number of filters is doubled or halved in each convolution layer for Type II and Type III, respectively. The performance comparison of different architectures is shown in Fig. 7, the average SINR of all evaluated architectures is given, and the x-axis indicates the number of total parameters of the network. We change the number of parameters by varying the depth of the network, the number of filters in each layer and the kernel size. We can see from the figure that the CV-FCNs where the number of filters is fixed to constant (Type I) obtain better results. In fact, the number of filters is doubled in each convolution layer for the famous CNN architectures such as VGG [29]. Due to the use of the pooling layer in VGG, the number of channels is doubled in each layer to ensure the information amount of the connected convolution layers does not differ too much. Since the pooling layer is not used in the CV-FCN to avoid signal distortion, the architecture where the number of filters is fixed to constant is more suitable.

Based on the conclusion, the CV-FCNs (Type I) with different numbers of total parameters are compared as shown in Fig. 7. The features of targets’ beat signals and interferences in the $t$-$f$ domain are relatively simple, a larger number of filters or a deeper network is unnecessary. It follows that the maximum average SINR is obtained using a model (Model A) with ten layers and a kernel size of 3$\times$3. Except for the last layer, the number of filters in each convolution layer is fixed to 16. Moreover, we compared the CV-FCNs with kernels of 3$\times$3 (Model A) and 5$\times$5 (Model B). Although the number of total parameters in Model B triples, it leads to an average SINR 0.43 dB lower than that by Model A. So, the kernel size of 3$\times$3 is used in the following experiments.

In recent years, researchers[15] have shown that introducing the residual connection can promote better backpropagation of gradient, avoiding the problems of gradient diminishing and explosion during training. To analyze the effect of residual connection for complex-valued networks, an additional residual connection is added between the input and output of the CV-FCN, resulting in a complex-valued residual network (CV-ResNet) used for comparison with CV-FCN. Except for one additional residual connection, the other layers and parameters remain unchanged. The performance comparison is shown in Fig. 8, and the result indicates that the CV-FCN has a better performance especially in low SINR scenarios. The beat frequency of the targets’ signals is constant, which results in the horizontal lines in the $t$-$f$ spectrum. The features of the targets’ beat signals can be extracted by the convolution filters of CV-FCN, while the filters in CV-ResNet deal with the features of interferences, which show inclined lines with different slopes, intensities, and durations in the $t$-$f$ spectrum and are relatively more complex. Different features extracted explain why the residual connection does not work well. However, the CV-ResNet may acquire better performance in other IM applications where signals are more complex than interferences.

In order to analyze the effect of complex-valued networks in the radar signal processing chain, we compared the CV-FCNs with their real-valued counterparts over a variety of network depths.

The real-valued FCN (RV-FCN) is constructed using the real-valued convolution layer and the ReLU activation function to replace the corresponding complex modules. In the complex-valued networks, the number of complex filters in each layer is the number of complex feature maps. It is also the effective number of feature maps for each of the real and imaginary parts. To obtain the same number of feature maps for performance comparison, the number of filters in each layer in RV-FCNs is twice that in CV-FCNs, and the other parameters remain unchanged (as shown in Fig. 4).

The performance comparison is shown in Fig. 9, where the CV-FCNs of different depths show a better performance in low SINR scenarios compared to the real-valued counterparts. In the interference mitigation task, it is more meaningful to consider the performance in low SINR scenarios. The detailed parameters and quantitative results of the CV-FCNs and RV-FCNs are shown in Table IV. When the SINR of interfered signals is between -40 dB and -20 dB, all the CV-FCNs with different depths show better results (in bold red font), and the SINRs of the beat signals recovered by CV-FCNs are on average 1.1 dB higher than that by RV-FCNs. Moreover, due to complex multiplication and half the number of filters in each layer in the CV-FCN, its parameter amount is half of the real-valued counterparts.

Next, we compared the optimal CV-FCN, whose total parameters are 37730, with the optimal RV-FCN whose total parameters are 84418. With the number of total parameters reduced by 55.3%, the CV-FCN achieves almost the same performance as the RV-FCN. In the computer’s memory, the optimal RV-FCN requires 1.04 megabytes of memory, while the CN-FCN requires only 525 kilobytes. Additionally, the SINR of recovered signals has improved from -0.54 dB to 0.93 dB when the SINR of the interfered signals is between -40 dB and -35 dB. This suggests that the CV-FCN can be better used in practical applications due to the limitation of the hardware memory and its superior performance in suppressing strong interferences.

In Section III-C, we proposed the prior-guided loss function based on the different distributions of targets’ beat signals and interferences in the spectrogram. In this part, we use the prior-guided loss function instead of MSE to train the obtained optimal CV-FCN (Model A) in Section V-B. The hyper-parameter $\lambda$ in equation (6) is used to make a trade-off between data consistency and prior knowledge. Note that the prior-guided loss function becomes the MSE when $\lambda=0$.

The performance of our proposed approach is compared with several state-of-the-art interference mitigation methods, including traditional signal processing approaches such as the Wavelet Denoising (WD) based method[5], Adaptive Noise Canceller (ANC) [4], CFAR-Z and CFAR-AC [30], and DL-based approaches such as CNN-based method [9] and ResNet-based method [15]. We used the simulated radar signals for test and quantitatively evaluated the performance of different methods. Fig. 14 shows the SINRs of the obtained signals after interference mitigation with different approaches. Due to the ResNet was designed to process the SAR images, in our experiments, the number of filters of the ResNet is set to half of that in [15] for FMCW radar signal processing. The CV-FCN is trained using the prior-guided loss function where $\lambda=400$ and the CNN and ResNet are trained using the MSE loss function.

The comparative results show that our proposed CV-FCN based prior-guided IM approach is obviously better than other methods. Specifically, the cfarZ and cfarAC use constant false alarm rate (CFAR) to detect the interference components of acquired beat signals in the time-frequency spectrum. The detection accuracy is determined by the selected parameters. Then cfarZ uses zeroing to mitigate detected interferences, which naturally removes targets’ beat signals at the same time. Different from cfarZ, cfarAC uses amplitude correction (AC) to reconstruct the beat signals removed by zeroing, which shows better performance than cfarZ. Besides, WD method can extract and remove the interferences in the wavelet domain, which shows a good performance in low SINR scenarios. In the ANC method, the negative half of the FFT spectrum is used as the input of its reference channel, and the filtering step size is manually adjusted. As described, the performance of the above traditional signal processing methods depends on a proper selection of a few manually adjustable parameters. Over a wide range of the SINR variations, their performance is not good as the selected DL-based methods¹¹1In principle, the range of SINRs of input signals can be divided into a few small segments. By tuning the related parameters of conventional methods over each small range of SINR of input signals, they could outperform the DL-based methods. However, selecting such a set of parameters is nontrivial in practice..

On the other hand, our proposed CV-FCN achieves better performance expecially in low SINR scenarios with only 12% the number of total parameters of the ResNet. The effect of residual connection has been verified and discussed in Section V-B, which is not suitable for extracting the feature of targets’ beat signals. The superior performance compared with other NNs shows the advantage of network architecture optimization by grid search, complex-valued representation in radar signal processing, and the prior feature offered by $L_{2,1}$ norm.