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 3.12
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 ignalthat 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...
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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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