Principles of Modern Radar - Volume 2 1891121537

476 CHAPTER 10 Clutter Suppression Using Space-Time Adaptive ProcessingWe assume x k/H0 ∼ CN (0, R k ), where R k = E [ x k/H0 x H k/H 0] is the null hypothesis covariancematrix. Using (10.54)–(10.55) in (10.53) givesSINR = wH k R sw kwk HR = σ ∣s2 ∣wk Hs s−t( f sp , ˜f d ) ∣ 2kw k wk HR (10.56)kw kWe further consider the relationship between SINR, P D , and P FA in Section 10.4.1.Additionally, note that we have not yet specified a choice for w k . Some choices includethe two-dimensional matched filter (which maximizes SNR), the STAP weight vector(which attempts to maximize SINR), and the displaced phase center antenna (DPCA)weight vector (which seeks to cancel the stationary clutter signal entirely).In the noise-limited case, R k = σ2n I NM . Thus, (10.56) becomesSINR = σ 2 s∣ w H k s s−t( f sp , ˜f d ) ∣ ∣ 2w H k R kw k= σ 2 sσ 2 n∣ w H k s s−t( f sp , ˜f d ) ∣ ∣ 2w H k w k(10.57)Equation (10.57) takes the same form as (10.23), so SINR → SNR. As previouslydiscussed,(the matched filter weight vector, w k = s s−t ( f sp , ˜f d ), yields max (SNR) =σ2s /σn2 ) MN, where MN is the space-time integration gain. Additionally, colored noise alwaysleads to degradation with respect to noise-limited performance, that is, SNR ≥ SINR.10.4.1 Performance MetricsIn this section we consider the following important performance metrics: probability ofdetection,P D ; probability of false alarm,P FA ; SINR loss; minimum detectable velocity(MDV); usable Doppler space fraction (UDSF); and improvement factor (IF).10.4.1.1 DetectionAn optimum detection statistic for the filter output y k = wk Hx k for the two hypotheses in(10.52), where x k/H0 ∼ CN(0, R k ), follows from the likelihood ratio test and appears as[1–3,14]H 1>|y k | v< T (10.58)H 0The performance of (10.58) is given by( )−β2P FA = exp T2∫ ∞ ( ( − u 2 + α 2) )P D = u expI 0 (αu) du (10.59)2β Twhere P FA is the probability of false alarm, P D is the probability of detection, β T is anormalized detection threshold, I 0 (·) is the modified zero-order Bessel function of thefirst kind, α equals the square-root of the peak output SINR, and β T = v T/ √ w H k R kw kis the normalized threshold. Using (10.56), and accounting for average signal power, we