2.1 Data and Noise Subspaces

By utilising (EVD), it is possible to split a square data matrix into subspace matrices. The most prominent eigenvalues result from environmental clutter and only a minor part is due to system noise. When only two subspaces are used (noise and data), the clutter is part of the data matrix and cannot be removed (Sabushimike et al., 2016; Shehab et al., 2019). Considering additive noise, a measured frame of a radar system A_meas is an M×N matrix, where N is the number of chirps and M is the number of time-steps. This matrix can be decomposed into

We assume that clutter represents the dominant part of the signal and so it is captured within the first few principal components PRCs. The PRCs are calculated by the PCA and contain the information that is required to reconstruct the initial signal. We assume that the PRCs contain most information about the signal. In contrast to the minor components MICs, which are the eigenvectors corresponding to the smallest eigenvalues. This leads to the representation (Xiaoming et al., 2016)

where q_i∈ℝⁿ are the eigenvectors of the data matrix A. We assume the clutter is contained in q−1 PRCs while the noise is contained in n−p MICs of the dataset. The clutter- and noise-free data is then obtainable by summing up the corresponding eigenimages

where $\mathrm{1}\le q\le p\le n$. The values for the lower and upper bounds of the eigenvalues, q and p, are set with respect to the data matrix. Since it can be assumed that most of the clutter is present in the first eigenvalue, it is mostly sufficient for naïve clutter removal to set q=2 (Jorgensen and Hansen, 2012). The same applies for p, in naïve background and clutter removal approaches, p can be simply set to the number of chirps that has been chosen for a frame, e.g., 32. In our extended approach, both values, p and q, are set dynamically, dependent on the measured signals. As we assume that A_filt is clutter-free and noise-free, the matrix only contains the signal that has been scattered by the to-be-detected target or targets. The clutter and noise is best suppressed with an adequate choice of q and p, in combination with interframe mean removal (IMR). Then, it is possible to detect both moving and temporarily static targets, as presented throughout the next sections. Please note that a given data matrix X does not have to be a square matrix and has the dimensions of M×N in the following. This is possible since the PCA calculates the eigenvalues via the correlation matrix R, this correlation matrix is a square matrix (Wold et al., 1987). Furthermore, the data matrix X and the subspace matrices A are real valued, in both simulations and measurements, since the employed radar system only allows to obtain the in-phase part of the received signal.

2.2 Principal Component Analysis (PCA)

The PCA is a widely used statistical method that is employed, e.g., for data reduction, classification of signals, and prediction in regression models (Wold et al., 1987; Jolliffe, 2002). In our application, however, we employ the PCA to remove clutter and noise. In general, PCA can be seen as an eigenvector-based multivariate data analysis method. Here, PCA moves the data X to a new coordinate system by using an orthogonal transformation. This new coordinate system is spanned by the PRCs. The PRCs are selected in a way that they are mutually uncorrelated. This is accomplished by so-called PCA whitening (Friedman, 1987; Kessy et al., 2018). It means that the total variance of the received signal X is distributed over all PRCs. So, the combination of the variances of all PRCs is the variance of the initial received signal X. We assume that the first PRC contains most of the variance of X, the second PRCs contains the second most variance of X, and so on. For our multistatic radar system, the data is arranged in a way that the N chirps are placed along the rows, and the M samples of each chirp are placed along the columns of X. The largest variance in the data generates the most significant pattern in the eventual range-Doppler map (RDM) (Kabourek et al., 2012). In our context, this is the clutter. The smallest variance is represented by the noise in the data and corresponds to the smallest PRCs, the so-called MICs (Kong et al., 2017). Therefore, by filtering the first PRCs and the MICs, clutter and noise can be suppressed. The suggested approach is as follows. In the first step the mean value is subtracted from the dataset,

Then the eigenvectors ${\mathit{w}}_{\text{1}},\mathrm{\dots },{\mathit{w}}_{\text{N}}$ of the square matrix ${\stackrel{\mathrm{‾}}{\mathbf{X}}}^{\text{T}}\stackrel{\mathrm{‾}}{\mathbf{X}}$ are used to build the transformation matrix and calculate the projected data

where $\mathbf{Y}\in {\mathbb{R}}^{M\times N}$ and $\mathbf{W}\in {\mathbb{R}}^{N\times N}$. For clutter and noise reduction the data is reconstructed with only a few PRCs. The filtered reconstruction can be written as

where y∈ℝ^M and w∈ℝ^N. For the second method, $\stackrel{\mathrm{‾}}{\mathbf{X}}$ is replaced in Eqs. (4) and (5) by a different mean removal method. We named this method interframe mean removal (IMR) in the following, according to

since it includes information about previous frames. This helps to prevent the undesired suppression of low-velocity objects. Here, depending on the current timestep i>0 and the corresponding data matrix X⁽ⁱ⁾, IMR is calculated as

