CFAR Detectors for MIMO Radars

Abstract

CFAR (Constant False-Alarm Rate) processors are useful for detecting radar targets in a background for which the parameters in the statistical distribution are not known. A variety of CFAR techniques such as CA (Cell Averaging), Go (Greatest Of), SO (Smallest Of), OS (Ordered Statistics) and ACMLD (Automatic Censored Mean-Level Detector) processors have been proposed for SISO (Single Input–Single Output) radars in a non-homogeneous background. In this paper, conventional CFAR algorithms including CA, SO, OS and ACMLD processors are generalized for MIMO (Multiple Input–Multiple Output) radars. The exact expressions for false-alarm probabilities of the proposed algorithms in a homogeneous background are presented. In addition, the detection performance of the proposed detectors is studied by means of simulation in the presence of interfering targets and also colored Gaussian clutter. Besides, the proposed CFAR processors are compared, and it is shown that the ACML-based algorithm is superior to the other investigated methods.

Appendices

Appendix 1

The False-alarm probability of a CFAR detector is determined by

$$ P_\mathrm{fa}=E_{Z} \bigl(P (x_{0}> \alpha z | z,H_{0} ) \bigr)=\int_{0}^{\infty }P (x_{0}>\alpha z | z,H_{0} )f_{Z} (z )\,dz $$

(58)

For CA-CFAR detector, x ₀ and z are defined by summation of MN and LMN exponential RVs, respectively (Eqs. (13) and (14)) and have Gamma distribution:

(59)

(60)

So we have

(61)

Substituting (60) and (61) in (58) and calculating the definite integral leads to

$$ P_\mathrm{fa}^{\mathrm{CA}}=\sum_{i=0}^{\mathit{MN}-1} \left (\begin{array}{c} \mathit{LMN}+i-1\\ i \end{array} \right )\frac{\alpha^{i}}{(1+\alpha)^{\mathit{LMN}+i}} $$

(62)

Appendix 2

The false-alarm probability can be determined by [11]:

$$ P_\mathrm{fa}=- \sum_{i}\bigl( \mathrm{Res} \bigl(\omega^{-1}\varPhi_{x_{0}|H_{0}} (\omega ) \varPhi_{z} (-\alpha\omega ) \bigr),\omega_{i}\bigr) $$

(63)

where $\varPhi_{x_{0}|H_{0}}$ and Φ _z (−αω) are given in (34) and (36). $\varPhi_{x_{0}|H_{0}}$ has a pole of order MN at $\omega_{0}=-\frac{1}{\mu}$, and residue at a pole of order n is defined by [13]:

$$ \mathrm{Res}\bigl(f(\omega),\omega_{0}\bigr)=\frac{1}{(n-1)!} \lim_{\omega \rightarrow \omega _{0}}\frac{d^{n-1}}{d\omega^{n-1}}\bigl(f(\omega) (\omega- \omega_{0})^{n}\bigr) $$

(64)

So, we have

(65)

Let ω=μz in (65)

(66)

This expression can be simplified to

(67)

Appendix 3

According to (23), the false-alarm probability of M-ACMLD for fixed k ₁,…,k _MN can be determined by [11]:

$$ P_\mathrm{fa} (k_{1},\dots,k_\mathit{MN} )=- \sum_{i}\bigl(\mathrm{Res} \bigl( \omega^{-1}\varPhi_{x_{0}|H_{0}} (\omega )\varPhi_{z} (-T_{\mathrm{Det}k_{1},k_{2},\dots ,k_\mathit{MN}}\omega ) \bigr), \omega_{i}\bigr) $$

(68)

In M-ACMLD the noise level estimates in different nodes are independent, So Φ _Z (.) of the total noise power is the product of the individual MGFs of z _j s defined in (53):

$$ \varPhi_{Z}(-T_{\mathrm{Det}k_{1},k_{2},\dots ,k_\mathit{MN}}z)=\prod_{j=1}^{MN} \left (\begin{array}{c} L\\ k_{j} \end{array} \right )\prod_{m=1}^{k_{j}} \bigl(-T_{\mathrm{Det}k_{1},k_{2},\dots ,k_\mathit{MN}}\mu{z}+d_{m}(k_{j}) \bigr)^{-1} $$

(69)

where

$$ d_{m} (k_{j} )=\frac{L-m+1}{k_{j}-m+1} $$

(70)

$\varPhi_{x_{0}|H_{0}}$ is also defined in (36) and has a pole of order MN at $\omega_{0}=-\frac{1}{\mu}$. Using (64) and substituting (36) and (69) in (68) we have

(71)

Using, ω=μz, this can be simplified to

(72)