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Unformatted text preview: 5 Even and Odd Functions
A function fe(t) is s aid t o b e a n e ven f unction of t if fe(t) = f e(t) (1.31) a nd a function fo(t) is said t o b e a n o dd f unction of t if
F ig. 1.22 Complex frequency plane. fo(t) =  fo(t)
of oscillation o f e st; t he real p art a ( the n eper frequency) gives information a bout
t he r ate of increase or decrease of t he a mplitude of e st. For signals whose complex
frequencies lie o n t he real axis (aaxis, where w = 0), the frequency of oscillation
is zero. Consequently these signals are monotonically increasing or decreasing exponentials (Fig. 1.21a). For signals whose frequencies lie on t he imaginary axis (jw
axis where a = 0), e "t = 1. Therefore, these signals are conventional sinusoids with
constant a mplitude (Fig. 1.21b). T he case s = 0 (a = w = 0) corresponds to a
constant (dc) signal because eOt = 1. For t he signals illustrated in Figs. 1.21c and
1.21d, b oth a a nd w are nonzero; t he frequency s is complex a nd does not lie on
either axis. T he signal in Fig. 1.21c decays exponentially. Therefore, a is negative,
a nd s lies to t he left of t he i maginary axis. In contrast, t he signal in Fig. 1.21d
grows exponentially. Therefore, a is positive, a nd s lies to t he r ight of t he imaginary axis. T hus t he splane (Fig. 1.21) can be differentiated into two parts: t he l eft
h alfplane ( LHP) corresponding t o e xponentially decaying signals and t he r ight
h alfplane ( RHP) corresponding t o exponentially growing signals. T he i maginary
axis separates t he two regions a nd corresponds to signals of constant amplitude.
An exponentially growing sinusoid e 2t cos (5t + e), for example, can be expressed as a s um of exponentials e (2+j5)t a nd e (2j5)t with complex frequencies
2 + j 5 a nd 2  j5, respectively, which lie in t he RHP. An exponentially decaying
sinusoid e  2t c os (5t + e) can be expressed as a sum of exponentials e (  2+j5)t a nd
e (  2j5)t w ith complex frequencies  2 + j 5 a nd  2  j5, respectively, which lie in
t he LHP. A c onstant a mplitude sinusoid cos ( 5t+e) c an be expressed as a sum of exponentials e j5t a nd e  j5t with complex frequencies ± j5, which lie on t he i maginary
axis. Observe t hat t he monotonic exponentials e ±2t a re also generalized sinusoids
w ith complex frequencies ±2. (1.32) An even function has t he s ame value a t t he i nstants t a nd  t for all values of t.
Clearly, fe(t) is s ymmetrical a bout t he vertical axis, as shown in Fig. 1.23a. O n t he
o ther hand, t he value of a n o dd function a t t he i nstant t is t he negative o f i ts value
a t t he i nstant  to Therefore, fo(t) is a ntisymmetrical a bout t he v ertical axis, as
depicted in Fig. 1.23b. 1 .51 Some Properties o f Even and Odd Functions E ven a nd o dd functions have t he following property:
even function x o dd function = o dd function
o dd function x odd function = even function
even function x even function = even function
T he proofs of these facts are trivial a nd fol...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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