Problem Set Solution 20: The Laplace Transform

20 The Laplace Transform Solutions to Recommended Problems S20.1 (a) The Fourier transform of the signal does not exist because of the presence of growing exponentials. In other words, x(t) is not absolutely integrable. For the case a = 1, we have that (b) (i) x(t)e -' (ii) = 3e'u(t) + 4e 2 u(t) Although the growth rate has been slowed, the Fourier transform still does not converge. For the case a = 2.5, we have that x(t)e-'' = 3e-0-5 t u(t) + 4e 0 5 tu(t) (iii) The first term has now been sufficiently weighted that it decays to 0 as t goes to infinity. However, since the second term is still growing exponen­ tially, the Fourier transform does not converge. For the case a = 3.5, we have that x(t)e -"' = 3e -' 5 t u(t) + 4e - 0 "'u(t) Both terms do decay as t goes to infinity, and the Fourier transform con­ verges. We note that for any value of a > 3.0, the signal x(t)e -' decays exponentially, and the Fourier transform converges. (c) The Laplace transform of x(t) is 3 X( s-2 4 7(s -7) s-3 (s - 2)(s - 3)' and its pole-zero plot and ROC are as shown in Figure S20.1. Im s plane --- -F/ Re 2 17 3 7 Figure S20.1 Note that if a > 3.0, s = a + jw is in the region of convergence because, as we showed in part (b)(iii), the Fourier transform converges. S20-1 Signals and Systems S20-2 S20.2 (a) X(s) = e -at u(t)e 't dt T Ls = + a The Laplace transform converges for Re~s} + a > 0, so o + a>0,0 or o> -a, as shown in Figure S20.2-1. Im s plane -4w 0 Re Figure S20.2-1 (b) X(s) = The Laplace transform converges for Re~s) + a > 0, so o + a>0,0 or o>-a, as shown in Figure S20.2-2. Im s plane Re Figure S20.2-2 (c) X(s) = -- e atu(-t)e - dt = -e (s+a)t dt = JW- s + a if Re~s) + a < 0, o + a < 0, , < -a. e(s+a) s + a­ 0 The Laplace Transform / Solutions S20-3 Figure S20.2-3 S20.3 (a) (i) Since the Fourier transform of x(t)e ~'exists, a = 1 must be in the ROC. Therefore only one possible ROC exists, shown in Figure S20.3-1. (ii) We are specifying a left-sided signal. The corresponding ROC is as given in Figure S20.3-2. Im s plane XRe -2 2 Figure S20.3-2 Signals and Systems S20-4 (iii) We are specifying a right-sided signal. The corresponding ROC is as given in Figure S20.3-3. Im s plane Re )X -2 2 Figure S20.3-3 (b) Since there are no poles present, the ROC exists everywhere in the s plane. a = 1 must be in the ROC. Therefore, the only possible ROC is that shown (c) (i) in Figure S20.3-4. (ii) We are specifying a left-sided signal. The corresponding ROC is as shown in Figure S20.3-5. Im s plane 0 -2 2 Figure S20.3-5 Re The Laplace Transform / Solutions S20-5 (iii) (d) (i) We are specifying a right-sided signal. The corresponding ROC is as given in Figure S20.3-6. a = 1 must be in the ROC. Therefore, the only possible ROC is as shown in Figure S20.3-7. s plane - Re Figure S20.3-7 (ii) We are specifying a left-sided signal. The corresponding ROC is as shown in Figure S20.3-8. Im 2 s plane Re -2 Figure S20.3-8 Signals and Systems S20-6 (iii) We are specifying a right-sided signal. The corresponding ROC is as shown in Figure S20.3-9. s plane Figure S20.3-9 Constraint on ROC for Pole-Zero Pattern