Fundamentals - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 5

ensure the convergence of the sums, and yet all practical finite energy signals verify
it. As a simple example, the signal below is of finite energy:

− at

() ( )

xt = Ae cos 2 πθ ft + t

t

≥ 0

= 0 < 0

(this type of damped exponential oscillatory waveform is a fundamental signal in the
analysis of linear systems that are invariant by translation).

, ,

Figure 1.1. Electromagnetic interference signal, and its decomposition

On the more complex example of Figure 1.1, we see a finite energy signal of the
form:

4

∑

() ( )

x t = x t − t

i i

i = 1

where the 4 components x i ( t ) start at staggered times { t i }. Its shape, even though
complex, is supposed to reflect a perfectly reproducible physical experiment.

Finite power signals will be, in practice, permanent signals, i.e. not canceling at
infinity. As a simple example:

( )

() () ()

xt = At sin ψ t t ∈ℜ

[1.5]

()

At <∞