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 waxis 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 rossproduct t erms arising
from t he o rthogonal signals are zero by...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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