# Introduction to Laplace Transform Analysis

The *one-sided Laplace transform* of a signal is defined
by

*unilateral*Laplace transform. There is also a

*two-sided*, or

*bilateral*, Laplace transform obtained by setting the lower integration limit to instead of 0. Since we will be analyzing only

*causal*

^{D.1}linear systems using the Laplace transform, we can use either. However, it is customary in engineering treatments to use the one-sided definition.

When evaluated along the axis (*i.e.*, ), the
Laplace transform reduces to the unilateral *Fourier transform*:

An advantage of the Laplace transform is the ability to transform signals which have no Fourier transform. To see this, we can write the Laplace transform as

*exponentially windowed*input signal. For (the so-called ``

*strict right-half plane*'' (RHP)), this exponential weighting forces the Fourier-transformed signal toward zero as . As long as the signal does not increase faster than for some , its Laplace transform will exist for all . We make this more precise in the next section.

## Existence of the Laplace Transform

A function has a Laplace transform whenever it is of*exponential order*. That is, there must be a real number such that

The Laplace transform of a causal, growing exponential function

Thus, the Laplace transform of an exponential is , but this is defined only for re.

## Analytic Continuation

It turns out that the domain of definition of the Laplace transform can be extended
by means of *analytic continuation* [14, p. 259].
Analytic continuation is carried out by expanding a function of
about all points in its domain of definition, and
extending the domain of definition to all points for which the series
expansion converges.

In the case of our exponential example

the Taylor series expansion of about the point in the plane is given by

where, writing as and using the chain rule for differentiation,

and so on. We also used the *factorial notation*
, and we defined the special cases
and
, as is normally done.
The series expansion of can thus be written

We now ask for what values of does the series Eq.(D.2)
*converge*? The value is particularly easy to
check, since

*not*converge for , no matter what our choice of might be. We must therefore accept the point at infinity for . This is eminently reasonable since the closed form Laplace transform we derived,

*does*``blow up'' at . The point is called a

*pole*of .

More generally, let's apply the *ratio test* for the convergence
of a geometric series. Since the th term of the series is

*pole*at .

The *analytic continuation* of the domain of Eq.(D.1) is now
defined as the *union* of the disks of convergence for all points
. It is easy to see that a sequence of such disks can
be chosen so as to define all points in the plane except at the
pole .

In summary, the Laplace transform of an exponential is

Analytic continuation works for any finite number of poles of finite
order,^{D.2} and for an infinite number of
distinct poles of finite order. It breaks down only in pathological
situations such as when the Laplace transform is singular everywhere
on some closed contour in the complex plane. Such pathologies do not
arise in practice, so we need not be concerned about them.

##
Relation to the *z* Transform

The Laplace transform is used to analyze *continuous-time*
systems. Its discrete-time counterpart is the transform:

In summary,

Note that the plane and plane are generally related by

*bandlimited to half the sampling rate*. As is well known, this condition is necessary to prevent

*aliasing*when sampling the continuous-time signal at the rate to produce , (see [84, Appendix G]).

## Laplace Transform Theorems

### Linearity

The Laplace transform is a *linear operator*. To show this, let
denote a linear combination of signals and ,

Thus, linearity of the Laplace transform follows immediately from the linearity of integration.

### Differentiation

The *differentiation theorem* for Laplace transforms states that

*Proof: *
This follows immediately from integration by parts:

since by assumption.

**Corollary: ***Integration Theorem*

Thus, successive time derivatives correspond to successively higher powers of , and successive integrals with respect to time correspond to successively higher powers of .

## Laplace Analysis of Linear Systems

The differentiation theorem can be used to convert differential
equations into *algebraic* equations, which are easier to solve.
We will now show this by means of two examples.

### Moving Mass

Figure D.1 depicts a free mass driven by an external force along
an ideal frictionless surface in one dimension. Figure D.2
shows the *electrical equivalent circuit* for this scenario in
which the external force is represented by a voltage source emitting
*volts*, and the mass is modeled by an *inductor*
having the value *Henrys*.

From Newton's second law of motion ``'', we have

Thus, given

- Laplace transform of the driving force ,
- initial mass position, and
- initial mass velocity,

*all*linear, time-invariant (LTI) systems. For nonlinear and/or time-varying systems, Laplace-transform analysis cannot, strictly speaking, be used at all.

If the applied external force is zero, then, by linearity of the Laplace transform, so is , and we readily obtain

^{D.3}Similarly, any initial velocity is integrated with respect to time, meaning that the mass moves forever at the initial velocity.

To summarize, this simple example illustrated use the Laplace transform to solve for the motion of a simple physical system (an ideal mass) in response to initial conditions (no external driving forces). The system was described by a differential equation which was converted to an algebraic equation by the Laplace transform.

### Mass-Spring Oscillator Analysis

Consider now the mass-spring oscillator depicted physically in Fig.D.3, and in equivalent-circuit form in Fig.D.4.

By Newton's second law of motion, the force applied to a mass equals its mass times its acceleration:

We have thus derived a second-order differential equation governing the motion of the mass and spring. (Note that in Fig.D.3 is both the position of the mass and compression of the spring at time .)

Taking the Laplace transform of both sides of this differential equation gives

To simplify notation, denote the initial position and velocity by and , respectively. Solving for gives

denoting the modulus and angle of the pole residue , respectively. From §D.1, the inverse Laplace transform of is , where is the Heaviside unit step function at time 0. Then by linearity, the solution for the motion of the mass is

If the initial velocity is zero (), the above formula reduces to and the mass simply oscillates sinusoidally at frequency , starting from its initial position . If instead the initial position is , we obtain

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