The sliding window Constant False Alarm Rate (CFAR) detector compares the Cell Under Test (CUT), denoted as $x_{\textrm{CUT}}$, with an estimate of the background noise level, denoted as $g({x_{1}}, {x_{2}}, \ldots, {x_{N}})$, multiplied by a positive constant $\tau$. The aim is to make a decision on the presence or absence of a target in the CUT as below

where the secondary data { ${x_1,\ldots,x_N}$ } is assumed to be independent and identically distributed (i.i.d.) random variables containing the background noise. The function $g(\cdot)$ maps the vector $[x_1, \ldots, x_N]$ to a non-negative real number. The hypothesis testing problem in target detection involves making a decision between $H_0$ (indicating that $x_{\textrm{CUT}}$ contains only noise measurements) and ${H_1}$ (indicating that $x_{\textrm{CUT}}$ includes a target component within the noise returns). The threshold multiplier $\tau > 0$ is used to control the required Probability of False Alarm ($P_{\textrm{fa}}$). By adjusting the value of $\tau$, the desired level of false alarms can be achieved. For brevity, we will use $g$ to represent $g({x_{1}}, {x_{2}}, \ldots, {x_{N}})$ throughout this text.

The CHA-CFAR detector was proposed as a new CFAR detector in [1], specifically designed for dense outlier situations. Unlike other methods that eliminate outliers through censoring (as seen in OS-CFAR and TM-CFAR), CHA-CFAR softens the effect of outliers by utilizing the harmonic mean and the Ordered Statistics (OS) principle to estimate the noise level, as shown in equation (2):

where ${x_{(k)}}$ represents the $k$ th sample of ordered statistics. In CHA-CFAR detector, $k$ represents the number of eliminated secondary data with the smallest amplitudes. It is suggested to choose small values for $k$. In the following, $k$ is set to 8. Indeed, the function $g$ reduces the impact of outliers instead of rejecting them. Consequently, unlike other existing methods, the CHA-CFAR detector does not require prior knowledge of the number of outliers [1].

As mentioned, OS-CFAR and TM-CFAR detectors are based on the Ordered Statistics principle. Also, many CFAR techniques such as the weighted amplitude iteration (WAI)-CFAR detector use iterative algorithms. Some CFAR detectors are summarized in Table 1 [1]. The superiority of CHA-CFAR is illustrated through simulations and real Synthetic Aperture Radar (SAR) images in the following figures.

Table 1. Summary of Various CFAR detectors Unified in (1).

In the following simulations, we use the results of the CA-CFAR detector in a homogeneous environment as an optimal bound, which called Ideal. Fig. 1 plots $P_{\textrm{d}}$ and $P{\textrm{fa}}$ of different detectors versus SCR (the ratio of target power or outliers power to the noise power). As expected, all detectors have the same performance in a homogeneous environment in Fig. 1 ($P_{\textrm{fa}} = 10^{-3}$ and $N = 32$).

Fig 1. $P_{\textrm{d}}$ and $P_{\textrm{fa}}$ plot versus SCR in a homogeneous environment.

In the presence of outliers, the CHA-CFAR detector outperforms other methods without knowing the number of high-amplitude outliers and rejecting them. To demonstrate the robustness of the CHA-CFAR detector to a high number of outliers, 15 interfering targets are located in the reference window. As can be seen, the CHA-CFAR detector significantly outperforms others in Fig. 2. Although the performance of the WAI-CFAR detector is better than that of TM- and OS-CFAR detectors, it has a higher processing load. From the computational complexity point of view, the CHA-CFAR detector only needs to censor a few samples with the lowest amplitude, and any additional or iterative processing is not necessary to handle the outliers.

Fig 2. $P_{\textrm{d}}$ and $P_{\textrm{fa}}$ plot versus SCR for 15 interfering targets.

We compare all detectors for a large vessel as an extended target using real SAR images in Fig. 3. As can be seen, the CHA-CFAR detector exhibits the best detection performance.

Fig 3. Detection performance comparison on extended target. SLI VV polarization SAR image is applied with $P_{\textrm{fa}} = 10^{−3}$ on TerraSAR-X image for different detectors.

To set the thresholds for CFAR detectors using Monte Carlo, follow these steps:

1. Open the main_interfering_target.m script and set the number of secondary data contaminated with outliers to zero (Number_Interfering_Target = 0).

2. Set the values for the false alarm probability ($P_{\textrm{fa}}$) and the number of secondary data $N$. The default values are pfa = 10^-3 and window_size = 32 (Note that N = window_size).

3. In the CFAR Parameters section of the script, adjust the variables to achieve the desired $P_{\textrm{fa}}$ for all detectors through trial and error.

After adjusting the CFAR parameters to achieve the desired $P_{\textrm{fa}}$, see all detectors have the same $P_{\textrm{fa}}$ when there are no outliers.

• Use the interfering_target.m script to observe outliers by adjusting the Number_Interfering_Target variable and observe its effect on the false alarm probability ($P_{\textrm{fa}}$) and detection probability ($P_{\textrm{d}}$) of different detectors.

• Employ the edge.m script to observe the impact of a sudden change in the power of background noise on the $P_{\textrm{fa}}$ and $P_{\textrm{d}}$ of different detectors.

• Utilize the real_SAR_data.m script to examine the $P_{\textrm{d}}$ results of different detectors on real SAR data. Select the area and push submit button to execute the script (Fig. 4).

  

Fig 4. The TerraSAR-X images is contains single-look complex images that were acquired from July 2008 to November 2009 over Barcelona, Spain.

[1] R. G. Zefreh, M. R. Taban, M. M. Naghsh, and S. Gazor, “Robust CFAR detector based on censored harmonic averaging in heterogeneous clutter,” IEEE Transactions on Aerospace and Electronic Systems, vol. 57, no. 3, pp. 1956–1963, Jun. 2021.