x(t) (i) (a) (b) (d) (c) Fourier transfor_ -2 < < 2 Entire s plane a> -2 a > 0 a< -2 Entire s plane o< -2 a < 0 o> 2 Entire s plane a> -2 a> 0 converges (ii) x(t) = 0, t > 10 (iii)x(t) =0 Table S20.3 S20.4 (a) For x(t) right-sided, the ROC is to the right of the rightmost pole, as shown in Figure S20.4-1. Im s plane Re Figure S20.4-1 The Laplace Transform / Solutions S20-7 Using partial fractions, 1 = 1 1 (s+ 1)(s + 2) s+ 1 s + 2' so, by inspection, x(t) = e -t u(t) - e 2t U(t) (b) For x(t) left-sided, the ROC is to the left of the leftmost pole, as shown in Figure S20.4-2. Im s plane 2u 1- Re Figure S20.4-2 Since X(S) = 1 - s +1 s+ 2 we conclude that x(t) = -e - t u(-t) - (-e 2- u(-t)) (c) For the two-sided assumption, we know that x(t) will have the form fi(t)u(- t) + fA(t)u(t) We know the inverse Laplace transforms of the following: 1 S+ 1 1 s + 2 e ~'u(t), I-e'u(-t), assuming right-sided, assuming left-sided, e- u(t), assuming right-sided, -e assuming left-sided -2'U(-t), Which of the combinations should we choose for the two-sided case? Suppose we choose x(t) = e -u(t) + (-e -2')u(-t) We ask, For what values of a does x(t)e -0' have a Fourier transform? And we see that there are no values. That is, suppose we choose a > -1, so that the first term has a Fourier transform. For a > -1, e - 2 te -'' is a growing exponen­ tial as t goes to negative infinity, so the second term does not have a Fourier transform. If we increase a, the first term decays faster as t goes to infinity, but Signals and Systems S20-8 the second term grows faster as t goes to negative infinity. Therefore, choosing a > -1 will not yield a Fourier transform of x(t)e -'. If we choose a 5 -1, we note that the first term will not have a Fourier transform. Therefore, we con­ clude that our choice of the two-sided sequence was wrong. It corresponds to the invalid region of convergence shown in Figure S20.4-3. Im s plane -0 '/ -2 -- Re Figure S20.4-3 If we choose the other possibility, x(t) = -e-u(-t) - e 2n(t), we see that the valid region of convergence is as given in Figure S20.4-4. Im splane -2 1 0 -, Re Figure S20.4-4 S20.5 There are two ways to solve this problem. Method 1 This method is based on recognizing that the system input is a superposition of eigenfunctions. Specifically, the eigenfunction property follows from the convolu­ tion integral y(t) = h(r)x(t - r) dr Now suppose x(t) = e". Then y(t) = f h(r)ea(" r)dr = e'" h(r)e -"dr The Laplace Transform / Solutions S20-9 Now we recognize that f h(r)e ~a' dr = H(s) ,s=a so that if x(t) = e ", then y(t) H(s) sale't, i.e., e a' is an eigenfunction of the system. Using linearity and superposition, we recognize that if x(t) = e -t/2 + 2et/3 then y(t) = e -t1 2H(s) s= -1/2 + 2e-' 13H(s) s= -1/3 so that for all t. y(t) = 2e -t/2 + 3e ~'/' Method 2 We consider the solution of this problem as the superposition of the response to two signals x 1 (t), x 2(t), where x 1 (t) is the noncausal part of x(t) and x 2 (t) is the causal part of x(t). That is, xi(t) = e -t/2U(-t) + 2e t /3 U( -), x 2(t) = e - t 2u(t) + 2e - t 3u(t) This allows us to use Laplace transforms, but we must be careful about the