PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 347

R.4.82 The most important time (t)–frequency (s) properties and relations of the unilateral

LT are given as follows (similar to the FT properties):

a. Linearity

Let

£[ft^11 ()]

Fs()and £[ ()]ft2 2Fs()

then

af t^11 ()af t2 2()←→ aF s^11 () aF s^22 ()

b. Time scaling

fat aF

s

()↔ a









1

c. Time shifting

ft t Fse

()() 0 ↔ st^0

d. s-Shifting

fte Fs s

()st^0 ↔ ( 0 )

TABLE 4.2

Transform Pairs

Signal/System—f(t) Transform—F(s)

δ(t)1

u(t)1/s

tu(t)1/s^2

tnu(t) n!/sn+^1

e−atu(t) 1/(s + a)

te−atu(t) 1/(s + a)^2

cos(ωat)u(t) s/(s + ω 02 )

sin(ωat)u(t) ω 0 /(s^2 – ω 02 )^

e−at cos(ωat)u(t)(s + a)/ ( (s + a)^2 + ω (^02) )
e−at sin(ωat)u(t)(s + ω 0 )/ ( (s + a)^2 +^ ω (^02) )
cosh(ω 0 t)u(t) s/(s^2 − ω 02 )
sinh(ω 0 t)u(t) ω 0 /(s^2 + ω 02 )
1
1
1
()!
()
n
nateut
−
t−^1
()sa− n
1
ab
eeutat bt
−
()()−
1
()()sasb−−
(1 − e at)u(t)
−
−
a
ss a ()
[1 − cos (w 0 t)]u(t)
w
ss a
0
2
()^22 +
[sin (w 0 t + θ)]u(t)
sw
sw
sin( ) + cos( )

• 0
  2 02
  [cos (w 0 t + θ)]u(t)
  sw
  sw
  cos( ) − sin( )


• 
  

  0
  2 02