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

# U i f signal r epresentation by f ourier series t his

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: ent o f a Vector A v ector is specified b y i ts m agnitude a nd i ts d irection. W e s hall d enote a ll vectors b y b oldface. F or e xample, x is a c ertain v ector w ith m agnitude o r l ength Ixl· F or t he t wo v ectors f a nd x s hown in Fig. 3.1, we define t heir d ot ( inner o r s calar) p roduct a s f· x = Ifllxl cos () (3.1) w here () is t he a ngle b etween t hese v ectors. U sing t his d efinition we c an e xpress t he l ength o f a v ector x a s 171 lxi, 172 3 Signal R epresentation by Orthogonal Sets 3.1 Signals a nd Vectors 173 Multiplying b oth sides by \x\ yields (3.2) Let t he c omponent of f along x be ex as depicted in Fig. 3.1. Geometrically t he c omponent o f f along x is t he p rojection of f o n x, and is o btained by drawing a perpendicular from the t ip of f on t he vector x , as illustrated in Fig. 3.1. W hat is t he m athematical significance of a component of a vector along another vector? As seen from F ig. 3.1, the vector f c an be expressed in terms o f v ector x as e\x\2 = \f\\x\ cos Therefore e= f· x f· x 1 e=--=-f·x x· X \x\2 (3.6) From Fig. 3.1, it is a pparent t hat w hen f a nd x a re perpendicular, or orthogonal, then f has a zero component along x; consequently, e = O. Keeping a n eye on Eq. (3.6), we therefore define f a nd x t o b e o rthogonal if t he i nner (scalar or dot) product of t he two vectors is zero, t hat is, if f· x = 0 3.1-2 ex F ig. 3 .1 x f (t) c::: ex(t) (3.3) However, this is not t he only way to express f in terms of x. Figure 3.2 shows two of t he infinite o ther possibilities. From Figs. 3.2a and 3.2b, we have In each of t hese t hree representations f is represented in t erms of x plus another vector called t he e rror v ector. I f we a pproximate f by ex, (3.5) (3.8) T he e rror e(t) in this approximation is e(t) = (3.4) fc::: e x Component o f a Signal T he c oncept of a vector component a nd o rthogonality can be extended t o sig· nals. Consider t he problem of approximating a real signal f (t) in terms of another real signal x (t) over a n interval [tl, t2J: Component (projection) of a vector along another vector. f =ex+e (3.7) f (t) - ex(t) { t l .:; t .:; t2 o otherwise (3.9) We now select some criterion for t he ' best a pproximation'. We know t hat t he signal energy is one possible measure of a signal size. For b est a pproximation, we need to minimize t he e rror s ignal-that is, minimize its size, which is i ts energy E e over t he interval [tl, t2J given by t he e rror in t he a pproximation is t he vector e = f - ex. Similarly, t he errors in approximations in Figs. 3.2a a nd 3.2b are e l a nd e2. W hat is unique a bout t he a pproximation in Fig. 3.1 is t hat t he e rror vector is the smallest. We c an now define m athematically t he c omponent of a vector f along vector x t o be ex where e is chosen t o m inimize t he l ength of the error vector e = f - ex. Note t hat t he r ight· h and side is a definite integral w ith t as t he d ummy variable. Hence, E e is a function of t he p arameter e ( not t) a nd E e is m inimum for some choice of e. To minimize...
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