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Unformatted text preview: s um of t he
energies of its orthogonal components C jXl(t), C2X2(t), C3X3(t), . ...
T he series on t he r ighthand side of Eq. (3.41) is called t he g eneralized
F ourier s eries of f (t) w ith respect t o t he s et { xn(t)}. W hen t he s et { xn(t)}
is such t hat t he e rror energy Ee  > 0 as N  > 0 0 for every member of some particular class, we say t hat t he s et { xn(t)} is complete on [tl, t 2J for t hat class of f (t),
a nd t he s et { xn(t)} is called a set of b asis f unctions or b asis s ignals. Unless
otherwise mentioned, in future we shall consider only t he class of energy signals.
Thus, when t he s et { xn(t)} is complete, we have t he e quality (3.41). O ne
s ubtle point t hat m ust be understood clearly is t he meaning of equality in Eq.
(3.41). The equality here is n ot a n equality in the o rdinary sense, but in t he sense
that the error energy, that is, the energy o f the difference between the two sides o f
Eq. (3.41), approaches zero. I f t he e quality exists in t he o rdinary sense, t he e rror
energy is always zero, b ut t he converse is n ot necessarily true. T he e rror energy can
approach zero even though e(t), t he difference between the two sides, is nonzero a t
some isolated instants. T he reason is t hat even if e (t) is nonzero a t such instants,
the area under e 2 (t) is still zero; thus t he Fourier series on t he r ighthand side of 3 Signal R epresentation by Orthogonal Sets 186 Eq. (3.41) may differ from f (t) a t a finite number of points. In fact, when f (t) has
a j ump d iscontinuity a t t = to, t he corresponding Fourier series a t to converges t o
t he m ean of f (to+) a nd f (to)·
I n Eq. (3.41), t he energy of t he l efthand side is E f ' a nd t he energy of t he
r ighthand side is t he s um of the energies of all t he orthogonal components.t T hus 00 = LC 2E
n (3.42) n n =l This equation goes under t he n ame of P arseval's t heorem. Recall t hat t he signal
energy (area u nder t he s quared value of a signal) is analogous t o t he s quare of t he
l ength of a vector in t he vectorsignal analogy. In vector space we know t hat t he
s quare of t he l ength of a vector is equal t o t he s um of the squares of t he l engths of
its orthogonal components. T he above equation (3.42) is t he s tatement of this fact
as it applies t o signals.
Generalization to Complex Signals
T he above results can be generalized t o complex signals as follows: A s et of
functions XI(t), X2(t), . .. , XN(t) is m utually orthogonal over t he interval [ tl, t2] if
m fn (3.43) m =n
I f t his set is c omplete for a certain class of functions, then a function f (t) in this class can be expressed as
(3.44)
where
C 1
n=  En 1t2 f(t)x~(t) dt (3.45) t, E quation (3.39) [or Eq. (3.45)] shows one interesting property of t he coefficients
of C I, C2, . .. , C N; t he o ptimum value of any coefficient in t he a pproximation (3.37) is
i ndependent o f t he number of terms used in t he approximation. For example, if we
have u...
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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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