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Unformatted text preview: f (t) ~ c x(t) (3.15) where f (t) a nd x (t) now can be complex functions of t. Recall t hat t he energy E x
of t he complex signal x (t) over a n interval [tl, t2J is
Ex = 1t2Ix(t)12 dt
t, In this case, b oth t he coefficient c a nd t he e rror csin t e(t) = f (t) - cx(t) so t hat the energy of the error signal is minimum.
In this case (3.16) are complex (in general). For the ' best' a pproximation, we choose c so t hat E e , t he
energy of t he e rror signal e(t) is minimum. Now, t2 and Ee = From Eq. (3.11), w e find 1 I f(t) - cx(t)12 dt (3.17) t, 1 t " f (t)sintdt=; [ J( " s intdt+"
c =;Jo ~ ~sin t represents the best approximation of f (t) by the function sin t, which will minimize the
error energy. This sinusoidal component of f (t) is shown shaded in Fig. 3.3. By analogy
with vectors, we say t hat the square function f (t) depicted in Fig. 3.3 has a component of
signal sin t and that the magnitude of this component is 4/1r. • = sin t ....... t, t, x (t) ........./ 21t ................•. 1t2 x I t •••. o tFor complex signals the definition is modified as in Eq. (3.20). Recall also t hat 2 " 4
- sintdt =:;;: (3.13) lu + vl 2 = (u + v )(u' + v ') = lul 2 + Ivl 2 + u 'v + u v' Using this result, we can, after some manipulation, rearrange Eq. (3.17) as (3.18) 3 Signal R epresentation by Orthogonal Sets 176 Ee = t
it,' If(t)1 2 dt - 1ffx
1 1t2 f(t)x*(t) dt 12 + 1cjE;, - ffx it.2 f(t)x*(t) dt 12
1 a Since t he first t wo t erms on t he r ight-hand side are independent of c, i t is clear t hat
Ee is minimized by choosing c so t hat t he t hird t erm on the right-hand side is zero.
Ex t2 f(t)x*(t) dt it. (3.19) In light of the above result, we need t o redefine orthogonality for t he complex case
as follows: Two complex functions XI(t) a nd X2(t) a re orthogonal over a n interval
( tl :S t :S t2) if
(3.20) or E ither e quality suffices. This is a general definition of orthogonality, which reduces
t o Eq. (3.12) w hen t he functions are real. 3.2 Signal Comparison: Correlation Section 3.1 has prepared the foundation for signal comparison. Here again, we
can benefit by considering t he concept of vector comparison. Two vectors f a nd x
are similar if f h as a large component along x. In other words, if c in Eq. (3.6) is
large, t he vectors f a nd x a re similar. We could consider c as a quantitative measure
of similarity between f a nd x. Such a measure, however, would be defective. T he
degree of similarity between f a nd x should be independent of t he lengths of f a nd
x. I f we double t he l ength of f, for example, t he degree of similarity between f a nd
x should not change. From Eq. (3.6), however, we see t hat doubling f doubles t he
value of c (whereas doubling x halves the value of c). O ur m easure is clearly faulty.
Similarity between two vectors is i ndicated by t he angle () between t he vectors. T he
smaller t he (), t he g reater t he similarity, a nd vice versa. T he degree of sim...
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