ABSTRACT

In this chapter we study energy and power spectra and their relations to signal duration, periodicity and correlation functions.

Let f (t) be an electric potential in volts applied across a resistance of R = 1 ohm. The total energy dissipated in such a resistance is given by

E =

{ f2 (t) /R

} dt. (12.1)

Since the resistance value is unity the dissipated energy may be also be referred to as normalized energy. In what follows we shall refer to it simply as the dissipated energy, with the implicit assumption that it is the energy dissipated into a resistance of 1 ohm. We recall Parseval’s theorem which states that if a function f (t) is generally complex

and if F (jω) is the Fourier transform of f (t) then

|f (t)|2dt = 1 2π

|F (jω)|2dω. (12.2)

The energy in the resistance may therefore be written in the form

E =

f2 (t) dt = 1

|F (jω)|2dω. (12.3)

The function |F (jω)|2 is called the energy spectral density, or simply the energy density, of f (t). It is attributed the special symbol εff (ω), that is,

εff (ω)=△ |F (jω)|2 . (12.4) We note that its integral is equal to 2π times the signal energy

E = 1

εff (ω) dω (12.5)

hence the name “spectral density.” Given two signals f1 (t) and f2 (t), where f1 (t) represent a current source and f2 (t) the

voltage that the current source produces across a resistance R of 1 ohm, the normalized cross-energy or simply cross-energy is given by

ˆ ∞

f1 (t)f2 (t) dt. (12.6)

f1 (t) f2 (t) dt = 1

F1 (−jω)F2 (jω) dω. (12.7)

If f1 (t) and f2 (t) are real

F1 (−jω) = F ∗1 (jω) , F2 (−jω) = F ∗ (jω) . (12.8) the cross-energy is therefore given by

f1 (t) f2 (t) dt = 1

F ∗1 (jω)F2 (jω) dω. (12.9)

The function εf1f2 (ω)=

△F ∗1 (jω)F2 (jω) (12.10)

is called the cross-energy spectral density. The cross-energy of the two signals is then given by

E = 1

εf1f2 (ω)dω. (12.11)

Example 12.1 Consider the ideal lowpass filter frequency response shown in Fig. 12.1.