ROCs. Now consider L{xi(t)}, where C{-} denotes the Laplace transform: -Lxi(t)} = Xi(s) - 1 s+ _ - 2 s+ , Re{s) < -1 3 1 2 Now since the response to x 1(t) is y 1(t) = £ '{XI(s)H(s)}, then Yi(s) -2 1 2 1)' (s+i)(s+ (s+1)(s+i) 3 -2 -3 2 s+1 s+i s+I 5 s+1 2 s+i 3 s+1' -1 < Res) < s+1' y 1 (t) = 5e - tu(t) + 2e -t /2 U(- t) + 3e -t /3 u(- t) 2' Signals and Systems S20-10 The pole-zero plot and associated ROC for Yi(s) is shown in Figure S20.5-1. Im s plane Re -1__u1 Figure S20.5-1 Next consider the response y 2 (t) to x 2(t): x 2(t) = e -t/2U(t) + 2e ~'3 u(t), s X2(s) Y2 (s) = Y2(s) = 1+1 S 1-IS- + 2 i, + 1 X 2(s)H(s) = s+i + 1 Re{s} > - (s + 2)(s + 1) 3 -3 s+i s+1' 2 s+1 + ' 2 (s + 1)(s+ y 2 (t) = -5e -'u(t) + 2e ~1 /2U(t) + 3e - 1)' t 3 / u(t) The pole-zero plot and associated ROC for Y2(s) is shown in Figure S20.5-2. s plane 3 2 FY Figure S20.5-2 Since y(t) = y 1 (t) + y 2(t), then y(t) = 2e t~ / 2 + 3e ~'/3 for all t S20.6 (a) Since X(s) = x(t)e -s dt The Laplace Transform / Solutions S20-11 and s = a + jw, then J = X(s) s=,+j x(t)e -"'e-" dt ­ We see that the Laplace transform is the Fourier transform of x(t)e -'' from the definition of the Fourier analysis formula. (b) x(t)e -" = 27r X[s) e'f' dw ,+jo -_ This result is the inverse Fourier transform, or synthesis equation. So x(t) = e 27r -oo X(s) =-I and letting s = a + jw yields ds x(t) = (+jw)t dw dw, j dw: = -2 1ei' X(s) l,4j.] ~ rj X(s)e' ds _-joo Solutions to Optional Problems S20.7 (a) X(s) = 1 , s + 1 Refs) > -1 Therefore, x(t) is right-sided, and specifically x(t) = e -u(t) 1 , + 1 Therefore, (b) X(s) = s Re{s) < -1 x(t) = -e- t u(-t) s (c) X(s) = S2 + 4' Refs) > 0 Since eisot . 1 S -j W 1 L s + jw 0 L{cos(wot )u(t)} = £-e*ot L{cos(wot)u(t)) 2 if X(s) = 12 1. + - e-"=- 1(1 . 22 Jo + 1=+ s+jo s + W2 2s then x(t) = cos(2t)u(t) Signals and Systems S20-12 (d) X(s) = s+1 2=s 2 + 5s + 6 x(t) x(t) s - s +1 = s2(s - s-i1 s-s -e - 2 tu(t) + 2e - 3 u(t) S2 _ s(s -1) s2 s-_1) -1 +-1 -1+ 1 + + s s s2 S(s S + + 1) 2 (S + = (h) X(s) = s 1 - tu(t) u(-t) (S + 1)2 (g) X(s) = < Re{s} < 1 1 t x(t) = -e 0 1) S1 x(t) = 2 -1 s+1 s + 3' s + 2 (s + 2)(s + 3) e - 2 tu(-t) - 2e - 3 u(-t) (e) X(s) (f) X(s) 2 -1 ,so + s + 2 s + 3 s+1 (s + 2)(s + 3) _ 1 < Res} > 3s - 3s (S + 1)2 3(s + 1) 3 (S + 1)2 (S + 1)2> 1)2 b(t) - 3e -'u(t) + 3te-'u(t) s + 1 (s + 1)2 + 4 Consider Y(s) = s 2 s + 4 - y(t) = cos(2t)u(t) from part (c) Now f(t)e ~ x(t) = e F(s + a), cos (2t)u(t) S20.8 The Laplace transform of an impulse ab(t) is a.Therefore, if we expand a rational Laplace transform by dividing the denominator into the numerator, we require a constant term in the expansion. This will occur only if the numerator has order greater than or equal to the order of the denominator. Therefore, a necessary con­ dition on the number of zeros is that it be greater than or equal to the number of poles. This is only a necessary and not a sufficient condition as it is possible to con­ struct a rational Laplace transform that has a numerator order greater than the The Laplace Transform / Solutions S20-13 denominator order and that does not yield a constant term in the expansion. For example, X(s) s2 + 1 s 1 , s which