Signal Statistics and Detection

First, a quick review of intervals. We define the following four cases: • The closed interval [a, b] = {t : a ≤ t ≤ b}; The half-open interval (a, b] = {t : a < t ≤ b}; • The half-open interval [a, b) = {t : a ≤ t < b}; The open interval (a, b) = {t : a < t < b}. If a or b is ±∞, use a half-open or open interval. Never write (even implicitly) “t = ∞”–this is uncouth! Now let x(t) be a continuous-time signal. We define the following basic properties of x(t):