Fourier series, Fourier series ,16-26 – HP 50g Graphing Calculator User Manual

Page 503

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Page 16-26

f(t) = U

o

⋅[1-(t-a)/(b-1)]⋅[H(t-a)-H(t-b)].

Examples of the plots generated by these functions, for Uo = 1, a = 2, b = 3, c
= 4, horizontal range = (0,5), and vertical range = (-1, 1.5), are shown in the
figures below:

Fourier series

Fourier series are series involving sine and cosine functions typically used to
expand periodic functions. A function f(x) is said to be periodic, of period T, if
f(x+T) = f(t). For example, because sin(x+2

π) = sin x, and cos(x+2π) = cos x,

the functions sin and cos are 2

π-periodic functions. If two functions f(x) and g(x)

are periodic of period T, then their linear combination h(x) = a

⋅f(x) + b⋅g(x), is

also periodic of period T. A T-periodic function f(t) can be expanded into a
series of sine and cosine functions known as a Fourier series given by

where the coefficients a

n

and b

n

are given by

=

+

+

=

1

0

2

sin

2

cos

)

(

n

n

n

t

T

n

b

t

T

n

a

a

t

f

π

π

=

=

2

/

2

/

2

/

2

/

0

,

2

cos

)

(

2

,

)

(

1

T

T

T

T

n

dt

t

T

n

t

f

T

a

dt

t

f

T

a

π

=

2

/

2

/

.

2

sin

)

(

T

T

n

dt

t

T

n

t

f

b

π

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