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

1 4n2 e j2nt 00 next we find t he transfer function

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: for t he r eader t o l earn t o t hink o f a signal from b oth o f t hese perspectives. In t he n ext c hapter, we shall see t hat a periodic signals also have t his d ual p ersonality. Moreover, we shall show t hat even LTI systems have this d ual p ersonality, which offers complementary insights into t he s ystem b ehavior. 3.8 223 A ppendix I t he form Xm(t)Xn(t) dt w ith m t- n vanish. Similarly, t he d erivative w ith r espect t o Ci of all t erms n ot c ontaining Ci is zero. For each i , t his o bservation leaves only two nonzero t erms i n Eq. (3.93): limitations o f the Fourier Series Method o f Analysis We have developed here a m ethod o f representing a periodic signal as a weighted s um o f e verlasting e xponentials whose frequencies lie along t he j w-axis in t he s plane. T his r epresentation ( Fourier series) is valuable in m any a pplications. However, as a tool f or a nalyzing linear systems, i t h as serious limitations a nd consequently has l imited u tility for t he following reasons: 1. T he F ourier series c an b e used only for periodic inputs. All practical i nputs a re a periodic ( remember t hat a p eriodic signal s tarts a t t = - 00). or -2 l t2 f (t)xi(t) dt + 2Ci lt2 h Xi 2(t) dt = tl ° i = 1 ,2, . .. ,n T herefore t2 l, 1 f (t)xi(t) dt Ci 2. T his t echnique c an b e a pplied readily t o a symptotically s table s ystems. I t c annot h andle so easily u nstable o r even marginally s table s ystems (see the footnote o n p. 219). = tl t Xi2(t)dt -1 1 = E- t2 f (t)xi(t) dt i = 1 ,2, . .. , N (3.94) , t1 tl T he first l imitation c an b e overcome by representing aperiodic signals in terms of everlasting exponentials. T his r epresentation c an b e achieved t hrough t he F ourier integral, which m ay b e considered as a n e xtension of t he F ourier series. We shall therefore use t he F ourier series as a s tepping s tone t o t he F ourier integral developed in t he n ext c hapter. T he s econd limitation c an b e overcome by using exponentials e st w here s is n ot r estricted t o t he i maginary axis, b ut is free t o t ake o n complex values. T his g eneralization l eads t o t he L aplace integral, discussed in C hapter 6 ( the L aplace t ransform). Derivation o f Eq. ( 3.40) 3 .8 S ubstitution o f Eqs. (3.36) a nd (3.94) in t his e quation yields A ppendix Appendix 3A: Derivation o f Eq. (3.39) T he e rror e (t) i n t he a pproximation (3.37) N e(t) = f (t) - L cnxn(t) (3.91) (3.95) (3.92) As N - t 0 0 for a complete orthogonal set, E e - -t 0, a nd t he e nergy of f (t) is e qual t o t he s um o f energies of all t he o rthogonal components q Xl (t), C2X2(t), C3X3(t), . ... n =l T he e rror e nergy E e is Ee = 1 t2 2 e (t) dt tl = 1 tl L CnXn(t) N t2 [ f (t) - 2 dt n =l Since Ee is a f unction of N p arameters q , C2, . .. , condition is C N, t o minimize E e, a necessary Appendix 3 8: Orthogonality o f the Trigonometric Signal S et C onsider a n i ntegral I defined by I= BE _e BCi = _B BCi lt2 [f(t) - L cnxn(t) ]2 dt = ° N tl i = 1 ,2, . .. , N (3.93) n =l W hen we e xpand t he i ntegrand, we find t hat all t he c ross-product t erms arising from t he o rthogonal signals are zero by...
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