Introduction to signals - Continuous and Discrete Time Signals and Systems - page 16

g 1 ( t )=3 sin(4 p t )+7 cos(3 p t ) g 2 ( t )=3 sin(4 p t )+7 cos(10 t )

10 10

8 8

2s

6 6

2s

4 4

2 2

0 0

−2 −2

−4 −4

−6 −6

−8 −8

−10 −10

−4 −3 −2 −1

−4 −3 −2 −1

(a) (b)

Fig. 1.8. Signals (a) g 1 ( t ) and (b) g 2 ( t ) considered in Example 1.5. Signal g 1 ( t ) is periodic with a
fundamental period of 2 s, while g 2 ( t ) is not periodic.

1.1.4 Energy and power signals

Before presenting the conditions for classifying a signal as an energy or a power
signal, we present the formulas for calculating the energy and power in a signal.
The instantaneous power at time t = t 0 of a real-valued CT signal x ( t )is
given by x 2 ( t 0 ). Similarly, the instantaneous power of a real-valued DT signal
x [ k ] at time instant k = k 0 is given by x 2 [ k ]. If the signal is complex-valued,
the expressions for the instantaneous power are modified to x ( t 0 ) 2 or x [ k 0 ] 2 ,
where the symbol represents the absolute value of a complex number.
The energy present in a CT or DT signal within a given time interval is given
by the following:

T 2

x ( t ) 2 d t in interval t

CT signals E ( T 1 , T 2 ) = = ( T 1 , T 2 ) with T 2 > T 1 ;

T 1

(1.10a)

N 2

x [ k ] 2 in interval k DT sequences E [ N 1 , N 2 ] = = [ N 1 , N 2 ] with N 2 > N 1 .

k = N 1

(1.10b)
The total energy of a CT signal is its energy calculated over the interval t =
[ −∞, ∞ ]. Likewise, the total energy of a DT signal is its energy calculated over
the range k = [ −∞, ∞ ]. The expressions for the total energy are therefore given
by the following:

∞

x ( t ) 2 d t ; CT signals E x = (1.11a)

−∞