1.3 Continuous-Time Signals 75
1.3.2 Even and Odd Signals
Symmetry with respect to the origin differentiates signals and will be useful in their Fourier analysis.
We have that an analog signalx(t)is called
n Evenwheneverx(t)coincides with its reflectionx(−t). Such a signal is symmetric with respect to
the time origin.
n Oddwheneverx(t)coincides with−x(−t)—that is, the negative of its reflection. Such a signal is
asymmetric with respect to the time origin.
Even and odd signals are defined as follows:
x(t) even : x(t)=x(−t) (1.4)
x(t) odd : x(t)=−x(−t) (1.5)
Even and odd decomposition: Any signaly(t)is representable as a sum of an even componentye(t)and an
odd componentyo(t):
y(t)=ye(t)+yo(t) (1.6)
where
ye(t)=0.5[y(t)+y(−t)] (1.7)
yo(t)=0.5[y(t)−y(−t)] (1.8)
Using the definitions of even and odd signals, any signaly(t)can be decomposed into the sum of an
even and an odd function. Indeed, the following is an identity:
y(t)=
1
2
[y(t)+y(−t)]+
1
2
[y(t)−y(−t)]
where the first term is the even and the second is the odd components ofy(t). It can be easily verified
thatye(t)is even and thatyo(t)is odd.
nExample 1.6
Consider the analog signal
x(t)=cos( 2 πt+θ) −∞<t<∞
Determine the value ofθ for whichx(t)is even and odd. Ifθ=π/4, isx(t)=cos( 2 πt+π/ 4 ),
−∞<t<∞, even or odd?
