Introduction to signals - Continuous and Discrete Time Signals and Systems - page 21

signal in electrical engineering is the thermal noise generated by a resistor. The
intensity of the thermal noise depends on the movement of billions of electrons
and cannot be predicted accurately.
The study of random signals is beyond the scope of this topic. We therefore
restrict our discussion to deterministic signals. However, most principles and
techniques that we develop are generalizable to random signals. The readers
are advised to consult more advanced topics for analysis of random signals.

1.1.6 Odd and even signals

A CT signal x e ( t ) is said to be an even signal if

x e ( t ) = x e ( − t ) . (1.16)

Conversely, a CT signal x o ( t ) is said to be an odd signal if

x o ( t ) =− x o ( − t ) . (1.17)

A DT signal x e [ k ] is said to be an even signal if

x e [ k ] = x e [ − k ] . (1.18)

Conversely, a DT signal x o [ k ] is said to be an odd signal if

x o [ k ] =− x o [ − k ] . (1.19)

The even signal property, Eq. (1.16) for CT signals or Eq. (1.18) for DT sig-
nals, implies that an even signal is symmetric about the vertical axis ( t = 0).
Likewise, the odd signal property, Eq. (1.17) for CT signals or Eq. (1.19) for
DT signals, implies that an odd signal is antisymmetric about the vertical axis
( t = 0). The symmetry characteristics of even and odd signals are illustrated
in Fig. 1.10. The waveform in Fig 1.10(a) is an even signal as it is symmetric
about the y -axis and the waveform in Fig. 1.10(b) is an odd signal as it is anti-
symmetric about the y -axis. The waveforms shown in Figs. 1.6(a) and (b) are
additional examples of even signals, while the waveforms shown in Figs. 1.6(c)
and (e) are examples of odd signals.
Most practical signals are neither odd nor even. For example, the signals
shown in Figs. 1.6(d) and (f ), and 1.8(a) do not exhibit any symmetry about
the y -axis. Such signals are classified in the “neither odd nor even” category.

Fig. 1.10. Example of (a) an
even signal and (b) an odd
signal.

x e ( t ) x o ( t )

5 5

5

t t

t t

−8 −6 −4 −2 0 246 −8 −6 −4 −2 0 2 46

(a) (b)

−5