Signals and Systems - Electrical Engineering

3.2 The Two-Sided Laplace Transform 167

LTI System

H(s)

x(t)=es^0 t y(t)=x(t) H(s 0 )

FIGURE 3.1

Eigenfunction property of LTI systems. The input of the system isx(t)=es^0 t=eσ^0 tej^0 tand the output of the

system is the same input multiplied by the complex valueH(s 0 )whereH(s)=L[h(t)]—that is, the Laplace

transform of the impulse responseh(t)of the LTI system.

abstract at the beginning, but after you see it applied here and in the Fourier representation later

you will think of it as a way to obtain a representation analogous to the impulse representation. You

will soon discover the importance of using complex exponentials, and it will then become clear that

eigenfunctions are connected with phasors that greatly simplify the sinusoidal steady-state solution

of circuits.

3.2.1 Eigenfunctions of LTI Systems

Consider as the input of an LTI system the complex signal

x(t)=es^0 t s 0 =σ 0 +j 0

for−∞<t<∞, and leth(t)be the impulse response of the system. According to the convolution

integral, the output of the system is

y(t)=

∫∞

−∞

h(τ)x(t−τ)dτ=

∫∞

−∞

h(τ)es^0 (t−τ)dτ

=es^0 t

∫∞

−∞

h(τ)e−τs^0 dτ=x(t)H(s 0 ) (3.1)

Since the same exponential at the input appears at the output,x(t)=es^0 tis called aneigenfunction^1

of the LTI system. The inputx(t)is changed at the output by the complex functionH(s 0 ), which is

related to the system through the impulse responseh(t). In general, for anys, the eigenfunction at the

output is modified by a complex function

H(s)=

∫∞

−∞

h(τ)e−τsdτ

which corresponds to the Laplace transform ofh(t)!

(^1) German mathematician David Hilbert (1862–1943) seems to be the first to use the German wordeigento denote eigenvalues and
eigenvectors in 1904. The wordeigenmeans own or proper.