Laplace Transform Pairs TABLE 9.2 LAPLACE TRANSFORMS OF ELEMENTARY FUNCTIONS Transform pair Signal Transform ROC 6(t) A/s Aes > 0 M(t) A/s M(s) ROC: Re(s) > 0 u(t) 1/s ROC: Re(s) > 0 u(-t) 1/s ROC: Re(s) < 0 e^(-at) u(t) A/(s + a) ROC: Re(s) > -a e^(-at) u(-t) A/(s + a) ROC: Re(s) < -a cos(wt) u(t) s/(s^2 + w^2) ROC: Re(s) > 0 sin(wt) u(t) w/(s^2 + w^2) ROC: Re(s) > 0 1 u(t) 1/s ROC: Re(s) > 0 e^(-at) sin(wt) u(t) w/( (s + a)^2 + w^2) ROC: Re(s) > -a e^(-at) cos(wt) u(t) (s + a)/( (s + a)^2 + w^2) ROC: Re(s) > -a sin(wt) u(-t) -w/(s^2 + w^2) ROC: Re(s) < 0 cos(wt) u(-t) (s)/(s^2 + w^2) ROC: Re(s) < 0 1 u(-t) -1/s ROC: Re(s) < 0 e^(-at) sin(wt) u(-t) -w/( (s + a)^2 + w^2) ROC: Re(s) < -a e^(-at) cos(wt) u(-t) (s + a)/( (s + a)^2 + w^2) ROC: Re(s) < -a 1 u(t) 1/s ROC: Re(s) > 0 M-n(t) 1/s^n ROC: Re(s) > 0 u(t) 1/s ROC: Re(s) > 0 u(t) A/s ROC: Re(s) > 0
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First, let's understand the table. It shows the Laplace Transform pairs, which means that for a given time-domain signal (function of time, t), it shows the corresponding Laplace Transform (function of complex variable s) and the Region of Convergence (ROC). Show more…
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Table of Laplace Transforms f(t)=mathcal{L}^{-1}{F(s)} F(s)=mathcal{L}{f(t)} f(t)=mathcal{L}^{-1}{F(s)} F(s)=mathcal{L}{f(t)} 1. 1 frac{1}{s} 2. e^{at} frac{1}{s-a} 3. t^n, n=1,2,3,... frac{n!}{s^{n+1}} 4. t^p, p > -1 frac{Gamma(p+1)}{s^{p+1}} 5. sqrt{t} frac{sqrt{pi}}{2s^{frac{3}{2}}} 6. t^{n-frac{1}{2}}, n=1,2,3,... frac{1cdot3cdot5cdots(2n-1)sqrt{pi}}{2^n s^{n+frac{1}{2}}} 7. sin(at) frac{a}{s^2+a^2} 8. cos(at) frac{s}{s^2+a^2} 9. t sin(at) frac{2as}{(s^2+a^2)^2} 10. t cos(at) frac{s^2-a^2}{(s^2+a^2)^2} 11. sin(at) - at cos(at) frac{2a^3}{(s^2+a^2)^2} 12. sin(at) + at cos(at) frac{2as^2}{(s^2+a^2)^2} 13. cos(at) - at sin(at) frac{s(s^2-a^2)}{(s^2+a^2)^2} 14. cos(at) + at sin(at) frac{s(s^2+3a^2)}{(s^2+a^2)^2} 15. sin(at+b) frac{s sin(b) + a cos(b)}{s^2+a^2} 16. cos(at+b) frac{s cos(b) - a sin(b)}{s^2+a^2} 17. sinh(at) frac{a}{s^2-a^2} 18. cosh(at) frac{s}{s^2-a^2} 19. e^{at} sin(bt) frac{b}{(s-a)^2+b^2} 20. e^{at} cos(bt) frac{s-a}{(s-a)^2+b^2} 21. e^{at} sinh(bt) frac{b}{(s-a)^2-b^2} 22. e^{at} cosh(bt) frac{s-a}{(s-a)^2-b^2} 23. t^n e^{at}, n=1,2,3,... frac{n!