does not have a constant term. Therefore a necessary condition is that the number of zeros equal or exceed the number of poles. S20.9 (a) x(t) = e -a'u(t), X(s) = s + a' a < 0, , and the ROC is shown in Figure S20.9-1. (b) x(t) = -e X(S) = a'u(-t), a > 0, 1, s -- a and the ROC is shown in Figure S20.9-2. Figure S20.9-2 Signals and Systems S20-14 (c) x(t) = e a'u(t), a > 0, 1 X(s)- s- a' and the ROC is shown in Figure S20.9-3. s plane Figure S20.9-3 (d) x(t) = e t a > 0, -a" I t = e -a u(t) + e a'u(- t), X(S) = 1 s+a + -1 -, s-a' and the ROC is shown in Figure S20.9-4. Im s plane -a Figure S2 0.9-4 (e) x(t) = u(t), X(s) = fe 0 -S dt=-, s and the ROC is shown in Figure S20.9-5. The Laplace Transform / Solutions S20-15 s plane Figure S20.9-5 (f) x(t) = btt - to), X(s) = fb(t - to)e -' dt = e ~sto and the ROC is the entire s plane. (g) x(t) = T0 a k t-kT ), k=0 X(s) = ( t 6(t - kT)e -s dt ak k=0 = Lake -skT = 1 k=0 with ROC such that ae-T < 1. Now a2 -2sr < 1 - 2 log a - 1 T s > - log a 2sT < 0 - (h) x(t) = cos (wot + b)u(t) Using the identity cos(a + b) = cos a cos b - sin a sin b we have that x(t) = cos b costwot)u(t) - sin b sin(wot)u(t) Using linearity and the transform pairs £ S cos(W0 t) 2 sin(wot) Wo 2 + W2 s2 +2 , 0 , Signals and Systems S20-16 we have X(s) = cos b s S X(s) =cosb - sin b 2o2 0x 0o S [s - (tan b)wo] s2 + W2 ' and the ROC is shown in Figure S20.9-6. s plane - Re --o0 Figure S20.9-6 (i) Consider xi(t) = sin(wot + b)u(t) = (sin wot cos b + cos wot sin b)u(t) Using linearity and the preceding sin w0t, cos wot pairs, we have X 1(s) = cos b X 1 (s) = s2 + S + sin b W2 s2 + [s + (cot b)wo] sin bW Using the property that f(t)e ~4'- F(s + a), we have X(s) = sin b [s + a + (cot b)wO] (s + a)2 + o' with the ROC as given in Figure S20.9-7. oC ' The Laplace Transform / Solutions S20-17 Im -a + jo span~e Re -(a + wo cot b)' -a -jcoo Figure S20.9-7 S20.10 (a) X(s) x(t)e-" dt = Consider t)e X 1(s) = S" dt Letting t = -t', we have x(t')e"' dt' X1(s) = =X(-s), but X 1(s) = X(s) since x(t) (b) X(s) = x(-t). Therefore, X(s) = X(-s). x(t)e -" dt = Consider X 1(s) -x(-t)e - dt, = -x(t'e"' dt' X 1(s) = =-X(S), but X 1(s) = X(s) since x(t) = -x(--t). Therefore, X(s) = -X(-s). (c) We note that if X(s) has poles, then it must be two-sided in order for x(t) = x(-t). (i) Ks s X(s) = ( (s + 1)(S - 1), -Ks X(-s) = (-s + 1)(-s - 1) so x(t) # x(- t). -Ks = # X(s), (s - 1)(s + 1) Signals and Systems S20-18 (ii) X(S) =K(s + 1)(s - 1) S K(-s + 1)(-s - 1 ) 0 X(s) -s Also, this pole pattern cannot have a two-sided ROC. X(-s) (iii) X(S) = K(s + j)(s - j) (s + 1)(s - 1)' K(-s +j)(-s -j) K(s - j)(s + j) S = =-~s (-s + 1)(-s - 1) (s - 1)(s + 1) so this can correspond to an even x(t). The corresponding ROC must be two-sided, as shown in Figure S20.10-1. X(-s) (iv) = _ This does not have any possible two-sided ROCs. (d) We see from the results in part (c)(i) that X(s) = -X(-s), so the result in part (c)(i) corresponds to an odd x(t) with an ROC as given in Figure S20.10-2. Im s plane Re Figure S20.10-2 Parts (c)(ii) and (c)(iv) do not have any possible two-sided ROCs. Part (c)(iii) is even, as previously shown, and therefore cannot be odd.