HP 49g+, Руководство пользователя

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