Introduction to signals - Continuous and Discrete Time Signals and Systems - page 17

∞

x [ k ] 2 .

DT sequences E x = (1.11b)

k =−∞

Since power is defined as energy per unit time, the average power ofaCT
signal x ( t ) over the interval t

= ( −∞, ∞ ) and of a DT signal x [ k ] over the

range k = [ −∞, ∞ ] are expressed as follows:

T / 2

1
T

x ( t ) 2 d t . =

CT signals P x lim

T →∞

(1.12)

− T / 2

K

1

x [ k ] 2 .

DT sequences P x = (1.13)

2 K + 1

k =− K

Equations (1.12) and (1.13) are simplified considerably for periodic signals.
Since a periodic signal repeats itself, the average power is calculated from one
period of the signal as follows:

t 1 + T 0

1
T 0

= 1
T 0

x ( t ) 2 d t x ( t ) 2 d t ,

CT signals P x = (1.14)

T 0 t 1

k 1 + K 0 − 1

1
K 0

1
K 0

x [ k ] 2 x [ k ] 2 , DT sequences P x = = (1.15)

k = K 0 k = k 1

where t 1 is an arbitrary real number and k 1 is an arbitrary integer. The symbols
T 0 and K 0 are, respectively, the fundamental periods of the CT signal x ( t ) and
the DT signal x [ k ]. In Eq. (1.14), the duration of integration is one complete
period over the range [ t 1 , t 1

+ T 0 ], where t 1 can take any arbitrary value. In
other words, the lower limit of integration can have any value provided that the
upper limit is one fundamental period apart from the lower limit. To illustrate
this mathematically, we introduce the notation ∫ T 0 to imply that the integration
is performed over a complete period T 0 and is independent of the lower limit.
Likewise, while computing the average power of a DT signal x [ k ], the upper
and lower limits of the summation in Eq. (1.15) can take any values as long as
the duration of summation equals one fundamental period K 0 .
A signal x ( t ), or x [ k ], is called an energy signal if the total energy E x has
a non-zero finite value, i.e. 0 < E x < ∞ . On the other hand, a signal is called
a power signal if it has non-zero finite power, i.e. 0 < P x < ∞ . Note that a
signal cannot be both an energy and a power signal simultaneously. The energy
signals have zero average power whereas the power signals have infinite total
energy. Some signals, however, can be classified as neither power signals nor as
energy signals. For example, the signal e 2 t u ( t ) is a growing exponential whose
average power cannot be calculated. Such signals are generally of little interest
to us.
Most periodic signals are typically power signals. For example, the average
power of the CT sinusoidal signal, or A sin( ω 0 t + θ ), is given by A 2 / 2 (see