Signals and Systems - Electrical Engineering

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

An inputx(t)=es^0 t,s 0 =σ 0 +j 0 , is called an eigenfunction of an LTI system with impulse responseh(t)if

the corresponding output of the system is

y(t)=x(t)

∫∞

−∞

h(t)e−s^0 t=x(t)H(s 0 )

whereH(s 0 )is the Laplace transform ofh(t)computed ats=s 0. This property is only valid for LTI systems—it

is not satisfied by time-varying or nonlinear systems.

Remarks

n You could think of H(s)as an infinite combination of complex exponentials, weighted by the impulse

response h(τ). One can use a similar representation for signals.

n Consider now the significance of applying the eigenfunction result. Suppose a signal x(t)is expressed as a

sum of complex exponentials in s=σ+j,

x(t)=

1

2 πj

σ∫+j∞

σ−j∞

X(s)estds

That is, an infinite sum of exponentials in s each weighted by the function X(s)/( 2 πj)(this equation is

connected with the inverse Laplace transform as we will see soon). Using the superposition property of LTI

systems, and considering that for an LTI system with impulse response h(t)the output due to estis H(s)est,

then the output due to x(t)is

y(t)=

1

2 πj

σ∫+j∞

σ−j∞

X(s)

[

H(s)est

]

ds=

1

2 πj

σ∫+j∞

σ−j∞

Y(s)estds

where we let Y(s)=X(s)H(s). But from Chapter 2 we have that y(t)is the convolution y(t)=[x∗h](t).

Thus, these two expressions are connected:

y(t)=[x∗h](t) ⇔ Y(s)=X(s)H(s)

The expression on the left indicates how to compute the output in the time domain, and the one on the

right shows how to compute the Laplace transform of the output in the frequency domain. This is the most

important property of the Laplace transform: It reduces the complexity of the convolution integral in time

to the multiplication of the Laplace transforms of the input X(s)and of the impulse response H(s).

Now we are ready for the proper definition of the direct and inverse Laplace transforms of a signal or

of the impulse response of a system.