y(t)= h(t)x(t-t)dt x(t)」 Linear convolution h(t).x(t) (a) Y(s)= H()X(s) Hs FIGURE 6.3 Representation of a time- invariant linear operator in(a)the time domain and(b)the s-domain. (Source: L.A. Cadzow and H F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N J. Prentice-Hall, 1985, P. 144. Time-Convolution Property The convolution integral signal y(t) can be expresse ()-_MG) where xdn) denotes the input signal, the h(t) characteristic signal identifying the operation process. The Laplace transform of the response signal is simply given by Y(s=H(S)X(s) where H(s)= [h(o] and x(s)=s [xn]. Thus, the convolution of two time-domain signals is seen to correspond to the multiplication of their respective Laplace transforms in the s-domain. This property may be envisioned as shown in Fig. 6.3 Time-Correlation Property The operation of correlating two signals x t) and y( n) is formally defined by the integral relationshil 9,(0)=x)y+rht The Laplace transform property of the correlation function % t)is in which the region of absolute convergence is given by max(-ox -,o,+)< Re(s)< min(-o+,Or-) c 2000 by CRC Press LLC© 2000 by CRC Press LLC Time-Convolution Property The convolution integral signal y(t) can be expressed as where x(t) denotes the input signal, the h(t) characteristic signal identifying the operation process. The Laplace transform of the response signal is simply given by where H(s) = + [h(t)] and X(s) = + [x(t)]. Thus, the convolution of two time-domain signals is seen to correspond to the multiplication of their respective Laplace transforms in the s-domain. This property may be envisioned as shown in Fig. 6.3. Time-Correlation Property The operation of correlating two signals x(t) and y(t) is formally defined by the integral relationship The Laplace transform property of the correlation function fxy(t)is in which the region of absolute convergence is given by FIGURE 6.3 Representation of a time-invariant linear operator in (a) the time domain and (b) the s-domain. (Source: J. A. Cadzow and H. F.Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 144. With permission.) y t( ) = - h()( x t )d -• • Ú t t t Y s( ) = H(s)X(s) f t t xy ( ) = + x(t) ( y t )dt -• • Ú Fxy ( )s = - X( s)Y(s) max(-s s - + , ) < Re( ) < min(-s s + , - ) x y y x s