HP 50g Graphing Calculator - Manual (Page 524)

Page 16-47
Properties of the Fourier transform
Linearity: If a and b are constants, and f and g functions, then F{a ⋅f + b⋅g} = a
F{f }+ b F{g}.
Transformation of partial derivatives. Let u = u(x,t). If the Fourier transform
transforms the variable x, then
F{ ∂u/∂x} = iω F{u}, F{∂ 2
u/ ∂x 2
} = - ω 2
F{u},
F{ ∂u/∂t} = ∂F{u}/∂t, F{∂ 2
u/ ∂t 2
} = ∂ 2
F{u}/ ∂t 2
Convolution: For Fourier transform applications, the operation of convolution is
defined as
The following property holds for convolution:
F{f*g} = F{f} ⋅F{g}.
Fast Fourier Transform (FFT)
The Fast Fourier Transform is a computer algorithm by which one can calculate
very efficiently a discrete Fourier transform (DFT). This algorithm has
applications in the analysis of different types of time-dependent signals, from
turbulence measurements to communication signals.
The discrete Fourier transform of a sequence of data values {x
j
}, j = 0, 1, 2, …,
n-1, is a new finite sequence {X
k
}, defined as
The direct calculation of the sequence X
k
involves n
2
products, which would
involve enormous amounts of computer (or calculator) time particularly for large
values of n. The Fast Fourier Transform reduces the number of operations to the
order of n
⋅log
2
n. For example, for n = 100, the FFT requires about 664
operations, while the direct calculation would require 10,000 operations. Thus,
∫ ⋅ ⋅ − ⋅ = . ) ( ) (
2
1
) )( * ( ξ ξ ξ
π
d g x f x g f
∑
−
=
− = ⋅ − ⋅ =
1
0
1 ,..., 2 , 1 , 0 ), / 2 exp(
1 n
j
j k
n k n kj i x
n
X π