}{(s-a)^{n+1}} 24. f(ct) frac{1}{c}F(frac{s}{c}) 25. u_c(t) = u(t-c) Heaviside Function frac{e^{-cs}}{s} 26. delta(t-c) Dirac Delta Function e^{-cs} 27. u_c(t)f(t-c) e^{-cs}F(s) 28. u_c(t)g(t) e^{-cs}mathcal{L}{g(t+c)} 29. e^{at}f(t) F(s-a) 30. t^n f(t), n=1,2,3,... (-1)^n F^{(n)}(s) 31. frac{1}{t}f(t) int_s^infty F(u)du 32. int_0^t f(v)dv frac{F(s)}{s} 33. int_0^t f(t- au)g( au)d au F(s)G(s) 34. f(t+T) = f(t) frac{int_0^T e^{-st}f(t)dt}{1-e^{-sT}} 35. f'(t) sF(s)-f(0) 36. f''(t) s^2F(s)-sf(0)-f'(0) 37. f^{(n)}(t) s^n F(s)-s^{n-1}f(0)-s^{n-2}f'(0)-cdots-sf^{(n-2)}(0)-f^{(n-1)}(0)
Adi S.
(a) Find the bilateral Laplace transform and ROC of the signal x(t) = 2e^{-4t}u(t+2) + 3e^{4t}u(-t) using the definition. (b) Find the bilateral Laplace transform and ROC of the signal x(t) = 7e^{4t}u(-t) - e^{2t}u(-t+1) using LT table and properties
Sri K.
6.1 Using the definition of the Laplace transform, find the Laplace transform of each of the following signals (including the corresponding ROC): (a) x(t) = e^(-at)u(t); (b) x(t) = e^(-a|t|); and (c) x(t) = [cos(w0t)]u(t). [Note: ∫ e^(ax) cos(bx)dx = 1/(a^2+b^2) (e^(ax)[acos(bx)+bsin(bx)]).] 6.2 Find the Laplace transform (including the ROC) of the following signals: (a) x(t) = e^(-2t)u(t); (b) x(t) = 3e^(-2t)u(t)+2e^(5t)u(-t); (c) x(t) = e^(-2t)u(t +4); (d) x(t) = ∫(t to -∞) e^(-2t) u(t)dt; (e) x(t) = -ae^(-at)u(-t +b) where a is a positive real constant and b is a real constant; (f) x(t) = te^(-3t)u(t +1); and (g) x(t) = tu(t +2). 6.3 Suppose that x(t) ↔ X(s) with ROC RX and y(t) ↔ Y(s) with ROC RY. Then, express Y(s) in terms of X(s), and express RY in terms of RX, in each of the cases below. (a) y(t) = x(at -b) where a and b are real constants and a ≠ 0; (b) y(t) = e^(-3t) [x(l) ∗ x(l)]|l=t-1; (c) y(t) = tx(3t -2); (d) y(t) = d/dt [x*(t -3)]; (e) y(t) = e^(-5t)x(3t +7); and (f) y(t) = e^(-j5t)x(t +3). 6.4 The Laplace transform X(s) of a signal x(t) is given by X(s) = (s+1/2)(s-1/2)/(s(s+1)(s-1)). Determine whether x(t) is left-sided, right-sided, two-sided, or finite duration for each of the following ROCs of X(s): (a) Re{s} < -1; (b) -1 < Re{s} < 0; (c) 0 < Re{s} < 1; and (d) Re{s} > 1. 6.5 Let X(s) denote the Laplace transform of x(t). Suppose that X(s) is given by X(s) = (s+1/2)/(s+1-j)(s+1+j)(s+2). Plot the ROC of X(s) if x(t) is (a) left-sided; (b) right-sided; (c) two-sided; (d) causal.
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