x(t) Differentiation x(t) X(s) Multiplication FIGURE 6.1 Equivalent operations in the (a)time-domain operation and(b) Laplace transform-domain operation. ( Source: J.A. Cadzow and H.E. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N J Prentice-Hall 985,P. 138. With permission. max (o+;o2)<Re)<min(2;σ2) where the pairs(ol; 02)< Re(s)< min(o o2)identify the regions of convergence for the Laplace transforms X(s) and X,(s), respectively. Time Domain Differentiation The operation of time-domain differentiation has then been found to correspond to a multiplication by s in the Laplace variable s domain. The Laplace transform of differentiated signal dx n)/dt is Furthermore, it is clear that the region of absolute convergence of dx( t)/dt is at least as large as that of x(o) This property may be envisioned as shown in Fig. 6.1 Time Shift The signal x(t-to) is said to be a version of the signal x n right shifted (or delayed) by to seconds. Right shifting (delaying)a signal by a to second duration in the time domain is seen to correspond to a multiplication by e-sro in the Laplace transform domain. The desired Laplace transform relationship [x(t-t0)=c-X(s) where X(s)denotes the Laplace transform of the unshifted signal x(t). As a general rule, any time a term of the form e-sto appears in X(s), this implies some form of time shift in the time domain. This most important property is depicted in Fig. 6.2. It should be further noted that the regions of absolute convergence for the signals x n)and xt- to) are ide Delay by o FIGURE 6.2 Equivalent operations in(a)the time domain and(b) the Laplace transform domain. ( Source: J-A Cadzow and H.F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N J. Prentice-Hall, 1985, P. 140. With© 2000 by CRC Press LLC where the pairs (s1 +; s + 2 ) < Re(s) < min(s– 1 ; s– 2 ) identify the regions of convergence for the Laplace transforms X1(s) and X2(s), respectively. Time-Domain Differentiation The operation of time-domain differentiation has then been found to correspond to a multiplication by s in the Laplace variable s domain. The Laplace transform of differentiated signal dx(t)/dt is Furthermore, it is clear that the region of absolute convergence of dx(t)/dt is at least as large as that of x(t). This property may be envisioned as shown in Fig. 6.1. Time Shift The signal x(t – t0) is said to be a version of the signal x(t) right shifted (or delayed) by t0 seconds. Right shifting (delaying) a signal by a t0 second duration in the time domain is seen to correspond to a multiplication by e–st 0 in the Laplace transform domain. The desired Laplace transform relationship is where X(s) denotes the Laplace transform of the unshifted signal x(t). As a general rule, any time a term of the form e–st 0 appears in X(s), this implies some form of time shift in the time domain. This most important property is depicted in Fig. 6.2. It should be further noted that the regions of absolute convergence for the signals x(t) and x(t – t0) are identical. FIGURE 6.1 Equivalent operations in the (a) time-domain operation and (b) Laplace transform-domain operation. (Source: J.A. Cadzow and H.F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 138. With permission.) FIGURE 6.2 Equivalent operations in (a) the time domain and (b) the Laplace transform domain. (Source: J.A. Cadzow and H.F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 140. With permission.) max(s s; ) Re( ) min(s s; ) + + < < - - 1 2 1 2 s + dx t dt sX s ( ) ( ) È Î Í ˘ ˚ ˙ = + [ (x t t )] e X(s) st - = - 0 0