Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 29

Eigenfunctions of LTI Analog Systems

If the input signal to an LTI analog system is x(t) = e jxt , then

y ð t Þ¼ h ð t Þ x ð t Þ¼ x ð t Þ h ð t Þ
¼ Z h ð k Þ e jx ð t k Þ dk ¼ e jxt Z

1 1

ð 1 : 24 Þ

h ð k Þ e jxk dk

1 1

If one defines H ð x Þ¼ R 1 1 h ð k Þ e jxk dk, then y(t) = e jx tH(x). That is, the
output of the system is just a scaled version of the input. In other words, e jx t is an
eigenfunction of the system, and the associated eigenvalue is H(x).

1.2.3.5 Stability of Analog LTI Systems-Frequency Domain

In a previous subsection system stability in the time domain was addressed. Here
system stability is considered in the framework of the complex frequency domain.
In this latter domain an analog system is typically characterized by its transfer
function, H(s) (i.e. by the Laplace transform of its impulse response h(t)). It can be
shown that any practical transfer function H(s), can be re-expressed as the ratio of
two polynomial functions of s: H(s) = N(s)/D(s). It can also be shown that an
analog system is BIBO stable if and only if [ 3 ]:

1. The degree of N(s) \ degree of D(s).
2. All poles (i.e., zeros of the denominator D(s)) are in the left half of the s-plane.

Example Plot the pole-zero diagram of the system

s ð s þ 1 Þ
ð s þ 2 Þ 2 ð s þ 3 Þ

H ð s Þ¼

and conclude whether the system is BIBO stable or not.

Solution: Zeros are located at s = 0, -1; poles are located at s =-2 (double) and
s =-3. The numerator polynomial is of a lower order than the denominator
polynomial, All poles are in the left half of the s plane. See the pole-zero diagram
in Fig. 1.19 . The system is therefore stable.

1.2.4 Signal Correlation and Its Applications

The correlation between two deterministic energy signals s(t) and r(t), is defined
by the integral: