Composite Test The detection statistics Constant False Alarm,e.g.,PrA=P(H1;Ho) Under Ho I(x)~N(0,02/N) PrA=P(T()>Y;Ho)=fH(t)dt=Q(Y/V2TN Po)-(P- Pp is a function of the unknown parameter A For each value of A,this detector is optimal Uniform Most Powerful(UMP) whxiong@uestc.edu.cn 11
whxiong@uestc.edu.cn Composite Test 11 The detection statistics Constant False Alarm, e.g., T(x) = 1 N Xx[n] PFA = P(H1; H0) Under H0 T(x) » N(0; ¾ 2 =N) PFA = P(T(x) > ° 0 ; H0) = Z 1 °0 fH0 (t)dt = Q ³ ° 0 = p ¾2 =N ´ ° 0 = r ¾2 N Q ¡1 (PFA) PD = P(T(x) > ° 0 ; H1) = Q Ã °0 ¡ A p ¾2 =N ! = Q µ Q ¡1 (PFA) ¡ q NA 2 =¾2 ¶ PD is a function of the unknown parameter A For each value of A, this detector is optimal Uniform Most Powerful (UMP)
Composite Test-two-sided test E.g.,Any value of DC in WGN with known noise statistics Ho xIn]wln] n=0,1..N-1 H xin]=A+wln] whxiong@uestc.edu.cn 12
whxiong@uestc.edu.cn Composite Test—two-sided test 12 E.g., Any value of DC in WGN with known noise statistics H0 x[n] = w[n] n=0,1…N-1 H1 x[n] = A + w[n]
Composite Test-two-sided test E.g.,Any value of DC in WGN with known noise statistics Ho xIn]wln] n=0,1..W-1 H xin]=A+wln] Following the NP rule,we have detection statistics T)=∑m whxiong@uestc.edu.cn 13
whxiong@uestc.edu.cn Composite Test—two-sided test 13 E.g., Any value of DC in WGN with known noise statistics H0 x[n] = w[n] n=0,1…N-1 H1 x[n] = A + w[n] Following the NP rule, we have detection statistics T(x) = 1 N Xx[n]
Composite Test-two-sided test E.g.,Any value of DC in WGN with known noise statistics Ho xIn]wln] n=0,1..W-1 H xIn]=A+wIn] Following the NP rule,we have detection statistics T)=∑m But,the threshold is different whxiong@uestc.edu.cn 14
whxiong@uestc.edu.cn Composite Test—two-sided test 14 E.g., Any value of DC in WGN with known noise statistics H0 x[n] = w[n] n=0,1…N-1 H1 x[n] = A + w[n] Following the NP rule, we have detection statistics T(x) = 1 N Xx[n] But, the threshold is different
Composite Test-two-sided test E.g.,Any value of DC in WGN with known noise statistics Ho xIn]wln] n=0,1..N-1 H xIn]=A+wIn] Following the NP rule,we have detection statistics T)=∑m But,the threshold is different 元>Y for A>0 Decide Hi if 元<-Y for A<O whxiong@uestc.edu.cn 15
whxiong@uestc.edu.cn Composite Test—two-sided test 15 E.g., Any value of DC in WGN with known noise statistics H0 x[n] = w[n] n=0,1…N-1 H1 x[n] = A + w[n] Following the NP rule, we have detection statistics T(x) = 1 N Xx[n] But, the threshold is different x¹> ° 0 Decide H1 if for A>0 x¹< ¡° 0 for A<0