Signal Processing and Linear Systems-B.P.Lathi copy

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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 (a-axis, 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 s-plane (Fig. 1.21) can be differentiated into two parts: t he l eft h alf-plane ( LHP) corresponding t o e xponentially decaying signals and t he r ight h alf-plane ( 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 (2-j5)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 ( - 2-j5)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 nti-symmetrical a bout t he v ertical axis, as depicted in Fig. 1.23b. 1 .5-1 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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