Chapter 9
Optimum Receivers and Signal Space Concepts 9.1
Problems
Problem 9.1 a. Given H1 , Z = N , so fZ (z|H1 ) = fN (n) |z=n = 10e−10z u (z) Given H2 , Z = S +N , where S and N are independent. Thus, the resulting pdf under H2 is the convolution of the separate pdfs of S and N : fZ (z|H2 ) =
Z
∞
−∞
fs (z − λ) fN (λ) dλ = 20
Z
z
e−2z e−8λ dλ, λ > 0
0
which integrates to the result given in the problem statement. b. The likelihood ratio is Λ (Z) =
¡ ¢ fZ (Z|H2 ) = 0.25 e8Z − 1 , Z > 0 fZ (Z|H1 )
c. The threshold is η=
1 P0 (c21 − c11 ) = P1 (c12 − c22 ) 3 1