The Bilateral Laplace Transform of a signal x(t) is defined as:

The complex variable s = σ + jω, where ω is the frequency

variable of the Fourier Transform (simply set σ = 0). The
Laplace Transform converges for more functions than the
Fourier Transform since it could converge off of the jω axis.
Here is a plot of the s-plane:
The xy-axis plane, where x-axis
is the real axis and y-axis is the
imaginary axis, is called as s-
plane. We will perform our
visualizations of Laplace
transform on s-plane

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 Region of Convergence (ROC)
• Similar to the integral in Fourier transform, the integral in
Laplace transform may also not converge for some
values of s.
• So, Laplace transform of a function is always defined by
two entities:
– Algebraic expression of X(s).
– Range of s values where X(s) is valid, i.e. region of convergence
(ROC).

bilateral and unilateral Laplace Bilateral and unilateral Laplace

Transforms Transforms
As we'll see, an important difference between the • If the ROC contains the jω- axis, then if x(t)
bilateral and unilateral Laplace Transforms is that you were used as an impulse response, the system
need to specify the region of convergence (ROC) for would be BIBO stable. If the boundary of the
the bilateral case.
ROC is the jω-axis (i.e. Re(s) > 0 or Re(s) <
We point out (without proof) several features of ROCs: 0), the system would be BIBO unstable.
• A right-sided time function (i.e. x(t) = 0, t < t0 where Taking the Laplace Transform is clearly a
t0 is a constant) has an ROC that is a right half-plane. linear operation:
• A left-sided time function has an ROC that is a left
half-plane. • L[ax1(t) + bx2(t)] = aX1(s) + bX2(s)
• A 2-sided time function has an ROC that is either a • where X1(s) is the Laplace Transform of x1(t)
strip or else the ROC does not exist, which means that and X2(s) is the Laplace Transform of x2(t
the Laplace Transform does not exist.

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The Inverse Bilateral Laplace Transform of X(s) is:

If we define x(t) to be 0 for t < 0, this gives us the

unilateral Laplace transform:

we needed to specify that Re(s) > 0. If this is not the

case, the integral would have not converged at the upper
limit of infinity.

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 Proof

or

Example 1 Find the Laplace Transform of x(t) = sin[b(t - 2)]u(t - 2)

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Example
Recall the equation for the voltage of an inductor:

If we take the Laplace Transform of both sides of this equation, we get:

which is consistent with the fact that an inductor has impedance sL

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Proof of the Differentiation Property:
1) First write x(t) using the Inverse Laplace Transform
formula:

2) Then take the derivative of both sides of the equation with

respect to t (this brings down a factor of s in the second term
due to the exponential):

3) This shows that x'(t) is the Inverse Laplace Transform of s

X(s):

The Differentiation Property is useful for solving differential

equations.

Recall the equation for the voltage of a capacitor turned on at time 0:

Additional Properties

If we take the Laplace Transform of both sides of this equation, we get

.
which is consistent with the fact that a capacitor has impedance

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 Example
Derive this:

Take the derivative of both sides of this equation with respect to s:

This is the expression for the Laplace Transform of -t x(t). Therefore,

Initial Value

(Given without proof)

Final Value

(Given without proof)

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 Example

Derive this:

Independent-Variable Transformation (for Unilateral Laplace Transform)

Plugging in the definition, we find the Laplace Transform of

x(at -b)
Let u = at - b and du = adt, we get:
Derive this:

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Example 1 Find y(t) where the transfer function H(s) and the input x(t)
are given. Use Partial Fraction Expansion to find the output y(t):

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Example 2 Find the transfer function H(s) for the differential equation.
Example 3 Now let the input to the system be x(t) = 5u(t).
Assume zero initial conditions.
y'(t) + 2y(t) = 3x'(t). Find y(t).

Example 4 Find the step response s(t) to

h(t) = e-tu(t)

t:

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Example 5 Find the output of an LTI system with impulse
response h(t) = ebtu(t) to an input x(t) = eatu(t), where a ≠ b.

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 Invertibility
You can find the inverse of a system using Laplace
Transforms. This is because:

Take the Laplace Transform of both sides of this

equation:

Therefore, the Laplace Transform of the inverse

system is simply

Example 7 Find the inverse system of

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