Signals and Systems - Electrical Engineering

80 C H A P T E R 1: Continuous-Time Signals

wherei(t)andv(t)are the current and voltage in the resistor. Theenergyin the resistor for an interval

[t 0 ,t 1 ], of durationT=t 1 −t 0 , is the accumulation of instantaneous power over that time interval,

ET=

∫t^1

t 0

p(t)dt=

∫t^1

t 0

i^2 (t)dt=

∫t^1

t 0

v^2 (t)dt

Thepowerin the intervalT=t 1 −t 0 is the average energy

PT=

ET

T

=

1

T

∫t^1

t 0

i^2 (t)dt=

1

T

∫t^1

t 0

v^2 (t)dt

corresponding to the heat dissipated by the resistor (and for which you pay the electric company).

The energy and power concepts can thus be easily generalized.

Theenergyand thepowerof an analog signalx(t)are defined for either finite or infinite-support signals as:

Ex=

∫∞

−∞

|x(t)|^2 dt (1.10)

Px= lim

T→∞

1

2 T

∫T

−T

|x(t)|^2 dt (1.11)

The signalx(t)is then said to be finite energy, or square integrable, whenever

Ex<∞ (1.12)

The signal is said to have finite power if

Px<∞ (1.13)

Remarks

n The above definitions of energy and power are valid for any signal of finite or infinite support, since a

finite-support signal is zero outside its support.

n In the formulas for energy and power we are considering the possibility that the signals might be complex

and so we are squaring its magnitude: If the signal being considered is real, this simply is equivalent to

squaring the signal.

n According to the above definitions, a finite-energy signal has zero power. Indeed, if the energy of the signal

is some constant Ex<∞, then

Px= lim

T→∞

Ex

2 T

= 0