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

A c onstant a mplitude sinusoid cos 5te c an be

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 t hat (1.33a) 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 because (1.34) 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)] (1.35a) a nd (1.35b) F ig. 1 .25 Representation of a system. T he s tudy of...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online