HP F2226A - 48GII Graphic Calculator User Manual page 522

integration of the form
known as the kernel of the transformation.
The use of an integral transform allows us to resolve a function into a given
spectrum of components. To understand the concept of a spectrum, consider
the Fourier series
f ) ( t
representing a periodic function with a period T. This Fourier series can be
f ( x ) = a
re-written as
0
A
n
for n =1,2, ...
The amplitudes A will be referred to as the spectrum of the function and will
n
be a measure of the magnitude of the component of f(x) with frequency f
n/T. The basic or fundamental frequency in the Fourier series is f
all other frequencies are multiples of this basic frequency, i.e., f
we can define an angular frequency, ω
where ω
is the basic or fundamental angular frequency of the Fourier series.
0
Using the angular frequency notation, the Fourier series expansion is written
as
f (
a
0
b
( ) κ ( ) , ) (
F s t s f
a
a a cos ω x
0 n n
n 1
∞
+ A ⋅ cos( ϖ x + φ
n n n
n = 1
2 2
= a + b , φ = tan
n n n
= 2nπ/T = 2π⋅f
n
x ) a A cos( ω
0 n
n 1
a cos ω x b
n n
n 1
.
The function κ(s,t) is
t dt
b sin ω x ,
n n
),
where
 
b
− 1    
n ,
a
 
n
= 1/T, thus,
0
= n⋅f . Also,
n 0
= 2π⋅ n⋅f = n⋅ω
n 0
x φ ).
n n
sin ω x
n n
Page 16-44
=
n
,
0