5-10] APPLICATION TO ANALYSIS OF MATCHED FILTER RADAR 275 



following quadratic function of m will always be greater than or equal to 

 zero: 



J^ [m/W + ^W]Vx = m'^ j^ nx)dx + 2m y^ Ax)g{x)^x +j^ g\x)dx ^ 0. 



(5-113) 

 This expression is represented by 



^m' + 25m + C ^ 0. (5-114) 



Because this polynomial is always greater than zero, the equation 



^m' + 25m + C = 0. (5-115) 



cannot have distinct real roots, and its discriminant must be less than or 

 equal to zero: 



B'-JC^O. (5-116) 



Substituting for yf, B, and C gives the real form of Schwarz's inequality: 



(/^VkW^^)' ^ /^ f{x)dx j\Kx)dx. (5-117) 



The absolute value of the product of two complex numbers is always less 

 than or equal to the product of their absolute values. Further, the square 

 of the absolute value of the integral of a complex function is always less than 

 or equal to the integral of the square of the absolute value of the integrand. 

 Combining these ideas, we note that when/(;c) and g{x) are complex. 



f 



f{x)g{x)dx 



i: 



L 



\f{x)g{x)\^dx ^ \f{x)\^\g{x)\^dx. (5-118) 



This immediately leads to the more general form which we need. Putting 

 |/(;c)| and |^(^)| in place of/{x) and g{x) in Equation 5-118: 



/ 



f(x)g(x)d> 



j\f{x)\'dxj\g{x)\^dx. (5-119) 



Substituting the right-hand side of this inequality for the numerator in 

 Equation 5-112. 



^j_^ \Sic.)\'dc.^j_^ |y(a;)lVc.. (5- 



^J_^D|y(co)|Va; 



120) 



The integrals involving the filter transfer function can be canceled: 



2^^(^)^/_J^M|Vco. (5-121) 



