Signals and Systems - Electrical Engineering

3.3 The One-Sided Laplace Transform 181

0 2 4

− 1

−0.5

0

0.5

1

t

y(t)=cos(10t) exp(−t)u(t)

− 2 0 2

− 10

− 5

0

5

10

σ

jΩ

(a)

(b)

x(t)=exp(−t)u(t)

0 2 4

0

0.5

1

− 2 0 2

− 1

−0.5

0

0.5

1

σ

jΩ

FIGURE 3.7
Poles and zeros of the Laplace transform of (a) causal signalx(t)=e−tu(t)and of (b) causal decaying signal
y(t)=e−tcos( 10 t)u(t).

nExample 3.5

In statistical signal processing, the autocorrelation functionc(τ)of a random signal describes the

correlation that exists between the random signalx(t)and shifted versions of it,x(t+τ)andx(t−

τ)for shifts−∞< τ <∞. Clearly,c(τ)is two-sided (i.e., nonzero for both positive and negative

values ofτ) and symmetric. Its two-sided Laplace transform is related to the power spectrum of

the signalx(t). Letc(t)=e−a|t|, wherea>0 (we replaced theτvariable fortfor convenience). Find

its Laplace transform. Determine if it would be possible to compute|C()|^2 , which is called the

power spectrum of the random signalx(t).

Solution

The autocorrelation can be expressed as

c(t)=c(t)u(t)+c(t)u(−t)

=cc(t)+cac(t)

wherecc(t)is the causal component andcac(t)the anti-causal component ofc(t). The Laplace

transform ofc(t)is then given by

C(s)=L[cc(t)u(t)]+L[cac(−t)u(t)](−s)