Autocorrelation Function The autocorrelation function of the signal x( t) is formally defined by p(t)= x(t)x(t+ t)dt The Laplace transform of the autocorrelation function is Φx(s)=X(-s)X(s) and the corresponding region of absolute convergence is max(-0, -, o, +)< Re(s)< min(-o +, o,-) Other Properties A number of properties that characterize the Laplace transform are listed in Table 6. 2. Application of these properties often enables one to efficiently determine the Laplace transform of seemingly complex time functions TABLE 6.2 Laplace Transform Properties a1x(n)+x2x2(1)a1X1(s)+a2x2(s) At least the intersection of the region of convergence of X,(s)an sX(s) Time different dx(t) At least o.< Re(s)and x,(s) Time convolution H(sX(s) h(t)x(r-t)dr At least the intersection of the region of convergence of H(s)and X(s) Time scaling frequency shift X(s +a) o-Re(a)< Re(s)<o.-Refa) Multiplication x()x2() Cr J,(u)x, (s-ud om)+o2< Re(s)<o+o. 0+0<C<0+o Time integration ∫otxo0mxo- At least o.<Ref Frequency d x(s) At least o,< Re(s)<o differentiation max(-o,o)<Re(s)< min(-o, 0._) x()y(t+2)d Autocorrelation X(-s)X(s max(-o,o)<Re(s)< min(-o,+,o_) function ∫2(0x+-3m Source: J. A Cadzow and H.E. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N J: Prentice-Hall, 1985. with permission. c 2000 by CRC Press LLC© 2000 by CRC Press LLC Autocorrelation Function The autocorrelation function of the signal x(t) is formally defined by The Laplace transform of the autocorrelation function is and the corresponding region of absolute convergence is Other Properties A number of properties that characterize the Laplace transform are listed in Table 6.2. Application of these properties often enables one to efficiently determine the Laplace transform of seemingly complex time functions. TABLE 6.2 Laplace Transform Properties Signal x(t) Laplace Transform Region of Convergence of X(s) Property Time Domain X(s) s Domain s+ < Re(s) < s– Linearity a1x1(t) + a2x2(t) a1X1(s) + a2X2(s) At least the intersection of the region of convergence of X1(s) and X2(s) Time differentiation sX(s) At least s+ < Re(s) and X2(s) Time shift x(t – t0) e –st0X(s) s+ < Re(s) < s– Time convolution H(s)X(s) At least the intersection of the region of convergence of H(s) and X(s) Time scaling x(at) Frequency shift e –atx(t) X(s + a) s+ – Re(a) < Re(s) < s– – Re(a) Multiplication (frequency convolution) x1(t)x2(t) Time integration At least s+ < Re(s) < s– Frequency differentiation (–t)k x(t) At least s+ < Re(s) < s– Time correlation X(–s)Y(s) max(–sx – , sy+) < Re(s) < min(–sx + , sy– ) Autocorrelation function X(–s)X(s) max(–sx –, sx+) < Re(s) < min(–sx + , sx– ) Source: J.A. Cadzow and H.F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985. With permission. f t t xx ( ) = + x(t) ( x t )dt -• • Ú Fxx ( )s = - X( s)X(s) max(-s s - , + ) < Re( ) < min(-s + , s - ) x y x s y dx t dt ( ) h(t)x(t - t)dt -• • Ú 1 * a * X s a Ê Ë Á ˆ ¯ ˜ s s + - < Ê Ë Á ˆ ¯ Re ˜ < s a 1 2 1 2 pj X u X s u d c j c j ( ) ( ) - • + • Ú - s s s s s s s s + + - - + + - - + < < + + < < + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Re( ) 1 2 1 2 1 2 1 2 s c x d t (t t ) Ú-• 1 0 s X s( ) for X( ) = d X s ds k k ( ) x(t) y(t + z)dt - • + • Ú x(t)x(t + z)dt - • + • Ú