2.3 Dynamic PCA (dPCA) Reconstruction

The first step of the dPCA approach is performed with an IMR process of the measurement in order to be able to detect temporary-static targets. When considering a moving target, it is reasonable to exclude the first PRC from the reconstruction. If the target is in a temporarily static state, however, this removal suppresses the target and makes it invisible. Therefore, the lower boundary of the eigenvalues q has to be chosen dynamically. Additionally, the upper boundary p has to be chosen dynamically to reduce the noise level. In order to define these reconstruction boundaries, p and q, of the dPCA, our algorithm firstly calculates the eigenvalues λ. Then the eigenvalue ratio

is calculated. Here, the first eigenvalue is related to the sum of all other eigenvalues. In Fig. 1 the trends of the ratio r are plotted for the raw data, the mean removal (MR) data, and the IMR data. The ratio for the first eigenvalue is considerably large, relative to the raw data. This is also the case for IMR, where the first few eigenvalues are considerably large. After conducting a simple MR to the raw data, the course of the ratio becomes flat. Throughout the experimental evaluation of the eigenvalues a decision threshold has been defined empirically. This decision threshold decides whether or not the first eigenvalue is included into the reconstruction as the lower boundary

By doing so, the undesired side effect of suppressing low-velocity objects can be prevented. To define the total number of eigenvalues that are included into the reconstruction, also the upper boundary p is chosen dynamically. Hereby, the gradient of the eigenvalues, as illustrated in Fig. 2, is calculated. Here, also an empirically evaluated decision threshold is set to −10. That means the upper boundary p includes all eigenvalues that have a gradient of below −10. We further implement an IMR, where the current mean is combined with a previous mean value that fades exponentially.

Figure 1Development of the ratio r of the eigenvalues according to Eq. (9) for 80 frames of one measurement series. Each frame consists of an M×N matrix (here, 256 time-steps × 32 chirps).

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Figure 2Development of the eigenvalue gradient for the first 20 eigenvalues of a 32 chirp measurement.

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2.4 Simulative and Experimental Setup

The dPCA as well as the standard PCA are employed in combination with a cell averaging constant false alarm rate (CA-CFAR) that handles the final detection of the objects. After implementing and testing the dPCA in a simulative environment, the method is now applied in real-life measurements. In the following, the hardware shown in Figs. 3 and 4 is in focus. Here, primarily the INRAS “Radarbook” K-Band radar system is employed, see Fig. 4. The Radarbook is a commercial rapid prototyping system with a modified frontend for our application (Inras GmbH, 2020). In order to maximise the coverage area of the radar system, omnidirectional eddy current suppressing (ECS) monopole antennas are employed for all channels (transmit as well as receiving). The transmitting channels are placed in the centre of the groundpad, the receiving channels are placed at the edges of the board, see Fig. 3.

Figure 3Top view of the antenna configuration and their spacing (d₁=34 cm, d₂=7 cm).

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Figure 4Photograph of the demonstrator setup. All hardware is safely placed between the wooden panels.

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2.5 Quality Evaluation Method

For the final evaluation of the filters, it is necessary to qualify them quantitatively. Therefore, the whole basis of this evaluation-tool, the so-called signal-to-clutter-ratio (SCR), is presented in the following. Analogous to the signal-to-noise-ratio (SNR), the SCR is the ratio of the signal power to the clutter power In a simulative environment it is rather simple to separate signal- and clutter-subspaces. For measurements in highly reflective environments, however, this becomes much more complex, so an approximation is necessary. The following approach is empirical and has grown throughout this work. It is based on analysing the RDMs that are calculated for each frame.

Figure 5(a) Measurement of a moving living object. All data is from the same single channel of the radar system. All of the investigated background removal methods can detect the person. The best results for background and clutter removal is obtained with IMR + dPCA. The dPCA starts hereby with the second eigenimage. (b) Measurement of a living object at rest. Standard MR and PCA fail to detect the object. With IMR + dPCA it is still possible to detect the object as the reconstruction of the dPCA includes the first eigenimage. As the first eigenimage is employed in (v), unsuppressed clutter remains at zero distance and zero velocity. Experimental investigation of the proposed clutter and background removal techniques in combination with the presented mean removal approaches.

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First, the RDM is analysed to detect a possible target. This can be performed with different approaches, for example by hand, with a certain threshold level, or by a 2D-CA-CFAR approach. The result of each variant should be the same: information about the cells that contain the required signal which is scattered by the to-be-detected object. The RDM values of the corresponding target location are then utilised to estimate the signal power. We assume that the RDM is a matrix $\mathbf{R}\in {\mathbb{R}}^{M\times N}$ and R(m,n) is the RDM value at position (m,n). The positions of the detected cells (k,l) are stored in the quantity $\mathbf{D}\subset {\mathbb{N}}_{+}^{\mathrm{2}}$. Then, the estimated signal power can be calculated by

where d is the amount of elements in D, which, on the other hand, is the number of detected cells. The estimation of the clutter (or more general, the background) power

can be performed similarly; here, all cells except the ones included in D are utilised. Consequently, our employed quality evaluation-tool is the SCR, defined by

This method is independent of an exact separation of signal and clutter. Therefore, it can be easily performed on any RDM in either a simulated or real world environment.

Figure 6Overview of the required computation times for all considered filter methods over a period of 80 successive frames. For a better overview, MR and IMR are illustrated with a smaller time axis in the upper plot. The indices of the PCA represent the number of eigenimages that are employed for the reconstruction, i.e., PCA₃₁ includes 31 eigenimages and requires more time for the computation than PCA₃ with 3 eigenimages.

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