Basis Functions and Signal Representation - Signal Processing of Random Physiological Signals - page 22

From the orthogonality condition, (4.12) is rewritten in terms of the coefficient and
lambda as given in (4.13),

N

E = a n ( a n λ

n )

(4.13)

n = 0

or as given in (4.14).

t 2

N

x 2 ( t ) dt a n λ

= = (4.14)

E n

n = 0

t 1

Since a n λ n is the energy in the n th basis function, each term of the summation is the

energy associated with its index, n th component of the representation. Thus, the TOTAL
energy of a signal is the sum of the energies of its individual orthogonal coefficients, which
is often referred to as “Parseval's theorem.”
Many functions are, or can be, made orthogonal over an interval, but it does
not mean that the function may be a desirable function to use as a basis function. As
engineers, we tend to use the trigonometric functions, sine and cosine, in many analytical
applications. Why are the sinusoids so popular? Three important facts about sinusoidal
expressions stand out:

(1) Sinusoidal functions are very useful because they remain sinusoidal after var-
ious mathematical operations (i.e., integration, derivative). Also, exponents
of sinusoidal functions can be expressed as exponents by Euler's identity
(cos e j θ ). θ + j sin θ =

(2) The sum or difference of two sinusoids of the same frequency remains a
sinusoid.

(3) This property combined with the superposition properties of linear systems
implies that representing a signal as a sum of sinusoids may be a very convenient
technique, which is used for periodic signals.