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Unformatted text preview: ilarity can
therefore be conveniently measured by cos (). T he larger t he cos (), t he g reater is t he
similarity between the two vectors. Thus, a suitable measure would be C n = cos (),
which is given by f ·x Cn 6. E xercise E S.2
Show t hat o ver a n interval (0 :S t :S 271"), t he ' best' a pproximation o f t he s quare s ignal f (t) in
Fig. 3.3 in t erms o f t he signal ejt is given b y -J.; e jt . Verify t hat t he e rror signal e (t) = f (t) - -J.;e jt
is o rthogonal t o t he signal e jt . \ l Energy of the S um o f Orthogonal Signals We know t hat t he s quare of t he length of a sum of two orthogonal vectors is
equal to t he s um of t he squares of the lengths of t he two vectors. Thus, if vectors
x a nd y are o rthogonal, a nd if z = x + y , t hen We have a similar result for signals. T he energy of t he s um of two orthogonal signals
is e qual t o t he s um of t he energies of t he two signals. Thus, if signals x (t) a nd y(t)
a re orthogonal over a n interval [tl, t2], a nd if z (t) = x (t) + y(t), t hen 177 3.2 Signal Comparison: Correlation = cos () = Ifllxl (3.23) We c an readily verify t hat t his measure is i ndependent of t he lengths of f a nd x .
This similarity measure Cn is known as t he c orrelation c oefficient. Observe t hat
Thus, the magnitude of C n is never greater t han unity. I f t he two vectors are aligned,
the similarity is m aximum (c n = 1). Two vectors aligned in opposite directions
have t he m aximum dissimilarity (c n = - 1). I f t he two vectors are orthogonal, t he
similarity is zero.
We use t he s ame argument in defining a similarity index (the correlation coefficient) for signals. We shall consider t he signals over t he e ntire time interval from
- 00 t o 0 0. To make c in Eq. (3.11) independent of energies (sizes) of f (t) a nd x (t),
we m ust normalize c by normalizing t he two signals t o have u nit energies. Thus,
the appropriate similarity index C n analogous to Eq. (3.23) is given by 1 00 Ez = Ex + Ey (3.21) We now prove t his r esult for complex signals of which real signals are a special case.
From Eq. (3.18) it follows t hat Cn = ~
1 V EjEx f (t)x(t) dt (3.25) - 00 Observe t hat multiplying either f (t) or x (t) by any constant has no effect on this
index. I t is i ndependent of t he size (energies) of f (t) a nd x (t). Using t he Schwarz
inequality,t we c an show t hat t he m agnitude of C n is never greater t han 1
T he l ast result follows from t he fact t hat because of orthogonality, t he two integrals
of t he p roducts x(t)y*(t) a nd x *(t)y(t) a re zero [see Eq. (3.20)J. T his result can be
extended to t he s um of any number of mutually orthogonal signals. t Schwarz i nequality s tates t hat for two real energy signals f (t) a nd x (t) (I: f (t)X(t)dt) 2 :S E fEx (3.25n) with equality if a nd only if x (t) = K f(t), w here K is a n a rbitrary c onstant. T here is also similar
inequality for complex signals. 3 Signal Representation by Orthogonal Sets 178 3.2 179 Si...
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