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

35a ci xi xi 1 2 f xi i xil 339a i 1

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: s um of t he energies of its orthogonal components C jXl(t), C2X2(t), C3X3(t), . ... T he series on t he r ight-hand side of Eq. (3.41) is called t he g eneralized F ourier s eries of f (t) w ith respect t o t he s et { xn(t)}. W hen t he s et { xn(t)} is such t hat t he e rror energy Ee - > 0 as N - > 0 0 for every member of some particular class, we say t hat t he s et { xn(t)} is complete on [tl, t 2J for t hat class of f (t), a nd t he s et { xn(t)} is called a set of b asis f unctions or b asis s ignals. Unless otherwise mentioned, in future we shall consider only t he class of energy signals. Thus, when t he s et { xn(t)} is complete, we have t he e quality (3.41). O ne s ubtle point t hat m ust be understood clearly is t he meaning of equality in Eq. (3.41). The equality here is n ot a n equality in the o rdinary sense, but in t he sense that the error energy, that is, the energy o f the difference between the two sides o f Eq. (3.41), approaches zero. I f t he e quality exists in t he o rdinary sense, t he e rror energy is always zero, b ut t he converse is n ot necessarily true. T he e rror energy can approach zero even though e(t), t he difference between the two sides, is nonzero a t some isolated instants. T he reason is t hat even if e (t) is nonzero a t such instants, the area under e 2 (t) is still zero; thus t he Fourier series on t he r ight-hand side of 3 Signal R epresentation by Orthogonal Sets 186 Eq. (3.41) may differ from f (t) a t a finite number of points. In fact, when f (t) has a j ump d iscontinuity a t t = to, t he corresponding Fourier series a t to converges t o t he m ean of f (to+) a nd f (to-)· I n Eq. (3.41), t he energy of t he l eft-hand side is E f ' a nd t he energy of t he r ight-hand side is t he s um of the energies of all t he orthogonal components.t T hus 00 = LC 2E n (3.42) n n =l This equation goes under t he n ame of P arseval's t heorem. Recall t hat t he signal energy (area u nder t he s quared value of a signal) is analogous t o t he s quare of t he l ength of a vector in t he vector-signal analogy. In vector space we know t hat t he s quare of t he l ength of a vector is equal t o t he s um of the squares of t he l engths of its orthogonal components. T he above equation (3.42) is t he s tatement of this fact as it applies t o signals. Generalization to Complex Signals T he above results can be generalized t o complex signals as follows: A s et of functions XI(t), X2(t), . .. , XN(t) is m utually orthogonal over t he interval [ tl, t2] if m fn (3.43) m =n I f t his set is c omplete for a certain class of functions, then a function f (t) in this class can be expressed as (3.44) where C 1 n= - En 1t2 f(t)x~(t) dt (3.45) t, E quation (3.39) [or Eq. (3.45)] shows one interesting property of t he coefficients of C I, C2, . .. , C N; t he o ptimum value of any coefficient in t he a pproximation (3.37) is i ndependent o f t he number of terms used in t he approximation. For example, if we have u...
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