Signals & Systems - Reference Tables 1 Table of Fourier Transform Pairs Function, f(t) Fourier Transform, F( ) Definition of Inverse Fourier Transform f t F( )ej td 2 1 ( ) Definition of Fourier Transform F() f (t)e j tdt f (t t0) F( )e j t0 f (t)ej 0t F 0 f ( t) ( ) 1 F F(t) 2 f n n dt The following table are useful for applying this technique. In other … Properties of Laplace transform: 1. – – Kronecker delta δ0(k) 1 k = 0 0 k ≠ 0 1 2. Solution. – – δ0(n-k) 1 n = k 0 n ≠ k z-k 3. s 1 1(t) 1(k) 1 1 1 −z− 4. s +a 1 e-at e-akT 1 1 1 −e−aT z− 5. 3 2 s t2 (kT)2 ()1 3 2 1 1 The inverse Laplace transform We can also deﬁne the inverse Laplace transform: given a function X(s) in the s-domain, its inverse Laplace transform L−1[X(s)] is a function x(t) such that X(s) = L[x(t)]. The ﬁnal stage in that solution procedure involves calulating inverse Laplace transforms. In this section we look at the problem of ﬁnding inverse Laplace transforms. Table of Laplace and Z-transforms X(s) x(t) x(kT) or x(k) X(z) 1. 18.031 Laplace Transform Table Properties and Rules Function Transform f(t) F(s) = Z 1 0 f(t)e st dt (De nition) af(t) + bg(t) aF(s) + bG(s) (Linearity) eatf(t) F(s a) (s-shift) f0(t) sF(s) f(0 ) f00(t) s2F(s) sf(0 ) f0(0 ) f(n)(t) snF(s) sn 1f(0 ) f(n 1)(0 ) tf(t) F0(s) t nf(t) ( 1)nF( )(s) u(t a)f(t a) e asF(s) (t-translation or t-shift) u(t a)f(t) e asL(f(t+ a)) (t-translation) Table 1: Table of Laplace Transforms Number f(t) F(s) 1 δ(t)1 2 us(t) 1 s 3 t 1 s2 4 tn n! It can be shown that the Laplace transform of a causal signal is unique; hence, the inverse Laplace transform is uniquely deﬁned as well. 6.2: Solution of initial value problems (4) Topics: † Properties of Laplace transform, with proofs and examples † Inverse Laplace transform, with examples, review of partial fraction, † Solution of initial value problems, with examples covering various cases. Table 1: A List of Laplace and Inverse Laplace Transforms Related to Fractional Order Calculus. The text has a more detailed table. Linearity: Lfc1f(t)+c2g(t)g = c1Lff(t)g+c2Lfg(t)g. 2. 2 1 s t kT ()2 1 1 1 − −z Tz 6. tn−1 L eat = 1 s−a L−1 1 s−a = eat L[sinat] = a s 2+a L−1 1 s +a2 = 1 a sinat L[cosat] = s s 2+a L−1 s s 2+a = cosat Diﬀerentiation and integration L d dt f(t) = sL[f(t)]−f(0) L d2t dt2 f(t) = s2L[f(t)]−sf(0)−f0(0) L dn … Table of Laplace Transforms Definition of Laplace transform 0 L{f (t)} e st f (t)dt f (t) L 1{F(s)} F(s) L{f (t)} Laplace transforms of elementary functions 1 s 1 tn 1! 6.3 Inverse Laplace Transforms Recall the solution procedure outlined in Figure 6.1. A List of Laplace and Inverse Laplace Transforms Related to Fractional Order Calculus 3 F(s) f(t) k s2+k2 coth ˇs 2k jsinkt 1 s e k=s J 0(2 p kt) p1 s e k=s p1 ˇt cos2 p kt p1 s … nding inverse Laplace transforms is a critical step in solving initial value problems. tedious to deal with, one usually uses the Cauchy theorem to evaluate the inverse transform using f(t) = Σ enclosed residues of F (s)e st. s n+1 L−1 1 s = 1 (n−1)! We get the solution y(t) by taking the inverse Laplace transform. First derivative: Lff0(t)g = sLff(t)g¡f(0). An abbreviated table of Laplace transforms was given in the previous lecture. To determine the inverse Laplace transform of a function, we try to match it with the form of an entry in the right-hand column of a Laplace table. TABLE OF LAPLACE TRANSFORM FORMULAS L[tn] = n! Example 1. In this course we shall use lookup tables to evaluate the inverse Laplace transform. Determine L 1fFgfor (a) F(s) = 2 s3, (b) F(s) = 3 s 2+ 9, (c) F(s) = s 1 s 2s+ 5. 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