Unformatted text preview: low directly from t he definition of odd
a nd even functions [Eqs. (1.31) a nd (1.32)J. Area
Because fe(t) is symmetrical a bout t he v ertical axis, i t follows from Fig. 1.23a
a fe(t) dt = 2
fe(t) dt t f Jo -a I t is also clear from Fig. 1.23b t hat i aa fo(t) dt = 0 (1.33b) These results can also be proved formally by using t he definitions in Eqs. (1.31) and
(1.32). We leave t hem as a n exercise for t he reader. 76 I ntroduction t o Signals and Systems 77 1.6 Systems T he function e -atu(t) a nd i ts even and odd components are illustrated in Fig. 1.24 . j (t) • o t-- (a) E xample 1 .8 Find the even and odd components of e jt .
From Eq. (1.34)
e jt = f .(t) + fort) where f.(t) 1
= 2 [ e ''t + e-''tl = cos t and t-- t-- (b) (c) F ig. 1 .24 Finding a n even and odd components of a signal. 1.5-2 Even and Odd Components o f a Signal Every signal f (t) can be expressed as a sum of even and odd components
f (t) = [J(t) f (-t)]
[J(t) - f(-t)] ! + '-...-' +! '-...-' e ven o dd 1 .6 Systems
As mentioned in Sec. 1.1, systems are used t o process signals in order t o modify
or t o e xtract additional information from the signals. A system may consist of
physical components (hardware realization) or may consist of an algorithm t hat
computes the o utput signal from t he i nput signal (software realization).
A system is characterized by its i nputs, i ts o utputs (or r esponses), a nd the
r ules o f o peration (or laws) adequate t o describe its behavior. For example, in
electrical systems, t he laws of operation are t he familiar voltage-current relationships for the resistors, capacitors, inductors, transformers, transistors, a nd so on,
as well as the laws of interconnection (i.e., Kirchhoff's laws). Using these laws,
we derive mathematical equations relating t he o utputs t o t he i nputs. These equations then represent a m athematical m odel of t he system. Thus a system is
characterized by its inputs, its outputs, a nd its mathematical model.
A system can be conveniently illustrated by a "black box" with one set of
accessible terminals where t he i nput variables ! I(t), h (t), . .. , f j(t) are applied and
another set of accessible terminals where the o utput variables YI (t), Y2(t), . .. , Yk(t)
are observed. Note t hat t he direction of t he arrows for t he variables in Fig. 1.25 is
always from cause t o effect. From the definitions in Eqs. (1.31) and (1.32), we c an clearly see t hat t he first
:omponent. o~ t he r ight-hand side is an even function, while the second component
IS. o dd. ThIs IS appare.nt from the fact t hat replacing t by - t in the first component
YIelds t he same functIOn. T he same maneuver in the second component yields the
negative of t hat component.
Consider t he function f (t) = e -atu(t)
Express~ng t his function as a sum of the even and odd components fe(t) a nd fort), we o btam f (t) = fe(t) + fort) where [from Eq. (1.34)]
(1.35b) F ig. 1 .25 Representation of a system. T he s tudy of...
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