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
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
Inverse Laplace Transform Given a transform function X(s) and its region of convergence, the procedure for finding the signal x(r)that generated that transform is called finding the inverse Laplace transform and is symbolically denoted as x(t)=[X( The signal x(t) can be recovered by means of the relationshi (t) X(s)e" ds In this integral, the real number c is to be selected so that the complex number c+ jo lies entirely within the region of convergence of X(s)for all values of the imaginary component o For the important class of rational Laplace transform functions, there exists an effective alternate procedure that does not necessitate directly evaluating this integral. This procedure is generally known as the partial-fraction expansion method Partial Fraction Expansion Method just indicated, the partial fraction expansion method provides a convenient technique for reacquiring the ignal that generates a given rational Laplace transform. Recall that a transform function is said to be rational if it is expressible as a ratio of polynomial in s, that X(s)= b(s b m-+…+bs+b A(s) s"+a-S"-+.+a,s+a The partial fraction expansion method is based on the appealing notion of equivalently expressing this rational transform as a sum of n elementary transforms whose corresponding inverse Laplace transforms (i.e. generating ignals)are readily found in standard Laplace transform pair tables. This method entails the simple five-step process as outlined in Table 6.3. A description of each of these steps and their implementation is now given . Proper Form for Rational Transform. This division process yields an expression in the proper form as given by A(s) Q(s)+ A(s) TABLE 6.3 Partial Fraction Expansion Method for Determining the Inverse Laplace Transform L. Put rational transform into proper form whereby the degree of the numerator polynomial is less than or equal to that of Il. Factor the denominator polynomial. lll. Perform a partial fraction expansion. IV. Separate partial fraction expansion terms into causal and anticausal components using the associated region of absolute convergence for this purpose Source: J.A. Cadzow and H.F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N J. Prentice-Hall, 1985, P. 153. with permission. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC Inverse Laplace Transform Given a transform function X(s) and its region of convergence, the procedure for finding the signal x(t) that generated that transform is called finding the inverse Laplace transform and is symbolically denoted as The signal x(t) can be recovered by means of the relationship In this integral, the real number c is to be selected so that the complex number c + jw lies entirely within the region of convergence of X(s) for all values of the imaginary component w. For the important class of rational Laplace transform functions, there exists an effective alternate procedure that does not necessitate directly evaluating this integral. This procedure is generally known as the partial-fraction expansion method. Partial Fraction Expansion Method As just indicated, the partial fraction expansion method provides a convenient technique for reacquiring the signal that generates a given rational Laplace transform. Recall that a transform function is said to be rational if it is expressible as a ratio of polynomial in s, that is, The partial fraction expansion method is based on the appealing notion of equivalently expressing this rational transform as a sum of n elementary transforms whose corresponding inverse Laplace transforms (i.e., generating signals) are readily found in standard Laplace transform pair tables. This method entails the simple five-step process as outlined in Table 6.3. A description of each of these steps and their implementation is now given. I. Proper Form for Rational Transform. This division process yields an expression in the proper form as given by TABLE 6.3 Partial Fraction Expansion Method for Determining the Inverse Laplace Transform I. Put rational transform into proper form whereby the degree of the numerator polynomial is less than or equal to that of the denominator polynomial. II. Factor the denominator polynomial. III. Perform a partial fraction expansion. IV. Separate partial fraction expansion terms into causal and anticausal components using the associated region of absolute convergence for this purpose. V. Using a Laplace transform pair table, obtain the inverse Laplace transform. Source: J. A. Cadzow and H.F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 153. With permission. x t( ) = + [X(s)] ±1 x t j X s e ds st c j c j ( ) = ( ) - • + • Ú 1 2p X s B s A s b s b s b s b s a s a s a m m m m n n n ( ) ( ) ( ) = = + + × × × + + + + × × × + + - - - - 1 1 1 0 1 1 1 0 X s B s A s Q s R s A s ( ) ( ) ( ) ( ) ( ) ( ) = = +