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

1 signals a nd vectors 173 multiplying b oth sides by

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Unformatted text preview: E ., a necessary condition is (b) (a) d Ee de -=0 or x F ig. 3 .2 Approximation of a vector in terms of another vector. Now, t he l ength of t he c omponent of f along x is IfI cos from Fig. 3.1. Therefore elxl = If\ cos e e. B ut i t is also elxl as seen :e [[2 [ J(t) - ex(t)J2 (3.10) dt] =0 E xpanding t he s quared t erm inside t he integral, we o btain 3 Signal Representation by Orthogonal Sets 1 74 3.1 Signals a nd Vectors From which we o btain [ (I) -2 t2 f (t)x(t) dt + 2c 1t l 1t2 175 ........................ ,..........•.....•... .•.....•..........•... x 2 (t) dt = 0 1_ tl o a nd [2 c= f (t)x(t) dt = t , t2 1 x2(t) dt ~ f (t)x(t) dt -1 . (3.11) ·······... u.~.~.······· F ig. 3.3 Approximation of square signal in terms of a single sinusoid. We observe a r emarkable s imilarity between t he b ehavior of vectors a nd signals, as indicated b y Eqs. (3.6) a nd (3.11). I t is e vident from these two parallel expressions t hat t he a rea u nder t he p roduct o f two signals corresponds t o t he i nner (scalar o r d ot) product o f t wo vectors. I n fact, t he a rea u nder t he p roduct o f f (t) a nd x (t) is called t he i nner p roduct o f f (t) a nd x(t), a nd is d enoted by ( I, x). T he e nergy o f a signal is t he i nner p roduct o f a signal w ith itself, a nd c orresponds t o t he v ector length square (which is t he i nner p roduct o f t he vector with itself). To s ummarize o ur discussion, if a signal f (t) is a pproximated by a nother signal x(t) as f (t) ~ cx(t) t hen t he o ptimum value o f c t hat minimizes t he energy of t he e rror signal in this approximation is g iven by Eq. (3.11). T aking o ur c lue from vectors, we say t hat a signal f(t) c ontains a component cx(t), w here c is given by Eq. (3.11). N ote t hat in vector terminology, cx(t) is t he p rojection o f f (t) o n x(t). C ontinuing with t he analogy, we say t hat if t he c omponent o f a s ignal f (t) o f t he form x(t) is zero ( that is, c = 0), t he signals f (t) a nd x(t) a re orthogonal over t he i nterval [t l , t2J. Therefore, we define t he real signals f (t) a nd x (t) t o b e orthogonal over t he interval [tl, t2J i ft t2 1 f (t)x(t) dt (3.12) =0 t, E xample 3 .1 For the square signal f (t) shown in Fig. 3.3 find the component in f (t) of the form sin t. In other words, approximate f (t) in terms of sin t Thus f (t) f (t) ~ (3.14) 1r Exercise E3.1 Show that over an interval ( -1r ::; t ::; 7 r), the 'best' approximation of the signal [ (t) = t in terms of the function sin t is 2 sin t. Verify that the error signal e(t) = t - 2 sin t is orthogonal to the signal sin t over the interval - 7r ::; t ::; 7r. Sketch the signals t and 2 sin t over the interval f::, - 7r : :; t ::; 3 .1-3 7r. 'V Orthogonality in Complex Signals So far we have restricted ourselves t o real functions of t. To generalize t he results t o complex functions of t, consider again t he p roblem of approximating a function f (t) by a function x(t) over a n interval ( tl :-:; t :-:; t2):...
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