Page 52 - 4811

Finding the Impulse Function Using Laplace Transform

Definition 3.1✓ If f(t) is continuous in a ≤ t ≤ b then we define
the function δ(t) by the integral equation

b b
∫ ∫
f(t)δ(t − t )dt = lim f(t)f (t − t )dt
0
0
ε
a ε→0 + a
The object δ(t) on the left is called the Dirac Delta function, or
just the delta function for short.
. . . . .

Finding the Impulse Function Using Laplace

Transform

For ε > 0 we can solve the initial value problem (3.2) using Laplace transforms. To
do this we need to compute the Laplace transform of f (t), given by the integral
ε
∫ ∫
∞ 1 ε 1 − e −εs
L[f (t)] = f (t)e −st dt = e −st dt = .
ε
ε
0 ε 0 εs
Note that by using L’Hôpital’s rule we can write

1 − e −εs
lim L[f (t)] = lim = 1, s > 0.
ε
ε→0 + ε→0 + εs
Now, to find y (t), we apply the Laplace transform to both sides of equation (3.1)
ε
and using the initial conditions we obtain

1 − e −εs
2
ms Y (s) + γsY (s) + kY (s) = .
ε
ε
ε
εs
Solving for Y (s) we find
ε
1 1 − e −εs
Y (s) = .
ε
2
ms + γs + k εs
+
Letting ε → 0 we find
1
Y (s) =
2
ms + γs + k
which is the transfer function of the system. Now inverse transform Y (s) to find
the solution to the initial value problem. That is,

( )
1
−1
y(t) = L = ϕ(t).
2
ms + γs + k
Now, impulse inputs are usually modelled in terms of delta functions. Thus,
knowing the Laplace transform of such functions is important when solving

51