Signals and Systems - Electrical Engineering

226 C H A P T E R 3: The Laplace Transform

3.6 What Have We Accomplished? Where Do We Go from Here?....................

In this chapter you have learned the significance of the Laplace transform in the representation of

signals as well as of systems. The Laplace transform provides a complementary representation to the

time representation of a signal, so that damping and frequency, poles and zeros, together with regions

of convergence, conform a new domain for signals. But it is more than that—you will see that these

concepts will apply for the rest of this part of the book. When discussing the Fourier analysis of signals

and systems we will come back to the Laplace domain for computational tools and for interpretation.

The solution of differential equations and the different types of responses are obtained algebraically

with the Laplace transform. Likewise, the Laplace transform provides a simple and yet very significant

solution to the convolution integral. It also provides the concept of transfer function, which will be

fundamental in analysis and synthesis of linear time-invariant systems.

The common thread of the Laplace and the Fourier transforms is the eigenfunction property of LTI

systems. You will see that understanding this property will provide you with the needed insight into

the Fourier analysis, which we will cover in the next two chapters.

Problems............................................................................................

3.1. Generic signal representation and the Laplace transform

The generic representation of a signalx(t)in terms of impulses is

x(t)=

∫∞

−∞

x(τ)δ(t−τ)dτ

Considering the integral an infinite sum of termsx(τ)δ(t−τ)(think ofx(τ)as a constant, as it is not a

function of timet), find the Laplace transform of each of these terms and use the linearity property to find

X(s)=L[x(t)]. Are you surprised at this result?

3.2. Impulses and the Laplace transform

Given

x(t)=2[δ(t+ 1 )+δ(t− 1 )]

(a) Find the Laplace transformX(s)ofx(t)and determine its region of convergence.

(b) Plotx(t).

(c) The functionX(s)is complex. Lets=σ+jand carefully obtain the magnitude|X(σ+j)|and the

phase∠X(σ+j).

3.3. Sinusoids and the Laplace transform

Consider the following cases involving sinusoids:

(a) Find the Laplace transform ofy(t)=sin( 2 πt)u(t)−sin( 2 π(t− 1 ))u(t− 1 ))and its region of conver-

gence. Carefully ploty(t). Determine the region of convergence ofY(s).

(b) A very smooth pulse, called the raised cosine,x(t)is obtained as

x(t)= 1 −cos( 2 πt) 0 ≤t≤ 1